Computer-readable media for determining defective patterns

Abstract

This document describes a method for training a configuration to predict whether a feature associated with an imaging substrate will be defective after etching the imaging substrate, and for determining etching conditions based on the trained model. The method includes: obtaining (i) a developed image of the imaging substrate at a given location, the developed image including a plurality of features, and (ii) an etched image of the imaging substrate at the given location via a metric; and training the configuration using the developed image and the etched image to determine the defect of a given feature among the plurality of features in the developed image. In one embodiment, the defect determination is based on comparing the given feature in the developed image with a corresponding etched feature in the etched image.

Description

Computer-readable media used for identifying defect patterns This invention relates to a technique for improving the determination of defective patterns in order to further improve the manufacturing process of the device. This technique can be used in conjunction with lithography equipment. Manufacturing semiconductor devices typically involves processing a substrate (e.g., a semiconductor wafer) using several manufacturing processes to form various features and multiple layers of the semiconductor device. These layers and features are typically fabricated and processed using processes such as deposition, lithography, etching, chemical mechanical polishing, and ion implantation. Multiple devices can be fabricated at different locations on the substrate, and then these devices are separated into individual devices. This device manufacturing process can be considered a patterning process. The patterning process may include patterning steps for transferring patterns from patterning devices to the substrate. Furthermore, one or more related patterning steps may be present, such as photoresist development using a developing apparatus, baking the substrate using a baking tool, etching the pattern onto the substrate using an etching apparatus, measurement / inspection of the transferred circuit pattern, etc. After exposure, the substrate may undergo other processes, such as post-exposure baking (PEB), development, hard baking, and measurement / inspection of the transferred circuit pattern. This array of processes serves as the basis for manufacturing individual layers of a device (e.g., an IC). The substrate can then undergo various processes, such as etching, ion implantation (doping), metallization, oxidation, and chemical-mechanical polishing, all aimed at refining the individual layers of the device. If several layers are required in the device, the entire process or its variations are repeated for each layer. Ultimately, the device will be present in each target portion of the substrate. In one embodiment, a method is provided to train a configuration to predict whether a feature associated with an imaging substrate will be defective after etching the imaging substrate. The method includes: obtaining, via a metric, (i) a developed image of the imaging substrate at a given location, the developed image including a plurality of features, and (ii) an etched image of the imaging substrate at the given location, the etched image including etched features corresponding to the plurality of features; and training the configuration using the developed image and the etched image to determine a defect of a given feature among the plurality of features in the developed image. In one embodiment, the defect determination is based on comparing the given feature in the developed image with a corresponding etched feature in the etched image. Furthermore, a method for determining etching conditions of an imaging substrate is provided. The method includes: obtaining a developed image of the imaging substrate and initial etching conditions to be used for etching the imaging substrate; determining, via a trained model, a failure rate of a feature associated with the imaging substrate, the failure rate indicating that the feature is defective after etching the imaging substrate; and modifying the initial etching conditions based on the failure rate to reduce the probability that the feature is defective after etching. Furthermore, a method is provided for determining an etching characteristic associated with an etching process. The method includes: obtaining via a measuring instrument: (i) an image-displayed image (ADI) of an imaged pattern at a given location on a substrate, the imaged pattern including a feature of interest and adjacent features adjacent to the feature of interest; and (ii) an etched image (AEI) of the imaged pattern at the given location on the substrate, the AEI including an etched feature corresponding to the feature of interest in the ADI; and determining, using the ADI and the AEI, a correlation between the etched feature associated with the feature of interest in the ADI and the adjacent features, the correlation characterizing the etching characteristic associated with the etching process. Furthermore, a method is provided for determining an etching condition associated with an etching process. The method includes: obtaining a correlation between an etched feature of interest in an etched image (AEI) and an adjacent feature in an etched image (ADI) associated with the etched feature of interest; and determining the etching condition associated with the etching process based on the correlation, such that the correlation remains within a target range. Furthermore, in one embodiment, a method is provided for generating an interpretation model configured to interpret a prediction generated by a trained model. The method includes: obtaining a dataset by executing the trained model, the dataset containing a plurality of predictions associated with a plurality of features in an analog image (ADI), the ADI including a feature of interest, each of the plurality of predictions being generated by the trained model; determining the distance between the plurality of features and each location of the feature of interest; assigning weights to each of the plurality of predictions based on the distances; and determining model parameter values ​​of the interpretation model based on the weighted predictions, such that a difference between one output of the interpretation model and the weighted predictions is reduced. In one embodiment, the model parameter values ​​indicate the contribution of each pixel of the ADI to the prediction associated with the feature of interest. Furthermore, in one embodiment, a method is provided for identifying the contribution of pixels in an acquired image to a prediction generated by a trained model. The method includes: obtaining (i) the acquired image (ADI) including a feature of interest using a metric tool, and obtaining (ii) an interpretation model configured to interpret a prediction associated with the feature of interest, the prediction being generated via the trained model; and applying the interpretation model to the ADI image to generate an interpretation map containing pixel values ​​quantifying the contribution of each pixel in the ADI image to the prediction of the feature of interest. In addition, in one embodiment, a computer program product is provided, comprising a non-transitory computer-readable medium having one set of instructions recorded thereon, which, when executed by a computer system, perform the aforementioned method. Furthermore, in one embodiment, a method is provided for generating a model for determining the failure rate of features in a developed image. The method includes: obtaining the developed image (ADI) of a substrate, the ADI comprising a plurality of features; generating a first portion of the model based on physical characteristic values ​​associated with a subgroup of features of the ADI; and generating a second portion of the model based on the first portion of the model and the physical characteristic values ​​associated with all of the plurality of features of the ADI, wherein the subgroup of features of the ADI differs from the other features of the ADI. Furthermore, in one embodiment, a system is provided for determining a portion of a feature that will fail after etching. The system includes: a metric for capturing an illustrated image (ADI) of a substrate at a given location, the ADI including a plurality of features; and a processor configured to execute a model for determining the failure rate of the plurality of features of the ADI after etching. The model is a combination of: (i) a first probability distribution function configured to estimate the distribution of physical characteristic values ​​of fault-free vias, and (ii) a second probability distribution function configured to determine the failure rate based on the physical characteristic values ​​of all the plurality of features of the ADI. Furthermore, in one embodiment, a non-transitory computer-readable medium is provided, comprising instructions that, when executed by one or more processors, cause operations including: obtaining an analog-displayed image (ADI) of a substrate, the ADI comprising a plurality of features; generating a first portion of a model based on physical property values ​​associated with a subgroup of the features of the ADI; and generating a second portion of the model based on the first portion of the model and the physical property values ​​associated with all of the plurality of features of the ADI, wherein the subgroup of features of the ADI differs from the other features of the ADI. Furthermore, in one embodiment, a method is provided for training a model configured to determine an etched image (AEI) feature based on an acquired image (ADI) feature, the method comprising: obtaining (i) a measurement of an ADI feature imaged on a substrate and (ii) a measurement of an etched image (AEI) feature corresponding to the measured ADI feature on the substrate having undergone an etching process; assigning a first set of variables to characterize the measured ADI feature and assigning a second set of variables to characterize the measured AEI feature; determining a correlation between a combination of the first set of variables of the measured ADI feature and a combination of the second set of variables of the measured AEI feature; and training the model based on the correlation by means of one or more sub-combinations of the first set of variables having correlation values ​​within a specified correlation threshold, the model being used to determine an AEI feature for an input ADI feature. Furthermore, in one embodiment, a metrology tool is provided, comprising: a beam generator configured to measure an ADI feature after imaging a substrate and an AEI feature after etching the substrate; and a processor. The processor is configured to: obtain a correlation between the measured ADI feature and a measured AEI feature corresponding to the measured ADI feature printed on a substrate undergoing an etching process, the correlation being based on a combination of variables characterizing how the measured ADI feature transforms into the AEI feature; and adjust settings of the metrology tool based on the correlation to improve the correlation, the settings being determined based on a derivative of the correlation with respect to each setting, the derivative indicating an improvement in the correlation for each setting of the metrology tool. Furthermore, in one embodiment, a method is provided for training a model configured to determine an etched image (AEI) based on an image after development (ADI), the method comprising: obtaining (i) an ADI on an imaging substrate and (ii) an etched image (AEI) after etching the imaging substrate; determining a correlation between a combination of a first set of variables of the ADI and a combination of a second set of variables of the AEI, the first set of variables and the second set of variables being grayscale values ​​of the ADI and the AEI, respectively; and training the model based on the correlation by means of one or more sub-combinations of the first set of variables having correlation values ​​within a specified correlation threshold, the model being used to determine an AEI for an input ADI. Furthermore, in one embodiment, a method is provided for determining an etched image (AEI) based on an etched image (ADI). The method includes: obtaining an ADI of a substrate; and determining an AEI by inputting the ADI into and outputting the ADI to a trained model, the trained model being trained based on a correlation between a first set of variables of a measured ADI and a second set of variables of a measured AEI, the correlation being within a specified correlation threshold. Furthermore, in one embodiment, a non-transitory computer-readable medium is provided, comprising instructions that, when executed by one or more processors, cause the determination of an etched image (AEI) feature based on an etched image (ADI) feature. The operations include: obtaining an ADI of a substrate; and determining an AEI by inputting and outputting the ADI to a trained model, the trained model being trained based on a correlation between a first set of variables of a measured ADI and a second set of variables of a measured AEI, the correlation being within a specified correlation threshold. Furthermore, in one embodiment, a non-transitory computer-readable medium is provided, comprising instructions that, when executed by one or more processors, cause operations to train a model configured to determine one of an etched image (AEI) feature based on an acquired image (ADI) feature, the operations including: obtaining (i) a measurement of one ADI feature imaged on a substrate and (ii) a measurement of one etched image (AEI) feature corresponding to the measured ADI feature on the substrate having undergone an etching process; assigning a first set of variables to characterize the measured ADI feature and assigning a second set of variables to characterize the measured AEI feature; determining a correlation between a combination of the first set of variables of the measured ADI feature and a combination of the second set of variables of the measured AEI feature; and training the model based on the correlation by means of one or more sub-combinations of the first set of variables having correlation values ​​within a specified correlation threshold, the model being used to determine one of the AEI features for an input ADI feature. Furthermore, in one embodiment, a non-transitory computer-readable medium is provided, comprising instructions that, when executed by one or more processors, cause operations to train a model configured to determine an etched image (AEI) based on an image after development (ADI), the operations including: obtaining (i) an ADI over an imaging substrate and (ii) an etched image (AEI) after etching the imaging substrate; determining a correlation between a combination of a first set of variables of the ADI and a combination of a second set of variables of the AEI, the first set of variables and the second set of variables being grayscale values ​​of the ADI and the AEI, respectively; and training the model based on the correlation by means of one or more sub-combinations of the first set of variables having correlation values ​​within a specified correlation threshold, the model being used to determine an AEI for an input ADI. For years, the computing power of electronic devices has followed a pattern of increasing power and reducing physical size. This pattern has been achieved by increasing the number of circuit components (transistors, capacitors, diodes, etc.) on each integrated circuit (IC) chip. For example, the IC chip in a smartphone can be as small as a human thumbnail and can contain more than 2 billion transistors, each smaller than 1 / 1000th the size of a human hair. Manufacturing an IC is a complex and time-consuming process, with circuit components on different layers and involving hundreds of individual steps. Even an error in a single step can lead to a problem with the final IC. Even a "fatal defect" can cause device malfunction. The goal of the manufacturing process is to improve the overall yield. For example, for a 50-step process achieving a 75% yield, each individual step must have a yield greater than 99.4%, and if the yield of an individual step is 95%, the overall process yield drops to 7%. The challenge that conflicts with high yield is maintaining the goal of rapid production schedules (e.g., output or the number of wafers processed per hour). High process yield and high wafer output can be affected by the presence of defects (especially when operator intervention is required to inspect for defects). Therefore, high-volume detection and identification of minute defects using inspection tools such as optical or electron microscopes (SEM) is essential for maintaining high yield and low cost. Because microscopes used for defect detection can only examine a portion of a wafer at a time, defect detection can be very time-consuming, thus reducing overall yield. For example, if every location on a wafer must be inspected to find a defect, wafer yield can be significantly reduced because inspecting every location on every IC on the wafer takes time. One approach to this problem is to use techniques that predict defect locations based on information obtained from a photolithography system used in IC wafer manufacturing. In one example, defect detection can be performed after imaging or processing (such as after etching). In another example, instead of inspecting every location on the wafer after etching to find a defect, potential defects can be predicted based on post-development procedures. In yet another example, a better model can be configured to more accurately predict potential failures after etching based on process output prior to the etching process. For example, this model includes a first part specifically associated with fault-free vias and a second part specifically associated with faulty vias. In one embodiment, the model is determined based on at least two measurements of the same structure (e.g., using SEM metric tools). The difference between the two SEM measurements can be used to build a model or classify defects in features prior to the etching process. The advantage of such defect prediction is the ability to adjust etching conditions or detect a significantly reduced number of locations, thereby enabling a corresponding reduction in detection time and an increase in wafer throughput. In another example, a correlation can be established, for example, between post-development and post-etching, so that the etching process can be controlled based on this correlation. The advantages of such correlation-based process control are effectively used to reduce defects after etching, thereby improving the yield of the patterning process. Figure 1 illustrates an illustrative lithography projection device 10A. The main components are: a radiation source 12A, which may be a deep ultraviolet excimer laser source or other types of sources including extreme ultraviolet (EUV) sources (as discussed above, the lithography projection device itself does not need to have a radiation source); illumination optics, which, for example, define partial coherence (represented as mean square deviation) and may include optics 14A, 16Aa, and 16Ab that shape the radiation from source 12A; a patterning device 18A; and a transmission optics 16Ac that projects an image of the pattern from the patterning device onto a substrate plane 22A. An adjustable filter or aperture 20A at the pupil plane of the projection optics can define the range of beam angles illuminating the substrate plane 22A. The maximum possible angle defines the numerical aperture NA of the projection optics as NA = n sin(Θmax), where n is the refractive index of the medium between the substrate and the last element of the projection optics, and Θmax is the maximum angle of the beam emitted from the projection optics that can still illuminate the substrate plane 22A. In a lithography apparatus, a source provides illumination (i.e., radiation) to a patterning device, and projection optics guide and shape the illumination onto a substrate via the patterning device. The projection optics may include at least some of components 14A, 16Aa, 16Ab, and 16Ac. An aerial image (AI) is the distribution of radiation intensity at a substrate level. A resist layer on the substrate is exposed, and the aerial image is transferred to the resist layer as a latent "resist image" (RI). The resist image (RI) can be defined as the spatial distribution of the solubility of the resist in the resist layer. A resist model can be used to calculate the resist image from the aerial image, an example of which can be found in U.S. Patent Application Publication No. 2009-0157360, the entire disclosure of which is incorporated herein by reference. The resist model is only concerned with the properties of the resist layer (e.g., the effects of chemical processes occurring during exposure, PEB, and development). The optical characteristics of a lithography device (e.g., the characteristics of the source, patterning device, and projection optics) define the aerial image. Since the patterning device used in a lithography device can be modified, it may be necessary to separate the optical characteristics of the patterning device from the optical characteristics of the rest of the lithography device, which includes at least the source and projection optics. In one embodiment, auxiliary features (sub-resolution auxiliary features and / or printable resolution auxiliary features) can be placed in the design layout based on how the design layout is optimized according to the method of the present invention. For example, in one embodiment, the method employs a machine learning-based model to determine the patterning device pattern. The machine learning model can be a neural network, such as a convolutional neural network, which can be trained in a certain way (e.g., as discussed in FIG3) to obtain accurate predictions at a faster rate, thus enabling full-chip simulation of the patterning process. A neural network can be trained (i.e., its parameters determined) using a set of training data. The training data can consist of a training sample set or a set of training samples. Each sample can be a pair containing an input object (usually a vector, which may be called a feature vector) and a desired output value (also called a supervision signal), or a pair consisting of the input object and the desired output value. The training algorithm analyzes the training data and adjusts the neural network's behavior by modifying its parameters (e.g., the weights of one or more layers) based on the training data. After training, the neural network can be used to map new samples. In determining the content context of a patterned apparatus pattern, the feature vector may include one or more characteristics of the design layout contained or formed by the patterned apparatus (e.g., shape, configuration, size, etc.), one or more characteristics of the patterned apparatus (e.g., one or more physical properties, such as size, refractive index, material composition, etc.), and one or more characteristics of illumination used in the lithography process (e.g., wavelength). The monitoring signal may include one or more characteristics of the patterned apparatus pattern (e.g., the critical dimension (CD), profile, etc. of the patterned apparatus pattern). Given the form as... One set of N training samples makes x i Let y be the feature vector of the i-th instance and y i Using the supervisory signal, train the algorithm to find the neural network. Let X be the input space and Y be the output space. A feature vector is an n-dimensional vector representing the numerical characteristics of an object. The vector space associated with these vectors is often called the feature space. Sometimes the following operations are convenient: using a scoring function... Let g be a value that returns the highest score for y. Let F denote the space of the scoring function. Neural networks can be probabilistic, where g adopts a conditional probability model. In the form of, or f, a joint probability model is adopted. There are two basic methods for selecting f or g: empirical risk minimization and structural risk minimization. Empirical risk minimization seeks the neural network that best fits the training data. Structural risk minimization includes a penalty function that controls for bias / variance trade-offs. For example, in one embodiment, the penalty function may be based on a cost function, which may be squared error, number of defects, edge placement error (EPE), etc. The function (or the weights within the function) may be modified to reduce or minimize the variance. In both cases, it is assumed that the training set contains independent and identically distributed pairs (x... i , y iOne or more samples, or composed of one or more samples. In one embodiment, a loss function is defined to measure how well the measurement function fits the training data. For training samples Predicted value The loss is Risk of function g Defined as the expected loss of g. This can be estimated from the training data. . In one embodiment, a machine learning model for the patterning process can be trained to predict, for example, the contours, patterns, CDs, and / or the contours, CDs, and edge placement (e.g., edge placement errors) of photoresist and / or etched images on the wafer. The goal of the training is to enable accurate prediction of, for example, the contours of printed patterns on the wafer, the intensity slope of aerial images, and / or CDs. The intended design (e.g., the wafer target layout to be printed on the wafer) is generally defined as a pre-OPC design layout that can be provided by a normalized digital file format such as GDSII or OASIS or other file formats. Figure 2 illustrates an exemplary flowchart for modeling and / or simulating the patterning process. As will be understood, these models may represent different patterning processes and need not include all the models described below. Source model 1200 represents the optical characteristics of the illumination of the patterning device (including radiant intensity distribution, bandwidth, and / or phase distribution). Source model 1200 may represent the optical characteristics of the illumination, including but not limited to numerical aperture settings, illumination standard deviation (σ) settings, and any specific illumination shape (e.g., off-axis radiating shapes, such as toroidal, quadrupole, dipole, etc.), where σ (or standard deviation) is the radial extent outside the illuminator. The projection optics model 1210 represents the optical characteristics of the projection optics (including changes in the radiation intensity distribution and / or phase distribution caused by the projection optics). The projection optics model 1210 may represent the optical characteristics of the projection optics, including aberrations, distortion, one or more refractive indices, one or more solid sizes, one or more solid dimensions, etc. Patterning device / design layout model module 1220 captures how design features are arranged in a pattern of a patterned device and may include a representation of detailed physical characteristics of the patterned device, such as described, for example, in U.S. Patent No. 7,587,704, which is incorporated herein by reference in its entirety. In one embodiment, patterning device / design layout model module 1220 represents the optical characteristics (including changes in radiation intensity distribution and / or phase distribution caused by a given design layout) of a design layout (e.g., a device design layout corresponding to features of integrated circuits, memory, electronic devices, etc.), which is a representation of the feature configuration on or formed by the patterned device. Because the patterned device used in a lithography apparatus can be modified, it is necessary to separate the optical characteristics of the patterned device from the optical characteristics of the rest of the lithography apparatus, which includes at least illumination and projection optics. The goal of simulation is often to accurately predict, for example, edge placement and CD, which can then be compared with the device design. Device design is typically defined as a pre-OPC patterned device layout and will be provided in a standardized digital file format such as GDSII or OASIS. The aerial image 1230 can be simulated using the self-generated model 1200, the projection optics model 1210, and the patterning device / design layout model 1220. The aerial image (AI) is the radiation intensity distribution at the substrate level. The optical properties of the lithography device (e.g., the properties of the illumination element, the patterning device, and the projection optics) determine the aerial image. The resist layer on the substrate is exposed by an aerial image, which is then transferred to the resist layer as a latent "resist image" (RI). The resist image (RI) can be defined as the spatial distribution of the solubility of the resist within the resist layer. The resist image 1250 can be simulated from the aerial image 1230 using a resist model 1240. An example of this can be found in U.S. Patent Application Publication No. 2009-0157360, the entire disclosure of which is hereby incorporated by reference. Resist models typically describe the effects of chemical processes occurring during resist exposure, post-exposure baking (PEB), and development to predict, for example, the profile of resist features formed on the substrate, and therefore are generally only related to these properties of the resist layer (e.g., the effects of chemical processes occurring during exposure, PEB, and development). In one embodiment, the optical properties of the resist layer (e.g., refractive index, film thickness, propagation, and polarization effect) can be captured as part of the projection optics model 1210. Therefore, generally speaking, the connection between the optical model and the photoresist model is the simulated aerial image intensity within the photoresist layer, which arises from the projection of radiation onto the substrate, refraction at the photoresist interface, and multiple reflections within the photoresist film stack. The radiation intensity distribution (aerial image intensity) is transformed into a latent "resist image" through the absorption of incident energy, which is further modified by diffusion processes and various loading effects. A sufficiently fast and efficient simulation method for whole-chip applications approximates the actual 3D intensity distribution within the resist stack using 2D aerial (and resist) images. In one embodiment, the resist image can be used as input to the pattern transfer post-processing model module 1260. The pattern transfer post-processing model 1260 defines the performance of one or more resist development processes (e.g., etching, development, etc.). Simulation of the patterning process can, for example, predict the contours, CDs, and edge placement (e.g., edge placement errors) in the resist and / or etched image. Therefore, the goal of the simulation is to accurately predict, for example, the edge placement of the printed pattern, and / or the intensity slope of the aerial image, and / or CDs. These values ​​can be compared with the expected design to, for example, correct the patterning process, identify locations where defects are predicted, etc. The expected design is typically defined as a pre-OPC design layout that can be provided using a normalized digital file format such as GDSII or OASIS, or other file formats. Therefore, the model formulation describes most (if not all) of the known physical and chemical methods of the overall manufacturing process, and each of the model parameters ideally corresponds to a different physical or chemical effect. Thus, the model formulation sets an upper limit on the degree of goodness to which the model can be used to simulate the overall manufacturing process. In patterning processes (such as photolithography, electron beam lithography, guided self-assembly, etc.), energy-sensitive materials (such as photoresist) deposited on a substrate typically undergo a patterning transfer step (e.g., via exposure). Following the patterning transfer step, various post-processes, such as resist baking, and removal processes, such as resist development and etching, are applied. These post-exposure steps or processes exert various effects on the substrate, giving the patterned layer or etching a structure (with dimensions different from the target size). The computational analysis of patterning processes employs predictive models that, when properly calibrated, can produce accurate predictions of the dimensions output from the patterning process. Post-exposure process models are typically calibrated based on empirical measurements. The calibration process includes running test wafers with different process parameters, measuring critical dimensions obtained after the post-exposure process, and calibrating the model to the measured results. In practice, well-calibrated models provide fast and accurate dimensional predictions, improving device performance or yield, enhancing process windows, or increasing design options. In one example, deep convolutional neural networks (CNNs) are used to model post-exposure processes with accuracy comparable to or better than models produced using conventional techniques, which often involve modeling using physical expressions or closed-form equations. Compared to traditional modeling techniques, deep learning CNNs reduce the reliance on process knowledge for model formation and increase the dependence on engineers' personal experience in model tuning. In short, a deep CNN model used for post-exposure processes consists of input and output layers, as well as multiple hidden layers such as convolutional layers, normalization layers, and pooling layers. The parameters of the hidden layers are optimized to give the minimum of the loss function. In an embodiment, the CNN model can be trained to model the behavior of any process or combination of processes associated with patterning processes. Random, opportunistic defects in the structure (also known as features) on a substrate are considered undesirable in lithography (e.g., EUV lithography). These defects can be identified after lithography imaging of the structure on the substrate or after etching an imaged substrate onto the substrate. The advantage of identifying and classifying defects after etching is that the image of the substrate is easier to interpret because it provides a direct correlation with the performance of the patterning process (e.g., yield). However, identifying defects after the lithography step provides a more direct measure of lithography performance. Several algorithms exist for classifying defects in structures (e.g., contact holes) after developing SEM images. For example, Fralilia software measures the substrate rather than the content on the SEM image. In another instance, Stochalis software analyzes CD-SEM images based on pixel brightness. CD-SEM stands for Critical Size Scanning Electron Microscopy, a specialized system used to measure the dimensions of fine patterns formed on semiconductor substrates. However, the criteria used for defect classification do not depend on the etching conditions used during etching, but rather on a common-sense interpretation of the SEM image. For example, error criteria could be the SEM contrast or critical size (CD) after developing the imaging substrate. In addition, CD-SEM can be used to measure CD using post-developed images (ADI) or post-etched images (AEI), and can also measure CD transfer. However, ADI SEM measurements damage the resist, which affects the CD after etching. Therefore, it is necessary to measure CD based on ADI and AEI at different locations, and to compare only the average CD, local CD uniformity (LCDU, standard deviation of CD), or CD distribution. A drawback of current defect (or fault) classification after photolithography is that the defect classification is independent of process or etching conditions, while the final defect (or fault) rate associated with the substrate does depend on these conditions. Defect classification can be calibrated based on the capture rate of programmed defects or by comparing defect rates before and after etching. However, it has been shown that programmed defects and random defects are statistically distinct (see publication P. De Bisschop, E. Hendrickx, "Stochastic effects in EUV lithography", Proc. SPIE 10583, Extreme Ultraviolet (EUV) Lithography IX, 105831K (19 March 2018); doi: 10.1117 / 12.2300541). Therefore, a good capture rate of programmed defects does not guarantee accurate results. Furthermore, it has been reported that defect rates before and after etching often do not match. For example, the reference P. De Bisschop & E. Hendrickx, "Stochastic effects in EUV lithography" (SPIE 2018), shows in Figure 9 that the failure rates of ADI and AEI can differ by 0.1 to 1000 times. As previously stated and now referring to Figure 3, SEM measurements following lithography can damage, for example, the resist layers 301 / 303 placed on the oxide layer 305. For instance, the resist layer 301 prior to SEM measurement shrinks to layer 303 after SEM measurement. In another case, carbon can be redeposited due to SEM irradiation of the resist 301. Therefore, taking two SEM measurements at the same location after both the lithography and etching steps can affect the CD measured by SEM. In one embodiment, the missing contact hole defect after etching is caused by a residual resist layer inside the contact hole. However, in this application, it has been recognized that the resist shrinks during SEM, but SEM electrons cannot completely remove the residual resist layer in the contact hole, see Figure 3. Furthermore, it has been recognized that carbon redeposition does not fill the complete hole, thus closing it. Therefore, these findings contradict conventional wisdom. Thus, SEM damage may affect CD, but not the failure rate of the structure associated with the pattern. Therefore, the method of this embodiment enables a more accurate determination of the failure rate, which is based on a one-to-one feature comparison between ADI and AEI, rather than using a failure rate determination based on the average CD. Figures 4A and 4B are flowcharts of a method for training a configuration to predict whether features associated with an imaging substrate will be defective after etching the imaging substrate. Procedure P401 relates to obtaining (i) a developed image 401 of the imaging substrate at a given location, the developed image including a plurality of features, and (ii) an etched image 402 of the imaging substrate at the given location, the etched image including etched features corresponding to the plurality of features, via a measuring instrument. In one embodiment, the model is a linear model or a machine learning model. In one embodiment, the empirical model varies with the physical properties of the features associated with the imaging substrate (e.g., after development). Examples of training based on the empirical model are illustrated in Figures 7A-7C, where the physical properties of the features after the development process (such as CD) are used as model variables. In one embodiment, the model is trained to identify a CD threshold (e.g., 16 nm) that can correctly classify more than 90% of potential defective features based on the ADI. In other words, features classified as defective in the ADI are more likely to fail after etching compared to features not classified as defective. In another example, the grayscale values ​​of the ADI are used to define the model. For example, the model is trained to identify grayscale thresholds that can correctly classify more than 90% of potential defective features based on the ADI. In one embodiment, Figure 6 illustrates training a CNN model. Examples of training based on CD, grayscale values, and the CNN model will be discussed later in this invention. In one embodiment, obtaining a developed image involves imaging a photomask pattern on a substrate via a patterning device; obtaining a developed substrate (e.g., resist developed) via an imaging substrate; aligning a measuring instrument (e.g., a SEM) to the developed substrate at a given location; and capturing an image of the developed substrate. In one embodiment, the developed image is a pixelated image, wherein the intensity value of a pixel indicates the presence of a feature on the substrate. For example, the intensity value of a pixel is a measure of the number of secondary electrons on the substrate. Secondary electrons may depend on: (i) the secondary electron yield of the material (e.g., the yield of a resist is higher than that of the underlying layer on the substrate), and (ii) the geometric arrangement that masks the intensity in the aperture, making it darker and causing the yield at the edge of the aperture to be higher than that at the center. For example, a white band can be seen around the aperture. In one embodiment, the metrology tool is an optical tool or an electron beam microscope. In another embodiment, the metrology tool is a photomask (SEM) (e.g., FIG. 28), and the ADI and AEI images are SEM images. In one embodiment, aligning the SEM with the imaging substrate or after etching the substrate is based on features addressed outside the FOV of the SEM. For example, the SEM can be configured to have built-in options for aligning addressed features, where the addressed features are associated with the location where a scan should be performed. In another instance, the unit cell of the pattern is larger than the nondeterministic nature of the SEM positioning system (e.g., for logic devices), so alignment can also be based on features in the image, using the pattern itself as a location marker. In one embodiment, including additional location markers within the FOV of the SEM for alignment purposes is undesirable because it affects the photolithography image, making the surrounding features unrepresentative. Additionally, adjusting the photomask pattern in the functional area of ​​the substrate used for metrology is undesirable. In one embodiment, obtaining a post-etched image involves etching an imaging substrate via an etching process having specified etching conditions; aligning a measuring instrument with the etched substrate at a given location; and capturing the post-etched image of the etched substrate. In one embodiment, the etching conditions include etchant composition, plasma gas parameters, etching rate, electromagnetic field, plasma potential, type of inductance or capacitance of the etching, temperature of the substrate, ion energy distribution, ion angular distribution, sputtering and redeposition rates, or a combination thereof. In one embodiment, alignment is never perfect, so the correlation between the ADI and AEI images is used to ensure correct alignment. However, a problem is that the addressing features used for alignment may be damaged or shifted due to the etching process, affecting addressing. Therefore, according to this embodiment, the AEI image is digitally shifted in discrete distances across all symmetrical directions relative to the ADI image, and the correlation between the ADI and AEI CDs is checked to determine which one has the largest correlation. In one embodiment, a significant maximum value exists due to small offsets (e.g., offsets of 1 or 2 tones). Procedure P403 involves training a configured model using developed image 401 and etched image 402 to determine defects of a given feature among a plurality of features in the developed image, wherein the determination of a defect is based on comparing the given feature in the developed image with a corresponding etched feature in the etched image. In one embodiment, a defect is characterized by at least one of the following: a binary determination of defective or non-defective; or the probability that a given feature is defective. In one embodiment, the training involves: aligning the developed image and the etched image based on the plurality of features; comparing (e.g., as shown in FIG. 5) each of the plurality of features in the developed image with a corresponding feature among the etched features in the etched image; determining, based on the comparison, whether a given etched feature in the etched image satisfies a defect condition; classifying the identified feature as defective in response to not satisfying the defect condition; and adjusting one of the model parameter values ​​of the model based on the defect of the identified feature. In one embodiment, the adjustment of the model parameter value includes adjusting the values ​​of the plurality of model parameters. In one embodiment, the defect condition refers to a desired constraint associated with the physical characteristics of a structure, wherein if the constraint is not met, the structure is considered defective. Referring to Figure 5, an example of determining defective features of a substrate based on a comparison between ADI and AEI is illustrated. In one embodiment, ADI and AEI are obtained using a measuring tool (e.g., the SEM in Figure 28 or the inspection tool in Figure 29). ADI is a developed image of the substrate obtained after further imaging of the substrate using a patterning device (e.g., a lithography device) following imaging (e.g., a pattern transfer step). Example: ADI includes an array of contact holes, such as holes at locations L1, L2, and L3. AEI is a post-etched image containing an array of holes corresponding to the contact holes in ADI. In this example, comparing the holes in AEI and ADI reveals holes missing in AEI. For instance, ADI includes holes at locations L1, L2, and L3 (without any defects). However, after the etching process, the AEI image shows missing holes at locations L1', L2', and L3', respectively, corresponding to locations L1 to L3, thus indicating a failure of the contact holes. In other words, the probability of defects in the holes at locations L1 to L3 of the ADI after etching the substrate. Therefore, the holes at locations L1, L2, and L3 of the ADI are classified as defective. Therefore, the present invention compares a hole at one location on the developed substrate with a hole at the same location on the etched substrate. Conversely, the prior art compares holes at different locations on the substrate after the development step and after the etching step, thus preventing a one-to-one comparison of features. Therefore, the present invention generates more accurate data related to defects associated with the substrate's structure. Consequently, models trained based on this data can more accurately predict defect features and allow for appropriate adjustments to patterning processes (e.g., etching processes) to improve patterning yield. In one embodiment, adjustments may include changing the focus or dose of the photolithography apparatus, or adjusting the chemical composition of the resist. In one embodiment, the model may be an empirical model trained based on defect conditions. In one embodiment, the defect condition is at least one of the following: grayscale values ​​in a closed contour within the etched image; and / or the physical properties of a given etched feature in the etched image. In one embodiment, the physical property is at least one of the following: the critical size of a given etched feature; and / or the displacement of a given etched feature relative to a given feature in the developed image. Figures 7A to 7C illustrate examples of defect classification based on CD (Content Defect). In one embodiment, the empirical model is based on a CD threshold value, where the CD threshold value determines whether a feature is likely to be defective. Figure 7C illustrates the CD distribution of contact holes in an ADI and the corresponding contact hole AEI, which may be defective (e.g., prone to failure). This distribution shows the CD values ​​of contact holes in the ADI, the CD values ​​of contact holes that are not faulty after etching, and the CD values ​​of holes that are faulty after etching. In Figure 7A, comparing ADI1 and AEI1 shows that hole CH1 is classified as faulty, which is accurate because CH1' is missing in AEI1. Furthermore, in Figure 7B, comparing ADI3 and AEI3 shows that hole CH3 is classified as fault-free, which is also correct because CH3' is not missing in AEI3. In one embodiment, the defective holes (e.g., missing in the AEI) are typically small in size relative to the desired size. For classification, a CD threshold value can be used on the ADI, where contact hole breaks are classified as potential defects in the AEI. For example, the CD threshold value Th1 is approximately 16 nm, so holes smaller than Th1 (e.g., 16 nm) can be classified as defective. In one embodiment, 91.3% of the holes were correctly classified. In one embodiment, the model is a machine learning model, such as a convolutional neural network. The model parameters are then weights, biases, or combinations thereof associated with one or more layers of the machine learning model (e.g., a CNN). Figure 6 illustrates an example CNN trained using ADI and AEI data (e.g., the ADI and AEI images in Figure 5) as input. Based on the input, as discussed herein, defect classification of features can be based on a comparison between ADI and AEI. The defect classification can then be the output of the CNN. In one embodiment, ADI and AEI can be feature vectors provided to the CNN. In one embodiment, a trained model (e.g., a CNN (e.g., Figure 5) and a CD-based threshold model (e.g., discussed in Figures 7A-7C) are further configured to predict a failure rate associated with a given pattern in the developed image. The failure rate indicates the rate of defect occurrence when the imaging substrate is etched using specified etching conditions. For example, the failure rate associated with a feature after etching can be determined based on the defect classification of the feature and the total number of features. For example, the failure rate of a feature (e.g., a contact hole) is the ratio of the total number of defective instances of the feature to the total number of features. In one embodiment, further configuration of the training model involves the following procedures. For example, procedure P405 involves classifying a plurality of patterns associated with the pattern of interest as defective or non-defective; procedure P407 involves determining the total number of defective patterns associated with the pattern of interest; and procedure P409 involves calculating the failure rate of the pattern of interest as the ratio of the total number of defective patterns to the total number of patterns in the plurality of patterns. In one embodiment, method 400 may further include adjusting etching conditions. An example implementation relates to the execution procedure shown in FIG4B. Program P411 involves imaging a desired pattern 411 on a substrate using a patterning device. The imaged substrate may be further developed (e.g., resist development) and post-processed (e.g., etched). Step P413 involves obtaining a developed image of the imaged pattern. Additionally, initial etching conditions 413 may be obtained. Step P415 involves using the developed image to execute a training model 403 to classify whether the desired pattern has defects after etching. Step P417 includes adjusting (or determining) the etching conditions 417 based on the classified defective patterns, such that the imaged pattern will be free of defects after etching. Figure 8 is a flowchart of a method for determining etching conditions or multiple etching conditions of an imaging substrate based on a failure rate associated with an imaging pattern. Step P901 involves obtaining a developed image 901 of the imaging substrate and initial etching conditions 902 for etching the imaging substrate. In one embodiment, the etching conditions include at least one of the following: etchant composition, plasma gas parameters, etching rate, electromagnetic field, plasma potential, type of inductance or capacitance of the etching, substrate temperature, ion energy distribution, ion angular distribution, sputtering and redeposition rates, or combinations thereof. Procedure P903 involves using the developed image 901 and initial etching conditions 902 to determine, through a trained model (e.g., training model 403), the failure rate of a feature associated with the imaged substrate, indicating whether the feature is defective after etching the imaged substrate. Thus, the trained model provides a failure prediction prior to actual failures that may occur after etching. Procedure P905 involves modifying the initial etching conditions 902 based on the failure rate to reduce the chance of defects appearing in features after etching. The modified etching conditions 905 can be further used to etch the imaged substrate, thereby improving the yield of the patterning process (e.g., reducing defects in features / structures on the substrate). In one embodiment, the etching conditions are modified into an iterative process. The iteration involves obtaining a relationship between a given etching condition and a given failure rate associated with a given feature. An etching model is executed using the developed image and the etching conditions to determine the etched image associated with the imaging substrate; based on the etched image, it is determined whether the given feature meets a defect condition; in response to not meeting the defect condition, another etching condition with a lower failure rate compared to the given failure rate is determined based on this relationship. In one embodiment, the defect condition of the feature is at least one of the following: omission of the feature; a displacement range associated with the feature; or a tolerance range associated with the critical size of the feature. In lithography, the structure printed on the substrate in resist is etched into the layers described below to manufacture the functional chip. Etching processes / steps can be used to smooth local variations in the CD (Curve Density) of features, resulting in reduced local CD uniformity (LCDU) after etching. One of the basic mechanisms by which etching reduces LCDU is the loading effect. The loading effect is the relationship between the etching rate and how a region on the substrate fills with structure. In one embodiment, the loading effect is that in densely populated regions of the substrate (e.g., regions with a high percentage of structure within a defined area compared to other areas on the substrate), the etching rate is lower than in less populated or relatively empty regions (e.g., regions with fewer building overlaps). Therefore, if a via or its adjacent via has a larger ADI via (e.g., due to local fluctuations), such as 1 nm larger than the average via, the etching rate will be slower. Slower etching may result in ADI vias smaller than the average via ADI by 1 nm. The physical causes of the loading effect are insufficient etchant, etching byproducts inhibiting etching, or both. Three example parameters related to etching load are: (i) the degree of load effect – i.e., the distance between features that influence each other (e.g., in nm). For example, values ​​between 40 and 100 nm. In one embodiment, the range is represented by the radius "R" in the etching rate equation below; (ii) the change in etching deviation in the relevant region whenever the average CD of adjacent features changes. The value of parameter (ii) depends on the average pattern density. The unit may be nm / nm, and example values ​​according to the invention may be between 0 and 0.75 nm / nm; preferably, between 0 and 0.75 nm / nm. (iii) the correlation coefficient between the size of adjacent holes in ADI and the size of etched holes in AEI. Example values ​​of the correlation may be between 0 and 0.2. In one embodiment, the etch load depends on the pattern density and has various length scales, ranging from the wafer scale to a portion of the wafer or a sub-resolution. In one embodiment, the pattern density is the portion of the area occupied by the structure in a given region around the structure or feature of interest. In one embodiment, the load effect can be on a length scale between 40 and 100 nm (e.g., 1 to 2 pitches). However, the invention is not limited to this range. In one embodiment, the load effect can be optimized from sub-resolution (e.g., 10 nm) to OPC tilt regions (e.g., regions with a radius of approximately 300 nm-1 μm) during etch optimization for a specific structure. The desired resist yield is ensured by tuning the target CD and the desired yield in the pattern transfer process. Existing etch optimization processes are lengthy and tedious. For example, manual optimization involves tuning the knobs of the etch equipment to achieve the desired grain yield. In one embodiment, the loading effect is a type of saturation. The loading effect (also known as loading behavior) may differ for each etch cycle. Specific etch techniques cycle between small differences in loading effect to reduce, for example, line edge roughness (LER) or line width roughness (LWR), or to improve local CD uniformity. For example, in addition to balancing sputtering, etching, and redeposition rates, the loading effect also has spatial and angular components to establish preferred etching for CD / pitch / workcycle characteristics varying in the X or Y direction. The angular component is controlled by electromagnetic field (EM), airflow design, or both. Once an etch cycle has consumed preferred material at spatial frequency and orientation based on the loading effect, the cycle will be effectively saturated. This saturation can be detected by a spectrometer. In one embodiment, data from an onboard optical spectrometer can be used to determine the composition of materials in the plasma as a means of detecting the end of each cycle. This can then be used to trigger the next cycle. In another embodiment, data from the spectrometer can be supplemented / replaced by data from an onboard laser interferometer, which can determine the thickness of material etched at specific locations / angles on the substrate. This invention relates to the quantification of etching characteristics such as short-range etch load effects or micro-loads. Currently, etch loads are characterized by analyzing test structures and modeling them in (OPC) software suites (e.g., Tachyon, Synopsis, Coventor, etc.) prior to etch optimization procedures. For this purpose, features with variable pitch and CD are printed and etched, and the etch rate is fitted to an empirical formula that accounts for the open region within a defined area (e.g., a circle of a specific radius) surrounding the point of interest. For example, an empirical model of the etch rate ER can be defined as follows: In the above ER model, ER is the etching rate associated with the etching process. ER nom The nominal etching rate is associated with the etching process. τ is the sensitivity to pattern density, and OA(R) is an open region within a circle of radius R. Parameters R, τ and ER nom These are the fitting parameters for the model. In advanced models, multiple radii can be accumulated, different convolutional filters can be applied, or directional dependencies can be incorporated. In one embodiment, the etch rate ER can be used to simulate (e.g., using convection software) etch deviations (e.g., the difference between ADI CD and AEI CD). Furthermore, the relationship between etch deviations, sensitivity to pattern density, and open areas can be modeled. ADI CD and AEI CD refer to the CD functions within ADI and AEI. In one embodiment, the pattern transfer process may include a combination of etching and (re)deposition. Examples of etching processes involve bulk sputtering of material and chemical etching. The sputtered material, the added gas composition, or a combination thereof can also ensure (re)deposition. In such processes, the set sheath voltage affects the ion angle and the μ-wave power density of the plasma / sputtering rate of the sputtering process. The sputtering rate of the material depends on the incident angle, ion velocity, and the material composition that allows for profile adjustment. For example, fluorine gas pressure determines redeposition in the etching process. In the etch optimization process, the required etch load and redeposition were not tuned based on the expected dependencies of the initial simulation, but rather primarily through physical intuition and experimental testing that scanned many etch parameters that affect some of the aforementioned effects. However, existing methods for estimating micro-loading effects have several drawbacks. For example, the measurements are not performed on the product structure necessary for the chip's functionality, but rather on the test structure. According to the present invention, the micro-loading effect strongly depends on the pattern density, therefore characterizing this loading effect on the target structure is more appropriate. Another drawback is that the short-range loading effect depends on the conditions in the focus-exposure matrix (FEM). This dependence cannot yet be quantified by existing methods. Figure 9 is a flowchart of a method for determining etching characteristics associated with an etching process. In one embodiment, etching characteristics are associated with the uniformity of the etched imaging substrate. For example, etching characteristics indicate that the substrate etches faster at the edges and slower at the center. In another example, etching characteristics indicate that the micro-loading effect refers to the etching rate depending on the local pattern density. In one embodiment, the etching rate refers to the etching depth per unit time, for example, 100 to 1000 angstroms per minute. The etching rate can be further used (by simulation using convener software) to determine etching deviations (e.g., the difference between ADI CD and AEI CD). For example, etch load refers to the difference between the etching rate associated with a given feature in a high-density region and the etching rate of the same feature in a low-density (isolated) region on the same wafer. One example is related to the local consumption of reactants. To compensate for etching characteristics such as loading effects, pressure, diffusion rate, etchant flux, etc., can be adjusted. The method of Figure 9 is discussed in further detail below. In one instance, metrology (e.g., AEI CD) can be performed midway through the etching process (if you have an iterative etching process) or after sequential permeation synthesis (SIS), a step that can be applied before etching. Note that the post-development and post-etching images are used as examples of different processes in the patterning process. However, the invention is not limited to post-development and post-etching. Those skilled in the art can apply the methods described herein to other processes related to the patterning process. For example, a correlation can be established between a first layer (e.g., a resist layer) and subsequent layers, and different processes (e.g., etching) can be performed on the first layer after processing it. The principles described herein apply to any etching and combination of layers of a patterned substrate (e.g., a first resist layer, a second resist layer, etc.). Procedure P1001 relates to obtaining, via a measuring instrument: (i) a developed image 1001 (ADI) of an imaged pattern at a given location on a substrate, the imaged pattern including a feature of interest and adjacent features adjacent to the feature of interest; and (ii) an etched image 1002 (AEI) of the imaged pattern at the given location on the substrate, the AEI including etched features corresponding to the feature of interest in the ADI. For example, the imaged pattern may be an array of contact holes at the center of the substrate. Within the array of contact holes, the feature of interest may be a contact hole at a specific coordinate (e.g., GDS coordinate). In one embodiment, the feature of interest is at least one of the following: a contact hole; a line; a line end; or a critical feature or a portion thereof. In one embodiment, adjacent features are at least one of the following: a plurality of contact holes in a defined direction relative to the feature of interest (e.g., see FIG. 10); and / or a plurality of lines having a defined spacing. In one instance, adjacent features may be line segments of the same line that are a certain distance away from the line segment of interest. In some embodiments, multiple instances of a feature of interest (or multiple different features) in an image and their neighbors can be used to establish a correlation between the feature of interest and its corresponding etched counterpart. In some embodiments, a feature of interest in multiple images (e.g., obtained at different locations such as the center, edge, or other radial distance on a substrate) can be used to establish a correlation coefficient. In another embodiment, multiple images that are very close to each other can be obtained from the perspective of the lithography apparatus. For example, multiple images are present at least in the same die. The CD may differ slightly at different wafer locations, which will then dominate the correlation coefficient. The instance distance between images can be 1 μm. Figure 10 illustrates an example of ADI, including feature 1040 of interest and adjacent features 1050a to 1050f, and an AEI image including only etched feature 1060 corresponding to feature 1040 of interest. In other words, during the patterning process, 1040 and 1060 are both at the same contact hole at different points (e.g., after imaging and after etching). In one embodiment, adjacent features 1050a to 1050f are adjacent to feature 1040 of interest. Adjacent features 1050a to 1050f are separated from feature 1040 of interest by a specified distance. In one embodiment, the specified distance also affects etching characteristics. For example, the closer the adjacent features are to feature 1040 of interest, the higher the etch load effect. In one embodiment, a portion of the area occupied by adjacent features 1050a to 1050f surrounding the feature of interest defines the pattern density. The larger the surface area of ​​the adjacent feature overlap, the greater the pattern density. As previously mentioned, pattern density affects the etching characteristics of the etching process (e.g., etch load effect). Procedure P1003 involves using ADI and AEI to determine the correlation 1005 between the etched features and adjacent features associated with the features of interest in ADI, which characterizes the etch characteristics associated with the etch procedure. In one embodiment, the correlation determination involves using multiple ADI images having a feature of interest. Therefore, the correlation determination may involve obtaining (i) multiple ADIs at multiple given locations on a substrate, each ADI having the same feature of interest (e.g., a contact hole with a CD of approximately 21 nm), and (ii) multiple AEIs at multiple given locations, each AEI having an etched feature of interest corresponding to the feature of interest (e.g., an etched contact hole with a CD of 20 nm). In one embodiment, the ADI CD is larger than the AEI CD of the feature of interest; for example, the ADI CD may be 21 nm, and the AEI CD may be 20 nm. A correlation can then be established between adjacent features of the feature of interest in each ADI and the etched feature of interest in each AEI. An example correlation function using CD is described below; however, similar functions can be established using other physical properties associated with the feature of interest (e.g., quantifiable measurements). In one embodiment, the correlation is a function of the average pattern density of adjacent features adjacent to the feature of interest. In one embodiment, the correlation between a feature etched in ADI and adjacent features depends on at least one of the following: the geometry of the feature of interest or adjacent features; and the geometry of the feature of interest; the geometry of auxiliary features or deviations associated with the feature of interest; the distance between the feature of interest and adjacent features; the distance along a line element; the critical dimension of at least one feature; the coordinates on the substrate associated with the feature of interest, adjacent features, and the etched feature of interest; auxiliary features or the absence of auxiliary features around the feature of interest (e.g., around the feature of interest refers to the end of a series of features including the feature of interest); or random variations in the expected location of an edge position associated with its expected feature. In one embodiment, the expected location of an edge (e.g., feature profile) refers to the average of GDS locations (e.g., in a design layout) or similar features. In one embodiment, correlations can be indirectly calculated based on patterning process parameters, such as scanner dose and focus, etching temperature, plasma gas parameters, etchant composition, electromagnetic field, plasma potential, inductive or capacitive etching, temperature, ion energy distribution, ion angular distribution, and parameters related to sputtering and redeposition rates. For example, indirect determination of the correlation involves tuning or simulating the patterning process by adjusting one or more of the aforementioned patterning process parameters. In one embodiment, the geometry of the feature can be a hole or a line. The correlation coefficient associated with a contact hole will differ from that of a line. For example, if the feature of interest is surrounded by a line, the load effect may decrease along the length of that line. Furthermore, an L-shaped feature may have a different correlation than a line because an L-shape has corners and is therefore affected differently by adjacent features than a line. In one embodiment, the correlation also depends on the critical dimensions of adjacent features. For example, the larger the critical dimensions of adjacent features, the greater the load effect (see Figure 11A). In one embodiment, the following equation is used to calculate the correlation: In the above equation, Let be the relevant vector, where CDAEI is the AEI CD of the feature of interest; CDADI i For the first The ADI CD of i neighbors, r is the correlation coefficient, and This is the correlation matrix. The above formula is an example and is not limited to correlation based on CD. As mentioned earlier, correlations can be calculated based on other physical properties associated with the feature of interest and adjacent features (e.g., geometry, distance, auxiliary features, etc., as mentioned earlier). In the example experiment, referring to Figure 10, a measurement tool (such as SEM) measured the exposure of 10 under seven conditions based on the focus-exposure matrix (FEM). 5 There are 6 contact holes. The contact holes are located on a hexagonal grid, so each contact hole (e.g., 1040) has 6 adjacent holes (e.g., 1050a to 1050f). Next, the exposed substrate is etched using an etching recipe (e.g., IMEC TITAN VIA etching). Furthermore, for example, a MATLAB script configured to provide organizational data for further use is used to determine the CD values ​​of the contact holes before and after etching. Assuming a simple linear relationship between the CD values ​​of the hole ADI and AEI, a correlation between ADI and AEI can be established. For example, the partial variance of the AEI CD explained by the ADI CD of contact hole 1040 is thus the square of the correlation coefficient given below (…). R 2 ): For the variance component of the AEI CD interpreted from the ADI CDs of adjacent contact holes 1050a to 1050f, a correlation vector is used. The instance vectors are as follows: Then you can use the following formula to perform the operation. : Examples are illustrated in Figures 11A and 11B. The relevant diagram (in Figure 11A) shows the correlation for the seven conditions in FEM. and The y-axis represents a portion of the variation in AEI CD relative to the average CD of the vias, which is explained by the ADI CD of the feature of interest itself or the ADI CD of its neighborhood. The y-axis can be a dimensionless number, which can be expressed as a percentage when multiplied by 100. The correlation plot shows that short-range etch load has the strongest effect for the largest CD, while it is much weaker for relatively small CDs. Short range can be, for example, within the FOV of the SEM. Therefore, short-range etch load does indeed depend on the pattern density. In addition, Figure 11B shows a negative correlation, indicating that the etch load effect also depends on the FEM conditions. In the current example of Figure 11B, for the condition with the largest average CD in Figure 11A, the AEI CD (Y-axis) is plotted relative to the weighted average (X-axis) of the adjacent ADI CDs. The variation in CD, represented by the gray area 1103, is due to random fluctuations in the patterning process, and line 1105 illustrates the moving average of the CD of the feature of interest. Line 1105 illustrates the negative correlation between the adjacent AEI CDs and ADI CDs. The negative correlation indicates that, for the FEM condition shown, the correlation between the ADI etched feature and adjacent features is relatively high, which is printed with a relatively large average CD. To clarify, the relatively large average CD is not a randomly large CD within the imaging pattern, but rather a CD associated with a condition or pattern having a larger pattern density, because the average CD of that condition or pattern is larger (e.g., if a scanner dose height is too high or a mask design with a large CD is used). In one embodiment, the method includes determining etching conditions associated with an imaged pattern based on a correlation and at a given radial distance between the center and the edge of the substrate, such that the correlation remains within a target range. In one embodiment, the etching conditions depend on at least one of the following: the location of the etched substrate, which is the radial distance between the center and the edge of the substrate (e.g., the center or edge of the substrate); other distances to the substrate or the region of interest on the substrate); an etching cycle; an etching chamber; the sequence of etching cycles and deposition steps; or tuning parameters associated with the etching chamber, the tuning being based on the correlation's sensitivity to changes in the tuning parameters. In one embodiment, etching conditions for an imaged pattern located at the center of the substrate are determined based on correlation, such that the correlation is within a target range. In another embodiment, the method includes determining etching conditions for an imaged pattern located at the edge of the substrate based on correlation, such that the correlation remains within a target range. Typically, even with the same pattern density, different etching conditions may be required at different locations on the substrate due to substrate thickness distribution, drift associated with the etching equipment, etc. In one embodiment, etching conditions include etchant composition, plasma gas parameters, etching rate, electromagnetic field, plasma potential, type of etching inductance or capacitance, substrate temperature, ion energy distribution, ion angular distribution, parameters associated with sputtering and redeposition rates, etching cycle parameters based on saturation effect, or combinations thereof. In one embodiment, the saturation effect is a loading effect, which can be used to determine the composition of the material in the plasma as a means of detecting the end of each cycle. This can be used to trigger the next cycle. In one embodiment, the etching conditions can be adjusted compared to ideal etching conditions. For example, ideal etching conditions can be adjusted in existing etching equipment (e.g., by adjusting parameters such as etchant composition, plasma gas parameters, etching rate, etc.) or in a design tool used to determine the etching conditions. The design tool allows adjustment of parameters, such as electromagnetic fields, capacitive or inductive etching, to keep the relevant parameters within the desired target range. In one embodiment, the method further includes generating a power spectral density of the correlation based on the correlation between the AEI CD and ADI CD (e.g., a correlation established using lines as features). The power spectral density indicates the magnitude and range of etching characteristic effects (e.g., loading effects). In one embodiment, the power spectral density can be calculated in a spatial domain (e.g., along the length of the line feature). For example, the power spectral density can be calculated by performing a Fourier transform on the correlation in the spatial domain, where the correlation is a continuous function of the distance between two points. An instance power density of the correlation of a line can indicate that the correlation is relatively high at smaller intervals between line segments and gradually decreases for larger intervals between line segments. Furthermore, appropriate etching conditions can be determined based on the power spectral density. For example, the etching recipe can be defined based on the magnitude of the loading effect along the line so that the correlation between the ADI and AEI of the line remains within a target range during the etching process. In one embodiment, the performance of the correlation used to monitor and control the patterning process can be improved, for example, by controlling the etching formula and etching conditions (e.g., tuning parameters), so that the correlation remains within a target range. For example, the etching chamber can be monitored based on the critical dimensional uniformity across the entire substrate or the CD difference between different features at different radii across the entire substrate. In one example, control involves determining the impact on CD based not only on an optional etching knob (e.g., an etching knob), but also on factors such as air pressure, power, DC, and temperature. The desired performance can then be monitored (e.g., whether the relevant parameters remain within target ranges). The benefit of doing so is that more substrate chips will be within specifications during final yield testing. Furthermore, the benefit of monitoring based on parameters is that it may be unnecessary to re-check the final yield, such as in electronic test vehicles or by rigorously inspecting millions of features. Figure 12 is a flowchart of method 1200, which determines etching conditions or a plurality of etching conditions associated with the etching process based on the correlation between the etched feature of interest and adjacent features in the ADI (discussed above). Method 1200 is used to monitor and control the etching process based on a relevant target range. In one embodiment, a relevant target range (e.g., 0 to 0.4) may be defined, and etching conditions may be defined such that the target range is met during or after the etching process. The target range may be constant across the entire etched substrate; however, the etching conditions may vary, for example, at the center and edges of the substrate. Method 1200 is further described in detail below. In one embodiment, etching conditions are determined such that the ranges of a plurality of parameters (e.g., including correlations) related to the loading effect are within desired specifications. For example, the effects of density fluctuations and variance portions explained by ADI neighbors are also within desired specifications. For example, the degree of loading effect: less than 100 nm; impact density range: 0.3 to 0.35 nm / nm. A portion of the variance is illustrated as being between 0.15 and 0.17. Procedure P1201 includes obtaining a correlation 1201 between an etched feature of interest (AEI) in the etched image and adjacent features (ADI) associated with the etched feature of interest in the developed image. In one embodiment, obtaining the correlation between the etched feature and adjacent features includes obtaining the correlation between the etched feature and a plurality of adjacent features. In one embodiment, the correlation is obtained according to the method of FIG9. For example, obtaining the correlation involves obtaining (i) a developed image (ADI) of the imaged pattern at a given location, including the imaged pattern of the feature of interest and adjacent features adjacent to the feature of interest, and (ii) an etched image (AEI) of the imaged pattern at a given location, the AEI including the corresponding etched feature of interest ADI; using the ADI and AEI to determine the correlation between the etched feature and adjacent features associated with the feature of interest in the ADI. Program P1203 includes etching conditions 1205 based on relevant determinations and the etching process, so that the relevant conditions are kept within the target range. In one embodiment, the determination of etching conditions depends on at least one of the following: the location of the etched substrate, either the center or the edge of the substrate; the etching cycle of the etching process; the etching chamber used in the etching process; the sequence of the etching cycle and deposition steps; or tuning parameters associated with the etching chamber, the tuning being based on a sensitivity to changes in the tuning parameters. In one embodiment, the tuning parameters include a plurality of tuning parameters. In one embodiment, determining etching conditions involves monitoring the CD difference between instances of the etched feature of interest across the substrate's CDU or at different radii. For example, etching conditions can be determined by changing an optional etching knob, such as by altering etching conditions (air pressure, power, DC, temperature, etc.), and evaluating their impact on CD. One advantage of determining etching conditions based on these factors is that, compared to existing methods, more grains will be within specifications during final yield testing. In one embodiment, the etching conditions or etching formulation can be described as having a start stage, an intermediate stage, and a finish stage. Each stage of etching can consist of one or more "mini" etching formulations, which together represent the etching formulation. In one embodiment, these "mini" etching formulations are used to fine-tune the results of the etching process (e.g., characterized by the CD of the etching feature of interest or yield). Therefore, different "mini" etching formulations with slightly different behaviors, such as, but not limited to, different loadings, can be applied to obtain the desired final result (e.g., CD or yield). In one embodiment, such fine-tuning of the etching formulation is achieved by tuning different plasma gas parameters, power settings, gas flow rate settings, etc. Please note that although a “micro” etching formulation can be defined, the etching process is typically based on the overall etching result, not on the workpiece that can be represented as the start, middle, or end of the etching process. For example, the entire etching process can be performed without interruption or cessation at the start, middle, or end of the etching process. In one embodiment, when multiple materials are being etched on a substrate, they can all be etched in a single chamber using a combined etching process. The etching formulation for the first material will include multiple steps, followed by another etching formulation for the second material. This method may involve a different set of multiple steps (including modifications to the gas, etc.). For multi-material etching, each material may require varying degrees of anisotropy, so the outline of the etched pattern is not a perfect replication from one material to another. These differences can lead to different associated results. It is possible (but not common) to stop etching between layers to observe the individual outlines of the multi-profile etching. In one embodiment, the etched profile can be characterized by the geometry of the etched features, such as the height, angle, and width associated with the etched features. In one embodiment, determining the etching conditions involves tuning the values ​​of adjustment parameters associated with a given etching chamber so that the correlation associated with a given imaging pattern remains within a target range. The measurements provided by the present invention (e.g., ADI CD and AEI CD) help to understand the etching process and can accelerate the etching optimization process even when operated manually. For example, optimization is performed based on the correlations obtained from the method in Figure 3. Because the product structure is measured, the resulting etching is better optimized for the most critical structures, thereby improving the yield of the patterning process. Machine learning models (e.g., neural networks, CNNs, DCNNs, etc.) are mostly black boxes. Even when trained using supervised learning (e.g., by human intervention), such black-box models may make incomprehensible predictions and take actions, such as adjusting process parameters (e.g., dosage / focus, etching formula) to improve patterning processes. Therefore, when evaluating actions based on predictions or choosing whether to deploy a new model, it is necessary to understand the reasons behind the predictions made by the trained model. In one embodiment, a white-box model of a patterning procedure may have lower accuracy compared to a black-box model. For example, a white-box model may predict with 91.3% accuracy; however, the predictions made by this model may be easily interpreted. For instance, by examining the CD values ​​of features, it is easy to understand a model that classifies features as defective or non-defective based on the CD values ​​of features in an ADI image. On the other hand, a black-box model (e.g., a CNN) may predict with higher accuracy (e.g., 95.8%) than a white-box model. However, the decisions of a black-box model are difficult to interpret. For example, it may not be easy to deduce the feature defects based on the prediction results. Therefore, in one embodiment, the user may choose to sacrifice accuracy for interpretability. Interpretability is a criterion used to improve the interpretation of predictions from black-box models. It provides the relationship between input variables and the predictions of the black-box model. For example, based on input variables (such as pixel values ​​of ADI images), this relationship provides a qualitative / quantitative understanding of the prediction results (e.g., whether the ADI function is defective or as previously mentioned). In this invention, the relationships between input variables (e.g., features in an ADI) can be explained via an interpretation model associated with the features of interest. In one embodiment, the interpretation model helps explain defects in a particular feature. For example, the interpretation model can identify portions of an ADI image that explain feature defects. In one embodiment, the interpretation model can be determined using methods such as Locally Interpretable Model Agnostic Interpretation (LIME), Principal Component Analysis (PCA), or discriminant analysis such as Linear Discriminant Analysis (LDA) or Quadratic Discriminant Analysis (QDA). Figure 15A illustrates an example flowchart of determining an interpretation model configured to identify relevant features of an ADI that explains the defect classification of any input ADI. Referring to Figure 15A, method 1530 includes procedures P1531 and P1533, which are discussed in detail below. Procedure P1531 includes obtaining (i) an image after development (ADI) of the imaging substrate at a given location and (ii) an image after etching (AEI) of the imaging substrate at a given location using a metrology tool. Procedure P1533 includes determining an interpretation model 1510 based on the ADI and AEI, wherein the interpretation model 1510 is configured to identify defective portions of features in the interpretable input ADI. In one embodiment, the interpretation model is determined by employing a locally interpretable model-agnostic interpretation method configured to be generated together with the interpretation model, wherein the interpretation model is configured to generate an interpretation map classifying the interpretable input ADI. An example of the LIME method is discussed below with reference to Figure 15B. In one embodiment, the determination interpretation model 1510 includes determining the correlation between the ADI and AEI; and using the correlation data to perform principal component analysis or discriminant analysis to determine eigenvectors whose eigenvalues ​​exceed a specified threshold. Furthermore, the determination includes projecting the input ADI onto the eigenvectors to calculate a classification value; and, in response to a classification value exceeding a specified threshold, identifying a portion of the input ADI as a defect in interpreting the functionality of the input ADI. Examples of the PCA method are further discussed in detail below. In one embodiment, PCA (and similarly, LDA or QDA) can be performed based on the correlation between variables of the ADI image. In one embodiment, the pixel intensity of the ADI image can be used to determine the correlation. For example, (and similarly, LDA or QDA) can be performed as follows. In this example, the grayscale values ​​or pixel intensities of the ADI image (e.g., ADI10, ADI20, or ADI30 in Figures 14A to 14C) can be represented as vectors. For example, an ADI image can be cropped into a 51×51 pixel size, resulting in a 51-pixel long image. 2 = 2601 vector. Based on this vector The correlation matrix of all ADI images can be calculated as follows: (For example, ADI10, ADI20 and ADI30 in Figures 14A to 14C): In the above equation, Vectors on all slices (e.g., ADI10, ADI20, and ADI30) The average value, and It is the covariance between two pixels in the image. In the relevant matrix The documentation specifies whether to perform PCA, LDA, or QDA. In PCA, the matrix is ​​operated on. The correlation matrix contains eigenvalues ​​and eigenvectors. These eigenvalues ​​can be significantly greater than 1, approximately 1, while some eigenvalues ​​are much less than 1. Higher eigenvalues ​​indicate a corresponding set of highly correlated variables. For example, referring to ADI10 (Figure 14A), the correlation matrix may indicate that pixels associated with contact hole F10 (the darker areas) are highly correlated. Therefore, the correlation matrix may indicate the presence of a contact hole. Alternatively, for ADI30 (Figure 14B), the correlation matrix may show a relatively low correlation between pixels of contact hole F30, which may indicate a faulty contact hole. Feature vectors corresponding to large eigenvalues ​​(e.g., greater than 1) indicate that the grayscale values ​​of these pixels change together, which can indicate contact hole or printed hole defects. These feature vectors corresponding to large eigenvalues ​​can be used to interpret model 1510. For example, all instances of a crop are projected onto a few feature vectors with large eigenvalues, and the presence of strong printed hole clusters and defects is examined in one of these directions. Feature vectors with strong clustering directions are observed to be relevant features for defect classification. By converting these feature vectors back to the form of a 51×51 crop, relevant features in the input ADI, such as defects or printing defects, can be interpreted. In another instance, LDA / QDA also identifies feature vectors that can automatically find the optimal projection direction to distinguish between printed and faulty holes. In one embodiment, a LIME method (e.g., method 1500 discussed with reference to Figure 15B) can be used. LIME is an interpretation technique that interprets the predictions of any classifier in an interpretable manner by locally learning an interpretable model around the prediction. Instances of interpretable models are discussed in Figure 13 below. Figure 13 illustrates the decision dataset used to train the instance interpretation model. The decision dataset can be obtained from the trained model (e.g., the CNN model in Figure 4A). For example, trained model 403 uses an ADI image including multiple features to predict defects in multiple features after etching. For example, after etching using a specific etching recipe, the ADI function will print whether it is defective or defect-free. In Figure 13, prediction regions R1 and R2 represent complex decision functions of a trained machine learning model (unknown to the interpretation model). In one embodiment, this decision involves predictions from trained model 403. Therefore, prediction regions R1 and R2 correspond to whether features in a given ADI image will be defective or defect-free after etching. In one embodiment, such prediction regions R1 and R2 are separated by nonlinear boundaries that cannot be well approximated by a single linear model. Therefore, a set of models can be defined, where each model can locally (e.g., around selected points) explain why a particular prediction is made. In one embodiment, point P0 (thick cross) is an instance to be interpreted by the interpretation model. For example, point P0 represents a feature of interest in an ADI image. According to one embodiment, a fitted line is used to interpret the interpretation model that interprets the prediction associated with point P0, wherein the fit is based on data in the vicinity of point P0. The data in the neighborhood of point P0 contains two categories, represented by regions R1 and R2. For example, the first set of points P1, P2, P3, P4, P6 represents the judgment of a trained machine learning model that the etched feature will be defective. The second set of points P10, P11, P12, P13, ..., P20 represents the judgment of a trained machine learning model that the etched feature will not be defective. In one embodiment, the method for determining the interpretation model includes sampling instances (e.g., P1 to P20), obtaining predictions using a trained machine learning model (e.g., 403), and weighting (e.g., from points P1 to P20 here) to interpret the predictions (e.g., features of interest) by the proximity of the samples to instance P0. Then, a model is fitted based on the weighted predictions using a fitting method. For example, a least-squares error-based fitting method may be used. According to one embodiment, the fitted model is referred to as the trained interpretation model. In one embodiment, the dashed line M1 represents a trained interpretation model that provides a local (not global) interpretation of point P0. For example, "local" refers to points near the illustrated instance. The dashed line M1 can also be referred to as the trained interpretation model M1. In other words, the trained interpretation model M1 provides a linear approximation of the nonlinear boundary around line M1 based on data points near point P0 and the nonlinear boundary B1. In one embodiment, the interpretation model M1 can be trained to reduce (e.g., minimize) a cost function, such as a function of the difference between the output of the interpretation model M1 and the predictions near point P0. This invention is not limited to a specific fitting method. Other data fitting methods, such as least squares, Gaussian fitting, minimum bias, etc., can be used. The example in Figure 13 illustrates a binary decision used to explain a concept. However, the decision may be a binary classification, or it may include multiple categories (e.g., based on probability, where multiple probability ranges correspond to multiple categories). The scope of this invention is not limited to binary decision-making. Figures 14A to 14C illustrate example results of applying an interpretation model (e.g., M1) to an ADI image that includes features of interest. In this example, ADI images ADI10, ADI20, and ADI30 include features of interest F10, F20, and F30, respectively. In one embodiment, a trained model (e.g., 403) associated with a procedure (e.g., an etching procedure) predicts whether a particular feature in the ADI will be printed with defects or without defects after etching. Figures 14A and 14B illustrate examples of features F10 and F20 in ADI images ADI10 and ADI20, respectively, which are predicted to be printed without defects. For example, a trained model (e.g., model 403 trained according to the method of Figure 4A) predicts that features in images ADI10 and ADI20 will be printed without defects. Figure 14C illustrates an example of feature F30 in ADI image ADI30, which is predicted to be printed with defects. For example, a trained model (e.g., model 403 trained according to the method of Figure 4A) predicts that features in images ADI10 and ADI20 will be printed without defects. However, as mentioned earlier, the training model 403 can be a machine learning model (such as a CNN or DNN), which contains a weighted network of neurons distributed across multiple layers and interconnected. Therefore, the rationale behind this prediction is not available. The rationale or explanation behind such predictions can be obtained through trained interpretation models (also called interpretation models). For example, for each feature of interest, an interpretation model can be trained according to Figure 13. For example, a first interpretation model M10 is trained to explain predictions related to the feature of interest F10 in ADI10. Similarly, a second interpretation model M20 is trained to explain predictions related to the feature of interest in ADI20, and a third interpretation model M30 is trained to explain predictions related to the feature of interest in ADI30. In one embodiment, interpretation models M10, M20, and M30 generate interpretation maps MAP10, MAP20, and MAP30, respectively, as shown in Figures 14A to 14C. The interpretation maps (e.g., MAP10, MAP20, and MAP30) display patches that explain the contribution of each pixel surrounding the feature of interest (e.g., F10, F20, and F30) to making a prediction (e.g., whether it is defective or not) in relation to the feature of interest. In one embodiment, the patches may have intensity values ​​indicating the influence of adjacent features (e.g., corresponding to points P1 to P20 in Figure 13) on the determination of whether the feature of interest will be defective or not after etching. For example, in the interpretation map MAP10, patch E1 (e.g., a positive pixel value) helps determine that feature F10 (in ADI10) will not be defective after etching, while patch E2 (e.g., a negative pixel value) helps determine that feature F10 (in ADI10) will be defective after etching. Similarly, in the interpretation map MAP20, patch E3 (e.g., a positive pixel value) helps determine that feature F20 (in ADI20) will not be defective after etching. Finally, in the interpretation map MAP30, patch E4 helps determine that feature F30 (in ADI30) will not be defective after etching, while patch E5 (e.g., a negative pixel value) helps determine that feature F30 (in ADI30) will be damaged after etching. The interpretation map or its pixel values ​​can be further used to take actions such as adjusting the patterning process recipe (e.g., the etching recipe) to improve the yield of the patterning process. In one embodiment, the ADI image and its corresponding interpreted image may be superimposed to generate a superimposed image, depending on the circumstances. For example, ADI10 and MAP10 may be superimposed to generate a superimposed image S10. Similarly, ADI20 and MAP20, and ADI30 and MAP30 may be superimposed to generate superimposed images S20 and S30, respectively. In one embodiment, the superimposed image or pixel values ​​may be further used to take actions, such as determining the composition of certain portions of the imaging substrate. Figure 15B is a flowchart of a method 1500 for determining an interpretation model associated with a feature of interest. The interpretation model is configured to interpret predictions related to the feature of interest. For example, if there are N features of interest, then N interpretation models can be determined—one for each feature of interest. As shown in Figures 13 and 14A to 14C, the interpretation model can generate an interpretation map of the feature of interest, which can then explain the contribution of the vicinity of the feature of interest to making predictions associated with that feature. Furthermore, based on the interpretation map, actions related to improving the patterning process (e.g., etching process) can be taken. For example, if the interpretation map includes patches that contribute relatively highly to predicting that a feature will be defective, the etching formula can be adjusted for that specific patch. According to the invention, method 1500 can be performed after training a model (e.g., 403) related to the pattern forming process to predict, for example, the future characteristics of any feature in a developed image. The future feature (also referred to as the prediction) can be a CD or a feature defect. For example, the trained model 403 can predict whether a feature in an ADI image will be printed with or without defects after etching using an etching recipe. Method 1500 is not limited to a specific prediction or classification associated with a feature. In the following procedures of method 1500, the instance of prediction is a feature defect. As previously stated, a defect can represent the probability of feature failure after etching. In an example, to illustrate the concept of this method, a defect can be visualized as binary, such as defective or defect-free. Procedure P1501 includes, for example, obtaining a training dataset by executing a training model 403 associated with a patterning process (e.g., an etching process). In one embodiment, the training dataset includes a plurality of predictions 1502 associated with a plurality of features in the neighborhood of the feature of interest 1501 in the after-development image (ADI), each of the plurality of predictions being made by the trained model 403. In one embodiment, for training purposes, the neighborhood of the feature of interest refers to the location of a feature surrounding the feature of interest 1501. For example, referring to FIG13, points P1 to P20 are near the point of interest P0. In one embodiment, obtaining a plurality of predictions 1502 includes executing a trained model to predict the features of each of a plurality of features in the neighborhood of the feature of interest 1501. In one embodiment, similar to the previously discussed procedure P403, obtaining the developed image involves imaging a photomask pattern on a substrate via a patterning device; obtaining a developing substrate (e.g., resist development) of the imaging substrate; aligning a measurement tool (e.g., the SEM in Figures 28 and 29) with the developing substrate at a given location (e.g., the location of the feature of interest); and capturing the image of the developed substrate. In one embodiment, the developed image may be obtained from a database storing measurement data (e.g., SEM images) of the substrate (e.g., the database of the computer system in Figure 30). In one embodiment, an ADI image including a plurality of features is provided as input to a training model 403. The training model then predicts, for example, defects in the plurality of features. In one embodiment, defects in features of a 1502 series ADI are predicted, where a defect indicates the probability that the feature will be defective after etching. In another embodiment, a feature of interest in a 1502 series ADI is predicted to be printed with or without defects after etching. Procedure P1503 includes determining the distance 1503 between each position of a plurality of features and the feature of interest. In one embodiment, distance 1503 is a linear distance between two positions, specifically the position L1 of the feature of interest 1501 and the position L2 of an adjacent feature. For example, referring to FIG13, the distance D1 (unlabeled) between P0 and P1, the distance D2 between P0 and P2, and so on. Referring again to Figure 15B, procedure P1505 includes assigning weights to each of a plurality of predictions based on distance 1503. In one embodiment, assigning weights to each prediction includes assigning a relatively higher weight to the prediction among the plurality of predictions if the associated distance is relatively small. In one embodiment, the weights may be integer values ​​between 0 and 1 or normalized values ​​such that the sum of the weights is 1. Referring to Figure 13, higher weights are assigned to points P1, P2, P3, P10, P11, P12, and P13 compared to points P4, P5, P14, P15, and P16. In other words, points near the point of interest P0 are considered to contribute more to making specific predictions related to the point of interest P0. For example, the weight of the feature corresponding to points P1, P2, P3, P10, P11, and P12 is 0.9, while the weight of the feature corresponding to points P4, P5, P15, and P16 is 0.1. Therefore, higher weights are assigned to predictions associated with features located near the feature of interest 1501 compared to features far from the feature of interest 1501. In one embodiment, weights can be assigned according to an exponential function, for example... ,in It is a function of the distance 1503. Referring again to Figure 15B, procedure P1507 includes determining the model parameter values ​​of the interpretation model 1510 based on the fit of the weighted prediction 1505, thereby reducing the difference between the output of the interpretation model 1510 and the weighted prediction 1505. In one embodiment, the model parameter values ​​explain the contribution of each pixel of ADI to making predictions related to the features of interest. In one embodiment, determining the model parameter values ​​of the interpretation model is an iterative procedure, including obtaining initial model parameter values ​​and weighted predictions; executing the interpretation model using the initial model parameter values ​​to generate initial outputs; determining the difference between the weighted predictions and the initial outputs; and adjusting the initial model parameter values ​​based on the difference to minimize the difference. In one embodiment, the interpretation model 1510 receives an ADI (Advanced Directional Analysis) including the feature of interest 1501 as input and produces an interpretation map 1520 as output. In one embodiment, the interpretation map 1520 indicates the contribution of the neighboring regions of the feature of interest 1501 to making predictions associated with the feature of interest 1501. In one embodiment, interpretation model 1510 is a linear model associated with the feature of interest in ADI. In one embodiment, the linear model is fitted to a complex number of predictions using linear regression with least squares error. Figure 13 illustrates an example of interpretation model M1. In one embodiment, the interpreted map 1520 is a pixelated image (e.g., MAP10, MAP20, and MAP30 in Figures 14A to 14C), and the model parameter values ​​are weights or values ​​assigned to each pixel of the pixelated image. In one embodiment, the interpreted map is a binary map, where each pixel is assigned a value of 0 or 1. In one embodiment, the binary map is generated by assigning 0 or 1 to each pixel value. A threshold value is exceeded, where 0 indicates that the feature of interest will be printed with defects after etching, and 1 indicates that the feature of interest will be printed without defects after etching. In one embodiment, the threshold value is a value above which the contribution is considered positive or beneficial to prediction, and vice versa. In one embodiment, the interpreted image 1520 is a color image, wherein a specific color (e.g., RGB value) is assigned based on model parameter values. After training the interpretation model 1510, it can be used to understand predictions associated with the feature of interest. For example, as shown in Figure 14A, an ADI image ADI10, including the feature of interest F10, can be input into the interpretation model 1510. The interpretation model then generates an interpretation map, such as MAP10. The interpretation map MAP10 includes patches E1 and E2, which visually interpret which parts of the area surrounding feature F10 contribute to predicting that feature F10 will be printed defect-free after etching. For example, patch E1 has a substantially larger area than E2, and therefore can be interpreted as E1 having a higher contribution. In one embodiment, the interpretation model can be optimized, for example, by modifying the metric settings and determining the portion of the ADI that best explains the quality of defect classification. For instance, an initial feature vector provides 92% classification accuracy, while after optimization, two feature vectors can be identified, improving the classification accuracy to 94%, or six feature vectors can improve the classification accuracy to 99% after optimization. An instance optimization procedure for determining the optimal parameters (e.g., relevant feature vectors) is described below. In one embodiment, during the optimization procedure, the metric settings, the number of feature vectors to be considered, or other settings can be changed. The optimized parameters (e.g., feature vectors) are interpretable classifications that can be applied to any input ADI. In one embodiment, a method for applying interpretation model 1510 is discussed with reference to Figure 16. Figure 16 is a flowchart of a method 1600 for identifying the contribution of the vicinity of a feature of interest to classification defects. Feature of interest. Method 1600 includes the procedures described below. Procedure P1601 includes obtaining a developed image 1601 (e.g., ADI10, ADI20, and ADI30 in Figures 14A-14C) including features of interest (e.g., features F10, F20, and F30) and an interpretation model associated with the features of interest (e.g., 1510 in Figure 15B). Procedure P1603 includes applying the interpretation model 1510 to ADI 1601 to produce an interpretation map 1610. In one embodiment, the interpretation map 1610 includes pixel values ​​that quantify the classification of each pixel of ADI 1601 to classify defects in the features of interest. As mentioned herein, in one embodiment, the interpretation model 1510 is a linear model associated with the feature of interest in ADI 1601. In one embodiment, the interpretation map 1610 is a pixelated image, wherein the weight of each pixel indicates the contribution of each pixel to the defect of the feature of interest in classification. In one embodiment, the interpretation map 1610 may be a binary graph, wherein each pixel is assigned a value of 0 or 1. In the above methods, the concept of the present invention is illustrated using post-development and post-etching images as examples. However, the methods discussed herein are not limited to such ADI and AEI images. Those skilled in the art can use any image obtained before or after a specific process (e.g., OPC, optical process, resist process, etching, chemical mechanical polishing, etc.) or a combination of processes related to the patterning process to perform the above methods. Next, a model is established to determine the relationship between the contribution of process formulations (e.g., optical process formulations, resist process formulations, etching formulations, etc.) to the probability of failure after process execution using such images. As mentioned earlier, numerous algorithms exist for classifying faults in developed contact holes based on SEM images. The criteria for misclassification can be based on common sense in interpreting SEM images. For example, a misclassification criterion might be low SEM contrast or a small critical size (CD). Furthermore, attempts have been made to estimate the failure rate based on certain criteria derived from the CD distribution of the contact holes. For example, criteria could be (i) contact hole faults below a focal-specific critical CD, or (ii) a fault component that is a function of the average CD minus three standard deviations of the CD, or variables including skew and kurtosis, also known as "tailed CD." The prediction of tailed CD can be empirical, depending on the patterning procedure used. Furthermore, the deviation from the prediction may depend on the focal point. In this invention, faulty and fault-free contact holes in AEI are considered to have different characteristics in ADI measurements. Therefore, as described in the examples, the ADI CD distribution can be decomposed into two independent distributions to illustrate the faulty and fault-free contact holes in AEI measurements. The relative contribution of these ADI CD distributions determines the proportion of faulty holes after etching. In one embodiment, the ADI CD distributions of faulty and fault-free vias after etching are different; however, these two CD distributions may overlap. Faults cannot be classified using CD thresholds independent of FEM. However, when the CD distributions of all contact vias are obtained under certain finite element conditions, they can be fitted as the sum of the two distributions, and the relative contribution of these distributions can predict the proportion of faulty vias after etching. Figure 17 is a flowchart of method 1700 for determining characteristic defects after etching based on a model consisting of a first part (e.g., a first CD distribution) and a second part (e.g., a second CD distribution). The model generated using method 1700 can be applied to improve patterning processes. For example, the model can be used to estimate the proportion of filled contact holes based on ADI measurements. The estimated filled holes can be used, for example, as follows. In a typical application, the estimated filled holes can be used during the ramp-up phase of a lithography process. For example, the lithography equipment can be tuned to reduce the number of filled contact holes. Examples of improvements include tuning the scanner dose and focus, or adding an additional filtration step to the resist. In another typical application, during the ramp-up process, the proportion of filled contact holes can be used to assess whether additional descaling or drilling should be used before etching to reduce the impact of filled contact holes. In yet another typical application, during HVM, the ADI CD distribution can be checked to see if the machine is still within specifications. Note that this will require a large number of contact holes per wafer. However, this kind of fitting may be feasible when combining all data over a certain time range (e.g., 1 day). The method for generating a model and predicting the characteristic components that may lead to failure is discussed in detail below

[1700] . Procedure P1701 includes obtaining an illustrated image (ADI) 1702 of the substrate, the ADI image 1702 including a plurality of features. In one embodiment, the ADI is an image of the printed circuit board obtained by means of a measuring instrument or from a database storing images of printed circuit boards. In one embodiment, the plurality of features of the ADI image include a plurality of holes, a plurality of pillars, a plurality of lines, or combinations thereof. Thus, in one embodiment, the feature portion of the ADI image classified as defective after etching includes at least one of the following: closed or missing holes after etching due to resist blocking the development of the holes; and defects caused by etching, merged holes after etching, necking of one of the plurality of lines; bridging lines, or combinations thereof. In one embodiment, the physical property may be the critical dimension (CD) of a feature in an ADI image, and the physical property threshold associated with the feature may be the CD threshold. For example, the CD of a contact hole is determined by calculating the surface area enclosed by a contour generated by a contour algorithm (e.g., CD obtained at multiple metric tool thresholds for each feature of interest), and then the diameter of the circle can be determined using the same surface area. In one embodiment, the physical property may be at least one of the following: the geometric mean of the CD of the feature, wherein the CD can be measured along a first direction (e.g., the x-direction) or a second direction (e.g., the y-direction); the directional CD of the feature of interest in the ADI image; the curvature variation of the feature of interest in the ADI image; or the CD obtained at multiple metric tool thresholds for each feature of interest. In one embodiment, the directional CD is at least one of the following: CD measured along the x-direction; CD measured along the y-direction; or CD measured along a desired angle. As described below, physical property values ​​(e.g., CD values) or subgroups thereof can be used to generate a model. In one embodiment, the physical characteristic may be a function of one or more physical characteristics. For example, the physical characteristic may be the square of the CD value. The invention is not limited to a specific physical characteristic. Those skilled in the art will understand that any physical characteristic that can be used to characterize a fault can be used here. Procedure P1703 includes generating a first part of model 1710 (e.g., a first probability distribution function (PDF1)) based on physical property values ​​(e.g., CD, EPE) associated with a subgroup SET1 of features of ADI image 1702. Procedure P1705 includes generating a second part of model (e.g., a second probability distribution function (PDF2)) based on the first part of the model and physical property values ​​(e.g., CD) associated with all features among a plurality of features. Subgroup of features of ADI image 1702. In one embodiment, subgroup SET1 of ADI image features is distinguished from other features of ADI image 1702. For example, subgroup SET1 may be features with CD values ​​higher than a specified threshold. In one embodiment, a truncated PDF is used to fit subgroup SET1. In one embodiment, a truncated PDF is used based on a threshold value (e.g., CD, EPE). The normalization of PDF is changed by using fitting parameters and other parameters. In one embodiment, the generation of the first and second parts of the model includes fitting a first probability distribution function PDF1 and a second probability distribution function PDF2, respectively, by maximizing the log-likelihood measure of the model. In one embodiment, model 1710 is a combination of the first probability distribution function PDF1 and the second probability distribution function PDF2. In one embodiment, the first probability distribution function PDF1 is configured to estimate the distribution of physical characteristic values ​​(e.g., CD) for fault-free features (e.g., fault-free holes). In one embodiment, a fault-free hole may indicate a very low probability of failure. For example, the failure rate is within a given range (e.g., 0 to 0.1). In one embodiment, the second probability distribution function PDF2 is configured to determine the failure rate based on the physical characteristic values ​​of all complex features of the ADI image. In one embodiment, model 1710 is a weighted sum of a first probability distribution function and a second probability distribution function. For example, the model is calculated as the total distribution of the weighted sums of the first function PDF1 and the second function PDF2 for faulty and fault-free conditions, respectively. In one embodiment, the first probability distribution function is a normal distribution (or a truncated normal distribution), which is composed of a cutoff value related to physical characteristics (e.g., The first position parameter describing the shift (e.g., mean) of the normal distribution and the first scaling parameter (e.g., δ) describing the expansion of the normal distribution are characterized. In one embodiment, as in Equation 1, the square of CD fits the normal distribution, while CD itself may fit another distribution (e.g., GEV). In one embodiment, the second probability distribution function is a generalized extremum (GEV) distribution, characterized by a second position parameter (μ) describing the offset of the GEV distribution, a second scaling parameter (σ) describing the diffusion, the shape of the GEV distribution, and a shape parameter (ξ) describing the shape of the GEV distribution. In the examples in this specification, the total distribution of Model 1710 or the fitted model is the sum of the normal distribution of the square of ADI CD and the generalized extreme value (GEV) distribution. For example, the total probability distribution function (PDF) can be given by the following equation: In the above formula, the variable x represents the physical characteristics of ADI, such as CD. This represents the cumulative probability of the tail of the GEV distribution. The parameter represents the normal distribution or the truncated normal distribution, and The parameter represents the GEV distribution. In one embodiment, the log-likelihood of the above PDF can be calculated using the following equation: Using the examples of the normal distribution and GEV distribution mentioned above, this method generates the model in two steps, as described below. If using Therefore, GEV can fit noise to a fault-free distribution, rather than fitting the tail of the distribution. Thus, assuming It is very small. Therefore, a two-step procedure (e.g., including P1703 and P1705) is used to generate the model (e.g., 1710). First, using a normal distribution truncated from directly below... Fit to Above. For example, Figure 16 illustrates an instance of the truncated normal distribution 1810, in which... The value is 15 nm. In other words, CD values ​​greater than 15 nm are used to fit the normal distribution. In the above formula, and The mean and standard deviation of the correlation for a normal distribution can be obtained by maximizing the log-likelihood. For example... and In one embodiment, the value of can be solved iteratively until the log-likelihood of the aforementioned PDF (CD) is maximized. Furthermore, in the above equation, This refers to the probability distribution function of the normal distribution, and This refers to the cumulative distribution function of the normal distribution function. In one embodiment, Can be initially selected as This can be achieved iteratively. Furthermore, after the second-step fitting (discussed below regarding GEV), it can be checked... Check if the predicted value of the faulty hole is below a certain threshold (e.g., less than 1%). If not, a larger value can be used. Repeat the procedure (e.g., P1703 and P1705) for values ​​(e.g., greater than 15 nm). In one embodiment, the total CD distribution is fitted to the distribution proposed in equation (1), maintaining... and It is fixed to the previously obtained value. , , , This can be determined by maximizing the log-likelihood. Any nonlinear programming solver can be used to implement this fitting procedure and equation. The nonlinear programming solver can find the specified minimum unconstrained multivariate function. In one embodiment, a choice can be made... To improve robustness. As discussed in the examples above, the generation of the model includes the physical properties of the feature subgroups (e.g., CD 2The squared values ​​of the features are fitted to a first probability distribution function (e.g., a normal distribution) to maximize a first log-probability value associated with the first probability distribution function. In one embodiment, the subgroup SET1 of features has physical property values ​​higher than the physical property threshold. The fitted first probability distribution function can then be combined with a second probability distribution function. Based on the combined distribution, a second probability distribution function can be fitted based on the physical property values ​​of all features of a complex set of features to maximize a second log-probability value associated with the combined distribution. In one embodiment, the relative weights of the second distribution are determined during the fitting process. Figure 18A is an example of fitting two probability distribution functions based on CD values ​​of features in ADI images. Dots represent defect-free holes (e.g., determined based on analysis of etching data), while crosses represent unqualified holes (e.g., based on etching data and CD values ​​less than the desired value). The truncated normal distribution 1810 (an example of the first probability distribution function) can be fitted using the CD values ​​of fault-free holes, where holes with CD values ​​above the CD threshold (e.g., 15 nm) are considered fault-free holes. Furthermore, the GEV distribution 1820 can be used for tailing (e.g., CD values ​​below 15 nm), and the total distribution 1801 can be fitted, for example, using Equation (1) above and all ADI CD values. In one embodiment, for overlapping regions (e.g., approximately 15 nm), both the normal distribution and the GEV distribution can have similar weights (e.g., as represented by Equation (1)). In one example, although for small CD values ​​and best-fit parameters, The value is much greater than ,but It may be close to 0. In one embodiment, as the CD value gradually decreases, the weight associated with the GEV distribution gradually increases. In one embodiment, fitting the first probability distribution function is an iterative process. The iterative process includes: (a) determining a first log-probability measure using given values ​​of the parameters of the first probability distribution function; (b) determining whether the first log-probability measure is maximized; and (c) in response to not maximizing, adjusting the values ​​of the parameters of the first probability distribution function based on the gradient, and performing steps (a) to (c). In one embodiment, the gradient is the first derivative of the first log-probability measure with respect to the parameters of the first probability distribution function. In one embodiment, fitting the second probability distribution function involves determining the values ​​of the parameters of the second probability distribution function and their weights based on maximizing the second log-probability metric, without modifying the values ​​of the parameters of the first probability distribution function. In one embodiment, fitting the second probability distribution function (e.g., 1820) is an iterative procedure. The iterative procedure includes: (a) obtaining a combined distribution of the fitted first probability distribution function and the second probability distribution function; (b) determining a second log-similarity measure by fixing the parameter values ​​of the fitted first distribution based on the combined distribution (e.g., 1801) and using given values ​​of the parameters of the second probability distribution function; (c) determining whether the second log-similarity measure is maximized; and (d) in response to not maximizing, adjusting the values ​​of the parameters of the second probability distribution function based on the gradient and performing steps (b) through (c). In one embodiment, the gradient is the first-order derivative of the second log-similarity measure with respect to the parameters of the second probability distribution function. In one embodiment, the combined probability distribution function (e.g., 1801) can be used as a model 1710 for predicting faults or failure rates of ADI characteristics. Referring to Figure 18B, which illustrates the first focal exposure matrix PW1 (focal point on the x-axis, dose on the y-axis), where the ADI LCDU plots the ADI vias leading to the fault-free AEI feature, and another focal exposure matrix PW2 associated with the etched ADI, which can include both faulty and fault-free features. Figure 18B also illustrates how the LCDU of the pattern changes with dose for faulty and fault-free AEI. There are significant differences between LCDU and the dose function for all vias and fault-free features. For example, curve 1851 illustrates LCDU as a function of dose for all vias, and curve 1853 illustrates LCDU as a function of dose for fault-free vias in the AEI. In one instance, when merged vias exist after etching, LCDU (curve 1851) increases with a higher dose. Similarly, when defect-free vias exist after etching, LCDU (curve 1853) decreases with a higher dose. This relationship indicates that the fitting parameters, such as the CD distribution, associated with faulty and fault-free features (e.g., contact vias) will be different. The fitted CD distribution can be used to determine the process window. In this embodiment, because model 1710 is a combination of the first and second distributions, model 1710 more accurately captures the relationship between LCDU and dose. Therefore, for example, the statistical parameters or characteristics of the fitted distribution 1710 can be used to more accurately determine the process window of the patterned process. In one embodiment, method 1700 may further include procedures P1711 and P1713 configured to determine the process window PW. In one embodiment, P1711 includes extracting statistical features of a fitted probability distribution 1710 (e.g., PDF1 in FIG. 17) related to fault-free characteristics. For example, the statistical features may be the mean, standard deviation, skewness, or other statistics related to contact holes printed on the substrate. In one embodiment, in procedure P1713, extracted statistical features from fitted distribution 1710 are used to determine the process window. For example, the process window includes a range of dose focus values ​​that allow features to be printed on the substrate with few or no defects (e.g., one defective feature out of one million features). An example method for determining the process window is described in U.S. Patent Application No. 62 / 980,068, filed February 21, 2020, which is incorporated herein by reference in its entirety. As previously described, method 1700 has several applications. Therefore, method 1700 can be further modified to include improved patterning processes. For example, method 1700 may further include: imaging a desired pattern of a plurality of additional features contained on another substrate using a patterning apparatus; obtaining a developed image of the imaged pattern; using the developed image to perform first and second probability distribution functions to classify a subset of features within the ADI as defects after etching; and adjusting etching conditions according to the classified features so that the etched imaged pattern does not fail. In another application, method 1700 can be further modified or used to tune the lithography process to reduce the failure rate of ADI features after etching, wherein the tuning includes adjusting the dosage, focus, or both. In yet another application, method 1700 can be used to determine whether additional filtration steps should be performed on the resist layer to reduce the failure rate of ADI features after etching. In yet another application, method 1700 can be used to determine whether additional descaling or pass-through steps should be performed to reduce the failure rate of ADI features after etching. In yet another application, method 1700 can be used to inspect ADI features during high-volume manufacturing to determine whether the lithography equipment meets specified printing standards. In yet another application, method 1700 can be used to rework certain substrates or multiple substrates prior to etching based on the failure rate. In one embodiment, the system can be configured to use a two-part model to determine a subset of features that will fail after etching based on ADI measurements. In one embodiment, the system includes a metrology tool (e.g., SEM of Figures 28 and 29) for acquiring a post-development image (ADI) of a substrate at a given location; the ADI includes multiple features; a processor (e.g., 104 of Figure 30) is configured to determine a failure rate based on the ADI. In one embodiment, the processor (e.g., 104) is configured to execute a model (e.g., 1710 of Figure 17) to determine the failure rate of a plurality of features of the ADI that will fail after etching. In one embodiment, the model is a combination of: (i) a first probability distribution function configured to estimate the distribution of physical characteristic values ​​of fault-free vias; and (ii) a second probability distribution function configured to represent the feature values ​​of all plurality of features of the failure rate ADI based on physical variables. In one embodiment, the system further includes a patterning device (e.g., Figures 1 and 31-34) configured to image a desired pattern comprising a plurality of features onto a substrate. A processor (e.g., 104) may be further configured to receive the ADI (Advanced Difference Indicator) of the imaged substrate via a metrology tool. A first probability distribution (e.g., fitted PDF1) and a second probability distribution (e.g., fitted PDF2) are performed to determine the failure rate of features of the ADI, and adjustments are made based on features with relatively high failure rates to reduce the failure rate of the patterning device functionality. In one embodiment, the processor (e.g., 104) may be configured to adjust the dose or focus via a knob / setting on the patterning device. In one embodiment, the processor (e.g., 104) may be further configured to: determine whether additional filtering steps should be performed on the resist layer to reduce the failure rate of ADI features after etching; and determine whether additional descaling or drilling steps should be performed to reduce the failure rate of ADI functions after etching; or check ADI functions during mass production to determine whether the lithography equipment meets specified printing standards. In one embodiment, the measurement tool (e.g., Figures 28 and 29) comprises a scanning electron microscope (SEM). The SEM can be configured to measure at least one of the following physical properties: the average CD of a plurality of instances of the feature of interest in the ADI; the directionality CD of the function of interest in the ADI; the curvature change of the feature of interest in the ADI; or the CD obtained at a plurality of measurement tool thresholds for each measurement feature. As mentioned herein, random defects (interchangeably referred to as flaws) can significantly impact the performance of EUV lithography. Faults can be identified after the lithography or etching steps. Numerous algorithms utilize SEM images to classify characteristic defects such as contact holes. The criteria for such defect classification are based on common-sense interpretations of SEM images. For example, fault criteria could be SEM contrast or critical size (CD). In one embodiment, the methods discussed above are based on an improved defect classification and fault prediction method provided by Analog Devices (ADI). Other methods are also provided to estimate the failure rate based on the CD distribution of contact holes. As mentioned earlier, existing methods have some limitations. For example, defect classification can be calibrated based on the capture rate of programmed defects or by comparing defect rates before and after etching. It has been shown that programmed defects are statistically different from random defects, for example, see the previously mentioned publication P. De Bisschop. As described above, the method presented in this paper provides improved defect classification based on training data from repeated SEM measurements of ADI and AEI at the same location. The method in this paper successfully classified, for example, 93.5% of holes under error-prone FEM conditions. Compared to defect classification, a general drawback of predicting failure rates is that less information is collected and defect classification cannot be visually assessed. TailCD (i.e., the trailing edge of the CD distribution) prediction is an empirical prediction, which may depend on the procedures performed on the substrate. In one embodiment, the defect classification method discussed herein utilizes the fact that, to a certain extent, the features of a fault (e.g., contact holes) appear differently in static ADI images (e.g., SEM images) after etching. In this invention, it has been observed that the differences in ADI images between faulty and non-faulty contact holes after etching are small, and in many cases almost imperceptible to the naked eye. It has also been observed through example experiments that, for faulty contact holes, ADI SEM damage (e.g., CD differences between first and second SEM "reproductions" or reproducibility of measurements within the same ADI) is much greater. In one embodiment, exposing the same location on the wafer two or more times to obtain two different SEM measurements is referred to as SEM "reproduction." Therefore, in one embodiment, a method is provided that uses dynamic SEM information to distinguish between faulty and non-faulty holes, or utilizes this information to improve fault prediction. As discussed herein (e.g., with respect to Figure 3), SEM measurements are performed after lithography damages the resist on the substrate to cause shrinkage or excess carbon redeposition onto the resist. Such damage affects the CD of features on the substrate measured by SEM, especially when performing repeat SEM measurements (e.g., taking two SEM images at the same location using an ADI). For example, in Figure 3, the missing contact hole defect after etching is caused by a layer of resist remaining inside the contact hole. Therefore, the geometry of the defective and non-defective holes after etching is different. Consequently, shrinkage and carbon redeposition may differ, resulting in significant differences between SEM replicas of the defective hole after etching. In one embodiment, a method is provided to determine feature defects based on reproduction measurements of the same ADI feature (e.g., in Figure 19). In one embodiment, the reproduction measurements are included at two SEM images of the ADI, from which different features for defective and non-defective contact holes can be determined. Figure 19 is a flowchart of a method 1900 for determining the defect attribute of a feature in an analog-to-dissolved image (ADI) according to one embodiment. In one embodiment, the defect attribute is whether the ADI feature is defective or the probability of failure associated with the ADI feature. Method 1900 determines defects based on a defect criterion, which may be, for example, a CD of a first image and a second image. Method 1900 includes the following procedures, which are discussed in detail below. Procedure P1901 includes exposing an ADI feature to an electron beam or charged particle beam to generate a first image of the ADI feature, which is a structure within a resist material. In one embodiment, exposure includes exposing multiple ADI features to generate multiple first images. For example, multiple frames (e.g., 4, 5, 6, ..., 50) of the SEM image may be captured corresponding to different locations of the ADI feature on the substrate. Procedure P1903 includes re-exposing the ADI feature to an electron beam or charged particle beam to generate a second image of the ADI feature. In one embodiment, the re-exposure step includes a plurality of ADI features to generate a plurality of second images. For example, multiple frames (e.g., 4, 5, 6, ..., 50) of SEM images may be captured at the same locations on the substrate captured in P1901 as the ADI features captured in P1901. In one embodiment, an electron beam is generated using a scanning electron microscope (SEM), and the first and second images are SEM images. In one embodiment, a first set of images of ADI features (e.g., contact holes in the resist) can be captured at different locations on the substrate. Furthermore, a second set of images of ADI (e.g., contact holes) can be captured at the same location on the substrate (for the first set of images). In one embodiment, SEM projects high-energy electrons (also known as electron beams) onto a resist that is a polymer, causing damage to the resist. For example, the resist may shrink, thus increasing the size of the ADI feature (e.g., contact hole). SEM may also deposit carbon deposits, thereby altering the CD of the ADI function. SEM measurements may have different effects on ADI features with different geometries because the amount of resist that electrons may react with varies. For example, referring to Figure 3, the geometry of the resist remaining in a contact hole will be different compared to another contact hole with no resist or relatively little resist. Thus, for a partially filled contact hole, electrons may react with the resist at the bottom and walls of the contact hole. Alternatively, for a contact hole with no resist at the bottom, electrons may react only with the resist walls of the contact hole. Thus, the damage to the resist in filled contact holes will differ from that in unfilled or less filled contact holes, resulting in different geometric changes after SEM measurements. In another example, the electron distribution in circular and elliptical contact holes will cause different SEM damage to the resist in circular contact holes compared to that in elliptical or other non-circular contact holes. This difference is why performing a first SEM measurement to measure ADI features followed by another SEM measurement will yield different results; for example, a second SEM image may have slightly different geometry for the same ADI features compared to the first SEM. Procedure P1905 includes determining the defect attributes of the ADI feature based on physical characteristics (e.g., CD) associated with the first and second images. For example, a first CD can be extracted from the first image, and a second CD can be extracted from the second image. In one embodiment, the physical characteristic is a critical size or pixel intensity associated with the ADI feature (e.g., a contact hole). In one embodiment, the defect attribute may be binary (e.g., whether the ADI feature is defective). In another embodiment, the defect attribute may be a failure probability associated with the ADI feature (e.g., characterized by a CD PDF mounted on faulty data). In one embodiment, determining the defect attribute involves extracting a first feature from a first image and extracting a second feature from a second image. Based on the difference between the first and second features, it is determined whether a defect metric has been violated; and in response to a violation of the defect metric, the ADI function is classified as defective. In one embodiment, the first image captured in the first exposure comprises a plurality of images of the ADI feature. Similarly, the second image re-exposure step comprises a plurality of images of the same ADI feature. For example, a measuring instrument such as a scanning electron microscope (SEM) uses a focused electron beam to scan an object (e.g., the ADI feature). To obtain a reliable image with as few artifacts as possible, the SEM may scan the object multiple times (e.g., eight times). The response of each scan is called a "frame," and an average image can be generated by averaging multiple "frames." Thus, multiple frames of the ADI feature (e.g., a first set of frames) can be obtained from the first exposure step, and other multiple frames of the ADI feature (e.g., a second set of frames) can be obtained from the second exposure. Therefore, in one embodiment, the average image can be used as the first image from the first exposure, and the second average image can be used as the second image from the re-exposure. In another scenario, all frames or subgroups of frames in the first set can be considered as the first image, and all frames or subgroups of frames in the second set can be considered as the second image. Therefore, differences (e.g., CD differences, intensity differences, etc.) can be determined between all frames (or subgroups) in the first set and all frames (or subgroups) in the second set. Those skilled in this art will understand that frames can be represented as pixelated images, with each pixel having a grayscale value. In one embodiment, the difference determination includes determining the difference between physical features (e.g., CDs) extracted from one or more frames of the first image and physical features (e.g., CDs) extracted from corresponding one or more frames. Further frames of the second image. For example, the first image may include four frames, and physical features (e.g., CDs) may be extracted from each of the four frames. For example, the extracted physical features may be CD1, CD2, CD3, and CD4. Similarly, after the re-exposure step, the second image may include four frames, and physical features (e.g., CDs) may be extracted from each of the four frames. For example, CD5, CD6, CD7, and CD8. Therefore, differences are calculated, for example, between CD1 and CD5, CD2 and CD6, CD3 and CD7, and CD4 and CD8. In one embodiment, the defect metric is a function of a first physical property (e.g., CD1) of an ADI feature in a first image and a second physical property (e.g., CD2) of an ADI feature in a second image. In another embodiment, the defect metric is a bilinear function, a trained machine learning model, or a second or higher-order polynomial. For example, a function of two or more variables that is linear with respect to each variable is called bilinear. The simplest example is f(x, y) = xy. In another example, the first and second images may include multiple frames, as discussed herein. Physical properties (e.g., CD1, CD2, CD3, CD4, CD5, CD6, CD7, CD8) can be extracted from each frame. In this case, a multivariate distribution can be used, where the multivariate distribution is a PDF of combinations or merged CD1 to CD8. In the examples above, a binary function is used as an example to illustrate the concept. However, the invention is not limited to binary functions. Those skilled in the art can modify the method to include multiple frames in the first and second images. Furthermore, depending on the number of frames used to represent the first and second images, the defect metric can be represented by a multivariate function. In one instance, the defect metric takes the form f(CD1, CD2) < 0, where CD1 is the CD in the first SEM image of the object, and CD2 is the CD image of the same object in the second SEM. An example of the defect metric is the function represented by row 2010 in Figure 20. Figure 20 shows, according to one embodiment, the CD value of SEM damage relative to the contact hole in FEM under error-prone conditions (e.g., below normal dose) (e.g., for 10). 5A graph showing the CD values ​​of contact holes. This graph shows that for CD values ​​smaller than the nominal CD, SEM damage is relatively high. In the current embodiment, SEM damage is defined by the difference between the first and second CD values ​​extracted from the first and second SEM images, respectively. As shown in Figure 20, this difference is relatively high for CD values ​​below 14 nm. SEM damage can be characterized using any of a variety of methods. The moving average curve of SEM damage data 2005 2015 further demonstrates that SEM damage can predict failure of ADI functional components (e.g., contact holes). For example, if the moving average of SEM damage associated with a specified CD value range exceeds a specified damage threshold (e.g., 3 nm), then the specified CD range is more likely to fail after the etching process. In one embodiment, SEM damage information characterized by, for example, CD1 and CD2 values ​​from a first SEM image and a second SEM image, can be used to determine defect classification criteria, such as curve 2010. In this embodiment, the defect metric used for defect classification can be represented by curve 2010. The defect metric 2010 serves as a CD defect threshold value for SEM damage, which is a function of the CD1 and CD2 values ​​of the same ADI feature obtained from the two SEM measurements. In one embodiment, the defect metric 2010 can be established based on defect data (e.g., faulty and fault-free contact holes after etching) and CD1 and CD2 values ​​extracted from the first and second SEM images, respectively. In one embodiment, the defect metric 2010 indicates that if the SEM damage associated with an ADI feature having a nominal CD value (characterized by CD1 and CD2 values) is higher than the SEM damage associated with another ADI feature having the same nominal CD value, then the ADI component with higher SEM damage is more likely to have a defective ADI component after etching. The defect metric 2010 also indicates the minimum CD threshold value, which can be classified as a defect even if the SEM damage is zero or close to zero and is below the ADI feature. In one embodiment, the defect metric 2010 may be a bilinear function fitted based on measurement data. It is understood that the bilinear function is given by way of example only and does not limit the scope of the invention. As discussed herein, other multivariate functions, such as those with 4 or 8 variables, may be used as defect metrics. For comparative purposes, the graph in Figure 20 also illustrates another defect criterion, 2020. Example defect criterion 2020 represents a constant CD threshold value independent of SEM damage. For instance, such a constant CD threshold value 2020 can be set as previously discussed with respect to Figures 7A and 7B. In the examples of Figures 7A and 7C, the CD threshold value is set based on fault data, where ADI features with CD values ​​below the CD threshold value are classified as possible faults, while ADI features with CD values ​​above the CD threshold value are classified as unlikely faults. Alternatively, compared to the simple single-value CD threshold value 2020, the defect metric 2010 based on SEM damage can classify defect features more accurately. In another instance, a bivariate probability density function can be used in the defect criteria. For example, the bivariate PDF can be a combined PDF or a merged PDF of two or more variables. In one embodiment, the bivariate PDF can be determined based on, for example, the first SEM image and the second SEM image discussed with respect to Figure 22. The bivariate PDF can determine the probability that the first measurement will measure CD1 and the second measurement will measure CD2. In another instance, the first SEM image and the second SEM can include multiple frames, as described herein. Physical characteristics (e.g., CD1, CD2, CD3, CD4, CD5, CD6, CD7, CD8) can be extracted from each frame. In this case, a multivariate distribution can be used, where the multivariate distribution is a combined or merged PDF of CD1 to CD8. In another instance, the defect metric can be a trained machine learning model (e.g., a CNN). The trained model can be obtained by training the machine learning model using a training dataset comprising: (i) multiple image pairs, each image pair including a first image and a second image of multiple ADI features, and (ii) etched images (AEI) of the substrate corresponding to the ADI function. In one embodiment, training the machine learning model is an iterative process involving (a) adjusting the parameters of the machine learning model such that the model determines the defect attribute of a given ADI feature based on a comparison between a first model and a second model; (b) determining whether the defect attribute determined by the model is within a specified range of the defect attributes of the AEI feature corresponding to the given ADI feature; and (c) in response to not being within the specified range, performing steps (a) and (b). In one embodiment, a gradient descent method may be used to determine the model parameter values ​​that cause the defect attribute determined by the model to converge to the defect attribute of the AEI feature. After training, the trained model can be used as a means of defect measurement or defect classification. Figure 21 illustrates an example of training a machine learning model 2100 using a training dataset TD1, TD2, ..., TDn, including a first image and a second image, as described above. In one embodiment, each piece of training data includes a pair of first images SEM1 and SEM2, and a reference AEI image (not shown). In one embodiment, the reference AEI image is used as ground truth, guiding the machine learning model to accurately classify the input as a possible fault or non-fault. In one embodiment, one or more ADI features (e.g., contact holes, lines, etc.) can be used to generate training data. For example, training data can be generated based on ADI features including multiple contact holes, multiple lines, and other specified structures. For each of the multiple contact holes and each of the multiple lines, a corresponding pair of SEM1 and SEM2 images can be obtained using SEM tools. Furthermore, for each of the multiple contact holes and each of the multiple lines, an AEI image can be obtained for use as ground truth. The output of the machine learning model training is to classify the training data as a possible failure (e.g., FailCH) or a unlikely failure (e.g., NotFailCH) of the ADI function, which may occur in the etched image (AEI). In one embodiment, the machine learning model may be a convolutional neural network or other existing neural networks trained using existing training algorithms (e.g., gradient sizing methods). The invention is not limited to a specific machine learning model. According to one embodiment, Figure 22 is a flowchart of a method 2200 for forming a model 2210 to determine the failure rate of features in a developed image. In an earlier embodiment, a model is formed to predict the failure rate based on the physical characteristics (e.g., CD) of ADI features by fitting the distribution of contact holes using the following probability density function: In the above formula, The vector represents the probability of failure after etching. Let be the parameter set of the distribution, and and This represents the probability density function for faulty and fault-free ADI characteristics. In earlier embodiments, a generalized extreme value (GEV) distribution was used for faulty vias, and a normal distribution was used for fault-free vias. . In another embodiment of the discussion regarding Figure 21, a model is formed based on one of the given ADI features of the SEM images. For example, the ADI CD distributions from the first and second SEM images are used to fit the above equation. PDF ( CD). In the equation above PDF ( In CD), the parameters provided for data fitting For the two distributions (and) and (related to) are different, but The values ​​should be the same. This reduces the parameter space used for fitting relative to the number of data points. Furthermore, the first and second SEM measurements can reveal the parameters of the first and second distributions. The relationship between them. This relationship can be used to further reduce the parameter space of the components. In another instance, the fitting procedure described above in method 2200 can be modified to fit using the following probability density function. CD 1 and CD 2. Combinatorial distribution: PDF ( CD 1, CD 2) represents the combined distribution, a vector The parameter set for each distribution determined by the fitting procedure, and The fault parameters are determined by the fitting procedure. The combined distribution reduces the model parameter space for fitting relative to multiple data points. Method 2200 is further discussed in detail below. Procedure P2201 includes obtaining (i) first measurement data 2201 related to an analog image (ADI) of a substrate, the ADI including multiple features, and (ii) second measurement data 2202 associated with the same ADI, obtained after the first measurement, using a measurement tool. For example, the measurement tool may be a SEM, and the measurement data may be data associated with a SEM image. In this example, a first SEM image of the ADI features is captured, followed by a second SEM image of the same ADI features. In one embodiment, the measurement data includes the physical properties of the ADI features in the SEM image. In one embodiment, the measurement data may be extracted as intensity values ​​associated with the ADI features in the SEM image. Program P2203 includes generating a model 2210 based on first measurement data 2201 and second measurement data 2202 to determine the failure rate of a feature of the ADI. In one embodiment, generating the model 2210 includes adjusting the values ​​of one or more model parameters such that the metric associated with the model 2210 is improved compared to the metric associated with the initial values ​​of the model parameters. In one embodiment, as discussed herein, the model 2210 may be further used to determine a process window for a patterned process based on the failure rate of a feature predicted by the model for a given first measurement of a given ADI and by the second measurement of a given ADI. In one embodiment, the generation of model 2210 includes fitting a failure rate parameter (e.g., using first measurement data 2201 and second measurement data 2202) to the model. The model is defined as a first probability density function (PDF) related to the failure rate parameter, and a second probability density function (PDF) related to the log-similarity of the model by maximizing the log-similarity of the model. In one embodiment, fitting the first probability density function includes determining the value of each model parameter of the first PDF and the second PDF by maximizing the log-similarity of the model. For example, as described above, the model may be... PDF ( CD 1, CD 2). In one embodiment, the first PDF (e.g., The second physical property (PDF) is characterized by a combined distribution of the first physical property, the second physical property, and the first set of model parameters. The first physical property is associated with ADI's first measurement data 2201, and the second physical property is associated with ADI's second measurement data 2202. (e.g., the second PDF...) It is characterized by another combination of the first physical property and the second physical property, and the second set of model parameters. In one embodiment, the first PDF may be a bivariate distribution characterized by: a first position parameter and a second position parameter describing the offset of the bivariate distribution; and a first scaling parameter and a second position parameter describing the expansion of the bivariate distribution. In one embodiment, the second PDF may be a generalized extremum (GEV) distribution, characterized by: a third and a fourth position parameter describing the offset of the GEV distribution; a third and a fourth scaling parameter describing the expansion of the GEV distribution; and a shape parameter (ξ) describing the shape of the GEV distribution. In one embodiment, the first measurement data 2201 is a first SEM image of ADI, and the second measurement data 2202 is a second SEM image of ADI. In another embodiment, the first measurement data 2201 includes a first physical characteristic value of a feature in the first SEM image of ADI, and the second measurement data 2202 includes a second physical characteristic value of a feature in the second SEM image of ADI. In one embodiment, model generation includes simultaneously fitting a first PDF and a second PDF to first and second measurement data. For example, the first PDF is fitted based on first physical property values ​​of multiple features in a first SEM image from ADI; the second PDF is fitted based on second physical property values ​​of the same multiple features in a second SEM image from ADI. In another embodiment, the first PDF and the second PDF are fitted simultaneously by maximizing a logarithmic similarity associated with the model. In one embodiment, fitting the first PDF and the second PDF is an iterative procedure, including: (a) determining a log-probability measure using given values ​​of parameters of the first PDF and the second PDF; (b) determining whether the log-probability measure is maximized; and (c) if the response is not maximized, adjusting the values ​​of a first set of model parameters and a second set of model parameters of the first PDF, as well as the failure rate parameter (e.g., ...) based on the gradient. ), and perform steps (a) to (c). In one embodiment, the gradient is the first derivative of the logarithmic approximation of the first model parameters, the second model parameters, and the failure rate parameters. In one embodiment, the adjusted model parameter values ​​make the failure rate parameters associated with the first PDF and the second PDF (e.g., The values ​​are the same. In one embodiment, method 2200 may further include determining the relationship between one or more of the model parameters in the first set of model parameters and the second set of model parameters based on the first measurement data 2201 and the second measurement data 2202. Method 2200 may further include modifying the first model parameter set according to the relationship and the second model parameter set to reduce the number of parameters in the first or second model parameter set. Method 2200 may further include generating model 2210 using the modified parameters and the first measurement data 2201 and the second measurement data 2202. In one embodiment, the physical property is the critical dimension (CD) of the feature. In another embodiment, the physical property is at least one of the following: the average CD of multiple instances of the feature of interest in the ADI; the directionality CD of the function of interest in the ADI; the curvature change of the feature of interest in the ADI; or the CD obtained for each function of interest under multiple metric tool limits. In one embodiment, the directionality CD is at least one of the following: a CD measured along the x-direction; a CD measured along the y-direction; or a CD measured along the desired angle. As discussed herein, the failure rate indicates a defect condition characterized by the physical properties of an ADI feature or its corresponding AEI feature. In one embodiment, the defect condition includes one or more of the following: omitted features; displacement ranges associated with features; or tolerance ranges associated with critical dimensions of an element. As discussed herein, performing SEM reconstruction measurements can double the SEM measurement time. Another option is to save several subgroups of SEM image frames, such as 2x8 or 4x4 frames. The SEM image can be a stack of eight frames, typically aligned to determine the average SEM image for ADI functionality. In this embodiment, SEM frames can be saved separately, and the first two to four frames can be analyzed from the saved SEM frames, all eight frames can be analyzed together, or all eight frames can be analyzed separately. Thus, eight very blurry SEM images (rather than a single average SEM image) provide more information than an average SEM image of multiple frames with the same measurement. In one embodiment, the charging effect caused by excess or lack of electrons on the measured substrate can affect the contrast of the SEM images, resulting in differences between SEM images. In one embodiment, it can be analyzed which measurement scheme provides the greatest added value compared to measuring more vias in the ADI of the measured substrate. Therefore, SEM measurement data can be organized according to method 2200 and further used to form model 2210. For example, method 2200 can be modified to form model 2210 based on various methods of obtaining SEM measurement data. For example, model 2210 can be formed by grouping 8 frames into 4x4 frames, where 4 frames are used to form the model and the other 4 frames are used to verify the predictive accuracy of the model. In another example, model 2210 can be formed by grouping SEM measurement data with similar charging effects. As discussed herein, this method (e.g., methods 1900 and 2200) offers several advantages. For example, during accelerated production (e.g., high-volume HVM manufacturing), model 2210 can be used in or associated with lithography equipment to adjust lithography parameters based on the model-predicted failure rate, thereby reducing the number of feature defects (e.g., filled contact holes). Examples of improvements include adjusting the scanner dose and focus, resist screening, additional resist filtration steps, or other lithography-related parameters. In another example, when model 2210 predicts a defect in an ADI feature, photoresist can be stripped or removed, reapplied, and re-exposed to prevent the predicted defect from occurring on the substrate after etching. In one embodiment, accurate defect classification based on ADI can help identify the root cause of AEI failures, such as those involving contact holes. Furthermore, for example, partially filled contact holes can be used to assess whether additional descaling or drilling should be performed prior to etching to mitigate the impact of the filled contact holes. In one embodiment, the methods described herein (e.g., 1700, 1900, 2200) may be included as instructions in a computer-readable medium (e.g., storage memory). For example, a non-transitory computer-readable medium includes instructions that, when executed by one or more processors, cause operations including: obtaining an analog-displayed image (ADI) of a substrate, the ADI comprising a plurality of features; generating a first portion of a model based on physical property values ​​associated with a subgroup of the features of the ADI; and generating a second portion of the model based on the first portion of the model and the physical property values ​​associated with all of the plurality of features of the ADI, wherein the subgroup of features of the ADI differs from the other features of the ADI. In one embodiment, the non-transitory computer-readable medium includes a model that is a combination of the following: (i) a first probability distribution function configured to estimate the distribution of physical characteristic values ​​of fault-free vias, and (ii) a second probability distribution function configured to determine the failure rate based on the physical characteristic values ​​of all of the plurality of features of the ADI. In one embodiment, the model is a weighted sum of the first probability distribution function and the second probability distribution function. In one embodiment, the first probability distribution function is a normal distribution characterized by a cutoff value related to physical characteristics, a first position parameter describing the offset of the normal distribution, and a first scaling parameter describing the extension of the normal distribution. In one embodiment, the second probability distribution function is a generalized extreme value (GEV) distribution characterized by a second position parameter (μ) describing the offset in the GEV distribution, a second scaling parameter (σ) describing the extension of the GEV distribution, and a shape parameter (ξ) describing the shape of the GEV distribution. In one embodiment, the non-transitory computer-readable medium includes instructions for generating a model, the instructions causing operations including: fitting the first probability distribution function based on the square of one of the physical property values ​​of the subgroup of features by maximizing a first log-probability measure associated with the first probability distribution function, wherein the subgroup of features has physical property values ​​higher than a physical property threshold; combining the fitted first probability distribution function with the second probability distribution function; and fitting the second probability distribution function and an associated relative weight based on the combined distribution based on the physical property values ​​of all the features in the plurality of features, such that a second log-probability measure associated with the combined distribution is maximized. In one embodiment, the non-transitory computer-readable medium includes instructions for iteratively fitting a first probability distribution function. The iteration includes: (a) determining a first log-similarity measure using given values ​​of the parameters of the first probability distribution function; (b) determining whether the first log-similarity measure is maximized; and (c) in response to not maximizing, adjusting the values ​​of the parameters of the first probability distribution function based on a gradient, and performing steps (a) through (c). The gradient is the first derivative of the first log-similarity measure with respect to the parameters of the first probability distribution function. In one embodiment, the non-transitory computer-readable medium includes instructions for fitting a second probability distribution function, including determining the values ​​and weights of the parameters of the second probability distribution function based on maximizing the second log-probability metric, without modifying the values ​​of the parameters of the first probability distribution function. In one embodiment, fitting the second probability distribution function is an iterative procedure. The iterative procedure includes: (a) obtaining a combined distribution of the fitted first and second probability distribution functions; (b) determining the second log-probability metric based on the combined distribution and keeping the values ​​of the parameters fitted to the first distribution fixed, using the given values ​​of the parameters of the second probability distribution function; and (c) determining whether the second log-probability metric is maximized; and (d) in response to not maximizing, adjusting the values ​​of the parameters of the second probability distribution function based on a gradient, and performing steps (b) through (c). The gradient is the first-order derivative of the second log-probability metric with respect to the parameters of the second probability distribution function. In one embodiment, a non-transitory computer-readable medium may also cause operations including: imaging a desired pattern comprising a plurality of features on a substrate via a patterning device; obtaining the developed image of the imaged pattern; performing the first probability distribution function and the second probability distribution function using the developed image to classify a portion of the features within the ADI as defective after etching; and adjusting an etching condition based on the classified features such that the imaged pattern will not fail after etching. In one embodiment, the non-transitory computer-readable medium may further cause operations including: tuning a lithography process to reduce the failure rate of ADI features after etching, wherein the tuning includes adjusting the dosage, focus, or both; determining whether an additional filtering step of a resist layer should be performed to reduce the failure rate of the ADI features after etching; determining whether an additional descaling or pass-through step should be performed to reduce the failure rate of the ADI features after etching; or during high-volume manufacturing, detecting ADI features to determine whether a lithography apparatus meets specified printing criteria. In one embodiment, a computer program is writable and implemented on a processing unit to perform the various steps of the methods discussed herein. For example, the computer program can be configured to simulate patterning processes (e.g., lithography, etching, resist development, etc.). Then, based on the simulation results, the various parameters can be calibrated according to the correlation between the results of different procedures discussed in the methods described herein (e.g., after resist development and after etch development). In one embodiment, simulation-based tuning or calibration can also provide insights during manual tuning of the etching process. For example, insights related to the effect of changes in process parameters on the correlation. As previously discussed, this invention describes a method for quantifying the short-range etch load effect on the CD (Content Detector) on features after etching. In one instance, the short-range etch load has an effect after the etched pattern. For example, the short-range load effect characterizes the influence of adjacent features (in ADI) on the target feature size after etching. For example, the size of a centrally located contact hole is affected by the contact hole itself and its adjacent holes. Furthermore, post-exposure processes can affect feature placement. In one embodiment, the etch load is characterized by analyzing the test structure and modeling it using the OPC process prior to etching optimization. For this purpose, the printing and etching have variable pitch and CD characteristics, and an empirical formula is used to fit the etch rate, which describes the open area within a specific circle around the point of interest. After developing an imaged pattern on a substrate, one or more post-exposure processes are performed to transfer the pattern onto the substrate. For example, the pattern transfer process can be considered a combination of etching and (re)deposition processes. The etching process is performed by solid sputtering of the material and chemical etching. The sputtered material and / or added gas components can also ensure (re)deposition. For example, sheath voltage affects the ion angle of the sputtering process, while U-wave power affects the density of the plasma / sputtering velocity. The sputtering rate of the material depends on the incident angle, ion velocity, and material composition that allows for fine-tuning of the pattern. For example, fluorine gas pressure determines the redeposition in the post-exposure process. In one method, the process involves first shrinking the feature ("CD") and then regrowing it in multiple cycles (load reduction). This reduces variations in the incoming CD. However, the competing process should be adjusted in all cycles to transfer the desired pattern onto the substrate. The methods described herein are discussed with respect to etching processes according to the invention. However, the invention is not limited to etching processes, and effects associated with other post-exposure processes can be determined herein. Some existing methods establish a relationship between ADI and AEI based on one or more parameters associated with the pattern. For example, for contact holes, relevant parameters include the change in CD before and after etching and contact edge roughness (CER). For high spatial frequencies, CER is sensitive to SEM shot noise. Therefore, the magnitude of CER depends on the average value of the pattern image (e.g., SEM image) used by the contouring algorithm. Pattern transfer (e.g., from ADI to AEI) tends to act as a convolution filter (e.g., smoothing), and it is unclear what CER-related changes will be included in the final pattern transfer. In another instance, for lines, it is unclear which spatial frequencies in the power spectral density of the pattern contour to be transferred are relevant. Therefore, it is further unclear how image averaging should be applied. In some cases, the relationship between line edge roughness (LER) (a measure of variability) and average length depends on the resist type. Therefore, variability measures between different resists may not be comparable. In embodiments of the invention, short-range etch load is quantified using a measurement-etch-measurement experiment, wherein the same contact holes are imaged using SEM before and after etching. The correlation between the size of the contact holes after etching and the size of the contact holes before etching is used as a measure of the intensity of the etch load. In one embodiment, the correlation between adjacent holes spanning different distances is used as a measure of the range associated with the local etch load. In one embodiment, ADI and AEI are SEM images with minimal features, which contain a relatively large amount of noise, for example, due to shot noise in the SEM image. Furthermore, the SEM image is a two-dimensional (2D) excitation map of a three-dimensional (3D) structure. It is currently unclear how to map the 2D information in the map back to the relevant 3D information. Additionally, not all fluctuations in the ADI of this feature are related to the predicted AEI feature. Therefore, it is unclear what to do regarding extending the short-scale details of the ADI and AEI features to those related to the quantization lithography process, and which contouring method should be used. In one embodiment, the method described herein (discussed later) addresses the effects associated with short-range loads and the arrangement of adjacent contact holes. In one embodiment, the placement of adjacent contact holes may be described relative to a reference grid or a grid associated with a design layout (e.g., provided in GDS format). In one embodiment, the placement is described with respect to the feature of interest. For example, moving adjacent contact holes relatively inward (e.g., closer to the feature of interest) will affect the CD of the central contact hole after etching. Therefore, not only the size of the surrounding contact holes will be affected, but also the position of the etched pattern. This method, such as FIG. 23A, provides a systematic approach to address this relationship. In one embodiment, the method for finding and quantifying the transfer of relevant contour features ADI and AEI after etching employs canonical correlation analysis. This method can be used both to extract relevant contour points and to quantify the transfer after etching. In one embodiment, the ADI and AEI data used in this method can be obtained by creating an addressable SEM image of the structure on the substrate after a lithography step (e.g., ADI). Next, the imaged substrate is processed using a process of interest (e.g., a specified etching process). In one embodiment, after etching, an SEM image at the same location is obtained based on the addressable SEM image. Furthermore, the SEM image ADI and AEI are aligned. In one embodiment, a contour extraction algorithm is used to determine the contours of features of interest in ADI and AEI. In one instance, the contour extraction algorithm uses contour points or pixel intensities to describe the contours. Furthermore, a decision coefficient is determined between contour points such as ADI and AEI. R 2 The correlation is related to the variance of the linear combination of AEI contour points. In one embodiment, the correlation explains the proportion of variance of the linear combination of AEI contour points by the linear combination of the feature itself and the ADI contour points of its adjacent features. In one embodiment, the correlation determination procedure generates eigenvalue equations to determine the ADI and AEI contours and their corresponding variances. R 2 The optimal linear combination is determined. The method is described in further detail with reference to Figure 23A. Figure 23A is a flowchart of method 2300 used to train a model configured to determine the etched image (AEI) based on the developed image (ADI). This method includes the following procedures P2301, P2303, P2305 and P2307, which are discussed in detail below. Procedure P2301 includes obtaining (i) measurements of ADI features 2301 imaged on the substrate, and (ii) measurements of post-etched image (AEI) features 2302. The measurements of AEI features 2302 correspond to the measured ADI features on the substrate after the etching process. For example, the same feature may be measured before and after the etching process. It is understood that this method is not limited to a single ADI image or a single AEI image, and multiple ADI and AEI images may be used. In the present discussion, these concepts are illustrated using ADI feature 2301 and AEI feature 2302 as examples. However, the present invention is not limited to ADI features or AEI features. In one embodiment, the entire ADI image and AEI image can be obtained without measuring specific ADI features and AEI features. Furthermore, in one embodiment, the AEI image is not limited to the post-etched image; any other image obtained after the post-exposure step of the patterning process may be used herein, and such image is within the scope of the present invention. In one embodiment, the measured ADI feature 2301 and the measured AEI feature 2302 are obtained by a metrology tool or simulation program configured to generate ADI and AEI images for inputting target features. In one embodiment, the metrology tool is configured to capture scanning electron microscopy (SEM) images of the substrate's ADI and AEI (e.g., FIG. 28). ADI includes ADI functionality, and AEI includes AEI functionality. In one embodiment, ADI includes images obtained from first and second SEM measurements of the ADI feature prior to etching. In one embodiment, the first SEM measurement of the ADI feature is obtained by exposing the imaging substrate via a SEM tool. The second SEM measurement of the ADI feature is obtained by re-exposing the same ADI feature of the imaging substrate via a SEM tool. Similarly, AEI includes images obtained from the results of first and second SEM measurements of the AEI feature, obtained by exposing and re-exposing the etched substrate. The etched substrate is obtained after etching the imaging substrate. Program P2303 includes assigning a first set of variables VADI1 to characterize the measured ADI feature 2301 and a second set of variables VAEI1 to characterize the measured AEI feature 2302. In one embodiment, the first set of variables VADI1 corresponds to a set of positions on the ADI profile of the measured ADI feature 2301, and the second set of variables VAEI1 corresponds to a set of positions on the AEI profile of the measured AEI feature 2302. In one embodiment, the pixel intensity (e.g., grayscale value) of the ADI image and the pixel intensity (e.g., grayscale value) of the AEI image can be used as a first set of variables VADI1 and a second set of variables A. Variable VAEI1. Procedure P2305 includes determining the correlation 2310 between the combination of a first set of variables VADI1 of the measured ADI feature 2301 and the combination of a second set of variables VAEI1 of the measured AEI feature 2302. The first set of variables VADI1 is a linear combination, a non-linear combination, or a machine learning model. In one embodiment, the combination of the first set of variables VADI1 is a weighted sum of the first set of variables VADI1. In one embodiment, correlation 2310 can be determined based on specification correlation analysis or other correlation determination methods. For example, combinations of such variables are considered as correlation variables to characterize the pattern transfer behavior from ADI to AEI. Weights can be positive or negative. In one embodiment, positive or negative indicates the direction of the variable, such as to the left or right of the variable value to be applied. In one embodiment, positive or negative can indicate the contraction or growth of the ADI feature. In one embodiment, the combination of the second set of variables VAEI1 or one or more sub-combinations is a linear combination, a non-linear combination, or a machine learning model. In one embodiment, the first set of variables VAEI1 corresponds to a set of positions on the ADI profile of the measured ADI feature 2301, and the second set of variables VAEI1 corresponds to a set of positions on the AEI profile of the measured AEI feature 2302. In one embodiment, the ADI feature includes the feature of interest and one or more adjacent features. In one embodiment, the first set of variables VADI1 includes a first subgroup of variables associated with the feature of interest and a second subgroup of variables associated with one or more adjacent features. In one embodiment, the combination is a weighted sum of the first subgroup of variables associated with the feature of interest and the second subgroup of variables associated with one or more adjacent features. In one embodiment, the weight assigned to the variable of the adjacent feature is relatively higher than the weight assigned to the variable of another adjacent feature that is far from the feature of interest. Figures 24A and 24B illustrate exemplary ADI and AEI features, respectively. In Figure 24A, the ADI includes the feature of interest, ADIF1, and adjacent features ADINF1 and ADINF2 surrounding ADIF1. The first adjacent feature ADINF1 is relatively closer to the feature of interest, ADIF1, than the second adjacent feature ADINF2. In one embodiment, a set of variables may be positions marked with numbers 1 to 8 on the contour of feature ADIF1. Similarly, another set of variables may be positions (points) on the contours of adjacent features ADINF1 and ADINF2. In one embodiment, the first set of variables, VADI1, may be one set of variables from ADIF1, ADINF1, and ADINF2 (e.g., contour points on the feature contour). In one embodiment, the second set of variables, VAEI1, may be positions (points) on the contour of the AEI feature AEIF1. In one embodiment, the AEI feature AEIF1 may be generated after etching the ADI feature ADIF1. In one embodiment, adjacent features ADINF1 and ADINF2 can influence the shape and size of the AEI feature AEIF1. Therefore, in one instance, a linear combination of variables associated with ADIF1 (e.g., contour points) (as discussed in method 2300 herein) can be determined to be best correlated, second best correlated, third best correlated, etc., with a linear combination of variables of AEIF1. In another instance, a linear combination of variables associated with ADIF1 and adjacent features ADINF1 and ANF2 (as discussed in method 2300 herein) can be determined to be best correlated (second best, third best, etc.) with a linear combination of variables of AEIF1. In one embodiment, determining correlation 2310 includes calculating correlation 2310 using the following given values: (i) a first set of parameters associated with a combination of first set of variables VADI1; and (ii) a second set of parameters associated with a combination of second set of variables VAEI1; determining whether correlation 2310 is maximized (or within a specified range); and in response to correlation 2310 not being maximized (or within a specified range), adjusting the given values ​​of the first set of parameters and the second set of parameters until correlation 2310 is maximized. In one embodiment, the adjustment of the given values ​​of the first set of parameters and the second set of parameters is performed until correlation 2310 is maximized (or within a specified range). In one embodiment, the first set of variables VADI1 associated with ADI (e.g., ADI features ADIF1, ADINF1, or ADINF2 in FIG24A, or pixel intensity of the ADI image) can be represented as a vector. ,and This represents one or more combinations of the first set of variables VADI1. In one instance, It can be represented as a matrix, and It contains more than one number. In this example, It is a scalar, and and Optimization was performed to maximize the correlation. The optimization procedure can provide a solution where the correlation is (locally) maximized, corresponding to multiple combinations found. In one embodiment, This represents the weights associated with each variable in the first set of variables, VADI1. In one embodiment, the second set of variables, VAEI1, associated with AEI (e.g., AEIF1 in Figure 24B, or pixel intensity of the AEI image), can be represented as a vector. ,and This represents one or more combinations of the second set of variables VAEI1. In one embodiment, This represents the weight associated with each variable in the first set of variables, VADI1. In one embodiment, the relevant determination 2310 involves and about and The optimization of mutual information (in one embodiment, maximization). In one embodiment, the optimization of mutual information can be determined based on analytical or numerical methods. In one embodiment, the eigenvalue equation can be used to maximize the correlation between the combination of variables of ADI and the combination of variables of AEI. In one embodiment, mutual information can be determined based on the probability density function of the combination of variables in space. In one embodiment, for example for a finite dataset, a normalized histogram can be used instead of calculating the probability density. An example method for estimating mutual information can be found in "Estimating mutual information" by A. Kraskov, H. Stogbauer, and P. Grassberger (Phys. Rev. E 69, 2004), the entire contents of which are incorporated herein by reference. In one embodiment, the combination can be a nonlinear combination of variables. In the nonlinear case, the relevant determination 2310 includes maximizing... and The correlation between nonlinear functions (e.g., R 2 These functions can be: and The explicit analytical expression can also be a neural network that generates scalars from vector inputs. For example, an instance method for determining correlation 2310 can be based on "Deep Canonical Correlation Analysis by Andrew 2013," which is incorporated herein by reference in its entirety. For example, the optimization procedure needs to maximize the coefficients in the neural network. In one instance, the following correlation can be used. R 2 Equations are used to perform calculations related to 2310: In the above formula, cov and var represents the covariance and variance of the variables. f is a parameter Predefined scalar functions ,and g is a parameter scalar function . f and Instances of g include linear functions ( ), quadratic function, higher-order polynomial, weights are and Machine learning networks. In the case of linear combinations, the following correlation is used. R 2 Equations are used to perform calculations related to 2310: In the above equation, This is the vector form of the first set of variables, VADI1. Corresponding to the first set of parameters, Includes one or more combinations of the first set of variables VADI1. This is the vector form of the second set of variables, VAEI1. Corresponding to the second set of parameters, Includes one or more combinations of the second set of variables VAEI1. R 2 The molecule represents and The covariance between them, and the denominator represents variance and The product of their variances. In one embodiment, the correlation determination 2310 includes... R 2 Compared to and With the derivative set to 0, the following expression is produced: The first equation (A) is the determination. R 2 Eigenvalues ​​and eigenvectors The eigenvalue equation. In the second equation (B), α is the scaling operator. In one embodiment, the relevant... R 2 and and The length is irrelevant. According to one embodiment, the above equation indicates the eigenvalue with the largest eigenvalue. Vector and corresponding vector This represents the best linear combination for transfer. Eigenvalues Maximum vector and corresponding vector This is a suboptimal linear combination for transfer, and so on. In one embodiment, although... and Let be a vector, and As a scalar, multiple solutions to equations A and B were determined during the optimization of the above equations (e.g., maximizing the correlation), thus obtaining different transformation modes. In one embodiment, the number of non-zero eigenvalues ​​is at most a vector. and The minimum length. Among these non-zero eigenvalues, only a finite number of eigenvalues ​​correspond to... R 2 Furthermore, its value is significantly greater than 0. Therefore, only a limited number of contour characteristics are considered as transfers after etching. Other combinations may be disregarded. In one embodiment, this is achieved by examining the vector... and This allows us to find the entity meaning of the transmitted variable and reduce the parameter space accordingly. In one embodiment, eigenvalue analysis can be used to determine the transfer characteristics for selected focal and dose conditions after etching. In one example, for the contact hole of interest and its six adjacent holes, the first set of variables VADI1 associated with ADI can be the distances to 16 profile points to the centroid of the contact hole. Therefore, the first set of variables VADI1 includes 16 • (1 + 6) = 112 variables. The second set of variables VAEI1 associated with AEI can be the distances from the 16 profile points to the center of the contact hole in the AEI of interest. Therefore, the second set of variables VAEI1 includes 16 variables. In one embodiment, examples of pattern transfer based on linear combinations of these variables are further illustrated in Figures 25A to 25F. In one embodiment, nonzero eigenvalues ​​and corresponding eigenvectors (e.g., obtained from the relevant correlation equations A and B above) are also analyzed and interpreted with reference to Figures 25A to 25F. Figures 25A to 25F illustrate the optimal propagation of the linear combination of the first set of variables, as determined by the eigenvalue equations (e.g., equations A and B above). In the upper right corner of each subplot, solid lines (e.g., referred to as WT_ADICH1, WT_ADICH2, WT_ADICH3, WT_ADICH4, WT_ADICH5, and WT_ADICH6) correspond to the weights of the relevant feature (e.g., ADICH1). In the upper right corner of each subplot, dashed lines (e.g., collectively referred to as WT_NH1, WT_NH2, WT_NH3, WT_NH4, WT_NH5, and WT_NH6, respectively) correspond to the weights of adjacent features surrounding the feature of interest (e.g., ADICH1). Figures 25A and 25B illustrate the linear combination of variables corresponding to the translation of a feature, e.g., the translation of a hole in the x and y directions. Figure 25C illustrates a linear combination of variables corresponding to the CD of a feature (e.g., AEI CD), which is affected by the CD of the central hole and its neighborhood (in ADI). Figures 25E and 25F illustrate linear combinations of variables corresponding to the elongation of a feature, for example, the elongation is determined by the elongation of the ADI hole located at the center and the size and displacement of adjacent holes. Figure 25D illustrates a linear combination of variables corresponding to the triangular relationship of a feature, for example, the translation of the AEI feature affected by the CD and adjacent features in the ADI. Referring to Figure 25A, the weight WT_ADICH1 of each profile point on the ADI feature ADICH1 is plotted relative to the angle (right figure). Also, the weight WT_NH1 of each profile point on adjacent ADI features (e.g., NCH) is plotted relative to the angle (right figure). The left figure illustrates an example arrangement of ADI contact holes in polar coordinates. In the arrangement of contact holes (left figure), the dashed profile rch1 corresponds to the desired feature or reference feature with zero-weight profile points. In this example, an inward offset relative to the dashed profile rch1 (e.g., solid profile ADICH1) corresponds to a negative weight, and an outward offset relative to the dashed profile (e.g., solid profile ADICH1) corresponds to a positive weight. Similarly, the bottom curve represents the AEI contact hole AEICH1 and its weight WT_AEICH1. In this example, referring to Figures 25A to 25F, a linear combination is obtained based on eigenvalue analysis. In this example, the largest eigenvalue obtained using the first set of variables (e.g., 112 variables) and the second set of variables (e.g., 16 variables) is... R 2 = 0.67 (see Figure 25A). The corresponding weights of the AEI variables represent the leftward movement of the contact hole, see Figure 25A (bottom left). In the ADI and AEI diagrams (left), the radius of the left edge (e.g., from 0º to 180º) has a positive weight, while the radius of the right edge (e.g., from 180º to 360º) has a negative weight. The solid contact hole AEICH1 (bottom left) in Figure 25A moves to the left relative to the reference contact hole RCH1 (dotted circle), with the weight WT_AEICH1 being a (negative) cosine, as shown in Figure 25A (bottom). In Figure 25A (top row), the corresponding weights of the ADI variables WT_ADICH1 and WT_NCH1 represent the leftward movement of contact hole ADICH1 (at the center) relative to the reference hole RCH1 (dashed circle), without affecting the adjacent hole NCH. Observation shows that the weight WT_NCH1 of the variable related to the adjacent contact hole NCH is approximately zero. Similarly, Figures 25B to 25F illustrate the linear combinations of variables (e.g., the first group of variables and the second group of variables), the corresponding ADI and AEI feature transformations, and the weights related to how the ADI transformation corresponds to the AEI transformation. According to one embodiment, Figures 25A and 25B illustrate, respectively, a first characteristic value (e.g., 0.67) and a second characteristic value (e.g., 0.64) corresponding to the combination of a first set of variables and a second set of variables describing the conversion of ADI and AEI characteristics. The first and second characteristic values ​​correspond to the translations of contact holes ADICH1 and ADICH2 in a first direction and a second direction, respectively. In this example, the translations are in the x and y directions. Figures 25A and 25B show that the translations in both directions can be measured equally well because of the relevant... R 2 They are approximately the same in both directions. According to one embodiment, Figure 25C illustrates a third feature vector corresponding to a combination of a first set of variables and a second set of variables that describe the CD of the ADI and AEI features, respectively. Furthermore, weights WT_ADICH3, WT_NCH3, and WT_AEI3 associated with the linear combination of variables are plotted. The feature vector related to AEI (e.g., the linear combination of the second set of variables) indicates that approximately equal weights are assigned to each variable in the combination of the second set of variables. For example, a variable could be the radius (or diameter) of a contact hole measured in different directions. The radius then corresponds to the CD of the AEI feature, and the average diameter length is equal to the average CD of the AEI feature. Similarly, the ADI feature vector (e.g., the linear combination of the first set of variables) corresponds to the CD of the ADI contact hole minus the average CD of adjacent holes with certain weighting factors. In one embodiment, further examination of the eigenvectors (Figure 25C) also reveals that the placement of adjacent contact holes around contact hole ADICH3 affects the CD of AEI feature AEICH3. To understand this effect, the CD of the AEI feature of interest (e.g., a contact hole located at the center in the AEI) is correlated with the CD of the ADI feature of interest (e.g., a contact hole located at the center in the ADI), as shown in Figure 26A. Figure 26A shows that there is a positive correlation between the two CD parameters of ADI and AEI, but the data correlation is quite large because the correlation... R 2 The value is 0.22. The second figure, Figure 26B, plots the CD of the AEI feature of interest relative to the average CD of its neighbors. This figure reveals a negative correlation. R 2The value is 0.14. This is the local etch load mentioned earlier. In other words, if all adjacent dimensions are relatively large, the etching process will produce many byproducts, and less etchant will be available for the feature of interest (e.g., a centrally located contact hole in an ADI). This localized etch load effect reduces the etching rate and makes the AEI contact hole smaller than the desired size. It should be emphasized that the correlations in the plotting data have been corrected according to the relevant provisions of this embodiment, such as the feature value equation. For example, the corrected correlation between the hole size and its adjacent dimensions is much smaller, for example, R 2 = 0.006 and is positive. According to one embodiment, Figures 25E and 25F illustrate the fourth and fifth feature vectors corresponding to the combinations of the first and second sets of variables, respectively, which describe the elongation (in both directions) of the ADI and AEI features. Furthermore, Figure 25E shows the weights WT_ADICH5, WT_NCH5, and WT_AEI5 associated with the linear combinations of variables. The weights WT_ADICH6, WT_NCH6, and WT_AEI6 associated with other linear combinations of variables are shown in Figure 25F. The elongation of contact holes AEICH5 and AEICH6 is affected by the elongation of contact holes ADICH5 and ADICH6, respectively. Additionally, the elongation rate is affected by the CD and displacement of adjacent holes of ADICH5 and ADICH6. In this example, since... R 2 The two corresponding eigenvalues ​​are almost equal, thus explaining the elongation in any direction well. However, due to R 2 = 0.34, approximately two-thirds of the AEI elongation may not be explainable by ADI measurements. Therefore, related R 2 The value indicates that other factors affect the elongation, or that the ADI or AEI elongation measurement is prone to noise. According to one embodiment, Figure 25D illustrates a sixth eigenvector corresponding to the combination of the first set of variables and the second set of variables, respectively describing the triangularity of the ADI and AEI features. Furthermore, Figure 25D plots the weights WT_ADICH4, WT_NCH4, and WT_AEI4 associated with the linear combination of variables. In this example, the weights associated with the ADI feature are the size and displacement of the adjacent holes surrounding feature ADICH4. R 2When the value is 0.08, it indicates that most triangulation relationships measured after etching may not be interpretable using ADI profiles. In one embodiment, other feature values ​​are below a specified relevant threshold value, for example, R 2 ≤0.01. Relevant values ​​below the specified threshold may indicate that ADI measurements may fail to interpret AEI measurements, and that the corresponding profile attributes (e.g., triangulation) of the measured ADI may be irrelevant to the predicted AEI profile. Program P2307 includes training model 2320 based on correlation 2310 by including one or more sub-combinations of a first set of variables VADI1, the correlation values ​​of these sub-combinations being within specified correlation thresholds, and model 2320 being used to determine the AEI function of the input ADI function. As discussed herein, one or more sub-combinations of the first set of variables VADI1 are linear combinations, nonlinear combinations, or machine learning models. In one embodiment, one or more sub-combinations of the first set of variables VADI1 are weighted sums of the first set of variables VADI1, where the weights are positive or negative values. In one embodiment, one or more sub-combinations of the second set of variables VAEI1 are linear combinations, nonlinear combinations, or machine learning models. In one embodiment, a sub-combination may differ from a mathematically defined "subgroup" and may be the average of all contour points. In one embodiment, one or more sub-combinations characterize the amount of deformation of the ADI profile of the measured ADI feature 2301 caused by processing performed on the measured ADI feature 2301. In one embodiment, the deformation is the difference between a given position of the ADI profile and a corresponding position of the AEI profile. In one embodiment, the deformation is characterized by a linear combination of a first set of variables VADI1. In one embodiment, the determination of model 2320 includes (a) determining whether a sub-combination of the first set of variables VADI1 and the correlation 2310 of that sub-combination are higher than a specified correlation threshold; (b) responding to the sub-combinations mentioned above, including the sub-combinations in model 2320; and (c) responding to the fact that the sub-combination is not mentioned above, selecting another sub-combination of the first set of variables VADI1, and repeating steps (a) to (c) for the specified number of iterations or until the sub-combination is accumulated. In one embodiment, the specified correlation threshold is greater than 0.01. For example, Figures 25A to 25F represent values ​​greater than or equal to 0.08. R 2 The selected combination of values. In one embodiment, one or more sub-combinations characterize one or more of the following: the measured translation of ADI feature 2301 in a specified direction; the critical dimension of the measured ADI feature 2301; the elongation of the measured ADI feature 2301 in a specified direction; and the triangulation of the measured ADI feature 2301. Examples of sub-combinations and corresponding transfer characteristics are discussed herein with reference to Figures 25A to 25F. The methods described above (e.g., method 2300) have some applications related to lithography processes, post-exposure processes, metrology equipment, and other applications involving image-based feature transformations, where the image is due to one or more processing of features. In one embodiment, method 2300 can be used to quantify the short-range etch load effect and feature placement. For example, a combination of a first set of variables VADI1 associated with adjacent features of interest in the ADI can be determined. Using method 2300, the effect of placement variations of adjacent features (e.g., features surrounding the feature of interest in the ADI) on the CD and the placement of the AEI feature can be combined. In one embodiment, eigenvalue analysis is used to quantify the effect of the next nearest neighbor and thereafter on the AEI profile. For example, the effect attributed to adjacent features within 180 nm of the feature of interest in the ADI can be used. In one embodiment, the combined variables characterizing the ADI CD, the displacement of each feature, and the elongation of the feature can be used to quantify the short-range etch load effect. Figures 27A and 27B illustrate the placement of AEI as interpreted by the feature of interest (e.g., the contact hole at the center of the pattern) and its adjacent features (e.g., NH1, NH2, and NH3). R 2 This is one part of the change. The radius of the influence circle surrounding the feature of interest in the ADI. For example, line 2710 illustrates the correlation between the x-position of the AEI and the distance from the feature of interest (radius = 0) in the ADI. R 2 The dashed line 2720 indicates the correlation with the y-position. R 2 Related to the y-position. In ADI, the distance from the feature of interest (radius = 0) increases. Line 2710 indicates that the feature of interest (or the first set of variables associated with it) explains approximately 62% of the variation in the x-position of the AEI feature and approximately 60% of the variation in the y-position of the AEI feature. Additionally, line 2710 indicates that the first adjacent feature NH1 (or a linear combination of subgroups of the first set of variables) explains 4.2% of the R 2 Variance (e.g., the difference between a point on the y-axis and NH1). Similarly, the next adjacent feature NH2 explains... R 2 The variance is 0.3%. Therefore, the variations explained by features further away are fewer than those explained by the feature of interest itself. Similarly, line 2720 represents a portion of the variation in y-position explained by the feature of interest (radius = 0), while NH1, NH2, and NH3 in ADI are further away. Figure 27B explains AEI CD. R 2 A portion of the variance is explained by the feature of interest (e.g., the contact hole at the center of the pattern) and adjacent features (e.g., NH1, NH2, and NH3). The radius of the influence circle surrounding the feature of interest in ADI. For example, line 2730 shows the correlation explained by all variables (e.g., all variables in the first group) as the distance from the feature of interest (radius 0) in the ADI increases. R 2 The dashed line 2740 shows the variables associated with CD. R 2 Line 2740 shows the correlation R explained as the distance from the feature of interest in the ADI increases (at radius = 0). 2 The amount. Line 2730 indicates that all variables of the feature of interest explain approximately 23.5% of the variation in AEI feature CD. Furthermore, line 2740 indicates a subgroup of variables associated with the feature of interest CD, explaining approximately 23.5% of the variation. Line 2730 also indicates that the first adjacent feature NH1 (or a linear combination of subgroups of the first set of variables) explains R. 2The variance is approximately 27% (e.g., the difference between a point on the y-axis and NH1). In this example, the adjacent feature NH1 comprises six features equidistant from the center feature (e.g., see Figures 25A to 25F). The adjacent feature NH2, which is farther from adjacent feature NH1, explains approximately 0.5%, while another adjacent feature NH3 explains an additional 0.7%. Similarly, line 2740 shows a portion of the variation in AEI feature CD, which is explained by the target feature (radius = 0), while features NH1, NH2, and NH3 in ADI are even farther away. In this example, the exact radius of influence affecting the AEI feature depends on the pattern density in ADI. Additionally, note that 11% of the variation in AEI CD (the difference between lines 2730 and 2740) is due to placement variations associated with adjacent features. Such positional variations account for one-fifth of the total variation explained. Therefore, Figures 27A and 27B illustrate examples of short-range etch load quantization. Thus, by employing this method, for example, using all variables associated with ADI, short-range etch load quantization can be improved (e.g., by 11%). In other words, this method can explain more variations and the causes of such variations than existing methods, for example, through adjacent features and transfer characteristics (e.g., placement, CD, translation, triangulation, etc.). The determined causes can be further used to reduce, for example, variations after development of CD. In other words, the method of the present invention can help determine the causes of CD and placement variations and how they transfer after development, and then predict variations after etching, deposition, or other exposure processes based on ADI. In one embodiment, another application of method 2300 may be monitoring process quality. For example, method 2300 also includes procedures for monitoring process quality based on a first set of variables of ADI characteristics and a selected combination of their sensitivity to focus points and exposure conditions; adjusting one or more program parameters to maintain process quality within a specified range. In one embodiment, monitoring includes measuring relevant ADI profile attributes related to the tip of the pattern (e.g., sub-combinations of variables of the first set of variables); adjusting one or more program parameters based on the measured sensitivity and correlation to improve the conversion of the ADI function from point-to-point functionality to AEI functionality. For example, the etching behavior of tip-to-tip structures is largely dependent on the shape of features in the resist, which is also sensitive to the focal point. Furthermore, when SEM is used to measure the shape of features in the resist, the resist shape change is represented by the waveform produced by the SEM. Using this method 2300, it is possible to directly assess which parameters of the SEM waveform are related to the efficiency of the tip-to-tip transfer process. These tip characteristics can then be closely monitored in high-volume manufacturing (HVM) of semiconductor wafers. Moreover, these characteristics can be used in (empirical) simulators of the etching process, thus accelerating process optimization within the process. In one embodiment, the SEM measurement formula can be modified to monitor the HVM process. In one embodiment, the SEM formula includes SEM tool settings for measuring the tip-to-tip structure after development and etching. In one embodiment, the SEM measurement formula remains the same for both ADI and AEI measurements to examine the tip-to-tip structure. Furthermore, a set of variables characterizing the tip-to-tip structure can be correlated to determine its transfer characteristics after etching. For example, whether the transfer is good (e.g., within threshold limits) and how much the transfer varies. The tip-to-tip structure may differ from, for example, the structure of a hole or wire because it is focal-sensitive and contains a lot of 3D information (e.g., CD variation over the entire height). Therefore, setting up an SEM formula to measure the tip-to-tip structure is not an easy task. According to one embodiment, the method 2300 described herein can be applied as follows: The tip-to-tip structure can be measured after development, followed by contour extraction. In one embodiment, ADI measurements can be performed under different SEM settings, and contour information can be extracted at each SEM threshold. Next, AEI measurements can be performed to analyze the corresponding AEI contour relative to the ADI contour. For example, the changes in ADI and AEI contours can be correlated to determine which SEM settings best explain the changes in AEI based on the ADI contour information. Thus, the SEM settings can be quantified by measuring the quality of the tip-to-tip structure according to the SEM settings, thereby enabling the use of SEM tools to accelerate the inspection process. In one embodiment, the procedure P2305 for determining the relevant variables is based on sparsity constraints. Sparsity constraints refer to excluding one or more of the first set of variables or the second set of variables. In one embodiment, any profile can be used and characterized by a first set of variables. For very detailed profiles (e.g., profiles characterized by a large number of profile points), most variations in the profile points will be determined, for example, by SEM. SEM may emit noise or small resist fluctuations. By optimizing the correlation (e.g., making R...) 2 Maximizing will only select relevant linear combinations of variables (e.g., the weighted sum of all variables), and combinations related to, for example, SEM noise will not be transferred. Furthermore, the selected combination provides the following information: after etching, other shapes (e.g., higher contour deformation frequencies) will not transfer and are therefore irrelevant to the quantization etching behavior. In one embodiment, sparsity constraints can be systematically introduced by setting sparsity constraints in the optimization. Sparsity constraints can be automatically set to zero for irrelevant data points using a set of equations. Sparsity constraints can be implemented by adding regularization terms to the optimization (e.g., including weights). α and (L1 norm of β). In one embodiment, method 2300 may further include adjusting the metrology tool settings based on correlation to cause relevant improvements. Examples of adjusting metrology tool settings have been discussed above with respect to tip-to-tip structures. However, the invention is not limited to specific structures. SEM settings can be determined for any other structure to be printed on the substrate, such as contact holes, lines, rectangles, or other features of interest. In one embodiment, the metrology tool settings include at least one of the following: electron beam intensity, incident angle, voltage contrast, SEM threshold, pixel size, scan speed, or frame count. In one embodiment, method 2300 may further include adjusting one or more parameters associated with the contour extraction algorithm to cause relevant improvements. In one embodiment, Figure 23C illustrates a flowchart of a method 2370 for optimizing a metrology formulation based on the correlation between ADI and AEI. For example, optimization involves perturbing settings related to the metrology tool or algorithm (e.g., number of frames, SEM voltage, threshold value) to maximize the correlation R. 2 In one embodiment, optimization is performed by perturbing only the metrology-related parameters to best measure local variability. In another embodiment, metrology formulation optimization includes not only perturbing the metrology-related parameters but also patterning the process-related parameters to best measure process variations. For example, parameters related to the patterning process may be overlay, average CD of the pattern, focus, dose, etc. In one embodiment, procedure P2371 includes acquiring ADI and AEI data 2372 without perturbing the process parameters. In one embodiment, procedure P2371 includes acquiring ADI and AEI data 2374 by perturbing process variables (e.g., overlay, CD, dose, focus). Procedure P2373 includes extracting contours from the ADI and AEI images (e.g., using a contour extraction algorithm). As previously discussed in method 2300, procedure P2305 can be executed to determine the correlation 2310 between ADI and AEI. Program P2377 contains a metric formula or parameter for determining the profile formula algorithm to cause relevant 2310 improvements (e.g., maximization). In one embodiment, system optimization of the SEM formulation and profiling algorithm can be performed as follows. For example, the procedure of method 2300, as described above (e.g., via the computer system of FIG. 30), is performed to determine the correlation between the measured ADI and AEI. Furthermore, the parameters of the SEM formulation or the profiling algorithm are perturbed to determine whether the perturbed parameters improve the correlation between ADI and AEI. R 2 By perturbation, the maximum correlation can be obtained. R 2 The SEM threshold values ​​can be set. For example, SEM threshold values ​​such as 30%, 50%, and 70% can be used for ADI and / or AEI measurements. In this example, using multiple threshold values ​​while measuring AEI characteristics can increase the correlation for elongation and triangular relationship transfer characteristics. R 2 Using multiple threshold values ​​to measure ADI may increase the correlation between shift and CD transfer characteristics. R 2 . Furthermore, the parameters in the contour algorithm related to contour extraction from SEM images can be changed. Additionally, multiple variables in the first set of variables (e.g., the radius of the shrinkage hole measured in different directions) can be changed to determine the relevant... R 2 Maximize the number of variables. If the number of variables can be 8 or 32, it can be determined that reducing the number of variables will significantly reduce the correlation. R 2 Therefore, 8 spokes may not be sufficient to describe the profile. Alternatively, increasing the number of variables to 32 may only slightly increase the correlation. R 2 Therefore, it can be concluded that, for a given contact hole size, 16 variables may be sufficient to describe the profile. In one embodiment, a series of process variations can be performed. For example, a series of overlay values ​​can be programmed by modifying the photomask pattern. For example, as shown in Figures 23D and 23E, photomask patterns 2390 and 295 include arrays of contact holes. Overlay can be programmed by moving the contact holes, or the average CD can be disturbed by increasing the size of the contact holes. For example, in photomask pattern 2390, contact hole 2391 is moved to the right from a desired position (dashed line). In photomask pattern 2395, the size of contact hole 2396 is increased relative to the desired size (dashed line). Photomask patterns 2390 or 2395 can be used to fabricate a solid photomask and image a substrate. Thus, for example, the displaced contact holes (corresponding to hole 2391) are imaged on the substrate. An ADI image of such an imaging substrate is captured using a metrological tool (e.g., SEM). Furthermore, the AEI of the etched substrate is etched and captured. Overlap is measured using image AEI from nearby locations under different overlap conditions (e.g., overlap within the range of -10 nm to 10 nm). Then, the average profile or unit pixel can be determined for each overlay condition of ADI and AEI. ADI and AEI data based on different overlay conditions are an example of data 2374. Once data 2374 is acquired, other procedures of method 2370 in FIG23C can be executed to determine the optimal parameters of the metrology formula (e.g., SEM settings or contour extraction algorithm parameters), for example, maximizing the correlation between ADI and AEI. As discussed herein, the parameters of the metrology formula can be, for example, electron beam intensity, incident angle, voltage contrast, SEM threshold, pixel size, scan speed, number of frames, or combinations thereof. In one embodiment, one or more parameters associated with the contour extraction algorithm can be modified. In one embodiment, as discussed herein, repeated SEM measurements for acquiring ADI and AEI are performed at the SEM location. In another embodiment, when using a test substrate to obtain data 2374, ADI and AEI may not be performed at the same location. Therefore, in this invention, metrology formulation optimization applies not only to local variability in measurement but also to process-related variations, such as iterations. In one embodiment, method 2300 also includes adjusting parameters associated with the resist process or etching process by using relevant simulated patterning and etching processes to make the yield of the patterning process greater than a specified yield threshold. In one embodiment, method 2300 also includes adjusting parameters related to the lithography process by using a relevant simulated patterning process to ensure that the performance metrics of the lithography device are within specified performance limits. In one embodiment, the parameters of the patterning process include dose or focus conditions set via the lithography device. In one embodiment, method 2300 can be modified to train a model configured to determine an etched image (AEI) based on an image after development (ADI). For example, correlation is determined based on the grayscale values ​​of pixels in the ADI and AEI. In one embodiment, the method includes: obtaining (i) the ADI of an imaged substrate and (ii) the etched image (AEI) after etching the imaged substrate; determining a correlation between a combination of a first set of variables of the ADI and a combination of a second set of variables of the AEI, the first set of variables and the second set of variables being grayscale values ​​of the ADI and AEI, respectively; and training the model based on the correlation by including one or more sub-combinations of the first set of variables having correlation values ​​within a specified correlation threshold, the model being used to determine the AEI of the input ADI. Therefore, programs P2305 and P2307 can be modified to treat the first set of variables as grayscale values ​​of pixels in the entire ADI and the second set of variables as grayscale values ​​of pixels in the entire AEI. In one embodiment, a metrology tool (e.g., FIG. 28) is provided, configured to adjust the metrology tool settings based on the correlation between ADI and AEI. In one embodiment, the metrology tool includes: a beam generator configured to measure ADI features after imaging a substrate and AEI features after etching the substrate; and a processor (e.g., processor 104) configured to determine settings based on the correlation between the measured ADI and the AEI obtained by the electron beam. In one embodiment, the processor can be configured to obtain a correlation between measured ADI features and measured AEI features, the measured AEI features corresponding to measured ADI features printed on a substrate undergoing an etching process. This correlation is based on a combination of variables characterizing how the measured ADI features are converted into AEI features. The processor can then adjust the settings of the metrology tool based on this correlation to improve the correlation. The settings can be determined based on the derivative of the correlation with respect to each setting, the derivative indicating the improvement of the correlation for each setting of the metrology tool. In one embodiment, the measuring tool is a scanning electron microscope (SEM). In one embodiment, the settings of the measuring tool include values ​​of at least one of the following: an electron beam intensity, an incident angle, a voltage contrast ratio, an SEM threshold, a pixel size, a scanning speed, or a number of frames. In one embodiment, the beam generator is an electron beam generator. Figure 23B is a flowchart of a method 2350 for determining an etched image (AEI) based on an etched image (ADI) using a trained machine learning model 2210 (Figure 22) or 2320 (Figure 23A). This method includes procedures P2352 and P2354, which are discussed in detail below. Procedure P2352 includes obtaining the ADI of the substrate. For example, the ADI can be obtained using a measurement tool such as SEM discussed herein. Procedure P2354 includes determining the AEI by inputting the ADI into a training model and outputting the ADI via training model 2210 or 2320. In one embodiment, for example, as discussed in Figures 22 and 23A, a training model is obtained by training based on the correlation between a first set of variables of the measured ADI and a second set of variables of the measured AEI. The correlation is within a specified correlation threshold. In one embodiment, as described above, the correlation is determined by using the following given values ​​to calculate the correlation: (i) a first set of parameters associated with a combination of first set of variables, and (ii) a parameter associated with a combination of second set of variables; determining whether the correlation is maximized; and in response to failure to maximize the correlation, adjusting the given values ​​of the first set of parameters and the second set of parameters until the correlation is maximized. In one embodiment, the trained model is obtained by: (a) determining whether the first set of variables, a sub-combination of the second set of variables, and the correlation of the sub-combination are higher than a specified correlation threshold; (b) in response that the sub-combination is higher, including the sub-combination in a model; and (c) in response that the sub-combination is not higher, selecting another sub-combination of the first set of variables, and repeating steps (a) to (c) for a specified number of iterations or until the sub-combination is exhausted. In one embodiment, as described above, the combination or one or more sub-combinations of the first set of variables are linear combinations, nonlinear combinations, or machine learning models. In one embodiment, the combination of the first set of variables is a weighted sum of the first set of variables, wherein the weights are positive or negative values. In one embodiment, the combination of the second set of variables is a linear combination, a nonlinear combination, or a machine learning model. In one embodiment, ADI includes ADI features, and AEI includes AEI features corresponding to the ADI features, which are determined by a trained model. In one embodiment, the trained model determines one or more of the following: translation of an ADI feature along a specified direction; critical dimensions of an ADI function; elongation of an ADI feature in a specified direction; triangular relationships of an ADI feature; or rotation of an ADI function. In one embodiment, the processor may further include instructions stored on the processor configured to adjust one or more parameters associated with the contour extraction algorithm based on correlation to cause correlation improvements. For example, as described above, 16 variables may be sufficient to obtain the correlation improvement associated with a given contact hole, as previously stated. In one embodiment, one or more of the variables represent one or more of the following: a translation of the measured ADI feature in a specified direction; a critical dimension of the measured ADI feature; an elongation of the measured ADI feature in a specified direction; a triangular relationship of the measured ADI feature; or a rotation of the measured ADI feature. In one embodiment, a non-transitory computer-readable medium is provided, comprising instructions that, when executed by one or more processors, cause operation of a program including methods 2200, 2300, or 2400 described above. In one embodiment, the non-transitory computer-readable medium may be implemented in a metrology instrument, computer hardware system, lithography device, or other system related to a patterning process. Such non-transitory computer-readable media improves the patterning process, measurement results, and overall yield of the patterning process. In one embodiment, the methods discussed above (e.g., methods 400, 900, 1700, 2200, or 2300) may be implemented via a processor (e.g., computer system 100 of 104). In one embodiment, a computer program product includes a non-transitory computer-readable medium having instructions recorded thereon, which, when executed by a computer, implement the methods discussed herein. In some embodiments, the inspection device may be a scanning electron microscope (SEM) that obtains an image of a structure (e.g., some or all of the structure of a device) exposed or transferred onto a substrate. Figure 28 depicts an embodiment of the SEM tool. A primary electron beam EBP emitted from an electron source ESO is converged by a condenser lens CL and then transferred through beam deflectors EBD1, E×B deflectors EBD2, and an objective lens OL to irradiate the substrate PSub on the substrate stage ST at the focal point. When the substrate PSub is irradiated by the electron beam EBP, secondary electrons are generated by the substrate PSub. These secondary electrons are deflected by the E×B deflector EBD2 and detected by the secondary electron detector SED. The two-dimensional electron beam image can be obtained by: synchronously detecting electrons generated from the sample, for example, by performing a two-dimensional scan of the electron beam by the beam deflector EBD1 in the X or Y direction or by repeatedly scanning the electron beam EBP by the beam deflector EBD1, and by continuously moving the substrate PSub on the substrate stage ST in the other of the X or Y directions. The signal detected by the secondary electronic detector (SED) is converted into a digital signal by an analog-to-digital (A / D) converter (ADC) and sent to the image processing system (IPU). In one embodiment, the image processing system IPU may have a memory (MEM) for storing all or part of the digital images to be processed by the processing unit (PU). The processing unit (PU) (e.g., specially designed hardware or a combination of hardware and software) is configured to convert or process the digital images into a dataset representing the digital images. Furthermore, the image processing system IPU may have a storage medium (STOR) configured to store the digital images and corresponding datasets in a reference database. A display device (DIS) may be connected to the image processing system IPU, allowing an operator to perform necessary operations on the device via a graphical user interface. As mentioned above, SEM images can be processed to extract the contours of the edges of objects representing a device structure. These contours are then quantified using metrics such as CD (cutoff distance). Therefore, images of device structures are typically compared and quantified using oversimplified metrics such as distance between edges (CD) or simple pixel differences between images. Typical contour models for detecting object edges in an image to measure CD use image gradients. In practice, these models rely on strong image gradients. However, in practice, images are often noisy and have discontinuous boundaries. Techniques such as smoothing, adaptive finiteness, edge detection, abrasion, and dilation can be used to process the results of image gradient contour models to address noisy and discontinuous images, but ultimately result in low-resolution quantization of high-resolution images. Therefore, in most cases, mathematical manipulation of images of device structures to reduce noise and automated edge detection leads to a loss of image resolution, thereby resulting in a loss of information. Thus, the result is, in summary, low-resolution quantization of an oversimplified representation of complex high-resolution structures. Therefore, there is a need for a mathematical representation of the general shape of a structure (e.g., circuit features, alignment marks, or metric target portions (e.g., grating features)) that retains resolution and describes the structure produced or intended to be produced using a patterning process, regardless of whether such structure is, for example, in a latent resist image, in a developed resist image, or a layer transferred onto a substrate, for example, by etching. In the context of lithography or other patterning processes, the structure may be a manufactured device or a portion thereof, and the image may be a SEM image of the structure. In some cases, the structure may be a feature of a semiconductor device (e.g., an integrated circuit). In this case, the structure may be referred to as a pattern or a desired pattern comprising a plurality of features of the semiconductor device. In some cases, the structure may be an alignment mark or a portion thereof (e.g., a grating of an alignment mark) used in an alignment measurement process to determine the alignment of one object (e.g., a substrate) with another object (e.g., a patterning device), or a measurement target or a portion thereof (e.g., a grating of a measurement target) used to measure parameters of the patterning process (e.g., overlap, focus, dose, etc.). In one embodiment, the measurement target is a diffraction grating used to measure (e.g., overlap). Figure 29 schematically illustrates another embodiment of the detection device. The system is used to detect a sample 90 (such as a substrate) on a sample stage 88 and includes a charged particle beam generator 81, a condenser lens module 82, a probe forming objective lens module 83, a charged particle beam deflection module 84, a secondary charged particle detector module 85, and an image forming module 86. A charged particle beam generator 81 generates a primary charged particle beam 91. A condenser lens module 82 focuses the generated primary charged particle beam 91. A probe forming objective module 83 focuses the condensed primary charged particle beam into a charged particle beam probe 92. A charged particle beam deflection module 84 scans the formed charged particle beam probe 92 on the surface of the region of interest on the sample 90, which is fixed to the sample stage 88. In one embodiment, the charged particle beam generator 81, the condenser lens module 82, and the probe forming objective module 83, or their equivalents, alternatives, or any combination thereof, together form a charged particle beam probe generator that generates the scanning charged particle beam probe 92. A secondary charged particle detector module 85 detects secondary charged particles 93 emitted from the sample surface immediately after being bombarded by a charged particle beam probe 92 (and possibly along with other reflected or scattered charged particles from the sample surface) to generate a secondary charged particle detection signal 94. An image forming module 86 (e.g., a computing device) is coupled to the secondary charged particle detector module 85 to receive the secondary charged particle detection signal 94 from the secondary charged particle detector module 85 and accordingly forms at least one scanned image. In one embodiment, the secondary charged particle detector module 85 and the image forming module 86, or their equivalents, alternatives, or any combination thereof, together form an image forming apparatus for forming scanned images from detected secondary charged particles emitted from the sample 90 by the charged particle beam probe 92. In one embodiment, monitoring module 87 is coupled to image forming module 86 of an image forming apparatus to monitor, control, etc., the patterning process, and / or to derive parameters for patterning process design, control, monitoring, etc., using scanned images of samples 90 received from image forming module 86. Therefore, in one embodiment, monitoring module 87 is configured or programmed to cause the execution of the methods described herein. In one embodiment, monitoring module 87 includes computing devices. In one embodiment, monitoring module 87 includes a computer program for providing the functionality described herein, encoded on a computer-readable medium forming or disposed within monitoring module 87. In one embodiment, similar to the electron beam detection tool of FIG28 used to detect a substrate using a probe, the electron current in the system of FIG29 is significantly larger than that of, for example, a CD SEM depicted in FIG28, making the probe spot large enough to enable fast detection. However, due to the larger probe spot, the resolution may not be as high as that of a CD SEM. In one embodiment, without limiting the scope of the invention, the detection device described above can be a single-beam device or a multi-beam device. SEM images from systems such as Figure 28 or Figure 29 can be processed to extract the contours of the edges of objects representing the device structure. These contours are then typically quantized at user-defined tangents via a metric such as CD. Therefore, images of the device structure are typically compared and quantified via metrics such as the inter-edge distance (CD) measured on the extracted contours or simple pixel differences between images. Figure 30 is a block diagram illustrating a computer system 100 that can assist in implementing the methods and processes disclosed herein. The computer system 100 includes a bus 102 or other communication mechanism for transmitting information and a processor 104 (or multiple processors 104 and 105) coupled to the bus 102 for processing information. The computer system 100 also includes main memory 106, such as random access memory (RAM) or other dynamic storage device, coupled to the bus 102 for storing information and instructions to be executed by the processor 104. The main memory 106 can also be used to store temporary variables or other intermediate information during the execution of instructions to be executed by the processor 104. The computer system 100 further includes read-only memory (ROM) 108 or other static storage device coupled to the bus 102 for storing static information and instructions for the processor 104. A storage device 110 (such as a magnetic disk or optical disk) is provided and coupled to the bus 102 for storing information and instructions. Computer system 100 may be coupled via bus 102 to a display 112 for displaying information to a computer user, such as a cathode ray tube (CRT), flat panel display, or touch panel display. Input device 114, including alphanumeric keypads and other keys, is coupled to bus 102 for transmitting information and command selections to processor 104. Another type of user input device is a cursor controller 116, such as a mouse, trackball, or cursor direction keys, for transmitting directional information and command selections to processor 104 and for controlling cursor movement on display 112. This input device typically has two degrees of freedom in two axes (a first axis (e.g., x) and a second axis (e.g., y)), allowing the device to specify a position in a plane. Touch panel (screen) displays may also be used as input devices. According to one embodiment, a portion of the program described herein may be executed by computer system 100 in response to processor 104 executing one or more sequences of instructions contained in main memory 106. Such instructions may be read into main memory 106 from another computer-readable medium (such as storage device 110). Execution of the sequence of instructions contained in main memory 106 causes processor 104 to perform the processing steps described herein. One or more processors in a multiprocessor configuration may also be employed to execute the sequence of instructions contained in main memory 106. In an alternative embodiment, hardwired circuitry may be used in place of or in combination with software instructions. Therefore, the description herein is not limited to any particular combination of hardware circuitry and software. As used herein, the term "computer-readable media" means any media that participates in providing instructions to processor 104 for execution. This media can take many forms, including but not limited to non-volatile media, volatile media, and transmission media. Non-volatile media include, for example, optical discs or magnetic disks, such as storage device 110. Volatile media include dynamic memory, such as main memory 106. Transmission media include coaxial cables, copper wires, and optical fibers, including lines containing bus 102. Transmission media can also take the form of sound waves or light waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media include, for example, floppy disks, flexible disks, hard disks, magnetic tapes, any other magnetic media, CD-ROMs, DVDs, any other optical media, punched cards, paper tapes, any other physical media with a perforated pattern, RAM, PROMs and EPROMs, FLASH-EPROMs, any other memory chips or cartridges, carrier waves as described below, or any other media that can be read by a computer. Various forms of computer-readable media may involve carrying one or more sequences of instructions to processor 104 for execution. For example, initially, such instructions may be carried on a disk of a remote computer. The remote computer may load the instructions into its dynamic memory and transmit the instructions via a modem over a telephone line. The modem at the local end of computer system 100 may receive data over the telephone line and convert the data into an infrared signal using an infrared transmitter. An infrared detector coupled to bus 102 may receive the data carried in the infrared signal and place the data on bus 102. Bus 102 carries the data to main memory 106, and processor 104 retrieves and executes the instructions from main memory 106. Instructions received from main memory 106 may be stored on storage device 110 before or after execution by processor 104, depending on the circumstances. Computer system 100 also ideally includes a communication interface 118 coupled to bus 102. Communication interface 118 provides bidirectional data communication coupled to network connection 120, which connects to local area network 122. For example, communication interface 118 may be an integrated Services Digital Network (ISDN) card or modem to provide data communication connectivity to a corresponding type of telephone line. As another example, communication interface 118 may be a Local Area Network (LAN) card to provide data communication connectivity to a compatible LAN. Wireless connectivity may also be implemented. In any such implementation, communication interface 118 transmits and receives electrical, electromagnetic, or optical signals carrying digital data streams representing various types of information. Network connection 120 typically provides data communication to other data devices via one or more networks. For example, network connection 120 may provide a connection to host computer 124 or to data devices operated by Internet Service Provider (ISP) 126 via local area network 122. ISP 126, in turn, provides data communication services via a global packet data communication network (now commonly referred to as the "Internet" 128). Both local area network 122 and Internet 128 use electrical, electromagnetic, or optical signals carrying digital data streams. Signals via various networks and signals on network connection 120 and via communication interface 118 (which carry digital data to and from computer system 100) are examples of carrier waves for transmitting information. Computer system 100 can send and receive messages (including code) via a network, network link 120, and communication interface 118. In an Internet example, server 130 can transmit requested code for an application via Internet 128, ISP 126, local area network 122, and communication interface 118. Such a downloaded application can provide, for example, lighting optimization as described in the embodiment. The received code can be executed by processor 104 upon receipt and / or stored in storage device 110 or other non-volatile storage for later execution. In this way, computer system 100 can obtain application code in carrier form. Figure 31 schematically depicts an exemplary lithography apparatus incorporating the techniques described herein. The apparatus includes: - an illumination system IL for modulating a radiation beam B. In this particular case, the illumination system also includes a radiation source SO; - a first object stage (e.g., a patterning device stage) MT, equipped with a patterning device holder for holding a patterning device MA (e.g., a magnifying mask), and connected to a first locator for accurately positioning the patterning device relative to a project PS; - a second object stage (substrate stage) WT, equipped with a substrate holder for holding a substrate W (e.g., a resist-coated silicon wafer), and connected to a second locator for accurately positioning the substrate relative to the project PS; - a projection system ("lens") PS (e.g., a refractive, reflective, or reflective-refractive optical system) for imaging the irradiated portion of the patterning device MA onto a target portion C (e.g., comprising one or more grains) of the substrate W. As described herein, the device is of the transmissive type (i.e., with transmissive patterning). However, in general, it can also be of the reflective type, for example (with reflective patterning). The device can use patterning devices of a different kind than those of classic photomasks; examples include programmable mirror arrays or LCD matrices. A radiation beam is generated by a source SO (e.g., a mercury lamp or excimer laser, laser-produced plasma (LPP) EUV source). For example, this beam is fed directly into the illumination system (illuminator) IL, either directly or after passing through an adjustment member such as a beam extender Ex. The illuminator IL may include an adjustment member AD for setting the outer radial range and / or inner radial range (typically referred to as σ_outer and σ_inner, respectively) of the intensity distribution in the beam. Additionally, the illuminator IL typically includes various other components, such as a beam concentrator IN and a condenser CO. In this way, the beam B irradiating the patterning device MA has the desired uniformity and intensity distribution in its cross-section. Regarding Figure 31, it should be noted that the source SO can be inside the housing of the lithography device (this is often the case when the source SO is, for example, a mercury lamp), but it can also be located away from the lithography device, with the resulting radiation beam being directed into the device (e.g., by means of a suitable guiding mirror); the latter scenario often occurs when the source SO is an excimer laser (e.g., based on KrF, ArF, or F). 2. The situation during laser treatment. The beam PB then intercepts the patterning device MA held on the patterning device stage MT. After traversing the patterning device MA, the beam B passes through the lens PL, which focuses the beam B onto the target portion C of the substrate W. The substrate stage WT can be accurately moved (e.g.) by means of a second positioning member (and an interferometric measurement member IF) to position the different target portions C within the path of the beam PB. Similarly, the first positioning member can be used to accurately position the patterning device MA relative to the path of the beam B, for example, after mechanical capture of the patterning device MA from the patterning device library or during scanning. Generally, the movement of the stages MT and WT is achieved by means of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which are not explicitly depicted in FIG31. However, in the case of a stepper (relative to a stepping scanning tool), the patterning device stage MT may be connected only to the short-stroke actuator, or may be fixed. The depicted tool can be used in two different modes: - In step mode, the patterning stage MT remains substantially fixed, and the entire patterning image is projected onto the target portion C in a single pass (i.e., a single "flash"). The substrate stage WT is then shifted in the x and / or y directions, allowing different target portions C to be irradiated by the beam PB; - In scanning mode, the essentially same situation applies, except that a given target portion C is not exposed in a single "flash". Alternatively, the patterning stage MT can move at a speed v in a given direction (the so-called "scanning direction," e.g., the y-direction) such that the projected beam B scans the patterning image; simultaneously, the substrate stage WT moves simultaneously in the same or opposite directions at a speed V = Mv, where M is the magnification of the lens PL (typically, M = 1 / 4 or 1 / 5). In this way, a relatively large target portion C can be exposed without compromising resolution. Figure 32 schematically depicts another exemplary lithography projection apparatus 1000, comprising: - a source collector module SO for providing radiation; - an illumination system (illuminator) IL configured to modulate a radiation beam B (e.g., EUV radiation) from the source collector module SO; - a support structure (e.g., a photomask stage) MT configured to support a patterning device (e.g., a photomask or a magnifying photomask) MA and connected to a first positioner PM configured to accurately position the patterning device; - a substrate stage (e.g., a wafer stage) WT configured to hold a substrate (e.g., a resist-coated wafer) W and connected to a second positioner PW configured to accurately position the substrate; and - a projection system (e.g., a reflective projection system) PS configured to project a pattern imparted to the radiation beam B onto a target portion C (e.g., comprising one or more dies) of the substrate W by means of the patterning device MA. As described herein, device 1000 is of the reflective type (e.g., using a reflective photomask). It should be noted that because most materials are absorptive in the EUV wavelength range, the patterning device can have a multilayer reflector comprising, for example, multiple stacks of molybdenum and silicon. In one example, the multi-stacked reflector has 40 pairs of molybdenum and silicon layers, each layer being a quarter wavelength thick. X-ray lithography can be used to produce even smaller wavelengths. Since most materials are absorptive at both EUV and X-ray wavelengths, the thin pieces of patterned absorbing material on the patterning device configuration (e.g., a TaN absorber on top of the multilayer reflector) define where features will be printed (positive photoresist) or not printed (negative photoresist). Referring to Figure 32, the illuminator IL source collector module SO receives an extreme ultraviolet radiation beam. Methods for generating EUV radiation include, but are not limited to, converting a material into a plasma state, having at least one element possessing one or more emission lines in the EUV range, such as xenon, lithium, or tin. In one such method (often referred to as laser-generated plasma (“LPP”)), plasma can be generated by irradiating a fuel (such as droplets, streams, or clusters of material having spectral emission elements) with a laser beam. The source collector module SO may be part of an EUV radiation system including a laser (not shown in Figure 32) used to provide a laser beam for exciting the fuel. The resulting plasma emits output radiation (e.g., EUV radiation), which is collected using a radiation collector disposed within the source collector module. For example, when a CO2 laser is used to provide a laser beam for fuel excitation, the laser and source collector module may be separate entities. In such cases, the laser is not considered a component of the lithography apparatus, and the radiation beam is transferred from the laser to the source collector module via a beam delivery system that includes, for example, suitable guide mirrors and / or beam extenders. In other cases, such as when the radiation source is a discharge-generating plasma EUV generator (often referred to as a DPP radiation source), the radiation source may be an integral part of the source collector module. An illuminator IL may include an adjuster for adjusting the angular intensity distribution of the radiated beam. Typically, at least the outer radial range and / or inner radial range (usually referred to as σ_outer and σ_inner, respectively) of the intensity distribution in the pupil plane of the illuminator can be adjusted. Additionally, the illuminator IL may include various other components, such as faceted field mirror devices and faceted pupil mirror devices. The illuminator can be used to adjust the radiated beam to achieve desired uniformity and intensity distribution in its cross-section. A radiation beam B is incident on a patterning device (e.g., a photomask) MA, which is held on a support structure (e.g., a photomask stage) MT and patterned by the patterning device. After reflection from the patterning device (e.g., the photomask) MA, the radiation beam B is transferred through a projection system PS, which focuses the radiation beam B onto a target portion C of a substrate W. The substrate stage WT can be accurately moved, for example, to position different target portions C within the path of the radiation beam B, by means of a second locator PW and a position sensor PS2 (e.g., an interferometric measuring device, a linear encoder, or a capacitive sensor). Similarly, a first locator PM and another position sensor PS1 can be used to accurately position the patterning device (e.g., the photomask) MA relative to the path of the radiation beam B. Patterning device alignment marks M1, M2 and substrate alignment marks P1, P2 can be used to align the patterning device (e.g., the photomask) MA and the substrate W. The described apparatus 1000 can be used in at least one of the following modes: 1. In stepping mode, the support structure (e.g., photomask stage) MT and the substrate stage WT remain substantially fixed while the entire pattern of the radiated beam is projected onto the target portion C in a single step (i.e., single static exposure). The substrate stage WT is then shifted in the X and / or Y directions to expose different target portions C. 2. In scanning mode, the support structure (e.g., photomask stage) MT and the substrate stage WT are scanned synchronously while the pattern of the radiated beam is projected onto the target portion C (i.e., single dynamic exposure). The speed and direction of the substrate stage WT relative to the support structure (e.g., photomask stage) MT can be determined by the magnification (reduction) and image inversion characteristics of the projection system PS. 3. In another mode, the support structure (e.g., photomask stage) MT remains substantially fixed to hold the programmable patterning device, and the substrate stage WT is moved or scanned while the pattern of the radiated beam is projected onto the target portion C. In this mode, a pulsed radiation source is typically used, and the programmable patterning device is updated as needed after each movement of the substrate stage WT or between consecutive radiation pulses during scanning. This operating mode can be readily applied to maskless lithography using programmable patterning devices, such as programmable mirror arrays of the type mentioned above. Figure 33 shows the device 1000 in more detail, which includes a source collector module SO, an illumination system IL, and a projection system PS. The source collector module SO is constructed and configured to maintain a vacuum environment within the enclosure structure 220 of the source collector module SO. EUV radiation emitting plasma 210 can be formed by generating a plasma radiation source through discharge. EUV radiation can be generated by a gas or vapor (e.g., xenon gas, lithium vapor, or tin vapor), wherein an extreme thermal plasma 210 is generated to emit radiation in the EUV range of the electromagnetic spectrum. For example, the extreme thermal plasma 210 is generated by a discharge that causes at least partial ionization of the plasma. For efficient radiation generation, Xe, Li, Sn vapor, or any other suitable gas or vapor at a partial pressure of, for example, 10 Pa may be required. In one embodiment, excited tin (Sn) plasma is provided to generate EUV radiation. Radiation emitted by the thermoplasm 210 is transmitted from the source chamber 211 to the collector chamber 212 via a gas barrier or contaminant trap 230 (also referred to in some cases as a contaminant barrier or foil trap) positioned in or behind an opening in the source chamber 211. The contaminant trap 230 may include a channel structure. The contaminant trap 230 may also include a gas barrier, or a combination of a gas barrier and a channel structure. As known in the art, the contaminant trap or contaminant barrier 230 further indicated herein includes at least a channel structure. Collector chamber 211 may include a radiation collector CO, which may be a so-called grazing incidence collector. The radiation collector CO has an upstream radiation collector side 251 and a downstream radiation collector side 252. Radiation traversing the collector CO may be reflected from a grating spectral filter 240 to be focused into a virtual source point IF along the optical axis indicated by the dotted line "O". The virtual source point IF is commonly referred to as the intermediate focus, and the source collector module is configured such that the intermediate focus IF is located at or near an opening 221 in the enclosure structure 220. The virtual source point IF is an image of the radiative emission plasma 210. Subsequently, radiation traverses the illumination system IL, which may include a faceted field mirror device 22 and a faceted pupil mirror device 24. The faceted field mirror device 22 and the faceted pupil mirror device 24 are configured to provide the desired angular distribution of the radiation beam 21 at the patterning device MA, and the desired uniformity of the radiation intensity at the patterning device MA. After the radiation beam 21 is reflected at the patterning device MA held by the support structure MT, a patterned beam 26 is formed, and the patterned beam 26 is imaged onto the substrate W held by the substrate stage WT via the projection system PS and reflective elements 28 and 30. More components than those shown may typically be present in the illumination optics unit IL and the projection system PS. Depending on the type of lithography equipment, the grating spectral filter 240 may be present. Additionally, more mirrors than shown in the figures may be present; for example, the projection system PS may contain one to six additional reflective elements than those shown in Figure 33. The collector optics CO illustrated in Figure 33 is depicted as a nested collector with grazing incidence reflectors 253, 254, and 255, serving as only one example of a collector (or collector mirror). The grazing incidence reflectors 253, 254, and 255 are arranged symmetrically around the optical axis O, and this type of collector optics CO is ideally used in combination with a discharge-generating plasma radiation source. Alternatively, the source-collector module SO can be part of the LPP radiation system shown in Figure 34. The laser LAS is configured to deposit laser energy onto a fuel such as xenon (Xe), tin (Sn), or lithium (Li), thereby generating a highly ionized plasma 210 with an electron temperature of tens of electron volts. High-energy radiation generated during the de-excitation and recombination of this plasma is emitted from the plasma, collected by near-normal incident collector optics CO, and focused onto an opening 221 in the enclosure structure 220. The embodiments can be further described using the following clauses: 1. A method for training a configuration to predict whether a feature associated with an imaging substrate will be defective after etching the imaging substrate, the method comprising: obtaining via a metric instrument (i) a developed image of the imaging substrate at a given location, the developed image including a plurality of features, and (ii) an etched image of the imaging substrate at the given location, the etched image including etched features corresponding to the plurality of features; and training the configuration using the developed image and the etched image to determine the defect of a given feature among the plurality of features in the developed image, wherein the defect determination is based on comparing the given feature in the developed image with a corresponding etched feature in the etched image. 2. The method of clause 1, wherein the model is an empirical model or a machine learning model, wherein the empirical model varies with a physical property of the feature associated with the imaging substrate. 3. The method of any one of clauses 1 to 2, wherein obtaining the developed image comprises: imaging a photomask pattern on a substrate via a patterning device; obtaining a developed substrate of the imaged substrate; aligning a measuring instrument at a given position to the developed substrate; and capturing an image of the developed substrate. 4. The method of any one of clauses 1 to 3, wherein obtaining the etched image comprises: etching the imaged substrate via an etching process having specified etching conditions; aligning the measuring instrument at a given position to the etched substrate; and capturing the etched image of the etched substrate. 5. The method of clause 4, wherein the etching conditions include etchant composition, plasma gas parameters, etching rate, electromagnetic field, plasma potential, type of inductance or capacitance of the etching, temperature of the substrate, ion energy distribution, ion angular distribution, sputtering and redeposition rates, or a combination thereof. 6. The method of any one of clauses 1 to 5, wherein the training comprises: aligning the developed image and the etched image based on the plurality of features; comparing each of the plurality of features in the developed image with a corresponding feature of one of the etched features in the etched image; determining, based on the comparison, whether a given etched feature in the etched image satisfies a defect condition; classifying the identified feature as defective in response to not satisfying the defect condition; and adjusting a model parameter value of the model based on the defect of the identified feature. 7. The method of clause 6, wherein the adjustment of the model parameter value comprises adjusting the values ​​of the plurality of model parameters. 8. The method of clause 6, wherein the defect condition is a physical characteristic of the given etched feature in the etched image. 9. The method of clause 8, wherein the physical characteristic is at least one of: a critical size of the given etched feature; or a displacement of the given etched feature relative to a given feature in the developed image.10. The method of any one of items 1 to 9, wherein the defect is characterized by at least one of: a binary decision of defective or non-defective; or a probability that the given feature is defective. 11. The method of any one of items 2 to 10, wherein the machine learning model is a convolutional neural network. 12. The method of item 11, wherein the model parameters are weights or biases associated with one or more layers of the machine learning model. 13. The method of item 11, wherein the model parameters being weights or biases includes the model parameters being both weights and biases. 14. The method of any one of items 1 to 13, wherein the measuring instrument is an optical microscope or an electron beam microscope. 15. The method of any one of items 1 to 13, wherein the measuring instrument is a scanning electron microscope (SEM), and the measurements are obtained from an SEM image. 16. The method of any one of clauses 1 to 15, wherein the trained model is further configured to predict a failure rate associated with a given pattern of the developed image, the failure rate indicating the rate of defect occurrence when the imaged substrate is etched using the specified etching conditions. 17. The method of any one of clauses 1 to 16, wherein the further configuration of the trained model comprises: classifying the plurality of patterns associated with a pattern of interest as defective or defect-free; determining the total number of defective patterns associated with the pattern of interest; and calculating the failure rate of the pattern of interest as a ratio of the total number of defective patterns to the total number of patterns among the plurality of patterns. 18. The method of any one of claims 1 to 17, further comprising: imaging a desired pattern on a substrate via a patterning device; obtaining a developed image of the imaged pattern; performing the training model using the developed image to classify whether the desired pattern will be defective after etching; and adjusting an etching condition based on the pattern classified as defective, such that the imaged pattern will be defect-free after etching. 19. A method for determining etching conditions of an imaged substrate, the method comprising: obtaining a developed image of the imaged substrate and initial etching conditions to be used for etching the imaged substrate; determining, via a trained model, a failure rate of a feature associated with the imaged substrate, the failure rate indicating that the feature is defective after etching the imaged substrate; and modifying the initial etching conditions based on the failure rate, such that the probability of the feature being defective after etching is reduced.20. The method of claim 19, wherein training the etching conditions is an iterative process, and wherein an iteration comprises: obtaining a relationship between a given etching condition and a given failure rate associated with a given feature; determining an etched image associated with the imaging substrate by performing an etching model using the developed image and the etching conditions, determining whether the given feature satisfies a defect condition based on the etched image; and in response to not satisfying the defect condition, identifying another etching condition associated with a failure rate lower than the given failure rate based on the relationship. 21. The method of any one of claims 19 to 20, wherein the defect condition of the feature is at least one of the following: a missing feature; a displacement range associated with the feature; or a tolerance range associated with a critical dimension of the feature. 22. A method for determining an etching characteristic associated with an etching process, the method comprising: obtaining, via a metric instrument: (i) an image after development (ADI) of an imaged pattern at a given location on a substrate, wherein the imaged pattern includes a feature of interest and adjacent features adjacent to the feature of interest; and (ii) an etched image (AEI) of the imaged pattern at the given location on the substrate, wherein the AEI includes an etched feature corresponding to the feature of interest in the ADI; and determining, using the ADI and the AEI, a correlation between the etched feature associated with the feature of interest in the ADI and the adjacent features, wherein the correlation characterizes the etching characteristic associated with the etching process. 23. The method of claim 22, wherein the feature of interest includes a plurality of features of interest. 24. The method of any one of claims 22 to 23, wherein the correlation varies with the density of one of the adjacent features adjacent to the feature of interest. 25. The method of any one of clauses 22 to 24, wherein the correlation between the etched feature and the adjacent features in the ADI depends on at least one of: the geometry of one of the features of interest or the adjacent features; the geometry of one of the auxiliary features or a deviation associated with the feature of interest; a distance between the feature of interest and the adjacent features; a distance along a line feature; a critical dimension of the feature; coordinates on the substrate associated with the feature of interest, the adjacent features, and the etched feature of interest; auxiliary features around the feature of interest or the absence of auxiliary features; or a deviation of an edge position associated with the feature of interest from its intended position. 26. The method of any one of clauses 22 to 25, wherein the correlation based on the critical dimension of a feature is calculated using the following equation: in Let be one of the related vectors, where CDAEI is the AEI CD of the feature of interest; CDADI iThe ADI CD of the i-th neighbor; and r is the correlation coefficient, and A correlation matrix. 27. The method of any one of clauses 22 to 26, wherein the feature of interest is at least one of: a contact hole; a line; or a line end. 28. The method of any one of clauses 22 to 27, wherein the adjacent features are at least one of: a plurality of contact holes in a defined orientation relative to the feature of interest; or a plurality of lines having a defined spacing. 29. The method of any one of clauses 22 to 28, further comprising: generating a power spectral density of the correlation in a spatial domain, the power spectral density indicating a magnitude of an etching characteristic effect and a range of a loading effect. 30. The method of any one of clauses 22 to 29, further comprising determining etching conditions associated with the imaged pattern based on the correlation and at a given radial distance between the center of the substrate and the edge of the substrate, such that the correlation is maintained within a target range. 31. The method of any one of claims 22 to 30, further comprising determining, based on the correlation, the etching conditions of the imaged pattern located at the center of a substrate, such that the correlation is within the target range. 32. The method of any one of claims 22 to 31, further comprising determining, based on the correlation, the etching conditions of the imaged pattern located at the edge of a substrate, such that the correlation is maintained within the target range. 33. The method of any one of claims 31 to 32, wherein the etching conditions include etchant composition, plasma gas parameters, etching rate, electromagnetic field, plasma potential, type of inductance or capacitance of the etching, temperature of the substrate, ion energy distribution, ion angular distribution, parameters associated with sputtering and redeposition rates, etching cycle parameters based on a saturation effect, or a combination thereof. 34. The method of any one of clauses 30 to 33, wherein the etching conditions depend on at least one of: a location of the substrate being etched, the location being the radial distance between the center of the substrate and the edge of the substrate; an etching cycle; an etching chamber; a sequence of etching cycles and deposition steps; or a tuning parameter associated with the etching chamber, the tuning being based on a sensitivity to a change in one of the tuning parameters. 35. The method of any one of clauses 30 to 34, wherein determining the correlation comprises: obtaining (i) a plurality of ADIs at a plurality of given locations on the substrate, each ADI having the same feature of interest, and (ii) a plurality of AEIs at the plurality of given locations, each AEI having the etched feature of interest corresponding to the feature of interest; establishing the correlation between adjacent features of the feature of interest in each ADI and the etched feature of interest in each of the AEIs.36. A method for determining an etching condition associated with an etching process, the method comprising: obtaining a correlation between an etched feature of interest in an etched image (AEI) and one of adjacent features of the etched feature of interest in an etched image (ADI); and determining the etching condition associated with the etching process based on the correlation, such that the correlation is kept within a target range. 37. The method of claim 36, wherein obtaining the correlation between the etched feature and the adjacent feature includes obtaining the correlation between the etched feature and a plurality of adjacent features. 38. The method of claim 36, wherein determining the etching condition depends on at least one of: a location of the etched substrate, the location being a radial distance between the center of the substrate and the edge of the substrate; an etching cycle of the etching process; an etching chamber for the etching process; a sequence of etching cycles and deposition steps; or a tuning parameter associated with an etching chamber tuning parameter associated with the etching chamber, the tuning being based on a sensitivity to changes in the tuning parameter. 39. The method of claim 38, wherein the tuning parameter comprises a plurality of tuning parameters. 40. The method of any one of claims 36 to 39, wherein determining the etching condition comprises: adjusting the value of the tuning parameter associated with a given etching chamber such that the correlation associated with the given imaged pattern remains within the target range. 41. The method of any one of clauses 36 to 40, wherein obtaining the correlation comprises: obtaining via a metric instrument: (i) the developed image (ADI) of the imaged pattern at a given location, the imaged pattern including a feature of interest and an adjacent feature adjacent to the feature of interest; and (ii) the etched image (AEI) of the imaged pattern at the given location, the AEI including the etched feature of interest corresponding to the feature of interest in the ADI; and determining the correlation between the etched feature and the adjacent feature associated with the feature of interest in the ADI using the ADI and the AEI. 42. A method for generating an interpretation model configured to interpret a prediction generated by a trained model, the method comprising: obtaining a dataset by executing the trained model, the dataset comprising a plurality of predictions associated with a plurality of features of an analog image (ADI), the ADI including a feature of interest, each of the plurality of predictions being generated by the trained model; determining a distance between the plurality of features and each location of the feature of interest; assigning weights to each of the plurality of predictions based on the distances; and determining model parameter values ​​of the interpretation model based on the weighted predictions such that a difference between an output of the interpretation model and the weighted predictions is reduced, wherein the model parameter values ​​indicate the contribution of each pixel of the ADI to the prediction associated with the feature of interest.43. The method of item 42, wherein obtaining the plurality of predictions comprises: performing the trained model to predict a characteristic of each of the plurality of features in the ADI, the characteristic indicating a defect in the plurality of features. 44. The method of item 43, wherein the defect of a given feature in the ADI indicates a probability that the given feature will be defective after etching. 45. The method of any of items 42 to 44, wherein the prediction is whether the feature of interest in the ADI will be printed defectively or defect-free after etching. 46. The method of any of items 42 to 45, wherein the assignment of equal weights to each prediction comprises: assigning a relatively higher weight to a prediction if the distance associated with one of the plurality of predictions is relatively small. 47. A method of any one of claims 42 to 46, wherein determining the model parameter values ​​of the interpretation model is a fitting process comprising: obtaining initial model parameter values ​​and the weighted predictions; executing the interpretation model using the initial model parameter values ​​to produce an initial output; and determining a difference between the weighted predictions and the initial output; and adjusting the initial model parameter values ​​based on the difference to minimize the difference. 48. A method of any one of claims 42 to 47, wherein the interpretation model receives the ADI including the feature of interest as input and produces an interpretation map as output, wherein the interpretation map indicates the contribution of a neighboring region of the feature of interest to making the prediction associated with the feature of interest. 49. A method of any one of claims 42 to 48, wherein the interpretation map is a pixelated image, and the model parameter values ​​are weights assigned to each pixel of the pixelated image. 50. The method of any one of items 42 to 49, wherein the interpreted map is a binary map, wherein each pixel is assigned a value of either 0 or 1. 51. The method of any one of items 42 to 50, wherein the binary map is generated by assigning each pixel a value of either 0 or 1 based on a pixel value exceeding a threshold value, wherein 0 indicates that the feature of interest will be printed defectively after etching, and 1 indicates that the feature of interest will be printed defect-free after etching. 52. The method of any one of items 42 to 51, wherein the interpreted map is a color image, wherein a specific color is assigned based on the model parameter values. 53. The method of any one of items 42 to 52, wherein the interpretation model is a linear model associated with the feature of interest in the ADI. 54. The method of any of items 42 to 53, wherein the linear model uses a linear regression to fit the complex number of predictions using a least square error.55. A method for identifying the contribution of pixels in an acquired image to a prediction generated by a trained model, the method comprising: obtaining (i) the acquired image (ADI) including a feature of interest using a metric tool, and obtaining (ii) an interpretation model configured to interpret a prediction associated with the feature of interest, the prediction being generated by the trained model; and applying the interpretation model to the ADI image to generate an interpretation map comprising pixel values ​​quantifying the contribution of each pixel in the ADI image to the prediction of the feature of interest. 56. The method of claim 55, wherein the interpretation model is a linear model associated with the feature of interest in the ADI. 57. The method of any one of claims 55 to 56, wherein the interpretation map is a binary map, wherein each pixel is assigned a value of either 0 or 1. 58. The method of any one of items 55 to 57, wherein the prediction is a defect of the feature of interest, and the prediction is performed via a trained model. 59. The method of any one of items 55 to 58, wherein the interpreted map is a binary map, wherein each pixel is assigned a value of either 0 or 1. 60. A method for generating a model for determining the failure rate of features in a developed image, the method comprising: obtaining the developed image (ADI) of a substrate, the ADI comprising a plurality of features; generating a first part of the model based on physical property values ​​associated with a subgroup of features of the ADI; and generating a second part of the model based on the first part of the model and the physical property values ​​associated with all of the plurality of features of the ADI, wherein the subgroup of features of the ADI differs from the other features of the ADI. 61. The method of claim 60, wherein the generation of the first part and the second part of the model comprises fitting a first probability distribution function and a second probability distribution function respectively by maximizing a log-probability metric of the model. 62. The method of claim 61, wherein the model is a combination of: (i) the first probability distribution function configured to estimate the distribution of physical characteristic values ​​(e.g., CD) of fault-free vias, and (ii) the second probability distribution function configured to determine the failure rate based on the physical characteristic values ​​of all the plurality of features of the ADI. 63. The method of claim 61, wherein the model is a weighted sum of the first probability distribution function and the second probability distribution function.64. The method of clause 61, wherein generating the model comprises: fitting the first probability distribution function based on the square of one of the physical property values ​​of the subgroup of features by maximizing a first log-probability measure associated with the first probability distribution function, wherein the subgroup of features has physical property values ​​higher than a physical property threshold; combining the fitted first probability distribution function with the second probability distribution function; and fitting the second probability distribution function and a relative weight associated therewith based on the combined distribution based on the physical property values ​​of all the features in the plurality of features, such that a second log-probability measure associated with the combined distribution is maximized. 65. The method of clause 61, wherein fitting the first probability distribution function is an iterative process comprising: (a) determining the first log-probability measure using given values ​​of the parameters of the first probability distribution function; (b) determining whether the first log-probability measure is maximized; and (c) in response to not maximizing, adjusting the values ​​of the parameters of the first probability distribution function based on a gradient, and performing steps (a) to (c) wherein the gradient is the first-order derivative of the first log-probability measure with respect to the parameters of the first probability distribution function. 66. The method of clause 61, wherein fitting the second probability distribution function comprises: determining the values ​​of the parameters of the second probability distribution function and their weights based on maximizing the second log-probability measure, without modifying the values ​​of the parameters of the first probability distribution function. 67. The method of claim 61, wherein fitting the second probability distribution function is an iterative process comprising: (a) obtaining a combined distribution of the fitted first probability distribution function and the second probability distribution function; (b) determining a second log-probability measure based on the combined distribution and keeping the equivalent values ​​of the parameters of the fitted first distribution fixed, using a given value of the parameters of the second probability distribution function; (c) determining whether the second log-probability measure is maximized; and (d) in response to not maximizing, adjusting the equivalent values ​​of the parameters of the second probability distribution function based on a gradient, and performing steps (b) to (d) wherein the gradient is a first derivative of the second log-probability measure with respect to the parameters of the second probability distribution function. 68. The method of any one of claims 61 to 67, wherein the first probability distribution function is a normal distribution characterized by: a cutoff value associated with the physical property, a first position parameter describing a shift of the normal distribution, and a first scaling parameter describing an expansion of the normal distribution. 69. The method of any one of clauses 61 to 68, wherein the second probability distribution function is a generalized extremum (GEV) distribution characterized by: a second position parameter (μ) describing a shift of the GEV distribution, a second scaling parameter (σ) describing an extension of the GEV distribution, and a shape parameter (ξ) describing the shape of the GEV distribution.70. The method of any one of claims 61 to 69, further comprising: imaging a desired pattern comprising a plurality of additional features on another substrate via a patterning apparatus; obtaining the developed image of the imaged pattern; performing the first probability distribution function and the second probability distribution function using the developed image to classify a portion of a feature within the ADI as defective after etching; and adjusting an etching condition based on the classified features such that the imaged pattern will not fail after etching. 71. The method of any one of claims 61 to 70, wherein the plurality of features includes a plurality of holes, a plurality of lines, a plurality of pillars, or a combination thereof. 72. The method of any one of claims 70 to 71, wherein the portion of the ADI feature classified as defective after etching includes at least one of: a closed hole formed after etching due to resist blocking the development of one of the holes; or a merged hole after etching; or a necking of one of the plurality of lines. 73. The method of any one of claims 60 to 72, further comprising: tuning a lithography process to reduce the failure rate of ADI features after etching, wherein the tuning includes adjusting dosage, focus, or both; determining whether an additional filtering step of a resist layer should be performed to reduce the failure rate of the ADI features after etching; determining whether an additional descaling or pass-through step should be performed to reduce the failure rate of the ADI features after etching; during high-volume manufacturing, detecting the ADI features to determine whether a lithography apparatus meets specified printing criteria; or reprocessing a specific substrate or a large number of substrates prior to etching based on the failure rate. 74. The method of any one of claims 60 to 73, wherein the ADI is an image of the printed substrate obtained via a measuring instrument or from a database storing images of a printed substrate. 75. The method of any one of claims 60 to 74, wherein the physical characteristic is a critical dimension (CD) of a feature, and the physical characteristic threshold value is a CD threshold value. 76. The method of any one of clauses 60 to 74, wherein the physical characteristic is at least one of: a geometric mean of the CD of a feature, wherein the CDs are measured along a first direction and a second direction in the ADI; a directional CD of the feature of interest in the ADI; a curvature variation of the feature of interest in the ADI; or a CD obtained for each feature of interest at multiple metric instrument limits. 77. The method of clause 76, wherein the directional CD is at least one of: a CD measured along the x-direction; a CD measured along the y-direction; or a CD measured along a desired angle. 78. The method of any one of clauses 60 to 77, further comprising: extracting statistical characteristics associated with the fault-free vias from the model; and determining a process window for a patterned process based on the statistical characteristics.79. A system for determining a portion of features that will fail after etching, the system comprising: a metric for capturing an illustrated image (ADI) of a substrate at a given location, the illustrated image including a plurality of features; and a processor configured to: execute a model for determining the failure rate of the plurality of features of the ADI after etching, wherein the model is a combination of: (i) a first probability distribution function configured to estimate the distribution of physical characteristic values ​​of fault-free vias, and (ii) a second probability distribution function configured to determine the failure rate based on the physical characteristic values ​​of all of the plurality of features of the ADI. 80. The system of claim 79, further comprising: a patterning device configured to image a desired pattern including the plurality of features on the substrate; and a processor configured to: receive the ADI of the imaged substrate via the measuring instrument; perform the first probability distribution and the second probability distribution to determine the failure rate of the features of the ADI; and, based on the features having a relatively high failure rate, tune the patterning device to reduce the failure rate of the features. 81. The system of claim 80, wherein the processor is configured to tune a dose or focus via a knob / set on the patterning device. 82. A system of claim 81, wherein the processor is further configured to: determine whether an additional filtering step of a resist layer should be performed to reduce the failure rate of the ADI features after etching; determine whether an additional descaling or pass-through step should be performed to reduce the failure rate of the ADI features after etching; or, during high-volume manufacturing, inspect the ADI features to determine whether a lithography apparatus meets specified printing criteria. 83. A system of any one of claims 79 to 82, wherein the metrology tool includes a scanning electron microscope (SEM) configured to measure at least one of the following physical properties: the average CD of one plurality of instances of a feature of interest in the ADI; the directionality CD of one feature of interest in the ADI; the curvature variation of one feature of interest in the ADI; or the CD obtained for each feature of interest at multiple metrology tool limits. 84. A non-transitory computer-readable medium comprising instructions which, when executed by one or more processors, cause operations including: obtaining an analog-displayed image (ADI) of a substrate, the ADI comprising a plurality of features; generating a first portion of a model based on physical property values ​​associated with a subgroup of features of the ADI; and generating a second portion of the model based on the first portion of the model and the physical property values ​​associated with all of the plurality of features of the ADI, wherein the subgroup of features of the ADI differs from the other features of the ADI.85. A non-transitory computer-readable medium as described in clause 84, wherein the model is a combination of: (i) a first probability distribution function configured to estimate the distribution of physical characteristic values ​​of fault-free vias, and (ii) a second probability distribution function configured to determine the failure rate based on the physical characteristic values ​​of all of the plurality of features of the ADI. 86. A non-transitory computer-readable medium as described in clause 85, wherein the model is a weighted sum of one of the first probability distribution function and the second probability distribution function. 87. A non-transitory computer-readable medium as described in clause 85, wherein generating the model comprises: fitting the first probability distribution function based on the square of one of the physical property values ​​of the subset of features by maximizing a first log-probability measure associated with the first probability distribution function, wherein the physical property values ​​of the subset of features are above a physical property threshold; combining the fitted first probability distribution function with the second probability distribution function; and fitting the second probability distribution function and a relative weight associated therewith based on the combined distribution based on the physical property values ​​of all the features in the plurality of features, such that a second log-probability measure associated with the combined distribution is maximized. 88. A non-transitory computer-readable medium as described in clause 85, wherein fitting the first probability distribution function is an iterative process comprising: (a) determining the first log-probability measure using given values ​​of the parameters of the first probability distribution function; (b) determining whether the first log-probability measure is maximized; and (c) in response to not maximizing, adjusting the values ​​of the parameters of the first probability distribution function based on a gradient, and performing steps (a) to (c), wherein the gradient is the first-order derivative of the first log-probability measure with respect to the parameters of the first probability distribution function. 89. A non-transitory computer-readable medium as described in clause 85, wherein fitting the second probability distribution function comprises: determining the values ​​and weights of the parameters of the second probability distribution function based on maximizing the second log-probability measure, without modifying the values ​​of the parameters of the first probability distribution function. 90. A non-transitory computer-readable medium as described in clause 85, wherein the fitting of the second probability distribution function is an iterative process comprising: (a) obtaining a combined distribution of the fitted first probability distribution function and the second probability distribution function; (b) determining a second log-probability measure based on the combined distribution and keeping the equivalent values ​​of the parameters of the fitted first distribution fixed, using a given value of the parameters of the second probability distribution function; and (b) determining whether the second log-probability measure is maximized; and (c) in response to not maximizing, adjusting the equivalent values ​​of the parameters of the second probability distribution function based on a gradient, and performing steps (b) to (c), wherein the gradient is the first-order derivative of the second log-probability measure with respect to the parameters of the second probability distribution function.91. A non-transitory computer-readable medium as described in any of clauses 85 to 90, wherein the first probability distribution function is a normal distribution characterized by: a cutoff value associated with the physical property, a first position parameter describing a shift of the normal distribution, and a first scaling parameter describing an extension of the normal distribution. 92. A non-transitory computer-readable medium as described in any of clauses 85 to 91, wherein the second probability distribution function is a generalized extremum (GEV) distribution characterized by: a second position parameter (μ) describing a shift of the GEV distr...

Claims

1. A non-transitory computer-readable medium comprising instructions stored thereon, which, when executed by one or more processors, cause operations including: exposing an ADI feature to a charged particle beam to generate a first image of the ADI feature, the ADI feature being a structure within a resist material; re-exposing the ADI feature to the charged particle beam to generate a second image of the ADI feature; and determining a defect attribute of the ADI feature based on a physical characteristic associated with the first image and the second image.

2. The non-transitory computer-readable medium as claimed in claim 1, wherein determining the defect attribute includes: extracting a first feature from the first image and extracting a second feature from the second image; determining whether a defect metric is exceeded based on a difference between the first feature and the second feature; and classifying the ADI feature as defective in response to exceeding the defect metric.

3. The non-transitory computer-readable medium as in claim 2, wherein the physical characteristic is a critical size or pixel intensity.

4. The non-transitory computer-readable medium as claimed in claim 2, wherein the defect measure varies with a first physical characteristic of one of the ADI features in the first image and a second physical characteristic of one of the ADI features in the second image.

5. Non-transitory computer-readable media as described in any of claims 2 to 4, wherein the defect measure is a multivariable function, a bilinear function, a trained machine learning model, or a second-order or higher-order polynomial.

6. The non-transitory computer-readable medium of claim 5, wherein the trained machine learning model is obtained by: training a machine learning model using a training dataset comprising: (i) a plurality of image pairs, each image pair comprising a first image and a second image of a plurality of ADI features; and (ii) an etched image (AEI) of a substrate corresponding to the ADI features.

7. A non-transitory computer-readable medium as claimed in any of claims 1 to 4, wherein the first image comprises a plurality of frames obtained from the first exposure, and the second image comprises a plurality of frames obtained from a re-exposure of the ADI feature.

8. The non-transitory computer-readable medium of claim 7, wherein the plurality of frames of the first image are captured corresponding to different locations of the ADI feature, and the plurality of frames of the second image are captured corresponding to locations of the ADI feature that are the same as those captured from the first exposure.

9. The non-transitory computer-readable medium of claim 8, wherein the determination includes: determining the difference between the physical characteristics associated with one or more frames of the first image and the corresponding one or more frames of the second image.

10. A non-transitory computer-readable medium as claimed in any of claims 1 to 4, wherein the ADI feature includes one of the contact holes in the resist material.

Images