System and method for tomography imaging

Abstract

A tomography system is provided. The system receives measurements at a frequency of a wavefield scattered by the internal structure of the object, recursively reconstructs an image of the internal structure of the object until a termination condition is met, and renders the reconstructed image. For a current iteration, a frequency is added to a previous frequency used during a previous iteration to produce a current frequency such that the added frequency is higher than a frequency present in the previous frequency, and a current image of the internal structure of the object is reconstructed, the current image minimizes the difference between the scattered wavefield measured at the current frequency and the wavefield synthesized from the current image. A wavefield synthesized from the current image is generated by a neural network operator. A previous image determined during a previous iteration initializes reconstruction of the current image.

Description

Technical Field

[0001] This invention relates to tomography, and more particularly to systems and methods for tomographic imaging to reconstruct images of the internal structure of an object by solving an inverse scattering problem using deep learning operators that model wave propagation physics. Background Technology

[0002] Objects such as the human body and rocks can be composed of one or more types of materials. For example, the human body can be composed of tissues, bones, skin, muscles, blood, water, etc. Each material in the object can be considered a dielectric. Understanding the spatial distribution of the dielectric constant within an object is important for many applications, such as microwave imaging, biological microscopy, medical imaging, through-wall imaging (TWI), infrastructure monitoring, and seismic imaging. In particular, determining the dielectric constant enables the visualization of the internal structure of an object and the characterization of its physical properties. For example, in microwave imaging, the dielectric constant provides information about the structure and properties of the materials within the object. In biological microscopy, the dielectric constant allows for the visualization of internal cellular structures in three dimensions. In TWI, the dielectric constant allows for the learning of the dielectric properties of walls and uses this information to compensate for delays in signal propagation through walls.

[0003] In a typical scenario, a transmitter emits a signal (such as an electromagnetic (EM), optical, or acoustic pulse) that propagates through an object, reflects off various structures and corresponding materials within the object, and reaches a receiver antenna array. The composition of the object is then visualized by digitally generating an image representing the distribution of the dielectric constant within the object. Examples of such images include a refractive index image of the materials within the object. However, the received signal is generated by multiple reflections and / or refractions of the emitted signal due to multiple scattering from structures or materials within the object and from structures or materials surrounding the object. Therefore, the reconstructed image of the object may include artifacts, depending on the type of materials present inside or around the object, which clutter the reconstructed image and fill it with noise. Furthermore, multiple scattering of the emitted signal affects the received signal in a non-linear manner, making image reconstruction even more difficult.

[0004] Therefore, it is necessary to overcome the above-mentioned shortcomings and reconstruct an image of the spatial distribution of the dielectric constant of the material in the object, so that the reconstruction takes into account the multiple scattering of the emitted signal propagating through the object. Summary of the Invention

[0005] Some embodiments of this disclosure aim to reconstruct images of the internal structure of an object probed with electromagnetic and / or acoustic waves of limited bandwidth to measure the scattered wave field around the object. The incident wave field propagating within the object induces multiple scattered waves at the boundaries of the material within the object. Therefore, the scattered waves contain information about the spatial distribution of material properties. Due to the non-invasive nature of imaging techniques used to determine the spatial distribution of material within and around an object, such imaging techniques have been applied in many fields, such as non-destructive testing, optical tomography, geophysical imaging, ground-penetrating radar, and medical imaging. For example, the spatial distribution of material properties in the reconstructed image can be represented by the distribution of the refractive index and / or dielectric constant of the material within the object.

[0006] Some implementations are based on the understanding that an image can be reconstructed from the spatial distribution of material properties by solving an inverse scattering problem. In this case, the input wavefield of the probe pulse is known, and it may be necessary to measure the scattered wavefield. The inverse scattering problem can be used to find the spatial distribution of the material's dielectric constant, which converts the known input wavefield into the received reflection of the scattered wavefield. However, the inverse scattering problem is ill-posed and difficult to solve.

[0007] Some implementations are based on the understanding that the drawbacks of the inverse scattering problem for reconstructing the internal structure of an object can be overcome by minimizing a cost function that indicates the difference between the scattered wavefield and the wavefield synthesized from an image of the object's internal structure. However, if the background medium in which the object resides has a complex or unknown structure, the synthesis of the wavefield may be inaccurate or infeasible. Therefore, the object of some embodiments of this disclosure is to provide methods and systems suitable for reconstructing the internal structure of an object when the background medium is complex or unknown.

[0008] Some implementations are based on the understanding that the type of background medium, though complex or unknown, belongs to a category of commonly observed background media from an application in which a system for imaging the internal structure of an object will be deployed. Therefore, some embodiments of this disclosure disclose determining a neural network operator trained based on training examples belonging to categories of commonly observed background media for different applications that can be deployed for imaging, to learn a mapping between the application and a corresponding synthesized wavefield of the corresponding background medium. The aim of embodiments of this disclosure is to deploy or utilize a trained neural network operator that learns a mapping for the inverse scattering problem for reconstructing the internal structure of an object, thereby improving the quality of the wavefield synthesized from images of the object's internal structure.

[0009] Some implementations are based on the understanding that the structure of a neural network operator can be analogous to the iterative Born approximation in physics describing wave propagation. The aim of this disclosure is to design neural network operators using multiple cascaded layers of Fourier Neural Operators (FNOs), where the parameters of the FNO within each layer are identical to those in all other layers; that is, all layers of the FNO are identical.

[0010] Some implementations are based on the understanding that the type of the object to be imaged belongs to a category of target objects typically observed in applications where systems will be deployed for imaging. Therefore, another objective of some implementations of this disclosure is to determine a neural network autoencoder model consisting of encoder and decoder branches. The neural network autoencoder model learns to map example objects from the target object category to low-dimensional latent space vectors and learns to back-map the low-dimensional latent space vectors back to the example objects. According to embodiments of this disclosure, the learned neural network autoencoder model is incorporated into the inverse scattering problem for reconstructing the internal structure of an object by limiting the number of object types that can be reconstructed by the system used for imaging to only the number of object types that can be generated by the decoder branch of the trained neural network autoencoder model.

[0011] Some implementations are based on the understanding that the input wavefield is scattered differently at different frequencies, thus complicating the scattered wavefield and making the cost function highly nonlinear with multiple local minima. Therefore, minimizing this cost function inverse scattering problem is a challenging one.

[0012] Some embodiments of this disclosure are also based on the understanding that the low-frequency components of the incident wave field can penetrate further into the material of the object and exhibit weaker interactions with the internal structure of the object compared to higher-frequency components. Due to these weaker interactions, fewer scattering events occur during the penetration of the low-frequency wave field within the material of the object. Therefore, the measurement mismatch cost function corresponding to the low frequency has fewer local minima compared to higher frequencies. While low-frequency measurements contain less spatial detail than higher-frequency measurements, they can be used to initialize the optimization of the inverse scattering problem using higher-frequency measurements.

[0013] To this end, some implementations are based on incremental frequency inversion methods. For example, in some implementations, the incremental frequency inversion method recursively reconstructs an image of the internal structure of an object until a termination condition is met. For the current iteration, the method adds frequencies to a previous set of frequencies used in previous iterations to produce a current set of frequencies, and reconstructs a current image of the object's internal structure that minimizes the difference between a portion of the scattered wavefield measured at the current set of frequencies and the wavefield synthesized from the current image. The added frequencies are higher than one or more frequencies present in the previous set of frequencies. The wavefield synthesized from the current image is generated by a neural network operator trained to learn a mapping between the applied and corresponding synthesized wavefields of the corresponding background medium. Furthermore, frequencies are incrementally added to the inverse scattering problem. Additionally, the reconstruction of the current image is initialized with previous images determined during previous iterations. This initialization incrementally guides the solution to the inverse scattering problem toward a global minimizer.

[0014] The incremental frequency inversion method does not require a smooth initial model to reconstruct the image. Instead, it derives such a model from low-frequency measurements. In effect, the incremental frequency inversion method allows for the reconstruction of images of an object's internal structure with realistic resolution and accuracy, without using prior information about the image.

[0015] Therefore, embodiments of this disclosure disclose a tomographic imaging system comprising: an input interface for receiving measurements at a frequency set of wavefields scattered by the internal structure of an object; a processor configured to recursively reconstruct an image of the internal structure of the object until a termination condition is met; and an output interface configured to render the reconstructed image. For a current iteration of recursively reconstructing the image, the processor is configured to add frequencies to a previous frequency set used during previous iterations to produce a current frequency set, wherein the added frequencies are higher than one or more frequencies present in the previous frequency set. Furthermore, the processor is configured to reconstruct a current image of the internal structure of the object that minimizes the difference between a portion of the scattered wavefield measured at the current frequency set and a wavefield synthesized from the current image. The wavefield synthesized from the current image is generated by a neural network operator, and the reconstruction of the current image is initialized by previous images determined during previous iterations.

[0016] Some implementations are based on the understanding that the architecture of the neural network operator includes a sequence of Fourier neural operator (FNO) modules with the same parameters enforced by training the neural network operator with machine learning.

[0017] The one or more processors are further configured to reconstruct an image of the object's internal structure by solving an optimization problem that minimizes the difference between a measurement of a portion of the scattered wave field and the synthesized wave field.

[0018] Some implementations are based on the understanding that the reconstructed image is a refractive index image of one or more materials inside an object.

[0019] Some implementations are based on the understanding that the reconstructed image includes the distribution of the dielectric constant of one or more materials within the object.

[0020] In some implementations, to reconstruct the current image, the one or more processors are further configured to simultaneously determine an update of the current image of the object's internal structure and an estimate of the scattered wave field inside the object, and based on the update of the current image and the estimate of the scattered wave field, to minimize the sum of the differences between the received reflections and the synthetic reflections of the reconstructed scattered wave field for each frequency in the current frequency set. The estimated scattered wave field has a structure corresponding to the current image at each frequency in the current frequency set.

[0021] In some implementations, to simultaneously determine a current image of the object's internal structure and an estimate of the scattered wave field within the object, the one or more processors are further configured to determine the estimate of the scattered wave field based on the current estimate of the image of the object's internal structure by simulating the interaction between a probe pulse and the scattered wave field generated by scattering the probe pulse using one or more materials within the object. The one or more processors are also configured to determine an estimate of an accompanying scattered wave field that compensates for a residual error between the received reflection and the composite reflection of the reconstructed scattered wave field. The one or more processors are further configured to use an adjoint equation of state to calculate an update to the current image of the object's internal structure based on the estimate of the scattered wave field, the estimate of the accompanying scattered wave field, and the residual error between the received reflection and the composite reflection of the reconstructed scattered wave field.

[0022] In some implementations, to simultaneously determine the current image and the scattered wavefield, the one or more processors are further configured to form a Born Fourier neural operator (FNO) that approximates the interaction of the probe pulse, the scattered wavefield generated by scattering the probe pulse using one or more materials within the object, and the refractive indices of the one or more materials within the object. The one or more processors are also configured to invert the Born FNO given an initialized current image to determine the Jacobian determinant of the scattered wavefield with respect to the current image based on the refractive indices of the one or more materials. The one or more processors are further configured to update the current image based on the refractive indices of the one or more materials by minimizing a cost function between the received reflections and the synthesized scattered wavefield in the set of frequency components, the synthesized scattered wavefield being obtained by combining the Jacobian determinant of the scattered wavefield with the quasi-Newtonian descent direction of the cost function relative to the refractive index image of the one or more materials. The one or more processors are further configured to project the refractive index image of the one or more materials onto a constrained total variational penalty function.

[0023] Some implementations are based on the understanding that the cost function between the received reflections in the set of frequency components and the synthetic reflections of the reconstructed scattered wavefield includes one or a combination of the following: the Euclidean distance between each of the received reflections and its corresponding synthetic reflection, the norm distance between each of the received reflections and its corresponding synthetic reflection, or the sum of the Euclidean distance and the barrier function. The barrier function penalizes updates to the refractive index image of the one or more materials having negative refractive indices, and the barrier function is the sum of exponential functions taken as negative powers of each refractive index in the refractive index image of the one or more materials.

[0024] Some implementations are based on the understanding that the refractive index image of the one or more materials is represented by a generator network that maps a low-dimensional latent space representation to the refractive index image of the one or more materials.

[0025] Some implementations are based on the understanding that the generator network is determined to be part of an autoencoder network, wherein the encoder network of the autoencoder network is configured to determine a low-dimensional representation of the refractive index image of the one or more materials in a low-dimensional latent space, and the generator network of the autoencoder network is configured to decode the latent space representation to reproduce the refractive index image of the one or more materials.

[0026] Some implementations are based on the understanding that the constrained total variational penalty function is subject to an upper bound. The one or more processors are further configured to initialize an upper bound for the current image and update it at the start of each iteration using a Newton's root-finding method that adds the square of the Euclidean distance between the received reflection and the composite reflection of the reconstructed scattered wavefield to a polar function of the constrained total variational function, which is applied to the product of the adjoint matrix of the Born FNO and the difference between the received reflection and the composite reflection of the reconstructed scattered wavefield.

[0027] Some implementations are based on the understanding that the Born FNO operator approximates the probe pulse, the scattered wave field generated by scattering the probe pulse using one or more materials within the object, and the interaction of the refractive indices of the one or more materials within the object. The structure of the Born FNO is determined by cascading multiple Fourier neural operator modules into a multilayer neural network.

[0028] Some implementations are based on the understanding that the object includes elements of underground infrastructure.

[0029] Some implementations are based on the understanding that the tomographic imaging system further includes: a set of transmitters configured to transmit one or more probe pulses toward the object; and a set of receivers configured to measure, at each frequency from the set of frequencies, one or a combination of reflection and refraction of the one or more probe pulses propagating through the object to generate measurements of the wave field. The one or more probe pulses include at least one of: electromagnetic waves or sound waves occupying a frequency band including the set of frequencies.

[0030] Some implementations are based on the understanding that the set of transmitters and the set of receivers are located on the same side of the object, such that the tomographic imaging system operates in a reflection mode.

[0031] Another embodiment discloses a tomographic imaging method that uses a processor coupled to stored instructions for implementing the method, wherein the instructions, when executed by the processor, perform the following steps of the method: receiving measurements at a set of frequencies of a wavefield scattered by the internal structure of an object; recursively reconstructing an image of the internal structure of the object until a termination condition is met; and rendering the reconstructed image. For the current iteration, the method includes adding frequencies to a previous set of frequencies used during previous iterations to produce a current set of frequencies, wherein the added frequencies are higher than one or more frequencies present in the previous set of frequencies. The method also includes reconstructing a current image of the internal structure of the object that minimizes the difference between a portion of the scattered wavefield measured at the current set of frequencies and a wavefield synthesized from the current image, wherein the wavefield synthesized from the current image is generated by a neural network operator, and wherein the reconstruction of the current image is initialized by a previous image determined during previous iterations.

[0032] Another embodiment discloses a non-transitory computer-readable storage medium having a program implemented thereon that can be executed by a processor to perform a method comprising: receiving measurements at a set of frequencies of a wave field scattered by the internal structure of an object; recursively reconstructing an image of the internal structure of the object until a termination condition is met; and rendering the reconstructed image. For the current iteration, the method includes adding frequencies to a previous set of frequencies used during previous iterations to produce a current set of frequencies, wherein the added frequencies are higher than one or more frequencies present in the previous set of frequencies. The method also includes reconstructing a current image of the internal structure of the object that minimizes the difference between a portion of the scattered wave field measured at the current set of frequencies and a wave field synthesized from the current image, wherein the wave field synthesized from the current image is generated by a neural network operator, and wherein the reconstruction of the current image is initialized by a previous image determined during previous iterations. Attached Figure Description

[0033] The currently disclosed embodiments will be further explained with reference to the accompanying drawings. The drawings shown are not necessarily drawn to scale, but generally focus on illustrating the principles of the currently disclosed embodiments.

[0034] [ Figure 1 ]

[0035] Figure 1 A block diagram of a tomographic imaging system for determining the internal structure of an object, according to an example embodiment, is shown.

[0036] [ Figure 2A ]

[0037] Figure 2AA schematic diagram illustrating the application of a tomographic imaging system according to an example embodiment is shown.

[0038] [ Figure 2B ]

[0039] Figure 2B A schematic diagram of the task of the inverse scattering problem according to an example implementation is shown.

[0040] [ Figure 3 ]

[0041] Figure 3 A block diagram is shown of a method for determining the internal structure of an object based on an example implementation.

[0042] [ Figure 4A ]

[0043] Figure 4A A schematic diagram of an autoencoder network of a neural network operator according to an example implementation is shown.

[0044] [ Figure 4B ]

[0045] Figure 4B The structure of a neural network operator according to an example implementation is shown.

[0046] [ Figure 4C ]

[0047] Figure 4C A schematic diagram illustrating the process for solving an inversion problem to reconstruct the internal structure of an object, according to an example implementation.

[0048] [ Figure 5 ]

[0049] Figure 5 A schematic diagram illustrating the principle of the incremental frequency inversion method according to an example implementation is shown.

[0050] [ Figure 6 ]

[0051] Figure 6 A block diagram is shown of a method for reconstructing a current image for different frequencies according to an example implementation.

[0052] [ Figure 7 ]

[0053] Figure 7 A block diagram is shown of a method for updating an image of the spatial distribution of the dielectric constant of a material for a current set of frequencies, according to an example embodiment.

[0054] [ Figure 8 ]

[0055] Figure 8A block diagram is shown of a method for updating the upper limit of the total variational penalty function according to an example implementation.

[0056] [ Figure 9 ]

[0057] Figure 9 An exemplary schematic diagram of a reflection setup according to an example implementation is shown.

[0058] [ Figure 10 ]

[0059] Figure 10 An example method for reconstructing an image is shown according to an example implementation.

[0060] [ Figure 11 ]

[0061] Figure 11 A block diagram of a tomographic imaging system according to an example embodiment is shown. Detailed Implementation

[0062] In the following description, numerous specific details are set forth for purposes of explanation in order to provide a thorough understanding of this disclosure. However, it will be apparent to those skilled in the art that this disclosure may be practiced without these specific details. In other instances, apparatus and methods are shown only in block diagrams to avoid obscuring this disclosure. Various changes to the function and arrangement of the elements may be contemplated without departing from the spirit and scope of the subject matter disclosed as set forth in the appended claims.

[0063] As used in this specification and claims, when combined with a list of one or more components or other items, the terms "for example," "like," and "such as," as well as the verbs "comprising," "having," "including," and their other verb forms, are each interpreted as open-ended, meaning that the list is not considered to exclude other additional components or items. The term "based on" means at least partially based on. Furthermore, it should be understood that the wording and terminology used herein are for descriptive purposes and should not be considered restrictive. Any headings used in this specification are for convenience only and have no legal or limiting effect.

[0064] Specific details are set forth in the following description to provide a thorough understanding of the embodiments. However, those skilled in the art will understand that embodiments can be practiced without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in block diagram form to avoid obscuring the embodiments with unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail to avoid obscuring the embodiments. Furthermore, the same reference numerals and names in the various figures indicate the same elements.

[0065] Some implementations aim to disclose a tomographic imaging system for reconstructing an image of an object, thereby identifying the object's internal structure. In one example, the tomographic imaging system is configured to reconstruct the image by solving an inverse scattering problem using a deep learning neural network operator that models the physics of wave propagation. In particular, the structure of the neural network operator can be analogous to the iterative Born approximation describing the physics of wave propagation. The neural network operator learns a mapping for the inverse scattering problem used to reconstruct the object's internal structure in order to improve the quality of the wavefield synthesized from the image of the object's internal structure.

[0066] Figure 1 A block diagram of a tomographic imaging system 100 for determining an image 102 of the internal structure of an object 104 according to an exemplary embodiment is shown. The image 102 of the internal structure of the object 104 can be represented, for example, by an image formed based on the refractive index of one or more materials present within the object 104, an image showing the spatial distribution of the dielectric constants of one or more materials within the object, or a combination thereof. The tomographic imaging system 100 (hereinafter referred to as System 100) includes and / or is operatively connected to at least one transceiver 106 to propagate a probe pulse of an incident wave field 108 through one or more materials of the object 104 and to receive a set of echoes 110 generated by the scattered pulses from different parts of the object 104 or one or more materials.

[0067] For example, transceiver 106 may include at least one transmitter that transmits probe pulses to object 104. The probe pulses of the incident wave field 108 are scattered by the material and generate a set of echoes 110. The probe pulses may be, for example, electromagnetic waves, sound waves, or light waves. The probe pulses may include, for example, one or a combination of microwave pulses, radar pulses, laser pulses, ultrasonic pulses, and acoustic pulses. Transceiver 106 may also include at least one receiver disposed at a predetermined position relative to the transmitter for receiving the set of echoes 110. According to some embodiments, system 100 may generate a two-dimensional or three-dimensional image 102 of object 104, wherein each location in image 102 provides a value of the dielectric constant of the corresponding portion of the material at that location.

[0068] The tomographic imaging system also includes a processor 112 operatively connected to a transceiver 106 to determine an image 102 based on the set of echoes 110. To reconstruct the image 102 of the object 104 regardless of multiple scattering, the processor 112 divides the reconstruction into several stages by operating on a set 118 of frequencies of the scattered wavefield. The processor 112 implements an incremental frequency inversion method 114 and uses a neural network operator 116 to reconstruct the internal structure of the object 104 by improving the quality of the synthetic wavefield from an image of the object's internal structure, effectively addressing the inverse scattering problem. For example, in some embodiments, the incremental frequency inversion method 114 recursively reconstructs the image 102 of the internal structure of the object 104 for higher frequencies until a termination condition is met. Combined with, for example... Figure 2A Describe the details of the application areas of System 100.

[0069] Figure 2A A schematic diagram 200 illustrates an application of system 100 according to an example embodiment. According to this example, system 100 is used in the measurement acquisition process of a ground-penetrating radar (GPR) application. It should be noted that this application of system 100 is exemplary and should not be construed as limiting. In other embodiments of this disclosure, system 100 can be used in applications such as human body imaging, geological exploration, and oil or mineral extraction.

[0070] For ground-penetrating radar (GPR) applications, operator 202 can move an imaging device 204 equipped with radar. Imaging device 204 may include a transmitter that transmits radar waves or probe pulses into ground 206 and a receiver that receives reflections from an object 104 located below ground 206 for data collection. Based on the received reflections of the scattered wave field, imaging device 204 can be configured to reconstruct an image of the object 104 that may be located below ground 206.

[0071] The object of embodiments of this disclosure is to provide a tomographic imaging system 100 that accurately reconstructs images of objects, such as objects beneath the ground. This reconstructed image can indicate the internal structure of the object, which indicates, for example, the object's type, properties, physical characteristics, and / or chemical properties. In this regard, the tomographic imaging system 100 can use the inverse scattering problem to reconstruct the image. Combined with... Figure 2B Describe the details of the image scattering problem.

[0072] Figure 2BA schematic block diagram 210 illustrates the task of the inverse scattering problem according to an example implementation. The inverse scattering problem aims to reconstruct an image of the internal structure 214 of the ground 206 by solving an optimization problem. The optimization problem is used to minimize the difference between measurements of the scattered wave field (i.e., the reflections received by the receiver of system 100) and the synthetic wave field (i.e., the synthetic wave field generated from the image of the internal structure of object 104).

[0073] In operation, a probe pulse, including the incident wave field 108, can be transmitted to the ground 206 via system 102. Furthermore, one or more objects (such as object 104) may be located beneath the ground 206. Based on the received scattered wave field, an image of the internal structure 214 or the material beneath the ground 206 and / or the internal structure of object 104 can be reconstructed. In this regard, an ill-posed inverse scattering problem is used to determine the subsurface structure 218, comprising the subsurface internal structure 214 of the ground 206, based on sparse surface measurements 216. The inverse scattering problem is optimized based on an optimization problem to determine the internal structure 214 of the ground 206, including, for example, a first layer 220 of the background medium and a second layer 222 of the background medium.

[0074] For example, the objective of the inverse scattering problem is to estimate the subsurface internal structure 214, f of ground 206 given ground-based data y associated with ground 206. In one example, the subsurface internal structure 214, f can be estimated based on the background media 220 and 222 and the dielectric constant distribution present beneath ground 206 and / or within object 104. Additionally, information about the source of the probe pulses, including the incident wave field 108 and the background media 220 and 222 of ground 206, is also necessary for calculating the incident scattered field and the Green's function. In one example, f can be estimated by solving an optimization problem. The optimization problem is defined as:

[0075]

[0076] in It is a forward modeling operator that maps the dielectric constant constraint f to the corresponding wave field measurement y at frequency ω according to equation (10). ω ,and It is a regularizer.

[0077] In GPR applications, it is assumed that there is a layered dielectric constant distribution ∈ bThe layered structure of the background media 220 and 222 of (x) is likely reasonable. However, the depth and dielectric constant of each layer of the background media 220 and 222 may be unknown. Furthermore, it is important to compute the Green's function for the layered background. Alternatively, if the free space is assumed to be the layered background, the integration domain can be restricted to a bounded region without carefully dealing with the boundaries, since the layered background structure extends beyond the computational domain. To overcome the aforementioned drawbacks associated with the conventional inverse scattering problem, the updated inverse scattering problem is used in conjunction with the neural network operator 116.

[0078] Figure 3 A block diagram 300 illustrates a method for determining the internal structure of an object 104 according to an example embodiment. This method can be implemented using a processor 112 of the system 100. For example, the method can be implemented by the processor 112 executing a program embodied on a non-transitory computer-readable storage medium.

[0079] The method includes, at 302, transmitting a probe pulse of an incident wave field 108 through object 104 to receive a set of echoes 110. In the example, processor 112 is configured to transmit the probe pulse of the incident wave field 108 through the material of object 104. For example, processor 112 receives pulses scattered by the material in the form of the set of echoes 110. The echoes 110 are generated by pulses scattered by different portions of object 104 or by the material.

[0080] The pulses of the incident wave field 108 span a frequency band comprising a set of frequencies. For example, this set of frequencies is the quantized product of the frequency bands of the pulses of the incident wave field 108. In one embodiment, quantization is performed at a resolution proportional to the length of time the echo 110 is sampled. This resolution allows for the reconstruction of the necessary details of the image 102 indicating the internal structure of the object 104, such that the frequency intervals of the quantization intervals are small enough that adjacent frequency intervals contain overlapping information about image details. In another embodiment, the resolution is selected based on the computing power of the processor 112. For example, a processor with limited memory will require larger quantization intervals, resulting in a smaller number of frequencies.

[0081] In the example, processor 112 may be configured to use an analog-to-digital converter to convert the received reflection 306 into a digital signal and record the amplitude and / or other properties of the digital signal. In some embodiments, processor 112 may be configured to organize the received reflection 306 into an ordered set 118 of frequencies of the scattered wave field. The ordered set 118 may include frequencies ordered from lowest to highest frequency.

[0082] Based on the principle of incremental frequency inversion 114, this method does not aim to jointly reconstruct the image for all frequencies at once. Instead, the method according to this disclosure reconstructs the image for a subset of frequencies, incrementally adding higher frequencies to the subset to further refine the image. Ultimately, the method processes all frequencies. Due to incremental frequency inversion 114, the previously determined image is used as a priori for the reconstruction of the next image to improve the quality of the image reconstruction. In this way, image 102 is reconstructed iteratively based on multiple frequencies of a frequency band.

[0083] To this end, at 304, the method includes incrementally adding one or more frequencies from the received reflections 306 to the current frequency set. Specifically, processor 112 can be configured to begin adding frequencies from lowest to highest to the current frequency set in different iterations. For example, processor 112 can be configured to initialize the current frequency set by placing the lowest frequency in the current frequency set. In the next iteration, a second low frequency can be added to the current frequency set, and so on.

[0084] Some implementations are based on the understanding that the inverse scattering problem for reconstructing the internal structure of object 104 can be solved by minimizing a cost function that indicates the difference between the received reflection and the reflection synthesized from an image of the internal structure of object 104. However, the input wavefield is scattered differently for different frequencies, thus complicating the scattered wavefield and making the cost function highly nonlinear with multiple local minima. Therefore, minimizing such a cost function in inverse scattering is a challenging problem.

[0085] However, some implementations are based on the understanding that the low-frequency components of the incident wave field can penetrate further into the material of object 104 and exhibit weaker interactions with the internal structure of object 104 compared to higher-frequency components. Due to the weaker interactions, fewer scattering events occur during the penetration of the low-frequency wave field. Therefore, the measurement mismatch cost function corresponding to the low frequency has fewer local minima compared to higher frequencies. Low-frequency measurements contain less spatial detail compared to higher-frequency measurements, but they can be used to initialize the optimization of the inverse scattering problem using higher-frequency measurements.

[0086] Some implementations are based on the understanding that it is easier to process each frequency individually and then reconstruct image 102 later by stitching together images of different frequencies in the frequency domain. However, this option is not suitable for reconstructing the internal structure of object 104 due to the interdependence of the scattered wave field on different frequencies. Therefore, some implementations reconstruct image 102 jointly for multiple frequencies. However, frequencies are added incrementally to gradually initialize the reconstruction until all frequencies are combined.

[0087] The method for reconstructing image 102 of the internal structure of object 104 continues further, at 308, reconstructing the current image of object 104. For a particular iteration, processor 112 is configured to update the current image based on the wavefield in the current frequency set and check the termination condition. For the current iteration, image reconstruction of the current image is performed using only the frequencies present in the current frequency set. To accurately reconstruct the current image of the internal structure of object 104, the method minimizes the difference between a portion of the scattered wavefield measured at the current frequency set and the wavefield synthesized from the current image.

[0088] At 310, a wavefield is synthesized for the current iteration. In some implementations, the method uses a current image of object 104 from a previous iteration (i.e., a previous image reconstructed during a previous iteration) to synthesize a wavefield compared to the received reflection 306. For example, a wavefield synthesized from the current image of a previous iteration is generated by a neural network operator 116. Details of the structure of the neural network operator 116 are provided in, for example... Figure 4B Provided by China.

[0089] At position 312, the error between the received reflection 306 and the synthesized wavefield is compared. Furthermore, it is determined whether the error is less than a threshold. When the error is determined to be less than the threshold, the current image generated at the current iteration is output as a reconstructed image 102 of the internal structure of object 104.

[0090] However, if the error is determined to be greater than a threshold, then at 314, the current image generated in the current iteration is updated in a way that minimizes the error. For example, some implementations use a different regularizer at 318 to update the current image. Examples of regularizers may include, but are not limited to, total variational regularizers and autoencoder constraint regularizers, such as... Figure 4A .

[0091] Additionally or alternatively, the method terminates when it is determined at 316 that all frequencies in the checked frequency set have been processed. For example, the method terminates the reconstruction process when all frequencies in the ordered frequency set 118 have already been included in the current frequency set. In subsequent iterations, the processor can be configured to add frequencies from the ordered frequency set 118 such that the added frequencies are higher than any frequencies in the previous frequency set.

[0092] Figure 4A A schematic diagram of an autoencoder network 400 of a neural network operator 116 according to an exemplary embodiment is shown. The autoencoder network 400 includes an encoder branch and a decoder branch. In particular, the autoencoder network 400 includes an encoder network 402 and a generator network 404 (or a decoder).

[0093] The autoencoder network 400 may include a deep learning model for transforming data from a high-dimensional space to a low-dimensional space. For example, the autoencoder network 400 may be configured to transform an image formed based on received reflection 306 or information related to the refractive index of one or more materials of object 104 to a lower-dimensional space. For example, the encoder network 402 may encode image data (regardless of its size) into a 1-D vector smaller than the image, i.e., in the lower-dimensional space. Furthermore, the vector can be decoded by the generator network 404 to reconstruct the original data, i.e., the image. The autoencoder network 400 learns an effective data representation, i.e., encoding to ignore signal noise by training the encoder network 402. Subsequently, converting the low-dimensional image to the original image via the autoencoder network 400 can achieve, for example, image denoising.

[0094] Encoder network 402 is configured to map the internal structure of object 104 to a low-dimensional latent space. In one example, encoder network 402 is configured to determine a low-dimensional representation of an image, such as the current image or a refractive index image 102 of one or more materials in object 104 within the low-dimensional latent space. Furthermore, generator network 404 is configured to map from the low-dimensional latent space to the internal structure of the object. For example, generator network 404 is configured to decode the latent space representation to reproduce the refractive index image 102 of one or more materials of object 104 or the current image. To this end, the refractive index image of one or more materials is represented by generator network 404, which maps the low-dimensional latent space representation to the refractive index image of one or more materials.

[0095] Since the inverse scattering problem in the GPR setting is highly ill-posed, the solution of generator network 404 is limited to a generative model. The lower-dimensional subspace of the range. Given a graph including dielectric constants {∈ i} i The training dataset of the training examples is used to train the generator network or generative model by solving the following equation.

[0096]

[0097] Where ε represents the output of encoder 402; This represents the output of generator network 404, i.e., the learned generative prior; and And φε are respectively The training parameters for ε.

[0098] The trained autoencoder network 400 learns a parametric model that maps a low-dimensional vector z to the target internal structure ∈ R_04 during optimization. The trained autoencoder network 400 learns to optimize z ∈ R_0.n Instead of ∈ ∈ R w×h Subsequently, the trained autoencoder network 400 learns to constrain the solution space of the spatial distribution of the dielectric constant map to approximate the target dataset distribution learned by the generator network 404.

[0099] The process of generating a synthetic wave field using neural network operator 116 requires knowledge of a forward model. However, the forward model becomes complex when the spatial distribution of the dielectric constant of the background medium (such as the dielectric constants of background media 220 and 222) is complex. To determine the forward model, the computational domain is discretized into a uniformly sampled 2D grid D. The input to the forward model is... Where ∈ is the total dielectric constant of the underground structure on the grid, and This represents the real and imaginary parts of the free-space response of the source on the grid at frequency ω, i.e., the incident field with respect to the free-space background medium. This is used to provide information about the frequency and source of the probe pulse. The output of the forward model includes the grid. The real and imaginary parts of the total field. Combined Figure 4B and Figure 4C The method of generating synthetic wave fields using neural networks is described.

[0100] Figure 4B The structure 406 of a neural network operator 116 according to an example implementation is shown. In the example, the architecture of the neural network operator 116 includes a series of Fourier neural operator (FNO) modules, depicted as FNO modules 408A, 408B, ... 408N (collectively referred to as FNO modules 408), wherein the same parameters are enforced by training the neural network operator 116 with machine learning. In this respect, the FNO modules 408 are configured to learn continuous functions via parameterizing an autoencoder network 400 in its function space. This allows the FNO modules 408 to be trained on one grid or training dataset and subsequently evaluated on another grid or training dataset.

[0101] In particular, the FNO module 408 can form a Born FNO 410. In operation, the Born FNO module 410 can approximate the interaction between the pulse and the scattered wave field. The scattered wave field can be generated by the scattering of the probe pulse by one or more materials within the object 104 and the refractive index of one or more materials within the object 104. For example, the Born FNO module 410 can be formulated based on the iterative Born approximation as follows:

[0102] when hour,

[0103] v e (x)=P ∈ (x,∈(x)) (3)

[0104]

[0105] Where P ∈ P in Q is the local transformation parameterized by the multilayer perceptron (MLP), n is the number of BFNO layers or FNO modules 408, and σ(·) is the rectified linear unit (ReLU) nonlinear function. Note that conventional FNOs have different R... i and W i That is, the parameters and weights of each layer. However, the FNO module 408 of the Born FNO 410 of this disclosure uses the same regularizer and a set of weights (e.g., R, W0, W1) for all layers, which is similar to the structure in the Iterative Born Approximation (IBA) formula, as defined in equation (10) below. In one example, a normalization scheme can be used to preprocess the training data and train the Born FNO module 410 with the normalized mean squared error, as formulated in equation (8) as follows:

[0106]

[0107] Where, φ BFNO It is the set of network parameters of BFNO module 410, ∈ j It is a dielectric constant map from the training dataset, and It is frequency ω ∈ j The ground-based total field. The set of network parameters determined for the BFNO module 410 can be used to generate a synthetic wave field based on the current image of the object's internal structure and the current set of probe pulse frequencies.

[0108] For example, the received reflections can form a current image, which can be discretized and uniformly sampled onto a 3D grid D. Furthermore, the 3D grid of the image can be provided as input to the Born FNO module 410. Additionally, the real part 412 and imaginary part 414 of the free-space response of the source on the grid at frequency ω can be provided as input to the Born FNO module. The Born FNO module 410 can be manipulated to output a grid of the scattered wave field. The real part 412 and imaginary part 414 of the total field can be reflected from one or more materials inside object 104. In the example, the Born FNO module 410 can be inverted to determine the Jacobian determinant of the scattered wave field with respect to the current image based on the refractive index of one or more materials. Combined with... Figure 4C Describe in detail the inversion problem of Born FNO module 410.

[0109] Figure 4CA schematic diagram 416 illustrates the process of solving an inversion problem to reconstruct the internal structure of object 104 according to an example embodiment. The inversion problem of reconstructing the internal structure of object 104 can be solved by the structure 406 of the neural network operator 116.

[0110] In the example, the optimization problem defined in equation (1) computes a latent space vector z, which generates an estimated internal structure of object 104 when passed through the generator network 404 of the autoencoder network 400. The estimated internal structure of object 104 can be generated as a dielectric constant distribution f. The estimated internal structure of object 104 and the incident wave field are input to the Born FNO module 410 to output a synthetic total wave field 420, L, including the predicted dynamics of the internal structure of object 104 and the GT measurement of the received reflection 306 matched with the measured / received wave field. The Born FNO module 410 can compute the internal structure of object 104 using optimization problem 418 and generator network 404 while minimizing the difference between a portion of the scattered wave field measured at the current frequency set and the wave field synthesized from the current image. In this regard, the latent space vector is updated using a stochastic gradient descent method to ensure that the synthetic total wave field matches the measured wave field, thereby minimizing the difference from the current image. Details of the incremental frequency inversion 114 of image 102 used to reconstruct the internal structure of object 104 are shown in, for example Figure 5 As described in the text.

[0111] Figure 5 A schematic diagram 500 illustrates the principle of the incremental frequency inversion method 114 according to an example implementation. The incremental frequency inversion method 114 (hereinafter referred to as method 114) recursively reconstructs an image of the internal structure of object 104 until a termination condition is met. For the current iteration, method 114 adds frequencies to a previous set of frequencies used during previous iterations to produce a current set of frequencies. Based on the current set of frequencies and an optimization problem, the neural network operator 116 of the Born FNO module 410 can reconstruct a current image of the internal structure of object 104 that minimizes a cost function 502, which indicates the difference between a portion of the scattered wavefield measured at the current set of frequencies and the wavefield synthesized from the current image.

[0112] At the global minimum of 518, minimizing the cost function 502 yields the reflectivity parameter 304 of the image of the internal structure of the material of object 104. However, the cost function 502 has a complex shape (more complex than...). Figure 5 (The simplified example is much more complex). Therefore, minimizing the cost function 502 may not produce a global minimum 518, but rather one of several local minimums 516. Method 114 solves the above problem.

[0113] For example, the frequency set of method 114 includes frequencies of 10MHz, 20MHz, 30MHz, 40MHz, and 50MHz. During the first iteration, the image is reconstructed by minimizing the cost function 506 only for the 10MHz frequency. For such frequencies, the reconstruction is more likely to find a global minimum 518. Next, method 114 adds frequencies higher than one or more frequencies from the previous frequency set. For example, method 114 adds a 20MHz frequency and minimizes the cost function 508 of the joint frequency of 10MHz and 20MHz near the minimum of function 506. Next, method 114 adds a 30MHz frequency and minimizes the cost function 510 of the joint frequency of 10MHz, 20MHz, and 30MHz near the minimum of function 508. Similarly, method 114 adds a 40MHz frequency to minimize the cost function 512, and finally sums all frequencies between 10MHz and 50MHz to minimize the total function 514 to find a global minimum 518 corresponding to the final image 102.

[0114] In this way, the frequency is added incrementally to the inverse scattering problem. Furthermore, the reconstruction of the current image is initialized with the previous image determined during previous iterations. This initialization incrementally guides the solution to the inverse scattering problem toward the global minimum of 518.

[0115] The incremental frequency inversion method 114 does not require a smooth initial model of the image to be reconstructed. Instead, the incremental frequency inversion method 114 derives such a model from low-frequency measurements. In fact, the incremental frequency inversion method 114 allows the reconstruction of the image 102 of the internal structure of the object 104 with actual resolution and accuracy without using prior information about the image.

[0116] Frequency inversion

[0117] According to some embodiments of this disclosure, the image reconstruction problem can be formulated as a frequency inversion problem solved by the incremental frequency inversion method 114. This is based on the understanding that a scattering model describes the relationship between the scattered wave field and the medium parameters of one or more materials of the object 104. The scattering model allows for the formulation of a discrete inversion problem to reconstruct the medium or one or more materials from the received set of reflections from the scattered wave field.

[0118] Specifically, the wave equation governs acoustic or electromagnetic scattering from a non-homogeneous medium in the time domain. Its equivalent representation in the frequency domain is the scalar Helmholtz equation. The integral form of the Helmholtz equation is, for example, the scalar Lippmann-Schwinger equation.

[0119] According to this embodiment, it can be assumed that It is the scattered wave field Ω in the spatial domain or region of interest. In this case, if we assume For parameters of moderate magnitude, the free-space Green's function can be derived from... The scalar Lippmann-Schwinger scattering equation is defined as follows:

[0120]

[0121] Among them, u in It transmits a probe pulse of the incident wave field 108 generated by the transmitter of transceiver 106, and k = 2π / λ is the wave number, where λ represents the wavelength. The medium parameter f(x) = (ε(x) - ε b ) is the relative permittivity, where ε(x) is the permittivity of object 104, and ε b The dielectric constants of background media 220 and 222 are assumed to be that of vacuum (ε). vacuum =1).

[0122] Continuing further, the Helmholtz equation The free-space Green's function is given by the following equation:

[0123]

[0124] Where r = ||x||, It is a Hankel function of the second kind, zeroth order, and d is the dimension Ω. The scattered wave field 606 or the received reflection 306 is then measured by the receiver of transceiver 106, resulting in the following data equation:

[0125]

[0126] in, Let denot be the Green's function of the receiver, and Γ be the receiver domain. The forward modeling problem involves calculating the green function of the emitted incident wave field 10⁸, u. in The probe pulse provides the synthetic reflection or synthetic wave field y, the medium parameters or dielectric constant constraints of the object 104 and the background medium 220 and 222f, and the Green's functions g and h.

[0127] In the discrete setup, for each transmitter illumination and wavenumber, the scattering equations and data equations simplify to the following set of linear equations:

[0128]

[0129] in, and v∈C N These are the input probe pulses for the scattered wave field 606 and the emitted incident wave field 108, respectively; N represents the number of grid points used for the discretization domain Ω; Indicates medium parameters, while and Let n be the Green's functions for the domain and receiver, respectively. rec If the number of receivers in the discretized receiver domain Γ of transceiver 106 is... It is the noiseless scattered wave field 606 measured at the receiver.

[0130] The forward modeling problem involves estimating the scattered wavefield 606,u via the inverse matrix A := I-Gdiag(f), where I denotes the identity operator. As the discretization dimension N increases, explicitly forming matrix A and its inverse becomes prohibitively expensive. Therefore, the inversion can be performed using a functional form of A and the conjugate gradient method (CG). The convergence of CG depends on the tuning of operator A, which is crucial for high wavenumbers and high scattering media (i.e., ||f||). ∞ (The large value) may become pathological.

[0131] Figure 6 A block diagram 600 of a method for reconstructing a current image 602 for different frequencies, according to an example embodiment, is shown. For different frequency combinations, i.e., based on the current frequency set used for the corresponding iteration, the method is executed multiple times. For example, in one iteration, the method is executed to reconstruct image 602 by minimizing a cost function 508 for two frequencies (e.g., 10MHz and 20MHz in the current frequency set). In the next iteration, the method is invoked to update images 618 at frequencies of 10MHz, 20MHz, and 30MHz by minimizing a cost function 510. In this iteration, image 618 is updated based on the previous reconstruction 602 of the image generated by minimizing the cost function 508.

[0132] Given a current estimate of the image, a synthetic reflection 604 is generated and compared with the received reflection 306. Both the synthetic reflection 604 and the received reflection 306 are determined for the current frequency set. Then, the difference between the synthetic reflection 604 and the received reflection 306 is minimized by simultaneously updating 608 the current image and the estimates of the scattered wavefields 610-612 (collectively referred to as the scattered wavefield 606 at each frequency in the current frequency set). The updated current image and scattered wavefields 610-612 are then used to minimize a cost function 616 for the difference between the synthetic reflection 604 and the received reflection 306. In some implementations, the cost function 616 is minimized by simultaneously estimating the current image and either the scattered wavefields 610-612 or 606 at each frequency according to the principles of the separated-state method. The minimization metric for the cost function 616 can be chosen from one or a combination of the Euclidean distance between the received reflection 306 and the synthetic reflection 604, the L1 norm distance between the received reflection 306 and the synthetic reflection 604, and the sum of the Euclidean distance and the barrier function. For example, the barrier function penalizes the refractive index image of one or more materials with negative refractive indices in the updated object 104, and the barrier function is the sum of exponential functions of negative powers of each refractive index in the refractive index image 602 of the material.

[0133] Figure 7 A block diagram 700 illustrates a method, according to an example embodiment, for updating an image of the spatial distribution of the dielectric constant of a material for a current frequency set. In this respect, the Born FNO module 410 approximates a partial differential equation (PDE) such that a neural network operator 116 characterizes the interaction between an emitted probe pulse 702 having an emitted wavefield of the current frequency set and a previous estimate of the image (i.e., a previous image 706 of the distribution of the dielectric constant of the material at all frequencies in the current frequency set). A function of the Born FNO module 410 is determined such that the product of the neural network operator 116 and the correct scattered wavefield equals the emitted probe pulse 702. Therefore, given the previous image 706 determined during previous iterations and a portion of the emitted wavefield of the emitted probe pulse 702 for the current frequency set, the function of the Born FNO module 410 is inverted 708 using a backpropagation method to estimate the scattered wavefield. The received reflection 306 is then compared with the synthesized reflection 604 to update the residual 704. The residual 704 is used to update the gradient and Jacobian matrix 710 of the image according to the adjoint state process. Then, the image is updated by modifying the latent space variables 712 through backpropagation, and the result is projected onto the constrained total variational penalty function 714 to update the image 618.

[0134] Inversion problem

[0135] To solve the optimization problem defined in equation (1), the learned forward model can be combined with the learned prior model, where, It is an isotropic total variational regularizer. It utilizes pre-trained... and The optimization problem can be solved, for example, using the standard ADAM optimizer for 1200 steps. It can be noted that the incremental frequency inversion method 114 can be used during optimization or solving of the optimization problem. In the example, the incremental frequency inversion method 114 can be implemented by including a batch of 10 frequencies every 120 update steps, i.e., [ω0, ..., ω...]. 10 [ω0, ..., ω] is used for the first 120 steps. 20 This is used for the next 120 steps, and so on. For example, prior knowledge. Alternatively, it can be done by updating during optimization, for example, after 300 steps. parameters Fine-tuning is performed, as shown in equation (2). This fine-tuning process improves the performance for inversion problems. The generalization performance is good. Similar techniques can be used for the inversion problem of Generative Adversarial Networks (GANs). Such a GAN function can be expressed as:

[0136]

[0137] The least-squares cost function 616 in equation (1) provides a natural separation across the frequency set. Furthermore, the topology of the non-convex cost function 616 changes dramatically across the frequency set and can be used to find a sequence of local minima that gradually lead to the global minimizer 518 of the inversion problem. With the introduction of higher-frequency wavefields, the cost function begins to exhibit many local minima that are further away from the global minimizer 518 compared to the lower-frequency wavefields.

[0138] Some implementations are based on the understanding that when designing the incremental frequency inversion method 114, the spatial distribution of the dielectric constant of the material of object 104 is updated sequentially as higher frequencies are included in the inversion. Given a scattered wave field n... f The received reflections 306 from the frequency set 118, the frequency components can be from 1 to n f Indexed in ascending order, the incremental frequency inversion method 114 iteratively estimates and reconstructs the current image 602 of object 104 from low frequency to high frequency, while maintaining the low-frequency cost function as a regularizer for high-frequency inversion. Therefore, instead of solving a single non-convex minimization problem in equation (1), some implementations solve n sequentially according to equation (12). f A minimization problem, where ω n From ω1 to Therefore, the sequence of solutions brings us closer to the global minimizer 518.

[0139] In the example, an approximate quasi-Newton method can be used to solve the number n of the minimization problem. f In this respect, the gradient 710 of the function of the optimization problem in equation (1) can be expressed with respect to the dielectric constant constraint f as:

[0140]

[0141] The scattered wave field u satisfies the PDE constraint. This gradient calculation is performed using the adjoint state method to simultaneously estimate the current image and scattered wave field at each frequency. The descent direction is then obtained by forming an approximation to the Hessian matrix using a finite-memory BFGS. The (t+1)th iteration of the neural network operator 116 is then given by:

[0142]

[0143] Where, γ t It is the step size calculated using backtracking search. It is an L-BFGS Hessian matrix, and It is an approximate operator of the total change (TV) norm constrained by τ. For each frequency batch, the neural network operator 116 is updated until the norm of gradient 710 decreases to a small value, such as less than a predefined threshold.

[0144] Total variational regularization

[0145] For example, functions The TV norm, with the help of a bounded function φ, is expressed as:

[0146]

[0147] In the example, the TV norm measures the total change of the derivative of the function of the neural network operator 116 over a finite field. As a result, regularization using the TV norm facilitates a piecewise constant approximation of the real neural network operator 116.

[0148] According to some implementations, TV regularization can be used in its constraint form, such that...

[0149]

[0150] Where δ(·) is the index function and τ is the constraint parameter. Let D be the finite difference operator for discretizing the gradient 710, and then TV(f) = ||Df||1.

[0151] To impose the TV norm constraint, one implementation defines the near-end operator as:

[0152]

[0153] It can be evaluated using the alternating direction method of the multiplier (ADMM).

[0154] Figure 8 A block diagram 800 illustrates a method for updating the upper bound 802, τ of the total variational penalty function 714 according to an example implementation. Combined with... Figure 6 and Figure 7 To explain the components Figure 8 .

[0155] Based on the previous image 706, the total variational penalty function 714 is evaluated to initialize the upper limit 802 of the total variational penalty function 714. The previous image 706 is also used to estimate the synthetic scattered wave field 614 at the current frequency set by multiplying the inverse of the PDE operator A by the incident wave field of the emitted probe pulse 702. The upper limit 802 may need to be updated based on information from the reflections 306 received at the current frequency coefficient set. Based on the current synthetic wave field 614 corresponding to the previous image 706, the residual 804,r is calculated. For example, the residual 804 is used to calculate the squared Euclidean distance 806. Furthermore, the residual 804 is used in conjunction with the adjoint matrix H of the forward operator 808; for example, the product of the residual 804 and the adjoint matrix of the forward operator 808 is used to generate the vector z = HT. r The polar coordinate function of the total variational penalty function 810 is applied to the vector z. The adjoint matrix of the polar coordinate function of the forward operator can be defined as a vector D. z The biinfinite mixing norm is obtained, where D is the discretized gradient operator. This result is then used to partition the squared Euclidean distance of the difference 812 806. The resulting quantity is then added to the initial value of the upper bound 802 814 to produce the updated upper bound 816.

[0156] Figure 9An exemplary schematic diagram 900 illustrates an application of system 102 according to an example implementation. According to this example, system 102 is used in a sensing setup for imaging underground infrastructure, such as for geological discoveries, excavations, etc. In this regard, a transmitter 902 and several receivers (collectively referred to below as receiver 904), depicted as receivers 904A, 904B, 904C, and 904D, are positioned above ground 906. Transmitter 902 is configured to emit a probe pulse 702 that propagates across ground 906 until it interacts with infrastructure objects (depicted as infrastructure objects 908A and 908B). Infrastructure objects 908A and 908B may include, for example, water pipes or solid structural beams, building foundations, etc. Infrastructure objects 908A and 908B can cause multiple wave scattering events. The interaction between the emitted probe pulse 702 and infrastructure objects 908A and 908B can result in a reflected wave field, which can be measured by receiver 904 as a received reflection 306. Based on the embodiments described in this disclosure, together with Figure 1 , Figure 2A , Figure 2B , Figure 3 , Figure 4A , Figure 4B , Figure 4C , Figure 5 , Figure 6 , Figure 7 and Figure 8 A system 100 having a neural network operator 116 including a Born FNO module 410 is configured to reconstruct an image depicting the internal structure of an underground infrastructure within a surface 906 by using the neural network operator 116 and an incremental frequency inversion method 114 to estimate the background medium (i.e., the dielectric constant of the soil inside the surface 906).

[0157] Figure 10 An example method 1000 for reconstructing an image, according to an exemplary implementation, is shown. The steps of method 1000 can be implemented by system 100. Figure 1 , Figure 2A , Figure 2B , Figure 3 , Figure 4A , Figure 4B , Figure 4C , Figure 5 , Figure 6 , Figure 7 , Figure 8 and Figure 9 To explain the components Figure 10 .

[0158] At 1002, measurements are received of the frequency set of the wave field scattered by the internal structure of object 104. These measurements may include, for example, information related to the received reflection 306. For example, probe pulse 702 may transmit the incident wave field 108 using transmitter 902. Receiver 904 of system 100 may receive the scattered wave field 606 of the received reflection 306.

[0159] At 1004, an image of the internal structure of object 104 is recursively reconstructed until a termination condition is met. The image is recursively reconstructed using an incremental frequency inversion method 114 and a neural network operator 116. In the example, the architecture of the neural network operator 116 comprises a sequence of Fourier neural operator (FNO) modules 408 with identical parameters, such as weights and regularizers, which can be enforced by training the neural network operator 116 using machine learning. To reconstruct the image, the neural network operator is configured to reconstruct the image of the internal structure of object 104 by solving an optimization problem that minimizes the difference between a measurement of a portion of the scattered wave field 606 (i.e., the scattered wave field 606 of the received reflection 306) and the synthesized wave field 614.

[0160] For the current iteration, the processor is configured to add frequencies to a previous set of frequencies used during previous iterations, which can be used to generate the previous image 706 to produce the current set of frequencies. Specifically, frequencies higher than those present in the previous set can be added to the previous set of frequencies to generate the current set of frequencies.

[0161] Subsequently, a current image of the internal structure of object 104 can be reconstructed, minimizing the difference between a portion of the scattered wave field 606 measured at the current frequency set and the synthetic wave field 614 from the current image. Specifically, the synthetic wave field 614 can be generated from the current image by neural network operator 116. The reconstruction of the current image is initialized by a previous image 706 determined during previous iterations.

[0162] In some cases, the update 618 of the current image of the internal structure of object 104 and the estimate of the scattered wave field 606 inside object 104 can be determined simultaneously. The estimate of the scattered wave field 606 can have the structure of object 104 corresponding to the current image at each frequency in the current frequency set. Furthermore, based on the update 618 of the current image and the estimate of the scattered wave field 606, the sum of the differences between the received reflection 306 and the synthesized reflection 604 of the reconstructed scattered wave field 606 for each frequency in the current frequency set is minimized.

[0163] In the example, to simultaneously determine the update 618 of the current image of the internal structure of object 104 and the estimate of the scattered wave field 606 within object 104, neural network operator 116 can use Born FNO module 410 to formulate an optimization problem to approximate the interaction between probe pulse 702 and scattered wave field 606, which is generated by scattering probe pulse 702 using one or more materials within object 104 and the refractive index of one or more materials within object 104. Furthermore, the optimization problem of Born FNO 410 can be inverted using the initialized current image to determine the Jacobian determinant of scattered wave field 606 with respect to the current image based on the refractive index of one or more materials. Subsequently, the current image 618 can be updated based on the refractive index of one or more materials by minimizing the cost function between the received reflections 306 in this set of frequency components 118 and the synthesized scattered wave field 614. The refractive index can be obtained by combining the Jacobian determinant of the scattered wave field 606 and the quasi-Newtonian descent direction of the cost functions (such as cost functions 506, 508, 510, 512, 514, and 616) with respect to the refractive index image of one or more materials. For example, the current image of the refractive index of one or more materials can be projected onto a constrained total variational penalty function 714 to generate an updated image 618.

[0164] This process of recursively generating updated images can be performed until a termination condition is met by adding higher-order frequencies to the existing frequency set and minimizing the difference between the received reflection 306 and the synthesized reflection 604. For example, the termination condition could be a time constraint.

[0165] At 1006, a reconstructed image 102 of the internal structure of object 104 is rendered. In this example, reconstructed image 102 may be rendered on a display associated with system 102. For example, reconstructed image 102 is a refractive index image of one or more materials within object 104. Alternatively, reconstructed image 102 may include the distribution (or spatial distribution) of the dielectric constant of one or more materials within object 104.

[0166] Therefore, the blocks of flowchart 1000 support combinations of means for performing a specified function and combinations of operations for performing a specified function. It will also be understood that one or more blocks of flowchart 1000, and combinations of blocks in flowchart 1000, can be implemented by a dedicated hardware-based computer system or a combination of dedicated hardware and computer instructions to perform the specified function.

[0167] Alternatively, system 100 may include means for performing each of the above operations. In this regard, according to an example embodiment, examples of means for performing operations may include, for example, a processor and / or device or circuitry for executing instructions or executing algorithms for processing information as described above.

[0168] When implementing the method 1000 disclosed herein, the final result generated by the system 100 is a tangible high-resolution image of the internal structure of an object, which depicts the spatial distribution of the dielectric constants of one or more materials within the object, for performing several operations, such as geological exploration, oil exploration, exploration of human or animal anatomy, etc.

[0169] Figure 11 A block diagram 1100 of a tomographic imaging system 100 according to some embodiments is shown. The tomographic imaging system 100 may include multiple interfaces for connecting the system 100 to other systems and devices. In this regard, a network interface controller (NIC) 1102 is configured to connect the system 100 to a network 1106 via a bus 1104 that connects the tomographic imaging system 100 to a sensing device (not shown). For example, the tomographic imaging system 100 includes a transmitter interface 1108 configured to command a transmitter 1110 to transmit a pulse wave or probe pulse 702. Using a receiver interface 1112 connected to a receiver 1114, the system 100 can receive a received reflection 306 of a scattered wave field 606 corresponding to the transmitted probe pulse 702. In some implementations, the tomographic imaging system 100 receives information 1116 about the scattered wave field and the transmitted probe pulse 702 via the network 1106.

[0170] The tomographic imaging system 100 includes an output interface 1118 configured to render a reconstructed image 102 as an object 104. For example, the output interface 1118 can display the reconstructed image 102 on a display device, store the image 102 in a storage medium, and / or transmit the image 102 via a network 1106. For example, the system 100 can be linked to a display interface via a bus 1104, which is adapted to connect the system 100 to a display device such as a computer monitor, camera, television, projector, or mobile device. The system 100 can also be connected to an application interface adapted to connect the system to devices for performing various tasks.

[0171] In some implementations, the tomographic imaging system 100 includes an input interface to receive measurements at a set of frequencies of the wavefield scattered by the internal structure of an object, such as rock within the ground or the internal structure of an infrastructure object. Examples of input interfaces include a NIC 1102, a receiver interface 1112, and a human-machine interface (HMI) 1120. The HMI 1120 within the system 100 connects the system 100 to a keyboard 1122 and a pointing device 1124, etc., wherein the pointing device 1124 may include a mouse, trackball, touchpad, joystick, pointing stick, stylus, or touchscreen, etc.

[0172] System 100 includes a processor 1126 configured to execute stored instructions stored in storage device 1128, and a memory 1130 storing instructions executable by processor 1126. Processor 1126 may be a single-core processor, a multi-core processor, a computing cluster, or any number of other configurations. Memory 1130 may include random access memory (RAM), read-only memory (ROM), flash memory, or any other suitable memory system. Processor 1126 may be connected to one or more input and output devices via bus 1104.

[0173] The instructions can implement method 114 and neural network operator 116 for incremental frequency inversion. The instructions may include trained neural network operator 116, a preconditioner for determining the frequency inversion problem, and a frequency set 118 for incremental frequency inversion 114.

[0174] The embodiments of the present invention described above can be implemented in any of a variety of ways. For example, the embodiments can be implemented using hardware, software, or a combination thereof. When implemented in software, the software code can execute on any suitable processor or set of processors, whether provided in a single computer or distributed across multiple computers. Such a processor can be implemented as an integrated circuit, wherein one or more processors are within an integrated circuit assembly. However, the processor can be implemented using circuitry of any suitable format.

[0175] Furthermore, embodiments of the present invention can be specifically implemented as methods, examples of which have been provided. The actions performed as part of a method can be ordered in any suitable manner. Therefore, embodiments in which actions are performed in a different order than those shown can be constructed, which may include performing some actions simultaneously, even if they are shown as sequential actions in the illustrative embodiments.

[0176] The use of ordinal terms such as “first” or “second” to modify a claim element in a claim does not imply any priority, precedence, or order of one claim element relative to another, or the chronological order of actions of the method, but is merely a label to distinguish one claim element with a specific name from another element with the same name (but using ordinal terms) to differentiate claim elements.

[0177] Although this disclosure has been described by way of example of preferred embodiments, it should be understood that various other adjustments and modifications can be made within the spirit and scope of the invention.

[0178] Therefore, the purpose of the appended claims is to cover all such variations and modifications that fall within the true spirit and scope of the invention.

[0179] Various implementations are applicable to two acquisition modes in backscattering: a transmission mode, where the transmitter and receiver are located on opposite sides of the material; and a reflection mode, where the transmitter and receiver are located on the same side of the object. Reflection setups often arise due to limitations allowing access only to one side of the material, such as in subsurface imaging. For this reason, reflection setups are effective in some applications, but lead to severely ill-posed reflection tomography scenarios where backscattering problems occur, because the received reflections contain far less spatial frequency information about the object. Therefore, one implementation aims to provide methods and systems suitable for reflection tomography.

Claims

1. A tomographic imaging system, the tomographic imaging system comprising: An input interface that receives measurements of the frequency set of the wave field scattered by the internal structure of an object; A processor configured to recursively reconstruct an image of the object's internal structure until a termination condition is met, wherein, for the current iteration, the processor is configured to: Add frequencies to the previous set of frequencies used in previous iterations to produce the current set of frequencies, wherein the added frequencies are higher than one or more frequencies present in the previous set of frequencies; and A current image reconstructing the internal structure of the object, the current image minimizing the difference between a portion of the scattered wavefield measured at the current frequency set and a wavefield synthesized from the current image, wherein the wavefield synthesized from the current image is generated by a neural network operator, and wherein the reconstruction of the current image is initialized by a previous image determined during the previous iteration; and An output interface configured to render a reconstructed image.

2. The tomographic imaging system according to claim 1, wherein, The architecture of the neural network operator includes a sequence of Fourier neural operator (FNO) modules with identical parameters, which are enforced by training the neural network operator using machine learning.

3. The tomographic imaging system according to claim 1, wherein, The processor is also configured to: An image of the internal structure of the object is reconstructed by solving an optimization problem that minimizes the difference between a measured value of a portion of the scattered wave field and the synthesized wave field.

4. The tomographic imaging system according to claim 1, wherein, The reconstructed image is a refractive index image of one or more materials inside the object.

5. The tomographic imaging system according to claim 1, wherein, The reconstructed image includes the distribution of dielectric constants of one or more materials within the object.

6. The tomographic imaging system according to claim 1, wherein, In order to reconstruct the current image, the processor is also configured to: Simultaneously, the update of the current image of the internal structure of the object and the estimate of the scattered wave field inside the object are determined, the estimated scattered wave field having a structure corresponding to the current image at each frequency in the current frequency set; as well as Based on the update of the current image and the estimation of the scattered wave field, for each frequency in the current frequency set, the sum of the differences between the received reflection and the synthetic reflection of the reconstructed scattered wave field is minimized.

7. The tomographic imaging system according to claim 6, wherein, In order to simultaneously determine a current image of the internal structure of the object and an estimate of the scattered wave field inside the object, the processor is further configured to: An estimate of the scattered wave field is determined based on a current estimate of an image of the internal structure of the object by simulating the interaction between the probe pulse and the scattered wave field generated by scattering the probe pulse using one or more materials inside the object. An estimate of the accompanying scattered wave field is determined, which compensates for the residual error between the received reflection and the composite reflection of the reconstructed scattered wave field; as well as Using the adjoint state equation, an update of the current image of the object's internal structure is calculated based on the estimation of the scattered wave field, the estimation of the adjoint scattered wave field, and the residual error between the received reflection and the synthetic reflection of the reconstructed scattered wave field.

8. The tomographic imaging system according to claim 7, wherein, In order to simultaneously determine the current image and the scattered wave field, the processor is configured to: A Born Fourier neural operator (FNO) is formed, which approximates the probe pulse, the scattered wave field generated by scattering the probe pulse using one or more materials within the object, and the interaction of the refractive indices of one or more materials within the object; Given an initialized current image, invert the Born FNO to determine the Jacobian determinant of the scattered wave field with respect to the current image based on the refractive index of one or more of the materials; Based on the refractive index of the one or more materials, the current image is updated by minimizing a cost function between the received reflections and the synthetic scattered wavefield in the set of frequency components, the synthetic scattered wavefield being obtained by using a stochastic gradient descent method determined by backpropagation of automatically generated gradients or by combining the Jacobian determinant of the scattered wavefield with the quasi-Newtonian descent direction of the cost function relative to the refractive index image of the one or more materials, or a combination thereof; and The refractive index image of the one or more materials is projected onto a constrained total variational penalty function.

9. The tomographic imaging system according to claim 8, wherein, The cost function between the received reflections in the set of frequency components and the synthetic reflections of the reconstructed scattered wavefield includes one or a combination of the following: the Euclidean distance between each of the received reflections and the corresponding synthetic reflection, the Wasserstein distance between each of the received reflections and the corresponding synthetic reflection, the norm distance between each of the received reflections and the corresponding synthetic reflection, or the sum of the Euclidean distance and the barrier function, wherein the barrier function penalizes updates to the refractive index image of the one or more materials having negative refractive indices, and the barrier function is the sum of exponential functions taken as negative powers of each refractive index in the refractive index image of the one or more materials.

10. The tomographic imaging system according to claim 8, wherein, The refractive index image of the one or more materials is represented by a generator network that maps a low-dimensional latent space representation to the refractive index image of the one or more materials.

11. The tomographic imaging system according to claim 10, wherein, The generator network is determined to be part of an autoencoder network, wherein the encoder network of the autoencoder network is configured to determine a low-dimensional representation of the refractive index image of the one or more materials in a low-dimensional latent space, and the generator network of the autoencoder network is configured to decode the latent space representation to reproduce the refractive index image of the one or more materials.

12. The tomographic imaging system according to claim 8, wherein, The constrained total variational penalty function is subject to an upper limit constraint, wherein the processor is further configured to: Initialize the upper limit of the current image; and The upper limit is updated at the beginning of each iteration using the Newton-FNO method, which adds the ratio of the square of the Euclidean distance between the received reflection and the composite reflection of the reconstructed scattered wave field to the polar function of the constrained total variational function, which is applied to the product of the adjoint matrix of the Born FNO and the difference between the received reflection and the composite reflection of the reconstructed scattered wave field.

13. The tomographic imaging system according to claim 8, wherein, The Born FNO operator approximates the probe pulse, the scattered wave field generated by scattering the probe pulse using one or more materials within the object, and the interaction of the refractive indices of the one or more materials within the object, wherein the structure of the Born FNO is determined by cascading multiple Fourier neural operator modules into a multilayer neural network.

14. The tomographic imaging system according to claim 1, wherein, The objects include components of underground infrastructure.

15. The tomographic imaging system according to claim 1, further comprising: A set of transmitters configured to transmit one or more probe pulses toward the object, wherein the one or more probe pulses comprise at least one of: electromagnetic waves or sound waves occupying a frequency band including the set of frequencies; and A set of receivers configured to measure, at each frequency from the set of frequencies, one or a combination of reflection and refraction of the one or more probe pulses propagating through the object to generate a measurement of the wave field.

16. The tomographic imaging system according to claim 15, wherein, The set of transmitters and the set of receivers are located on the same side of the object, such that the tomographic imaging system operates in reflection mode.

17. A tomographic imaging method, the tomographic imaging method comprising: Receives measurements of the frequency set of the wave field scattered by the internal structure of an object; Recursively reconstructing images of the internal structure of the object until a termination condition is met, wherein, for the current iteration, the tomographic imaging method includes: Add frequencies to the previous set of frequencies used in previous iterations to produce the current set of frequencies, wherein the added frequencies are higher than one or more frequencies present in the previous set of frequencies; and A current image reconstructing the internal structure of the object, the current image minimizing the difference between a portion of the scattered wavefield measured at the current frequency set and a wavefield synthesized from the current image, wherein the wavefield synthesized from the current image is generated by a neural network operator, and wherein the reconstruction of the current image is initialized by a previous image determined during the previous iteration; and Render the reconstructed image.

18. The tomographic imaging method according to claim 17, wherein, The architecture of the neural network operator includes a sequence of Fourier neural operator (FNO) modules with identical parameters, which are enforced by training the neural network operator using machine learning.

19. The tomographic imaging method according to claim 17, further comprising: An image of the internal structure of the object is reconstructed by solving an optimization problem that minimizes the difference between a measured value of a portion of the scattered wave field and the synthesized wave field.

20. A non-transitory computer-readable storage medium having a program implemented thereon, executable by a processor to perform a method comprising: Receives measurements of the frequency set of the wave field scattered by the internal structure of an object; Recursively reconstruct an image of the internal structure of the object until a termination condition is met, wherein, for the current iteration, the method includes: Add frequencies to the previous set of frequencies used in previous iterations to produce the current set of frequencies, wherein the added frequencies are higher than one or more frequencies present in the previous set of frequencies; and A current image reconstructing the internal structure of the object, the current image minimizing the difference between a portion of the scattered wavefield measured at the current frequency set and a wavefield synthesized from the current image, wherein the wavefield synthesized from the current image is generated by a neural network operator, and wherein the reconstruction of the current image is initialized by a previous image determined during the previous iteration; and Render the reconstructed image.

Images