Systems and methods for tomographic imaging

Abstract

A tomographic imaging system is provided. The system receives measurements at frequencies of a wavefront scattered by an internal structure of an object, recursively reconstructs an image of the internal structure of the object until a termination condition is met, and draws the reconstructed image. In the current iteration, a current frequency is created by adding a frequency to the previous frequency used in the previous iteration, such that the added frequency is higher than the frequency present in the previous frequency, and a current image of the internal structure of the object is reconstructed that minimizes the difference between the scattered wavefront measured at the current frequency and the wavefront synthesized from the current image. The wavefront synthesized from the current image is generated by a neural network operator. The previous image determined in the previous iteration initializes the reconstruction of the current image.

Description

[Technical Field]

[0001] The present invention relates to tomography, and more particularly to a tomographic imaging system and method for reconstructing an image of an object's internal structure by solving an inverse scattering problem using deep learning operators that model the physics of wave propagation. [Background technology]

[0002] Objects, such as the human body or rocks, may consist of one or more types of materials. For example, the human body may consist of tissue, bone, skin, muscle, blood, water, etc. Each material within an object may be considered a dielectric. Knowledge of the spatial distribution of permittivity within an object is important for many applications, including microwave imaging, biomicroscopy, medical imaging, through-the-wall imaging (TWI), infrastructure monitoring, and seismic imaging. In particular, determining the permittivity allows visualization of an object's internal structure and characterization of its physical properties. For example, in microwave imaging, the permittivity provides the structure and properties of materials within an object. In biomicroscopy, the permittivity allows visualization of internal cellular structures in three dimensions. In TWI, the permittivity allows knowledge of the dielectric properties of walls and the ability to use this information to compensate for the delay of signals propagating through the walls.

[0003] In a typical scenario, a transmitter emits a signal, such as an electromagnetic (EM), optical, or acoustic pulse. The signal propagates through an object, reflecting off various structures and corresponding materials within the object and propagating to a receiver antenna array. The object's composition is then visualized by numerically generating an image representing the distribution of dielectric constant within the object. Such an image may include an image of the refractive index of materials within the object. However, the received signal results from multiple reflections and / or refractions of the transmitted signal due to multiple scattering from structures or materials within and surrounding the object. As a result, the reconstructed image of the object will contain artifacts that vary depending on the type of material present within or around the object, cluttering the reconstructed image and making it noisy. Additionally, multiple scattering of the transmitted signal has nonlinear effects on the received signal, making image reconstruction more difficult.

[0004] Therefore, there is a need to overcome the aforementioned drawbacks and reconstruct an image of the spatial distribution of the permittivity of materials within an object such that the reconstruction takes into account multiple scattering of a transmitted signal propagating through the object. Summary of the Invention

[0005] An objective of some embodiments of the present disclosure is to reconstruct an image of the internal structure of an object probed with finite-bandwidth electromagnetic and / or acoustic waves to measure the scattered wavefront around the object. An incident wavefront propagating inside the object induces multiple scattered waves at material boundaries within the object. As a result, the scattered waves contain information about the spatial dispersion of material properties. The non-invasive nature of imaging techniques for determining the spatial dispersion of materials within and around an object has led to the application of such imaging techniques in numerous fields, such as non-destructive testing, optical tomography, geophysical imaging, ground-penetrating radar, and medical imaging. For example, the spatial dispersion of material properties in the reconstructed image could be represented by the dispersion of the refractive index of the material(s) within the object and / or the dielectric constant of the material(s) within the object.

[0006] Some embodiments are based on the recognition that an image may be reconstructed based on the spatial distribution of material properties by solving an inverse scattering problem. In such a case, the input wavefront of a probing pulse would be known, and the scattered wavefront would have to be measured. The inverse scattering problem may be used to find the spatial distribution of the dielectric constant of a material that transforms the known input wavefront into a received reflection of the scattered wavefront. However, the inverse scattering problem is an ill-posed problem and difficult to solve.

[0007] Some embodiments recognize that the shortcomings of the inverse scattering problem for reconstructing the internal structure of an object may be addressed by minimizing a cost function that describes the difference between a wavefront synthesized from an image of the object's internal structure and the scattered wavefront. However, synthesis of the wavefront may be inaccurate or infeasible when the background medium in which the object is located has a complex or unknown structure. Therefore, it is an object of some embodiments of the present disclosure to provide methods and systems suitable for reconstructing the internal structure of an object when the background medium is complex or unknown.

[0008] Some embodiments recognize that the type of background medium, even if complex or unknown, belongs to a class of background media commonly observed from the application field in which the system for imaging the internal structure of an object is deployed. As a result, some embodiments of the present disclosure disclose determining a neural network operator trained to learn a mapping between the application field and multiple corresponding composite wavefronts of one corresponding background medium based on training examples belonging to the class of commonly observed background media for different application fields in which the system for imaging may be deployed. An objective of embodiments of the present disclosure is to develop or utilize a trained neural network operator that learns a mapping to an inverse scattering problem for reconstructing the internal structure of an object to improve the quality of the wavefront synthesized from an image of the internal structure of the object.

[0009] Some embodiments are based on the recognition that the structure of a neural network operator may be similar to the iterated Born approximation, which describes the physics of wave propagation. An objective of embodiments of the present disclosure is to design a neural network operator using multiple connected layers of Fourier neural operators (FNOs), where the parameters of the FNOs in each layer are the same in all other layers, i.e., all layers of the FNOs are the same.

[0010] Some embodiments recognize that the object types to be imaged belong to classes of object types commonly observed in the application field in which the system for imaging is deployed. Therefore, another objective of some embodiments of the present disclosure is to determine a neural network autoencoder model consisting of an encoder branch and a decoder branch. The neural network autoencoder model learns to map example objects from the object class to low-dimensional latent space vectors, and learns to inversely map the low-dimensional latent space vectors to the example objects. According to embodiments of the present disclosure, the trained neural network autoencoder model is incorporated into an inverse scattering problem for reconstructing the internal structure of an object by limiting the number of object types that the system for imaging can reconstruct to those that can only be generated by the decoder branch of the trained neural network autoencoder model.

[0011] Some embodiments are based on the recognition that an input wavefront is scattered differently for different frequencies, making the scattered wavefront complex and resulting in a highly nonlinear cost function with multiple local minima. Therefore, the inverse scattering problem of minimizing such a cost function is a challenging problem.

[0012] Some embodiments of the present disclosure are further based on the understanding that low-frequency components of an incident wavefront can penetrate deeper into the material of an object and exhibit weaker interaction with the object's internal structure compared to high-frequency components. Due to the weaker interaction, less scattering occurs during the penetration of the low-frequency wavefront into the material of the object. As a result, the cost function that disagrees with measurements corresponding to low frequencies has fewer local minima compared to high frequencies. Low-frequency measurements contain less spatial detail compared to high-frequency measurements, but can serve to initialize the optimization of the inverse scattering problem with high-frequency measurements.

[0013] To this end, some embodiments are based on an incremental frequency inversion method. For example, in some embodiments, the incremental frequency inversion method recursively reconstructs an image of the object's internal structure until a termination condition is met. In the current iteration, the method adds frequencies to the previous frequency set used in the previous iteration to create a current frequency set, and reconstructs a current image of the object's internal structure that minimizes the difference between the portion of the scattered wavefront measured in the current frequency set and the wavefront synthesized from the current image. The added frequencies are higher than any of the one or more frequencies present in the previous frequency set. The wavefront synthesized from the current image is generated by a neural network operator trained to learn a mapping between the application field and multiple corresponding synthesized wavefronts in a corresponding background medium. Additionally, frequencies are incrementally added to the inverse scattering problem. Additionally, the previous image determined in the previous iteration initializes the reconstruction of the current image. This initialization progressively guides the solution of the inverse scattering problem toward a global minimization function.

[0014] Incremental frequency inversion does not require a smooth initial model of the image to be reconstructed. In contrast, incremental frequency inversion derives such a model from low-frequency measurements. In essence, incremental frequency inversion allows images of the internal structure of an object to be reconstructed with practical resolution and accuracy without the need for prior information about the image.

[0015] Accordingly, an embodiment of the present disclosure provides a tomographic imaging system. The tomographic imaging system includes an input interface for receiving measurements at a frequency set of a wavefront scattered by an 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. In a current iteration of the recursive image reconstruction, the processor is configured to create a current frequency set by adding frequencies to a previous frequency set used in a previous iteration, the added frequencies being 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 wavefront measured at the current frequency set and a wavefront synthesized from the current image. The wavefront synthesized from the current image is generated by a neural network operator, and the previous image determined in the previous iteration initializes the reconstruction of the current image.

[0016] Some embodiments are based on the understanding that the architecture of a neural network operator comprises a series of Fourier Neural Operator (FNO) modules with identical parameters that are enhanced 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 measurements of portions of the scattered wavefront and the composite wavefront.

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

[0019] Some embodiments are based on the understanding that the reconstructed image comprises the dispersion of the dielectric constant of one or more materials within the object.

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

[0021] In some embodiments, to simultaneously determine a current image of the object's internal structure and an estimate of the scattered wavefront within the object, the one or more processors are further configured to determine an estimate of the scattered wavefront based on the current estimate of the image of the object's internal structure by simulating the interaction of a probing pulse with a scattered wavefront resulting from scattering the probing pulse off one or more materials within the object. The one or more processors are further configured to determine an estimate of an adjoint scattered wavefront that compensates for a residual error between the received reflections and a composite reflection of the reconstructed scattered wavefront. The one or more processors are further configured to calculate an update of the current image of the object's internal structure based on the estimate of the scattered wavefront, the estimate of the adjoint scattered wavefront, and the residual error between the received reflections and the composite reflection of the reconstructed scattered wavefront using an adjoint state equation.

[0022] In some embodiments, to simultaneously determine the current image and the scattered wavefront, the one or more processors are further configured to form a Born-Fourier neural operator (FNO) that approximates the interaction of a probing pulse, a scattered wavefront resulting from scattering the probing pulse off one or more materials within the object, and the refractive index of one or more materials within the object. The one or more processors are further configured to invert the Born-FNO for the current image after a given initialization and determine a Jacobian matrix of the scattered wavefront for the current image based on the refractive index of the one or more materials. The one or more processors are further configured to update the current image based on the refractive index of the one or more materials by minimizing a cost function of the received reflections in the set of frequency components and a composite scattered wavefront obtained by combining the Jacobian matrix of the scattered wavefront for the image of the refractive index of the one or more materials with a quasi-Newton descent direction of the cost function. The one or more processors are further configured to project the image of the refractive index of the one or more materials onto a constrained total variation penalty function.

[0023] Some embodiments are based on the understanding that the cost function between the received reflections in the set of frequency components and the composite reflections of the reconstructed scattered wavefront comprises one or a combination of the Euclidean distance between each of the received reflections and the corresponding composite reflection, the normed distance between each of the received reflections and the corresponding composite reflection, or the sum of the Euclidean distance and a barrier function, where the barrier function penalizes updating the refractive index image of one or more materials with a negative refractive index, and the barrier function is a sum of exponential functions taken to negative powers of each refractive index in the refractive index image of the one or more materials.

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

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

[0026] Some embodiments are based on the understanding that the constrained total variation penalty function is constrained by an upper bound. The one or more processors are further configured to initialize the upper bound for the current image and update the upper bound at the start of every iteration using a Newton root-finding method that adds a ratio of the squared Euclidean distances between the received reflection and the composite reflection of the reconstructed scattered wavefront and a polar function of the constrained total variation function applied to the product of the difference between the received reflection and the composite reflection of the reconstructed scattered wavefront and the adjoint of the Born FNO.

[0027] Some embodiments are based on the understanding that a Born-FNO operator approximates the interaction of a probing pulse, a scattered wavefront resulting from scattering the probing pulse off one or more materials within the object, and the refractive index of one or more materials within the object. The structure of the Born-FNO is determined by concatenating multiple FNO modules into a multi-layer neural network.

[0028] Some embodiments are based on the understanding that the object comprises an element of underground infrastructure.

[0029] Some embodiments are based on the understanding that a tomographic imaging system comprises a set of transmitters configured to transmit one or more probing pulses into an object, and a set of receivers configured to measure reflection and / or refraction of the one or more probing pulses propagating through the object at each frequency from a set of frequencies to generate a wavefront measurement, wherein the one or more probing pulses comprise at least one of electromagnetic waves or acoustic waves occupying a frequency band that includes the set of frequencies.

[0030] Some embodiments are based on the understanding that, for a tomographic imaging system to operate in a reflection mode, the set of transmitters and the set of receivers are positioned on the same side of the object.

[0031] Another embodiment discloses a tomographic imaging method using a processor coupled to stored instructions for implementing the method. The instructions, when executed by the processor, perform method steps, including receiving measurements at a frequency set of a wavefront scattered by an 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. In a current iteration, the method includes creating a current frequency set by adding frequencies to a previous frequency set used in a previous iteration, the added frequencies being higher than one or more frequencies present in the previous frequency set. The method further includes reconstructing a current image of the internal structure of the object that minimizes a difference between a portion of the scattered wavefront measured at the current frequency set and a wavefront synthesized from the current image, the wavefront synthesized from the current image being generated by a neural network operator, and the previous image determined in the previous iteration initializes the reconstruction of the current image.

[0032] Yet another embodiment discloses a non-transitory computer-readable storage medium carrying a program executable by a processor to perform a method. The method includes receiving measurements at a frequency set of a wavefront scattered by an 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. In the current iteration, the method includes creating a current frequency set by adding frequencies to a previous frequency set used in the previous iteration, the added frequencies being higher than one or more frequencies present in the previous frequency set. The method further includes reconstructing a current image of the internal structure of the object that minimizes a difference between a portion of the scattered wavefront measured at the current frequency set and a wavefront synthesized from the current image, the wavefront synthesized from the current image being generated by a neural network operator, and the previous image determined in the previous iteration initializes the reconstruction of the current image. [Brief explanation of the drawings]

[0033] The presently disclosed embodiments will be further described with reference to the accompanying drawings, in which the drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments.

[0034] [Figure 1] 1 shows a block diagram of a tomographic imaging system for determining an image of an internal structure of an object, according to an exemplary embodiment; [Figure 2A] 1 shows a schematic diagram of an application of a tomographic imaging system, according to an exemplary embodiment; [Figure 2B] 1 illustrates a schematic diagram of the task of the inverse scattering problem, according to an exemplary embodiment; [Figure 3] 1 shows a block diagram of a method for determining an image of an internal structure of an object, according to an exemplary embodiment; [Figure 4A] 1 illustrates a schematic diagram of an autoencoder network of neural network operators, according to an exemplary embodiment; [Figure 4B] FIG. 2 illustrates the structure of a neural network operator, according to an exemplary embodiment. [Figure 4C] 1 shows a schematic diagram of a process for solving an inverse problem to reconstruct the internal structure of an object, according to an exemplary embodiment; [Figure 5] 1 shows a schematic diagram of the principle of incremental frequency inversion, according to an exemplary embodiment; [Figure 6] FIG. 1 shows a block diagram of a method for reconstructing one current image for different frequencies according to an exemplary embodiment. [Figure 7] 1 illustrates a block diagram of a method for updating an image of the spatial distribution of the dielectric constant of a material for a current frequency set, according to an exemplary embodiment. [Figure 8] FIG. 1 shows a block diagram of a method for updating an upper bound on a total variation penalty function, according to an example embodiment. [Figure 9] 1 shows an exemplary schematic diagram of a reflection setup, according to an exemplary embodiment; [Figure 10] FIG. 2 illustrates an exemplary method for reconstructing an image, according to an exemplary embodiment. [Figure 11] 1 shows a block diagram of a tomographic imaging system according to an exemplary embodiment; DETAILED DESCRIPTION OF THE INVENTION

[0035] In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. However, it will be apparent to those skilled in the art that the present disclosure may be practiced without these specific details. In other instances, devices and methods are shown only in block diagram form in order to avoid obscuring the disclosure. Various changes may be made in the function and arrangement of elements without departing from the spirit and scope of the subject matter as set forth in the appended claims.

[0036] As used in this specification and the appended claims, the terms "for example," "for instance," and "such as," and the verbs "comprising," "having," and "including" and other verb forms thereof, when used in conjunction with a list of one or more components or other items, should each be construed as open-ended, meaning that the list should not be considered to exclude other, additional components or other items. The term "based on" means based, at least in part, on. Furthermore, it should be understood that the phraseology and terminology used herein are for descriptive purposes and should not be regarded as limiting. The headings used herein are for convenience only and do not have any legal or limiting effect.

[0037] Specific details have been set forth in the above description to provide a thorough understanding of the embodiments. However, those skilled in the art will understand that these embodiments may be practiced without these specific details. For example, systems, processes, and other elements of the disclosed subject matter may be illustrated in block diagrams as components to avoid obscuring the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be illustrated without unnecessary detail to avoid obscuring the embodiments. Furthermore, like reference numerals and symbols in the various drawings refer to like elements.

[0038] It is an object of some embodiments to disclose a tomographic imaging system for reconstructing an image of an object such that the internal structure of the object is identified. 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 may resemble an iterative Born approximation that describes the physics of wave propagation. To improve the quality of the wavefront synthesized from the image of the object's internal structure, the neural network operator learns a mapping to the inverse scattering problem to reconstruct the object's internal structure.

[0039] 1 shows a block diagram of a tomographic imaging system 100 for determining an image 102 of an internal structure of an object 104, according to an exemplary embodiment. The image 102 of the internal structure of the object 104 may 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 constant of one or more materials within the object, or a combination thereof. The tomographic imaging system 100 (hereinafter referred to as system 100) comprises and / or is operably connected to at least one transceiver 106 for propagating a probing pulse of an incident wavefront 108 through one or more materials of the object 104 and for receiving an echo set 110 resulting from scattering of the pulse by different portions or one or more materials of the object 104.

[0040] For example, the transceiver 106 may include at least one transmitter that transmits a probing pulse to the object 104. The probing pulse in the incident wavefront 108 is scattered by the material to generate an echo set 110. The probing pulse may be, for example, an electromagnetic wave, an acoustic wave, or an optical wave. The probing pulse may include, for example, one or a combination of a microwave pulse, a radar pulse, a laser pulse, an ultrasound pulse, and an acoustic pulse. The transceiver 106 may also include at least one receiver positioned at a predetermined location relative to the transmitter for receiving the echo set 110. According to some embodiments, the system 100 may create a two-dimensional or three-dimensional image 102 of the object 104, where each location in the image 102 provides a value of the permittivity of a corresponding portion of the material corresponding to that location.

[0041] The tomographic imaging system also includes a processor 112 operatively connected to the transceiver 106 for determining the image 102 based on the echo set 110. To reconstruct the image 102 of the object 104 despite multiple scattering, the processor 112 divides the reconstruction into several stages by performing operations on a frequency set 118 of the scattered wavefront. The processor 112 implements an incremental frequency inversion method 114 and utilizes a neural network operator 116 to reconstruct the internal structure of the object 104 by improving the quality of the composite wavefront from the image of the object's internal structure to sequentially and efficiently solve the inverse scattering problem. For example, in some embodiments, the incremental frequency inversion method 114 recursively reconstructs the image 102 of the object's 104 internal structure for high frequencies until a termination condition is met. Further details about the application of the system 100 are described, for example, in conjunction with FIG. 2A .

[0042] 2A shows a schematic diagram 200 of an application of system 100, according to an exemplary embodiment. In this example, system 100 is used in a ground-penetrating radar (GPR) measurement acquisition process. It should be noted that this application of system 100 is exemplary and should not be considered limiting. In other embodiments of the present disclosure, system 100 may be used in applications such as human body imaging, geological exploration, and oil or mineral extraction.

[0043] For ground penetrating radar (GPR) applications, an operator 202 may move a radar-equipped imager 204. The imager 204 may include a transmitter that sends radar waves or probing pulses into the ground 206 and a receiver that receives reflections from objects 104 located below the ground 206 for data collection. Based on the received scattered wavefront reflections, the imager 204 may be configured to reconstruct an image of the object 104, which may be located below the ground 206.

[0044] An object of embodiments of the present disclosure is to provide a tomographic imaging system 100 that accurately reconstructs images of objects, such as objects located underground. Such reconstructed images may show, for example, the internal structure of the object, indicating the type, nature, physical properties, and / or chemical characteristics of the object. In this regard, the tomographic imaging system 100 may use an inverse scattering problem to reconstruct the image. More details about the image scattering problem are described in conjunction with FIG. 2B.

[0045] 2B shows a schematic block diagram 210 of the task of the inverse scattering problem, according to an exemplary embodiment. The inverse scattering problem aims to reconstruct an image of the internal structure 214 of the ground 104 by solving an optimization problem. The optimization problem is used to minimize the difference between the measured scattered wavefront, i.e., the reflections received by the receivers of the system 100, and the synthetic wavefront, i.e., the synthetic wavefront generated from the image of the internal structure of the object 104.

[0046] During operation, a probing pulse comprised of an incident wavefront 108 may be transmitted through the system 102 to the ground 206. One or more objects, such as the object 104, may also be located below the ground 206. Based on the received scattered wavefront, an image of the internal structure 214 or material below the ground 206 and / or the internal structure of the object 104 may be reconstructed. In this regard, an ill-posed inverse scattering problem determines the subsurface structure 218 comprised of the subsurface internal structure 214 of the ground 206 from the 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, which is comprised of, for example, a first layer of background medium 220 and a second layer of background medium 222.

[0047]

number

[0048]

number

[0049] 3 shows a block diagram 300 of a method for determining an image 102 of an internal structure of an object 104, according to an example embodiment. The method may be implemented using the processor 112 of the system 100. For example, the method may be implemented by the processor 112 executing a program embodied on a non-transitory computer-readable storage medium.

[0050] The method includes, at 302, transmitting a probing pulse of the incident wavefront 108 to penetrate the object 104 and receiving a set of echoes 110. In one example, a processor 112 is configured to transmit the probing pulse of the incident wavefront 108 to penetrate a material of the object 102. For example, the processor 112 receives the pulse scattered by the material in the form of a set of echoes 110. The echoes 110 result from the pulse being scattered by different portions or different materials of the object 104.

[0051] The pulses of the incident wavefront 108 span a frequency band comprising a frequency set. For example, the frequency set is the product of the quantization of the frequency bands of the pulses of the incident wavefront 108. In one embodiment, the quantization is performed at a resolution proportional to the period over which the reverberations 110 are sampled. This resolution makes it possible to reconstruct the necessary details of the image 102 showing the internal structure of the object 104, such that the frequency separation of the quantization bins is small enough that adjacent frequency bins contain overlapping information about the image details. In another embodiment, the resolution is selected based on the computational capabilities of the processor 112. For example, a processor with limited memory may require larger quantization bins, resulting in a smaller number of frequencies.

[0052] In one example, the processor 112 may be configured to convert the received reflections 306 to digital signals using an analog-to-digital converter and record the amplitude and / or other characteristics of the digital signals. In some embodiments, the processor 112 may be configured to organize the received reflections 306 into an ordered set of scattered wavefront frequencies 118. The ordered set of frequencies 118 may include frequencies ordered from lowest frequency to highest frequency.

[0053] According to the principle of incremental frequency inversion 114, the method does not aim to jointly reconstruct one image for all frequencies at once. In contrast, the method according to the present disclosure reconstructs one image for a subset of frequencies, gradually adding higher frequencies to the subset to further refine the image. Eventually, the method processes all frequencies. With incremental frequency inversion 114, a previously determined image serves as a precursor to the reconstruction of the next image, improving the quality of the image reconstruction. In this way, the image 102 is iteratively reconstructed based on multiple frequencies in the frequency band.

[0054] To that end, at 304, the method includes progressively adding one or more frequencies from the received reflections 306 to the current frequency set. In particular, the processor 112 may be configured to begin adding frequencies to the current frequency set in different iterations from lowest to highest. For example, the processor 112 may be configured to initialize the current frequency set by placing the lowest frequency in the current frequency set. In the next iteration, the second lowest frequency may be added to the current frequency set, and so on.

[0055] Some embodiments are based on the recognition that the inverse scattering problem for reconstructing the internal structure of an object 104 may be addressed by minimizing a cost function that describes the difference between the received reflection and a reflection synthesized from an image of the internal structure of the object 104. However, the input wavefront is scattered differently for different frequencies, complicating the scattered wavefront and making the cost function highly nonlinear with multiple local minima. Therefore, the inverse scattering problem of minimizing such a cost function is a challenging problem.

[0056] However, some embodiments are based on the understanding that low-frequency components of an incident wavefront can penetrate further into the material of the object 104 and exhibit weaker interaction with the internal structure of the object 104 than higher-frequency components. As a result of the weaker interaction, fewer scattering events occur during penetration of the low-frequency wavefront. As a result, the measurement mismatch cost function corresponding to low frequencies has fewer local minima compared to higher frequencies. While low-frequency measurements contain less spatial detail compared to high-frequency measurements, they can serve to initialize the optimization of the inverse scattering problem with high-frequency measurements.

[0057] Some embodiments realize that it is easier to process each frequency separately and then jointly reconstruct the image 102 by stitching images of different frequencies in one frequency domain. However, this option is not suitable for reconstructing the internal structure of the object 104, since the scattered wavefronts at different frequencies are independent. To do this, some embodiments jointly reconstruct one image 102 for multiple frequencies. However, these frequencies are added incrementally, gradually initializing the reconstruction until all frequencies are concatenated.

[0058] Continuing further in the method for reconstructing an image 102 of the internal structure of an object 104, at 308, a current image of the object 104 is reconstructed. For a particular iteration, the processor 112 is configured to update the current image based on the wavefront in the current frequency set and check for an end condition. For the current iteration, image reconstruction of the current image is performed using only frequencies present in the current frequency set. To accurately reconstruct the current image of the internal structure of the object 104, the method minimizes the difference between the portion of the scattered wavefront measured at the current frequency set and the wavefront synthesized from the current image.

[0059] At 310, the wavefront for the current iteration is synthesized. In some embodiments, the method uses a current image of the object from the previous iteration 104, i.e., the previous image reconstructed during the previous iteration, to synthesize a wavefront that is compared to the received reflections 306. For example, the synthesized wavefront from the current image of the previous iteration is generated by a neural network operator 116. The structural details of the neural network operator 116 are shown, for example, in FIG. 4B.

[0060] At 312, the error between the received reflections 306 is compared to the composite wavefront. A determination is further made as to whether the error is less than a threshold. If the error is determined to be less than the threshold, the current image produced in the current iteration is output as the reconstructed image 102 of the internal structure of the object 104.

[0061] However, if the error is determined to be greater than a threshold, the current image generated in the current iteration is updated at 314. The current image is updated to minimize the error. For example, some embodiments use a different regularizer to update the current image at 318. Regularizers include, but are not limited to, a total variation regularizer such as that shown in FIG. 4A and an autoencoder constrained regularizer.

[0062] Additionally or alternatively, the method ends when a determination is made to see if all frequencies in the frequency set have been processed at 316. For example, the method ends the reconstruction process when all frequencies in the ordered frequency set 118 have been included in the current frequency set. In subsequent iterations, the processor may 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.

[0063] 4A shows a schematic diagram of an autoencoder network 400 of the neural network operator 116, according to an example embodiment. The autoencoder network 400 comprises an encoder branch and a decoder branch. In particular, the autoencoder network 400 comprises an encoder network 402 and a generator network 404 (or decoder).

[0064] The autoencoder network 400 may include a deep learning model that transforms 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 the received reflections 306 or information about the refractive index of one or more materials of the object 104 into a low-dimensional space. For example, the encoder network 402 may encode the image data, whatever its size, into a 1D vector, i.e., a low-dimensional space, that is smaller in size than the image. The vector may then be decoded to reconstruct the original data, i.e., the image, by the generator network 404. The autoencoder network 400 learns an efficient data representation, i.e., encoding, by training the encoder network 402 to ignore signal noise. The subsequent transformation of the low-dimensional image back to the original image by the autoencoder network 400 may enable, for example, image denoising.

[0065] The encoder network 402 is configured to map the internal structure of the object 104 into a low-dimensional latent space. In one example, the encoder network 402 is configured to determine a low-dimensional representation of an image, such as an image 102 of the refractive index of one or more materials in the current image or object 104 in the low-dimensional latent space. Further, the generator network 404 is configured to map from the low-dimensional latent space to the internal structure of the object. For example, the generator network 404 is configured to decode the latent space representation to recreate a current image of the refractive index of one or more materials in the image 102 or object 104. To do this, the image of the refractive index of the one or more materials is represented by the generator network 404, which maps the low-dimensional latent space representation to the image of the refractive index of the one or more materials.

[0066]

number

[0067]

number

[0068]

number

[0069] 4B illustrates the structure 406 of the neural network operator 116 according to an exemplary embodiment. In one example, the architecture of the neural network operator 116 comprises a series of Fourier neural operator (FNO) modules, denoted as FNO modules 408A, 408B, ..., 408N (collectively referred to as FNO modules 408), having identical parameters that are enhanced by training the neural network operator 116 with machine learning. In this regard, the FNO modules 408 are configured to learn a continuous function by parameterizing the autoencoder network 400 in that function space. This allows the FNO modules 408 to be trained on one mesh or training data set and subsequently trained on another mesh or training data set.

[0070] In particular, the FNO module 408 may form a Born FNO 410. In operation, the Born FNO module 410 may approximate the interaction of a probing pulse with a scattered wavefront. The scattered wavefront may result from scattering the probing pulse off one or more materials within the object 104 and the refractive indices of the one or more materials within the object 104. For example, the Born FNO module 410 may be formulated based on the iterative Born approximation as follows:

number

[0071]

number

[0072] 4C shows a schematic diagram 416 of a process for solving an inverse problem to reconstruct the internal structure of the object 104, according to an example embodiment. The inverse problem to reconstruct the internal structure of the object 104 may be solved by the structure 406 of the neural network operator 116.

[0073]

number

[0074] 5 shows a schematic diagram 500 of the principle of the incremental frequency inversion method 114, according to an exemplary embodiment. The incremental frequency inversion method 114 (hereinafter referred to as method 114) recursively reconstructs an image of the internal structure of the object 104 until a termination condition is met. In the current iteration, method 114 creates a current frequency set by adding frequencies to the previous frequency set used in the previous iteration. Based on the current frequency set and the optimization problem, the neural network operator 116 of the Born FNO module 410 may reconstruct a current image of the internal structure of the object 104 that minimizes a cost function 502 that indicates the difference between the portion of the scattered wavefront measured at the current frequency set and the wavefront synthesized from the current image.

[0075] At the global minimum 518, the minimized cost function 502 produces reflectance parameters 304 of an image of the internal structure of the material of the object 104. However, the shape of the cost function 502 is complex (much more complex than the shape of the simplified example of FIG. 5). Therefore, minimizing the cost function 502 will produce one of multiple local minima 516 rather than the global minimum 518. The method 114 can address the aforementioned problem.

[0076] For example, the frequency set of method 114 includes frequencies of 10, 20, 30, 40, and 50 MHz. During a first iteration, the image is reconstructed by minimizing cost function 506 for only the 10 MHz frequency. For such frequencies, the reconstruction is likely to find a global minimum 518. Next, method 114 adds frequencies higher than one or more frequencies in the previous frequency set. For example, method 114 adds a 20 MHz frequency to minimize cost function 508 for the joint frequencies of 10 MHz and 20 MHz near the local minimum corresponding to function 506. Next, method 114 adds a 30 MHz frequency to minimize cost function 510 for the joint frequencies of 10 MHz, 20 MHz, and 30 MHz near the local minimum corresponding to function 508. Similarly, the method 114 adds the 40 MHz frequency and minimizes the cost function 512, and finally adds all frequencies from 10 to 50 MHz and minimizes the global function 514 to find the global minimum point 518 corresponding to the final image 102.

[0077] In this way, frequencies are progressively added to the inverse scattering problem. In addition, the previous image determined in the previous iteration initializes the reconstruction of the current image. This initialization progressively guides the solution of the inverse scattering problem towards a global minimization function 518.

[0078] The incremental frequency inversion method 114 does not require the reconstruction of a smooth initial model of the image. In contrast, the incremental frequency inversion method 114 derives such a model from low-frequency measurements. In short, the incremental frequency inversion method 114 allows the reconstruction of an image 102 of the internal structure of an object 104 with practical resolution and accuracy without requiring the use of prior information about the image. Frequency Inversion

[0079] According to some embodiments of the present disclosure, the image reconstruction problem may be formulated as a frequency inversion problem solved by an incremental frequency inversion method 114. The scattering model is based on the recognition that it describes the relationship between the scattered wavefront and the medium parameters of one or more materials of the object 104. The scattering model allows for the formulation of a discrete inverse problem to reconstruct the medium or one or more materials from a set of received reflections of the scattered wavefront.

[0080] Specifically, the wave equation governs acoustic or electromagnetic scattering from inhomogeneous media in the time domain. The equivalent expression in the frequency domain is the scalar Helmholtz equation. An integral form of the Helmholtz equation, for example, is the scalar Lippmann-Schwinger equation.

[0081]

number

[0082]

number

[0083] In the discrete setting, the scattering and data equations simplify to the following system of linear equations for each transmitter illumination and wavenumber:

number

[0084]

number

[0085] FIG. 6 shows a block diagram 600 of a method for reconstructing 602 a current image for different frequencies according to an exemplary embodiment. This method is executed multiple times for different combinations of frequencies, i.e., based on the current frequency set for the corresponding iteration. For example, in one iteration, the method is executed to reconstruct 602 an image by minimizing the cost function 508 for two frequencies, e.g., 10 MHz and 20 MHz of the current frequency set. In the next iteration, the method is invoked to update 618 the image for frequencies of 10 MHz, 20 MHz, and 30 MHz by minimizing the cost function 510. In this iteration, the image is updated 618 based on the previous reconstruction 602 of the image obtained by minimizing the cost function 508.

[0086] For a given image, a composite reflection 604 is generated and compared to the received reflection 306. Both the composite reflection 604 and the received reflection 306 are determined for the current frequency set. The difference between the composite reflection 604 and the received reflection 306 is then minimized by simultaneously updating (608) the current image and an estimate of the scattered wavefront 610-612 (collectively referred to as the scattered wavefront 606) at each frequency in the current frequency set. The updated current image and scattered wavefront 610-612 are then used to minimize a cost function 616 of the difference between the composite reflection 604 and the received reflection 306. In some implementations, the cost function 616 is minimized by simultaneously estimating the scattered wavefront 610-612 or 606 at each frequency and the current image according to the principles of the mutually exclusive states method. The metric of the minimized cost function 616 may be selected from one or a combination of the Euclidean distance between the received reflection 306 and the composite reflection 604, a normed distance between the received reflection 306 and the composite reflection 604, and a sum of the Euclidean distance and a barrier function. For example, the barrier function penalizes updating the refractive index image of one or more materials in the object 104 that have a negative refractive index, and the barrier function is a sum of exponential functions taken to the negative power of each refractive index in the material refractive index image 602.

[0087] FIG. 7 shows a block diagram 700 of a method for updating an image of the spatial dispersion of the permittivity of a material for a current frequency set, according to an exemplary embodiment. In this regard, the Born FNO module 410 approximates a partial differential equation (PDE) such that the neural network operator 116 characterizes the interaction of the transmitted probing pulse 702 with the transmit wavefront of the current frequency set and the previous estimate of the image, i.e., the previous image 706 of the dispersion of the permittivity of the material at all frequencies in the current frequency set. The Born FNO module 410 function is determined such that the product of the neural network operator 116 with the normal scattered wavefront is equal to the transmitted probing pulse 702. To estimate the scattered wavefront, the Born FNO module 410 function is inverted 708 for the previous image 706 determined in a given previous iteration and the portion of the transmit wavefront of the transmitted probing pulse 702 for the current frequency set. The received reflection 306 is then compared to the synthetic reflection 604 to update the residual 704. The residuals 704 are used to update the image gradients and Jacobian 710 according to an adjoint state procedure. The image is then updated by modifying the latent space variables 712 using backpropagation, and the result is projected 618 onto a constrained total variation penalty function 714 to update the image. inverse problem

[0088]

number

[0089] The least-squares cost function 616 in equation (1) allows for a natural separation between frequency sets. Additionally, the topology of the non-convex cost function 616 may be exploited to find a series of local minima that vary significantly between frequency sets and gradually lead to the inverse problem's global minimization function 518. As high-frequency wavefronts are introduced, the cost function begins to exhibit many local minima that are farther away from the global minimization function 518 than the low-frequency wavefronts.

[0090]

number

[0091]

number

[0092]

number

[0093] In one example, the TV norm measures the phase change of the derivative of the function of the neural network operator 116 over a finite domain. As a result, normalization with the TV norm facilitates interval-constant approximation of the true neural network operator 116.

[0094] According to some embodiments, the TV normalization method can be used in its constrained form as follows.

number

[0095] To impose the TV-norm constraint, one embodiment defines a proximity operator as follows:

number

[0096]

number

[0097]

number

[0098] FIG. 9 illustrates an exemplary schematic diagram 900 of an application of system 102, according to an exemplary embodiment. According to this example, system 102 is utilized in a sensing setup for imaging underground infrastructure, for example, for geological discovery, excavation, and the like. In this regard, one transmitter 902 and several receivers, represented as receivers 904A, 904B, 904C, and 904D (collectively referred to hereinafter as receivers 904), are positioned above ground level 906. Transmitter 902 is configured to emit probing pulses 702 that propagate through ground level 906 until they interact with infrastructure objects, represented as infrastructure objects 908A and 908B. Infrastructure objects 908A and 908B may include, for example, water pipes or solid structural beams, electrical conduits, building foundations, and the like. Infrastructure objects 908A and 908B produce multiple wave scattering events. Interaction of the transmitted probing pulse 702 with infrastructure objects 908A and 908B may result in a reflected wavefront that may be measured by the receiver 904 as received reflections 306. Based on the embodiments described in this disclosure in conjunction with Figures 1, 2A, 2B, 3, 4A, 4B, 4C, 5, 6, 7, and 8, the system 100 having a neural network operator 116 including a Born FNO module 410 is configured to reconstruct an image of an underground infrastructure representing the internal structure within the ground surface 906 by estimating the dielectric constant of the background medium, i.e., the soil within the ground surface 906, using the neural network operator 116 and the incremental frequency inversion method 114.

[0099] Figure 10 illustrates an example method 1000 for reconstructing an image, according to an example embodiment. Steps of method 1000 may be implemented by system 100. Figure 10 will be described in conjunction with elements of Figures 1, 2A, 2B, 3, 4A, 4B, 4C, 5, 6, 7, 8, and 9.

[0100] At 1002, measurements are received at a set of frequencies of a wavefront scattered by an internal structure of the object 104. These measurements may include, for example, information about the received reflection 306. For example, the probing pulse 702 may transmit the incident wavefront 108 using the transmitter 902. A receiver 904 of the system 100 may receive the scattered wavefront 606 of the received reflection 306.

[0101] At 1004, an image of the internal structure of the 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 one example, the architecture of the neural network operator 116 comprises a series of Fourier neural operator (FNO) modules 408 having identical parameters, such as weights and regularizers, which may be enhanced by training the neural network operator 116 with machine learning. When reconstructing the image, the neural network operator is configured to reconstruct the image of the internal structure of the object 104 by solving an optimization problem that minimizes the difference between the composite wavefront 614 and a measurement of a portion of the scattered wavefront 606, i.e., the portion of the scattered wavefront 606 of the received reflection 306.

[0102] In the current iteration, the processor is configured to add frequencies to the previous frequency set used during the previous iteration, which may be used to create the previous image 706 to create the current frequency set. In particular, one or more frequencies higher than those present in the previous frequency set may be added to the previous frequency set to generate the current frequency set.

[0103] A current image of the internal structure of the object 104 may then be reconstructed that minimizes the difference between the portion of the scattered wavefront 606 measured at the current frequency set and the composite wavefront 614 from the current image. In particular, the composite wavefront 614 may be generated from the current image by the neural network operator 116. The previous image 706 determined during the previous iteration initializes the reconstruction of the current image.

[0104] In certain cases, a current image update 618 of the internal structure of the object 104 and an estimate of the scattered wavefront 606 inside the object 104 may be determined simultaneously. The estimate of the scattered wavefront 606 may have the structure of the object 104 corresponding to the current image at each frequency of the current frequency set. Furthermore, the sum of the differences between the composite reflection 604 and the received reflection 306 of the reconstructed scattered wavefront 606 for each frequency of the current frequency set is minimized based on the current image update 618 and the estimate of the scattered wavefront 606.

[0105] In one example, to simultaneously determine an update 618 of the current image of the internal structure of the object 104 and an estimate of the scattered wavefront 606 within the object 104, the neural network operator 116 may use the Born FNO module 410 to create an optimization problem to approximate the interaction of the probing pulse 702, the scattered wavefront 606 resulting from scattering the probing pulse 702 off one or more materials within the object 104, and the refractive index of one or more materials within the object 104. The Born FNO 410 optimization problem may then be inverted using the initialized current image to determine a Jacobian matrix of the scattered wavefront 606 for the current image based on the refractive index of the one or more materials. The current image may then be updated 618 based on the refractive index of the one or more materials by minimizing a cost function between the composite scattered wavefront 614 and the received reflections 306 in the set of frequency components 118. The refractive index may be obtained by combining the quasi-Newton descent direction of a cost function, such as cost functions 506, 508, 510, 512, 514, and 616, on the image of the refractive index of one or more materials with the Jacobian matrix of the scattered wavefront 606. For example, the current image of the refractive index of one or more materials may be projected onto a constrained total variation penalty function 714 to generate an updated image 618.

[0106] This process of recursively generating an updated image may be performed by adding higher order frequencies to the existing frequency set and minimizing the difference between the received reflection 306 and the composite reflection 604 until a termination condition is met. For example, the termination condition may be a time limit.

[0107] At 1006, a reconstructed image 102 of the internal structure of the object 104 is rendered. In one example, the reconstructed image 102 may be rendered on a display associated with the system 102. For example, the reconstructed image 102 is an image of the refractive index of one or more materials within the object 104. Alternatively, the reconstructed image 102 includes a dispersion (or spatial dispersion) of the dielectric constant of one or more materials within the object 104.

[0108] Thus, the blocks of flowchart 1000 correspond to combinations of means for performing the specified functions and combinations of acts for performing the specified functions. It should also be understood that one or more blocks of flowchart 1000, and combinations of blocks of flowchart 1000, can be implemented by a dedicated hardware-based computer system that performs the specified functions or a combination of dedicated hardware and computer instructions.

[0109] Alternatively, system 100 may comprise means for performing each of the operations described above. In this regard, according to an exemplary embodiment, means for performing operations may include, for example, a processor and / or a device or circuitry for executing instructions or for executing an algorithm for processing information as described above.

[0110] In implementing the method 1000 disclosed herein, the end result generated by the system 100 is a tangible, high-resolution image of the internal structure of an object representing the spatial distribution of the dielectric constant of one or more materials within the object for performing several tasks, such as geological exploration, petroleum exploration, and human or animal anatomical exploration.

[0111] 11 shows a block diagram 1100 of a tomographic imaging system 100 according to some embodiments. The tomographic imaging system 100 may include multiple interfaces that connect the system 100 to other systems and devices. In this regard, a network interface controller (NIC) 1102 is configured to connect the system 100 via a bus 1104 to a network 1106 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 instruct a transmitter 1110 to emit a pulsed wave or probing pulse 702. Using a receiver interface 1112 connected to a receiver 1114, the system 100 may receive a received reflection 306 of a scattered wavefront 606 corresponding to the transmitted probing pulse 702. In some implementations, the tomographic imaging system 100 receives information 1116 about the scattered wavefront and the transmitted probing pulses 702 over the network 1106 .

[0112] Tomographic imaging system 100 includes an output interface 1118 configured to render a reconstructed image 102 of object 104. For example, output interface 1118 may display reconstructed image 102 on a display device, store image 102 on a storage medium, and / or transmit image 102 over network 1106. For example, system 100 may be linked through bus 1104 to a display interface adapted to connect system 100 to a display device such as a computer monitor, a camera, a television, a projector, or a mobile device, among other things. System 100 may also be connected to an application interface adapted to connect the system to devices for performing various tasks.

[0113] In some implementations, the tomographic imaging system 100 may include an input interface to receive measurements at a frequency set of a wavefront scattered by an internal structure of an object, such as a rock or an infrastructure object within the earth. The input interface may include a NIC 1102, a receiver interface 1112, and a human-machine interface device (HMI) 1120. The HMI 1120 in the system 100 connects the system 100 to a keyboard 1122 and a pointing device 1124, among others, where the pointing device 1124 may include a mouse, a trackball, a touchpad, a joystick, a pointing stick, a stylus, or a touchscreen.

[0114] System 100 includes a processor 1126 configured to execute instructions stored in storage 1128 and a memory 1130 that stores instructions executable by processor 1126. Processor 1126 may be a single-core processor, a multi-core processor, a computer cluster, or any number of other configurations. Memory 1130 may include RAM (random access memory), ROM (read-only memory), flash memory, or other suitable memory systems. Processor 1126 may be connected to one or more input devices and one or more output devices via bus 1104.

[0115] The instructions may implement a method for incremental frequency inversion 114 and a neural network operator 116. The instructions may include a trained neural network operator 116 for determining preconditions for the frequency inversion problem and a frequency set 118 for incremental frequency inversion 114.

[0116] The embodiments of the present disclosure described above may be implemented in any of numerous ways. For example, embodiments may be implemented using hardware, software, or a combination thereof. If implemented in software, the software code may be executed on any suitable processor or collection of processors, whether located on a single computer or distributed among multiple computers. Such a processor may be implemented as an integrated circuit including one or more processors on a single integrated circuit component. However, a processor may be implemented using circuitry in any suitable format.

[0117] Additionally, embodiments of the present invention may be implemented as described methods. The actions performed as part of the method may be ordered as appropriate. Thus, embodiments may be constructed that perform actions in an order different from the illustrated order, and may include performing some actions simultaneously even though the illustrated embodiments show sequential actions.

[0118] Furthermore, the use of ordinal numbers such as "first," "second," etc. to modify claim elements in the appended claims does not, by itself, imply a preference, precedence, or order in which one claim element precedes another, nor does it imply a chronological order in which actions within a method should be performed; rather, it merely serves to distinguish claim elements (however ordinal numbers are used) as a label to distinguish one named claim element from another named claim element.

[0119] Although the present disclosure has been described by way of examples of preferred embodiments, it will be understood that various other adaptations and modifications may be made therein without departing from the spirit and scope of the invention.

[0120] Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.

[0121] Various embodiments are suitable for two acquisition modes in inverse scattering: a transmission mode, in which transmitters and receivers are located on opposite sides of a material, and a reflection mode, in which transmitters and receivers are located on the same side of an object. Reflection setups are generally driven by limitations that allow access to only one side of a material, such as in subsurface imaging. While this is effective for some applications, it leads to reflection tomography scenarios where the inverse scattering problem is severely ill-posed because the received reflection contains a much smaller amount of spatial frequency information about the object. Therefore, it is an objective of one embodiment to provide a method and system suitable for reflection tomographic imaging.

Claims

1. 1. A tomographic imaging system comprising: an input interface for receiving measurements at a set of frequencies of a wavefront scattered by an internal structure of the object; a processor configured to recursively reconstruct an image of the internal structure of the object until a termination condition is met, wherein in a current iteration, the processor: and configured to create a current frequency set by adding frequencies to a previous frequency set used in a previous iteration, the added frequencies being higher than one or more frequencies present in the previous frequency set; and the processor further: configured to reconstruct the current image of the internal structure of the object that minimizes the difference between a portion of the scattered wavefront measured at the current frequency set and a wavefront synthesized from a current image, the wavefront synthesized from the current image being generated by a neural network operator, and a previous image determined during the previous iteration initializing the reconstruction of the current image; The tomographic imaging system further comprising an output interface configured to render the reconstructed image.

2. 2. The tomographic imaging system of claim 1, wherein the neural network operator architecture comprises a series of Fourier Neural Operator (FNO) modules with identical parameters that are enhanced by training the neural network operator with machine learning.

3. The processor further comprises:

2. The tomographic imaging system of claim 1, configured to reconstruct the image of the internal structure of the object by solving an optimization problem that minimizes the difference between the measurements of the portion of the scattered wavefront and the composite wavefront.

4. The tomographic imaging system of claim 1 , wherein the reconstructed image is an image of the refractive index of one or more materials within the object.

5. The tomographic imaging system of claim 1 , wherein the reconstructed image comprises a dispersion of the dielectric constant of one or more materials within the object.

6. To reconstruct the current image, the processor further comprises: and configured to simultaneously determine an update of the current image of the internal structure of the object and an estimate of the scattered wavefront inside the object, the estimate of the scattered wavefront having a structure corresponding to the current image at each frequency of the current frequency set, the processor further comprising:

2. The tomographic imaging system of claim 1, configured to minimize the sum of the differences between received reflections and composite reflections of reconstructed scattered wavefronts for each frequency of the current frequency set based on the update of the current image and the estimation of the scattered wavefront.

7. To simultaneously determine the current image of the internal structure of the object and the estimate of the scattered wavefront inside the object, the processor further comprises: determining an estimate of the scattered wavefront based on a current estimate of the image of the internal structure of the object by simulating an interaction of a probing pulse with the scattered wavefront resulting from scattering the probing pulse off one or more materials within the object; determining an estimate of an adjoint scattered wavefront that compensates for residual errors between the received reflections and the composite reflection of the reconstructed scattered wavefront; 7. The tomographic imaging system of claim 6, configured to use an adjoint equation of state to calculate the update of the current image of the internal structure of the object based on the estimate of the scattered wavefront, the estimate of the adjoint scattered wavefront, and the residual error between the received reflection and the composite reflection of the reconstructed scattered wavefront.

8. To simultaneously determine the current image and the scattered wavefront, the processor: forming a Born-Fourier neural operator (FNO) that approximates the interaction of the probing pulse, the scattered wavefront resulting from scattering the probing pulse off the one or more materials within the object, and the refractive index of the one or more materials within the object; inverting the Born FNO for a given initialized current image and determining a Jacobian matrix of the scattered wavefront for the current image based on the refractive indices of the one or more materials; updating the current image based on the refractive index of the one or more materials by minimizing a cost function of the received reflections in the set of frequency components and a synthetic scattered wavefront obtained by either using a stochastic gradient descent method determined by backpropagation of automatically generated gradients or by combining the Jacobian matrix of the scattered wavefront with respect to the image of the refractive index of the one or more materials and a quasi-Newton descent direction of the cost function, or a combination thereof; The tomographic imaging system of claim 7 , configured to project the image of the refractive index of the one or more materials onto a constrained total variation penalty function.

9. 9. The tomographic imaging system of claim 8, wherein the cost function between the received reflections in the set of frequency components and the composite reflection of the reconstructed scattered wavefront comprises one or a combination of a Euclidean distance between each of the received reflections and the corresponding composite reflection, a Wasserstein distance between each of the received reflections and the corresponding composite reflection, a normed distance between each of the received reflections and the corresponding composite reflection, or a sum of a Euclidean distance and a barrier function, wherein the barrier function penalizes updating the image of the refractive index of the one or more materials with a negative refractive index, and the barrier function is a sum of exponential functions taken to negative powers of each refractive index in the image of the refractive index of the one or more materials.

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

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

12. The constrained total variation penalty function is constrained by an upper bound, and the processor further initializing the upper bound for the current image; 9. The tomographic imaging system of claim 8, configured to update the upper bound at the start of every iteration using a Newton root-finding method that adds the ratio of the squared Euclidean distance between the received reflection and the composite reflection of the reconstructed scattered wavefront and a polar function of the constrained total variation function applied to the product of the difference between the received reflection and the composite reflection of the reconstructed scattered wavefront and the adjoint of the Born FNO.

13. 9. The tomographic imaging system of claim 8, wherein the Born-FNO operator approximates the interaction of the probing pulse, the scattered wavefront resulting from scattering the probing pulse off the one or more materials within the object, and the refractive index of the one or more materials within the object, and the structure of the Born-FNO operator is determined by concatenating multiple Fourier neural operator modules into a multi-layer neural network.

14. The tomographic imaging system of claim 1 , wherein the object comprises an element of underground infrastructure.

15. a set of transmitters configured to transmit one or more probing pulses into the object, the one or more probing pulses comprising at least one of electromagnetic waves or acoustic waves occupying a frequency band that includes the set of frequencies; 10. The tomographic imaging system of claim 1, further comprising 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 probing pulses propagating through the object to generate the measurement of the wavefront.

16. 16. The tomographic imaging system of 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 a reflection mode.

17. 1. A tomographic imaging method comprising: receiving measurements at a set of frequencies of a wavefront scattered by an internal structure of the object; and recursively reconstructing an image of the internal structure of the object until a termination condition is met, wherein in a current iteration, the tomographic imaging method comprises: and generating a current frequency set by adding frequencies to a previous frequency set used in a previous iteration, the added frequencies being higher than one or more frequencies present in the previous frequency set, the tomographic imaging method further comprising: reconstructing the current image of the internal structure of the object that minimizes the difference between a portion of the scattered wavefront measured at the current frequency set and a wavefront synthesized from a current image, the wavefront synthesized from the current image being generated by a neural network operator, and a previous image determined during the previous iteration initializing the reconstruction of the current image; A tomographic imaging method further comprising rendering the reconstructed image.

18. 20. The tomographic imaging method of claim 17, wherein the neural network operator architecture comprises a series of Fourier Neural Operator (FNO) modules with identical parameters that are enhanced by training the neural network operator with machine learning.

19. 20. The tomographic imaging method of claim 17, further comprising reconstructing the image of the internal structure of the object by solving an optimization problem that minimizes the difference between the measurements of the portion of the scattered wavefront and the composite wavefront.

20. A non-transitory computer-readable storage medium carrying a program executable by a processor to perform a method, the method comprising: receiving measurements at a set of frequencies of a wavefront scattered by an internal structure of the object; and recursively reconstructing an image of the internal structure of the object until a termination condition is met, wherein in a current iteration, the method comprises: and creating a current frequency set by adding frequencies to a previous frequency set used in a previous iteration, the added frequencies being higher than one or more frequencies present in the previous frequency set, the method further comprising: reconstructing the current image of the internal structure of the object that minimizes a difference between a portion of the scattered wavefront measured at the current frequency set and a wavefront synthesized from a current image, the wavefront synthesized from the current image being generated by a neural network operator, and a previous image determined during the previous iteration initializing the reconstruction of the current image, the method further comprising: a non-transitory computer-readable storage medium comprising: a computer readable storage medium;

Images