Method for object reflectance tomography in inhomogeneous background media.

Abstract

An embodiment of the present disclosure discloses a tomography system for reconstructing an image of an internal structure of an object. An incident wave field is transmitted to an object present in a background domain that embeds the object. The incident wave field is scattered by the object into multiple scattered wave fields. The incident and scattered wave fields are measured as a total wave field. The total wave field propagates through a computational domain and a residual domain within the background domain defined by a cross-domain operator and a residual measurement operator. The total wave field is used for image reconstruction. The image is reconstructed by solving an optimization problem corresponding to the computational domain. The optimization problem is iteratively solved to minimize the difference between the total wave field and a wave field synthesized from the reconstructed image using a measurement operator and a Green's function operator. The reconstructed image is output via an output interface.

Description

[Technical field]

[0001] The present disclosure relates to tomography, and more particularly to a tomography system for reconstructing an image of the internal structure of an object. [Background technology]

[0002] In tomography of an object, it would be desirable for many applications to generate high-resolution images of an object located within a heterogeneous background material. These applications may include microwave imaging, biomicroscopy, medical imaging, through-the-wall imaging (TWI), infrastructure monitoring, and seismic imaging. The image may correspond to an image of the refractive index of the material inside the object. Typically, to generate an image, a wave field, such as an electromagnetic (EM) wave field, light or acoustic pulse, may be transmitted to the object by a transmitter. The wave field may be scattered by various structures inside the object and further propagate to an array of antennas at a receiver. The scattered wave field captured at the receiver can be used to generate an image representing a visualization of the distribution of the dielectric constant of the material of the object. The scattered wave field contains information about the spatial distribution corresponding to the dielectric constant of the object or of a region of interest corresponding to the object. The dielectric constant provides a visualization of the internal structure of the object and a characterization of the physical properties of the object.

[0003] As an example, in microwave imaging, permittivity provides information about the structure and properties of an object. In biomicroscopy, permittivity allows internal cellular structures to be visualized in three dimensions. In TWI, permittivity allows learning about the dielectric properties of walls that can be used to compensate for the delay of signals propagating through the walls.

[0004] However, it may be difficult to generate a high-resolution image of an object, especially from a non-homogeneous background material. For example, the non-homogeneous background material may cause multiple scattering of the wave field. The multiple scattering wave field may result in an inaccurate image of the object due to the small information of the spatial distribution contained therein. In some cases, the image may be generated inaccurately due to position ambiguity in the array of the receiver's antenna. In some cases, the computational domain corresponding to the object in the non-homogeneous background material may also affect the reconstruction of the image. For example, a large size of the computational domain for image reconstruction may increase the computational complexity, which may not be efficient and feasible.

[0005] Therefore, there is a need for a technical solution to overcome the above challenges, and more specifically, to reconstruct images of the internal structure of an object located within a heterogeneous background material in an efficient and feasible manner. Summary of the Invention

[0006] In tomography, images may be affected by spatially dependent absorption and scattering properties of media, particularly inhomogeneous background reflectance media. Objects may be embedded in inhomogeneous background reflectance media (hereafter referred to as inhomogeneous background media). Objects with unknown internal structure may be visualized in a reconstructed image of the object. The reconstructed image may correspond to at least one of an image of the refractive index of materials inside the object and a distribution of the dielectric constant of materials inside the object.

[0007] To reconstruct an image, an incident wave field, such as an electromagnetic or acoustic wave with a finite bandwidth, is transmitted into an inhomogeneous background medium in which an object is embedded. The incident wave field may interact in a complex manner with the material properties of the object and the inhomogeneous background medium. When the incident wave field of an electromagnetic or acoustic wave propagates inside the object and the inhomogeneous background medium, multiple scattered waves are caused by the inhomogeneous background medium on the material boundaries inside the object. In particular, due to the material properties, the multiple waves are scattered in different ways for different frequencies.

[0008] These scattered waves contain information about the spatial distribution of material properties that can be used to reconstruct an image of the internal structure of the object. However, multiple waves that scatter differently for different frequencies due to material properties may provide little spatial frequency information about the object. This may result in an inaccurate internal structure image of an object embedded in a heterogeneous background medium. Multiple scattered waves for different frequencies also result in a nonlinear cost function with multiple local minima, making image reconstruction more complicated.

[0009] Various embodiments can be implemented for two acquisition modes in inverse scattering. The two acquisition modes include a transmission mode and a reflection mode. In the transmission mode, the transmitter and receiver are placed on both sides of the target object, such as an object embedded in a heterogeneous background medium. In the reflection mode, the transmitter and receiver are placed on the same side of the object. The reflection setup is generally caused by the limitation of only being able to access one side of the material, such as in the case of subsurface imaging. In some applications, the reflection setup may be useful. However, in a reflection tomography scenario, the problem of inverse scattering can be unique because the measured reflection may contain very little spatial frequency information about the object. This may result in an inaccurate reconstructed image of the object.

[0010] Therefore, an objective of some embodiments is to reconstruct an image of an unknown internal structure of an object embedded in a heterogeneous background medium. Also, an objective of some embodiments is to reduce the complexity of the image reconstruction. In some embodiments, an image can be reconstructed by formulating a scattering model that describes the relationship between the scattered wave field and the parameters of the heterogeneous background medium. To that end, a wave propagation integral equation is formulated incorporating background knowledge, i.e., knowledge of the heterogeneous background medium. The incorporation of background knowledge can reduce the computational complexity of the image reconstruction by reducing the size of the reconstruction domain. The reconstruction domain for the image reconstruction is the computational domain in which the object in the heterogeneous background medium exists. The computational domain exists in the background domain in the heterogeneous background medium. Therefore, a computational domain smaller than the background domain reduces the computational complexity for the image reconstruction.

[0011] Therefore, some embodiments are based on the realization that an image of an object can be reconstructed by solving an optimization problem that corresponds to the object's computational domain, which is subject to boundary conditions governed by operators. The operators include cross-domain operators and residual measurement operators that define the propagation of the total wave field. The total wave field is a combination of the incident and scattered wave fields received as measurements at a receiver with an array of antennas.

[0012] The cross-domain operator defines the propagation of the total wave field from the computational domain into the residual domain and back from the residual domain into the computational domain. The residual domain is that portion of the inhomogeneous background medium located outside the computational domain that is in the background domain. In particular, the cross-domain operator is a mapping function for the total wave field in the computational domain from each grid point in the computational domain to each corresponding grid point in the residual domain.

[0013] The residual measurement operator defines the propagation of the total wave field from the computational domain into the residual domain and back to the receiver. In particular, the residual measurement operator is a mapping function for the total wave field in the computational domain from each grid point in the computational domain to each position of the corresponding one or more antennas of the receiver.

[0014] In some embodiments, the wave propagation integral equation can be formulated by performing a series of integration operations. The series of integration operations includes a first operation, a second operation, and a third operation. The first operation performs an integration for each grid point in the residual domain over each grid point in the computational domain with a Green's function centered on the grid point in the residual domain. The second operation calculates an updated total wave field resulting from the interaction of the total wave field in the residual domain with a background medium inside the residual domain. The third operation performs an integration for each grid point in the computational domain over each grid point in the residual domain with a Green's function centered on the grid point in the computational domain.

[0015] Some embodiments are based on the understanding that measurements can be managed in the presence of high reflectance characteristics of inhomogeneous background media. Therefore, a reformulated wave propagation integral equation incorporated with background knowledge can be coupled to an incremental frequency inversion framework, which can present image reconstruction as a series of constrained nonlinear least-squares sub-problems.

[0016] The incremental frequency framework estimates a model of the dielectric constant of the object. Estimating the dielectric constant model of the object by the incremental frequency framework eliminates the need for a model for the image of the object to be reconstructed. This model is updated iteratively as the inversion of the measurements from low to high frequencies includes more high frequency measurements. In the inversion, a cost function of the low frequency wave field is used as a regularizer for the high frequency wave field. The iterative model update solves for a solution closer to the global minimum solution.

[0017] In some embodiments, an optimization problem corresponding to the computational domain of the object is transformed into another optimization problem with free-space boundary conditions for image reconstruction. The transformation of the optimization problem with free-space boundary conditions may include combining a background wave field at each grid point in the background domain and a measurement of the background wave field at the receiver resulting from the propagation of the incident wave field into the background domain, and simultaneously calculating a differential wave field with the unknown internal structure of the object by forming a partial differential equation (PDE) operator. The PDE operator discretizes the PDE that describes the interaction between the cross-domain operator, the residual measurement operator, the measurement operator, the Green's function operator, the incident wave field, the background wave field, the differential wave field, and the unknown internal structure of the object.

[0018] The PDE operator can be inverted with respect to the current image of the object using an inversion operation to determine a matrix, such as the Jacobian matrix of the scattered wave field with respect to the current image of the object. The inversion operation can be performed using a preconditioned conjugate gradient method. The preconditioned conjugate gradient method uses a deep learning network to compute a preconditioner for the PDE operator. The deep learning network is trained to determine an estimate of the scattered wave field from the incident wave field for the reconstructed image.

[0019] The current image of the object is updated through an update operation that minimizes a cost function between the measurements of the full wave field and a total wave field synthesized by combining the Jacobian of the scattered wave field and a quasi-Newtonian descent direction of the cost function for the current image of the object. The cost function may include one or a combination of the following: a Euclidean distance between the measurements of the full wave field and the synthesized total wave field, a 1-norm distance between the measurements of the full wave field and the synthesized total wave field, and a sum of the Euclidean distance and a barrier function that is a sum of exponential functions that penalizes updating the current image of the object.

[0020] After updating the current image, the current image of the object is projected onto a constrained total change penalty function that regularizes the updated image so that it remains consistent with the lower frequency measurements.

[0021] Accordingly, one embodiment discloses a tomography system. The tomography system comprises a transmitter configured to transmit an incident wave field to an object, the object being embedded in a heterogeneous reflectance background medium with a known internal structure and existing in a background domain within the heterogeneous reflectance background medium. The tomography system comprises a receiver configured to receive a measurement of the total wave field. The total wave field is a combination of the incident wave field and the wave field scattered by the unknown internal structure of the object and the known internal structure of the background domain. The object exists in a computational domain that is smaller than the background domain that is inside the heterogeneous reflectance background medium. The tomography system comprises a memory configured to store a cross-domain operator and a residual measurement operator. The cross-domain operator defines the propagation of the total wave field from the computational domain of the object with the unknown internal structure of the object into the residual domain and back to the computational domain. The residual domain exists in a portion of the background domain that is located outside the computational domain. The residual measurement operator defines the propagation of the total wave field from the computational domain into the residual domain and back to the receiver. The tomography system includes a processor configured to reconstruct an image of an unknown internal structure of the object from the measurements by solving an optimization problem corresponding to a computational domain of the object subject to boundary conditions of the stored cross-domain operator and residual measurement operator. The optimization problem is iteratively solved to minimize a difference between the measurements of the total wave field and a wave field synthesized from the reconstructed image using the measurement operator and the Green's function operator. The tomography system further includes an output interface configured to render the reconstructed image.

[0022] Accordingly, another embodiment discloses a tomography method, the method being combined with instructions stored therein for implementing the method, the instructions performing the steps of the method when executed by a processor. The method includes transmitting an incident wave field to an object, the object being embedded in a heterogeneous reflectance background medium having a known internal structure and residing in a background domain within the heterogeneous reflectance background medium. The method includes receiving measurements of a total wave field. The total wave field is a combination of the incident wave field and the wave field scattered by the unknown internal structure of the object and the known internal structure of the background domain. The object is residing in a computational domain smaller than the background domain, which is inside the heterogeneous reflectance background medium. The method includes reconstructing an image of the unknown internal structure of the object from the measurements by solving an optimization problem corresponding to the computational domain of the object subject to boundary conditions of a cross-domain operator and a residual measurement operator. The cross-domain operator defines the propagation of the total wave field from the computational domain of the object with the unknown internal structure of the object into the residual domain and back to the computational domain. The residual domain is residing in a portion of the background domain located outside the computational domain. The residual measurement operator defines the propagation of the total wave field from the computational domain into the residual domain and back to the receiver. An optimization problem is iteratively solved to minimize the difference between the measurement of the total wave field and the wave field synthesized from the reconstructed image using the measurement operator and the Green's function operator. The method further includes generating the reconstructed image via an output interface.

[0023] Therefore, another embodiment discloses a non-transitory computer-readable storage medium including a program executable by a processor to perform a method. The method includes transmitting an incident wave field to an object, the object being embedded in a heterogeneous reflectance background medium having a known internal structure and residing in a background domain within the heterogeneous reflectance background medium. The method includes receiving a measurement of the total wave field. The total wave field is a combination of the incident wave field and the wave field scattered by the unknown internal structure of the object and the known internal structure of the background domain. The object is residing in a computational domain smaller than the background domain, which is inside the heterogeneous reflectance background medium. The method includes reconstructing an image of the unknown internal structure of the object from the measurement by solving an optimization problem corresponding to the computational domain of the object subject to boundary conditions of a cross-domain operator and a residual measurement operator. The cross-domain operator defines the propagation of the total wave field from the computational domain of the object having the unknown internal structure of the object into the residual domain and back to the computational domain. The residual domain is residing in a portion of the background domain located outside the computational domain. The residual measurement operator defines the propagation of the total wave field from the computational domain into the residual domain and back to the receiver. An optimization problem is iteratively solved to minimize the difference between the measurement of the total wave field and the wave field synthesized from the reconstructed image using the measurement operator and the Green's function operator. The method further includes generating the reconstructed image via an output interface. [Brief description of the drawings]

[0024] [Figure 1] 1 shows a diagram of a non-uniform reflectance background medium and an object embedded in the non-uniform reflectance background medium, according to some embodiments of the present disclosure. [Diagram 2] FIG. 1 illustrates components of a tomography system according to some embodiments of the present disclosure. [Diagram 3] FIG. 2 illustrates an example of a setup for reconstructing an image of an object by a tomography system according to some embodiments of the present disclosure. [Figure 4]FIG. 2 illustrates a process for rendering a reconstructed image of an internal structure of an object, according to some embodiments of the present disclosure. [Diagram 5] FIG. 1 illustrates an incremental frequency inversion framework according to some embodiments of the present disclosure. [Figure 6] FIG. 1 illustrates a graphical representation of a cost function between total wave field measurements and a combined total wave field according to some embodiments of the present disclosure. [Figure 7A] FIG. 2 illustrates a representation of a reconstructed image of an object embedded in a heterogeneous background medium, according to some embodiments of the present disclosure. [Figure 7B] FIG. 2 illustrates a representation of a synthesized background wave field in the background domain according to some embodiments of the present disclosure. [Figure 8] 1 shows a flow diagram of a method for reconstructing an image of an internal structure of an object according to some embodiments of the present disclosure. [Figure 9] 1 shows a block diagram of a tomography system according to some embodiments of the present disclosure. [Figure 10] FIG. 1 illustrates a use case for implementation of a tomography system according to some embodiments of the present disclosure. [Figure 11] 11A-11C illustrate use cases for implementation of a tomography system according to some other embodiments of the present disclosure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0025] For purposes of explanation, numerous specific details are set forth in the following 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, the apparatus and methods are shown in block diagram form solely to avoid obscuring the present disclosure.

[0026] As used in the specification and claims, the terms "for example," "for instance," and "such as," as well as the verbs "comprising," "having," "including," and other forms of these verbs, 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 further components or items. The term "based on" means based at least in part on. Furthermore, it should be understood that the style and terminology used herein are for purposes of illustration and should not be considered limiting. Any headings used herein are for convenience only and should not be considered to have any legal or limiting effect.

[0027] The present disclosure provides a method and system for tomography of objects in a non-homogeneous reflectance background medium. The method and system provide a computationally efficient solution to the tomography problem by reducing the size of the computational problem associated with the reconstruction of images using techniques for tomography in a non-homogeneous reflectance background medium. The implementation of the efficient solution is described in conjunction with the description provided in the various embodiments below.

[0028] 1 illustrates a diagram of a heterogeneous reflectance background medium 100 and an object 102 embedded in the heterogeneous reflectance background medium 100, according to some embodiments of the present disclosure. The heterogeneous reflectance background medium 100 (hereinafter referred to as heterogeneous background medium 100) may be composed of various materials with high scattering or reflectance properties that may cause poor reconstruction of an image of the unknown internal structure of the object 102. Some examples of the heterogeneous background 100 may include ground components, wall components, etc., in which the object 102, such as a pipeline, may be embedded.

[0029] An image of the unknown internal structure of the object 102 can be reconstructed by a tomography system, which will now be further described with reference to FIG.

[0030] 2 illustrates components of a tomography system 200 according to some embodiments of the present disclosure. The tomography system 200 includes a transmitter 202, a receiver 204, a memory 206, a processor 208, and an output interface 210. The processor 208 is operatively connected to the transmitter 202, the receiver 204, the memory 206, and the output interface 210 via a bus (not shown).

[0031] The transmitter 202 is configured to transmit an incident wave field to an object, such as object 102, embedded in a non-homogeneous reflectivity background medium, such as the non-homogeneous background medium 100. The non-homogeneous background medium 100 has a known internal structure, and the object 102 resides in a background domain within the non-homogeneous background medium 100. The transmitter 202 may be represented as a collection of one or more transmitters in the tomography system 200.

[0032] The receiver 204 is configured to receive a measurement of the total wave field. The total wave field is a combination of the incident wave field and the wave field scattered by the unknown internal structure of the object 102 and the known internal structure of the background domain in the inhomogeneous background medium 100. The object 102 is in a computation domain, which is smaller than the background domain and is inside the background domain of the inhomogeneous background medium 100. The receiver 204 may be represented as a collection of one or more receivers of the tomography system 200. The one or more receivers may correspond to an array of antennas. In some example embodiments, the transmitter 202 and the receiver 204 may be embodied as a transceiver that propagates waves through the object 102 embedded in the inhomogeneous background medium 100 and receives scattered waves of the waves propagated by the material of the object 102 and the material of the inhomogeneous background medium 100.

[0033] The memory 206 is configured to store a cross-domain operator 206A and a residual measurement operator 206B. The cross-domain operator 206A is an operator that defines the propagation of the total wave field from the computational domain of the object 102 with the unknown internal structure of the object 102 into the residual domain and back to the computational domain. The residual domain is in a part of the background domain that is located outside the computational domain. The residual measurement operator 206B is an operator that defines the propagation of the total wave field from the computational domain into the residual domain and back to the receiver 204.

[0034] The processor 208 is configured to reconstruct an image of the unknown internal structure of the object 102 from the measurements by solving an optimization problem corresponding to the computational domain of the object 102 subject to the stored boundary conditions of the cross-domain operator 206A and the residual measurement operator 206B. The optimization problem is solved iteratively to minimize the difference between the measurements of the total wave field and the wave field synthesized from the reconstructed image using the measurement operator and the Green's function operator. The reconstructed image is output via the output interface 210.

[0035] In some example embodiments, the tomography system 200 may be set based on a transmission mode. In the transmission mode, the transmitter 202 and the receiver 204 may be disposed on both sides of the inhomogeneous background medium 100. In some cases, there may be a limitation that only one side of the inhomogeneous background medium 100 can be accessed. As an example, applications such as ground penetrating radar and seismic imaging may have a limitation that only one side of the inhomogeneous background medium 100 can be accessed. In such a case, the tomography system 200 may be set based on a reflection mode. In the reflection mode, the transmitter 202 and the receiver 204 may be disposed on the same side of the inhomogeneous background medium 100. Such a setting of the tomography system 200 based on the reflection mode for reconstructing an image of the object 102 is shown in FIG. 3.

[0036] 3 illustrates an example setup 300 for reconstructing an image 302 of an object 102 by a tomography system 200 according to some embodiments of the present disclosure. As shown in Fig. 3, the tomography system 200 is arranged in a reflection mode, with the transmitter 202 and the receiver 204 located on one side of the inhomogeneous background medium 100, i.e., above the ground in which the object 102 is embedded. The reflection mode estimates the spatial distribution of the dielectric constant of the object 102.

[0037] The transmitter 202 transmits an incident wave field 304 through the inhomogeneous background medium 100. The incident wave field 304 may include electromagnetic, acoustic, or light waves, such as one or a combination of microwave, radar, laser, ultrasound, and acoustic pulses. The incident wave field 304 propagates through the inhomogeneous background medium 100 and passes through the object 102. The incident wave field 304 is scattered by the object 102 and due to the reflectivity properties of the inhomogeneous background medium 100 into a plurality of wave fields 306, hereinafter referred to as scattered wave fields 306.

[0038] In some example embodiments, the receiver 204 may be located at a predetermined position relative to the transmitter 202 to receive the scattered wave field 306. The receiver 204 measures the incident wave field 304 and the scattered wave field 306 as a total wave field. To do so, the receiver 204 may measure only the total wave field. The scattered wave field 306 is calculated by subtracting the incident wave field 304 (which is generally known) from the measured total wave field. The total wave field propagates through different portions of the material, including the object 102, in the inhomogeneous background medium 100. The different portions include a background domain 308, a calculation domain 310, and a residual domain 312 (shown with diagonal lines). The background domain 308 is a region of interest or spatial domain that is large enough to attenuate the incident wave field 304 propagating between the transmitter 202 and the receiver 204. As such, the background domain 308 can be considered to be configured to attenuate the incident wave field 304 propagating between the transmitter 202 and the receiver 204. As the incident wave field 304 propagates through the background domain 308, there is a small effect on measurements of the total wave field from the propagation of the total wave field outside the background domain 308. Thus, measurements from the propagation of the wave field outside the background domain are not affected. The computational domain 310 is a smaller region within the background domain 308 in which the object 102 is located. The residual domain 312 is the portion of the background domain that is outside the computational domain 310, shown as shaded in FIG. 3 .

[0039] The propagation of the full wave field from the computational domain 310 into the residual domain 312, and from the residual domain 312 back into the computational domain 310, is defined by the cross-domain operator 206A. The cross-domain operator 206A corresponds to a mapping function for the full wave field that maps each grid point in the computational domain 310 to each corresponding grid point in the residual domain 312.

[0040] The total wave field also propagates from the computational domain 310 into the residual domain 312 and back to the receiver 204. The propagation of the total wave field from the computational domain 310, the residual domain 312 to the receiver 204 is defined by the residual measurement operator 206B. The residual measurement operator 206B is a mapping function of the total wave field in the computational domain that maps from each grid point in the computational domain to each position of the corresponding one or more antennas in the receiver 204.

[0041] The receiver 204 transmits the measurements of the total wave field to a processor 208 which reconstructs an image 302 of the internal structure of the object 102. The reconstructed image 302 represents a visualization of the internal structure of the object 102, which may be a two-dimensional (2D) or three-dimensional (3D) representation. The reconstructed image 302 may include information of the refractive index of materials within the object 102, the distribution of the dielectric constant of materials within the object 102, and combinations thereof. Each location, such as a pixel value, in the reconstructed image 302 may provide a dielectric constant value for the portion of the material of the object 102 that corresponds to that location.

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[0043] As such, the image 302 is reconstructed by solving an optimization problem corresponding to the computational domain 310. The optimization problem of the computational domain 310 is solved using the boundary conditions of the cross-domain operator 206A and the residual measure operator 206B. The optimization problem is solved iteratively, which will be further explained with reference to FIG.

[0044] 4 illustrates a process 400 for rendering a reconstructed image 302 of an internal structure of an object 102, according to some embodiments of the present disclosure. The process 400 is performed by the processor 208 of the tomography system 200. The process 400 begins with step 402 of transmitting an incident wave field 304 to the object 102 embedded in the inhomogeneous background medium 100. As the incident wave field 304 propagates through the object 102, a scattered wave field 306 is reflected by the material properties of the object 102. These transmitted incident wave field 304 and scattered wave fields 306 are measured by the receiver 204 as a total wave field measurement in step 404.

[0045] In step 406, the wave propagation integral equation is formulated through a series of integration operations by processing all the measurements. The series of operations includes a first operation of performing an integration with a Green's function for each grid point in the residual domain 312. The Green's function can be centered on a grid point in the residual domain 312 over each grid point in the computational domain 308. A second operation then calculates an updated total wave field resulting from the interaction of the total wave field in the residual domain 312 with the material properties of the inhomogeneous background medium 100 inside the residual domain 312. In a third operation, another integration operation is performed. This integration operation is performed with a Green's function for each grid point in the computational domain 308. The Green's function can be centered on a grid point in the computational domain 308 over each grid point in the residual domain 312.

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[0061] In step 408, the image 302 is reconstructed by solving an optimization problem corresponding to the computational domain 310 according to equation (12). The optimization problem corresponding to the computational domain 310 can be solved based on the inverse problem described in FIG.

[0062] At step 410, the reconstructed image 302 is output via the output interface 210. In some example embodiments, the reconstructed image 302 may correspond to an image of the refractive index of materials within the object. In some other example embodiments, the reconstructed image 302 may include a distribution of the dielectric constant of materials within the object.

[0063] In some embodiments, the reformulated wave propagation integral equation embedded in the parameters of the background domain can be coupled to an incremental frequency inversion framework that iteratively solves an optimization problem for the reconstruction of the image 302, which will now be further described with reference to FIG.

[0064] 5 illustrates an incremental frequency inversion framework 500 of the tomography system 200 for the image reconstruction 408 step of the method 400 for the object 102, according to some embodiments of the present disclosure. In the incremental frequency inversion framework 500, a measured total wave field 502 and a synthesized background wave field 504 are compared. The measured total wave field 502 corresponds to a total wave field measurement including the incident wave field 304 and the scattered wave field 306. The synthesized background wave field 504 is obtained by transformation of an optimization problem with free-space boundary conditions, such as the scalar Lippmann-Schwinger equation (Equation (4)).

[0065] A wave field difference 506 is calculated by comparing the measured total wave field 502 with the synthesized background wave field 504. The wave field difference 506 is calculated together with the unknown internal structure of the object 102 by forming a PDE operator that discretizes the PDEs that describe the interactions between the cross-domain operator 206A, the residual measurement operator 206B, the measurement operator, the Green's function operator, the incident wave field 304, the synthesized background wave field 504, the difference wave field 506, and the unknown internal structure of the object 102.

[0066] After calculating the wavefield difference 506, an inversion operation 508 is performed by the processor 208. The inversion operation 508 solves the PDE constrained optimization problem of equation (12). The inversion operation 508 solves the optimization problem of equation (12) by inverting the PDE operator for the current image 510, i.e., for a model of the internal structure of the object 102. In some embodiments, the current image 510 may be initialized by minimizing the difference between the portion of the scattered wavefield 306 measured at the current frequency content (e.g., 10 MHz, or 10-20 MHz) and the wavefield synthesized from the current image 510.

[0067] Additionally, the PDE operator is inverted to find a Jacobian, which is the determinant of the matrix of the scattered wave field 306 with respect to the current image 510 of the internal structure of the object 102. This matrix includes the first partial derivatives of the PDE. In some embodiments, the inversion operation to invert the PDE operator is performed using a preconditioned conjugate gradient method, which computes a preconditioner for the PDE operator using a deep learning network trained to find an estimate of the scattered wave field from the incident wave field 304.

[0068] In some example embodiments, the deep learning network may include a U-net architecture. The U-net architecture has a U-shaped structure that skips connections between layers of the same dimension in the deep learning network. The skipped connections allow the deep learning network to compute the preconditioner of the PDE operator in an efficient and robust manner. Therefore, the preconditioner can be computed in the presence of noise. In some example embodiments, the deep neural network-based preconditioner may be trained for a particular dimension of an image (e.g., image 302) of an object (e.g., object 102) and a particular configuration of the transmitter 202 and receiver 204. In addition, the deep learning network may be trained to approximate the inverse of the PDE operator from the incident wave field 304, apply the adjoint of the PDE operator defined assuming an arbitrary image of the material of the object 102, and output an estimate of the scattered wave field 306.

[0069] Furthermore, the incremental frequency inversion framework 500 iteratively solves an optimization problem (i.e., equation (11)) for image reconstruction of the object 102. Therefore, the image 302 may be reconstructed as a series of constrained nonlinear least-squares sub-problems. Therefore, the frequency components (n f The total wave field measurement measured by the receiver 204, including f The frequency components are indexed in increasing order, i.e., from low to high frequencies. For example, f ) index may vary from a 10 MHz index to a 50 MHz index.

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[0071] The update operation 512 updates the current image 510 of the object 102 by minimizing a cost function between the measured total wave field 502 and the synthesized total wave field. The cost function between the measurements of the total wave field 502 and the synthesized total wave field may include one of the following: a Euclidean distance between the total wave field 502 and the synthesized total wave field, a 1-norm distance between the total wave field 502 and the synthesized total wave field, and a combination of a sum of the Euclidean distance and a barrier function. The barrier function is a sum of exponential functions that penalize the update operation 512 of the current image 510.

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[0073] In some example embodiments, the gradient of the cost function may be calculated using the adjoint-state method, which calculates the gradient of the frequency components (n f ) simultaneously estimates the current image 510 and the scattered wave field 306 at each frequency. After computing the gradients, the quasi-Newton descent directions are obtained by forming an approximation to the Hessian matrix using the limited-memory Broydon-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The Hessian matrix is ​​the matrix of a quadratic PDE of the cost function, and the limited-memory BFGS algorithm iteratively solves the optimization problem of Equation (13).

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[0075] Therefore, the updated current image 510 is projected onto a constrained total-change penalty function 514. The total-change penalty function 514 uses the total-change norm of the gradients for regularization of the update operation 508.

[0076] For each frequency batch in equation (13), the current image 510 is updated until a termination condition 516 is met. In some example embodiments, the termination condition 516 may be met when the total change norm of the gradients decreases to a small value, which triggers an update termination operation 518.

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[0080] As a result, regularization with the total variation penalty function 514 promotes a piecewise constant approximation of a true model of the object 102 , such as a reconstructed image 520 of the object 102 .

[0081] A cost function showing the difference between the measured total wave field 502 and the synthesized total wave field at different frequencies is shown in FIG.

[0082] FIG. 6 illustrates the frequency content of the wave field (n f 6 shows a graphical representation 600 of a cost function corresponding to the frequency components (n f ) range from 10 MHz index to 50 MHz index, and the reflectance parameter (c) ranges from 0 to 120. The cost function is the difference between a measurement of the total wave field (e.g., measured total wave field 502) and a synthesized total wave field obtained from a model of the object 102, e.g., from the current image 510. The frequency components (n f) include a cost function 602 for a frequency of 10 MHz, a cost function 604 for a joint frequency of 10 to 20 MHz, a cost function 606 for a joint frequency of 10 to 30 MHz, a cost function 608 for a joint frequency of 10 to 40 MHz, and a cost function 610 for a joint frequency of 10 to 50 MHz.

[0083] Using the incremental frequency inversion framework 500, a cost function for reconstructing an image of the object 102 is iteratively minimized by the processor 208 according to equation (12). As an example, in a first iteration, the cost function 602 is minimized for the reconstruction of an image, such as the current image 510 of the object 102. The reconstruction of the image 510 based on minimizing the cost function at a frequency of 10 MHz results in a global minimum 612. The global minimum 612 guides the subsequent minimization of cost functions 604, 606, 608, and 610 in finding corresponding solutions in the vicinity of the global minimum 612.

[0084] Therefore, in the next iteration, i.e., the second iteration, a frequency of 20 MHz (higher than the frequency of 10 MHz) is added to minimize the cost function 604 in the vicinity of the global minimum 612 corresponding to the cost function 602. Similarly, in the third iteration, a frequency of 30 MHz is added and the cost function 606 for frequencies of 10, 20, and 30 MHz is minimized in the vicinity of the minimum corresponding to the cost function 604. In the fourth iteration, a frequency of 40 MHz is added and the cost function 608 for the joint frequencies of 10, 20, 30, and 40 MHz is minimized in the vicinity of the minimum corresponding to the cost function 606. In the fifth iteration, a frequency of 50 MHz is added and the cost function 610 for frequencies of 10, 20, 30, 40, and 50 MHz is minimized in the vicinity of the minimum corresponding to the cost function 608. The minimization of the cost function 610 outputs a global minimum 610A and a local minimum 610B.

[0085] An example of a reconstructed image 520 of the object 102 using the incremental frequency inversion framework 500 is shown in FIG. 7A.

[0086] 7A shows a representation 700A of a reconstructed image 702 of an object 102 embedded in a heterogeneous background medium 100 according to some embodiments of the present disclosure. The reconstructed image 702 is an example of an image 520 rendered by the tomography system 200 using the incremental frequency inversion framework 500. The reconstructed image 702 visualizes the distribution of dielectric constant within the object 102, which describes the internal structure of the object 102, as well as features of the physical properties of the object 102.

[0087] As mentioned above, during the transformation of the optimization problem to correspond to the free space condition, a background wave field is synthesized, such as background wave field 504. Such a background wave field 504 is shown in FIG.

[0088] FIG. 7B illustrates a representation 700B of a total target scene structure 704 in which an object is embedded in a heterogeneous background medium, according to some embodiments of the present disclosure. The total target scene structure 704 may be reconstructed based on the iterative updates of the incremental frequency inversion framework 500. As an example, the reconstructed scene structure may be updated as a synthesized scene structure 706 when the frequency is incremented to 10 MHz. As a result, a synthesized scene structure 708 may be obtained when the frequency is incremented from 10 MHz to 100 MHz. Similarly, a synthesized scene structure 710 may be obtained when the frequency is incremented from 10 MHz to 800 MHz. The update of the synthesized scene structure 704 may be terminated when a termination condition 516 is met.

[0089] 8 illustrates a flow diagram of a method 800 for reconstructing an image (e.g., image 700A) of an object (e.g., object 102) embedded in a heterogeneous background medium (e.g., heterogeneous background medium 100) according to some embodiments of the present disclosure. In some example embodiments, method 800 is performed by processor 208 of tomography system 200. Method 800 begins at step 802 and ends at step 808.

[0090] At step 802, method 800 includes transmitting an incident wave field (e.g., incident wave field 304) to an object (e.g., object 102) embedded in an inhomogeneous reflectance background medium (e.g., inhomogeneous background medium 100) having a known internal structure and residing in a background domain (e.g., background domain 308) within the inhomogeneous reflectance background medium. The background domain is large enough to attenuate the incident wave field propagating between the transmitter and receiver through the background domain, resulting in a negligible effect on measurements of the full wave field from propagation of the full wave field outside of the background domain.

[0091] In step 804, the method 800 includes receiving a measurement of a total wave field. The total wave field is a combination of the incident wave field and the wave field scattered by the unknown internal structure of the object and the known internal structure of the background domain. The object is in a computational domain (e.g., computational domain 310) that is inside and smaller than the background domain of a heterogeneous reflectance background medium.

[0092] In step 806, the method 800 includes reconstructing an image of the unknown internal structure of the object from the measurements by solving an optimization problem corresponding to the computational domain of the object subject to the boundary conditions of a cross-domain operator (e.g., cross-domain operator 206A) and a residual measurement operator (e.g., residual measurement operator 206B). The cross-domain operator defines the propagation of the full wave field from the computational domain of the object with the unknown internal structure of the object into the residual domain (e.g., residual domain 312) and back to the computational domain. The cross-domain operator is a mapping function of the full wave field in the computational domain from each grid point in the computational domain to each corresponding grid point in the residual domain. The residual domain exists in a portion of the background domain located outside the computational domain. The residual measurement operator defines the propagation of the full wave field from the computational domain into the residual domain and back to the receiver. The optimization problem is iteratively solved to minimize the difference between the measurement of the total wave field (e.g., total wave field 502) and the measurement of the synthesized wave field (e.g., synthesized background wave field 504) from the reconstructed image using a measurement operator and a Green's function operator. The residual measurement operator is a mapping function of the total wave field in the computational domain from each grid point in the computational domain to each position of the corresponding one or more antennas of the receiver. In some embodiments, the optimization problem corresponding to the computational domain of the object, which is subject to the boundary conditions of the cross-domain operator and the residual measurement operator, is transformed into another optimization problem with free-space boundary conditions (i.e., equation (4)). The transformation of the optimization problem into an optimization problem with free-space boundary conditions includes combining the background wave field in the background domain (e.g., synthesized background wave field 504) and the measurement of the background wave field 504 at the receiver. This transformation also includes calculating the differential wave field (e.g., wave field difference 506) with the unknown internal structure of the object by forming a difference PDE operator. The PDE operator discretizes a PDE that describes the interactions between operators such as cross-domain operators, residual measurement operators, measurement operators, Green's function operators, wave fields such as incident wave field, background wave field, difference wave field, and the unknown internal structure of the object.The PDE operator for the current image of the object (e.g., current image 510) is inverted by an inversion operation (e.g., inversion operation 508) to find the Jacobian of the scattered wave field for the current image of the object. After the inversion operation, an update operation (e.g., update operation 512) is performed to update the current image. The update operation is performed until an ending condition (e.g., ending condition 516) is met. The updated current image of the object is then projected onto a constrained total change penalty function (e.g., constrained total change penalty function 514).

[0093] At step 808, method 800 includes generating a reconstructed image, such as reconstructed image 702, via an output interface (e.g., output interface 210). In some example embodiments, the reconstructed image may include an image of the refractive index of materials within object 102, a distribution of dielectric constants of materials within object 102, and combinations thereof.

[0094] 9 illustrates a block diagram of a tomography system 900 according to some embodiments of the present disclosure. The tomography system 900 corresponds to the tomography system 200. The tomography system 900 includes a processor 902 configured to execute stored instructions for image reconstruction of the object 102. The processor 902 may be a single-core processor, a multi-core processor, a computing cluster, or any number of other configurations. The processor 902 may be connected to one or more input and output devices via a bus 930.

[0095] In some embodiments, instructions for image reconstruction may be stored in memory 904. Memory 904 may include random access memory (RAM), read only memory (ROM), flash memory, or any other suitable memory system. In some example embodiments, the instructions may include instructions for executing the incremental frequency inversion framework 500. The instructions may also include a trained deep learning module for determining frequency components of the incremental frequency inversion framework 500 and a preconditioner for the frequency inversion problem.

[0096] The tomography system 900 may include several interfaces for connecting to other systems and devices. As an example, the tomography system 900 includes a network interface controller (NIC) 924. The NIC 924 is adapted to connect the tomography system 900 to a network 920 via a bus 930 to connect the tomography system 900 to a sensing device. For example, the tomography system 900 includes a transmitter interface 908 configured to instruct a transmitter 910 to transmit an incident wave field, such as the incident wave field 304. Using a receiver interface 916 connected to a receiver 918, the tomography system 900 can receive a scattered wave field, such as the scattered wave field 306 corresponding to the transmitted incident wave field 304. In some implementations, the tomography system 900 receives measurements of a total wave field 932 for the incident wave field 304 and the scattered wave field 306 via the network 920.

[0097] The tomography system 900 also includes an output interface 912 configured to output the reconstructed image 702. For example, the output interface 912 can display the reconstructed image 702 on a display device, store the image 702 on a storage medium, and / or transmit the image 702 over a network 920. For example, the tomography system 900 can be coupled via a bus 930 to a display interface adapted to connect the tomography system 900 to an output device 914. The output device 914 can include, but is not limited to, a computer monitor, a camera, a television, a projector, a mobile device, a tablet, a phablet, and the like. The tomography system 900 can also be connected to an application interface adapted to connect the tomography system 900 to equipment for performing various tasks.

[0098] In some implementations, the tomography system 900 includes an input interface for receiving measurements at frequencies (e.g., indexed frequency components in the range of 10 MHz to 50 MHz) of the wave field (e.g., the scattered wave field 306) scattered by the internal structure of the object 102. Examples of input interfaces include a NIC 924, a receiver interface 916, and a human machine interface (HMI) 922. The human machine interface 922 in the tomography system 900 connects to a keyboard 926 and a pointing device 928. The pointing device 928 may include a mouse, a trackball, a touchpad, a joystick, a pointing stick, a stylus, or a touch screen, among others.

[0099] FIG. 10 illustrates a use case 1000 for implementation of the tomography system 900 according to some embodiments of the present disclosure. In one example scenario, the tomography system 900 can be used for underground or subsurface imaging by a remote sensing device 1002. The remote sensing device 1002 may hover over an area of ​​interest on the ground for imaging. The remote sensing device 1002 may include a transceiver or sensor for transmitting waves (e.g., electromagnetic waves) that propagate through the ground. The transmitted waves may interact with objects buried in the ground. The objects may include pipelines, beams as structures, wirelines, etc. that can scatter the waves transmitted from the sensor of the remote sensing device 1002. These scattered waves are received by the remote sensing device 1002 via a sensor. The remote sensing device 1002 may further transmit the scattered waves to the tomography system 900 via a network 1006. The network 1006 is one example of a network 920. The tomography system 900 processes the scattered and transmitted waves to return an accurate reconstructed image of the internal structure of the embedded object, making the internal structure visible.

[0100] The tomography system 900 synthesizes a background wave field for image reconstruction by solving an optimization problem using free-space boundary conditions that incorporate knowledge corresponding to parameters such as ground reflectivity, dielectric constant, etc., so that the reconstructed image is accurate. In addition, the tomography system 900 iteratively reconstructs the image, resulting in an accurate reconstructed image of the internal structure of the buried object. The tomography system 900 also returns the reconstructed image within a feasible time, since the image is reconstructed in a reconstruction domain (i.e., the computational domain 310) that is reduced in size from the actual background space in the earth.

[0101] In some cases, a suspicious object may be buried underground. By way of example, the buried object may include an explosive. Such an explosive may be detected by the remote sensing device 1002 using the tomography system 900.

[0102] Similarly, in another example scenario, the tomography system 900 may be implemented in a vehicle navigation application. The vehicle 1004 may correspond to an autonomous vehicle, a semi-autonomous vehicle, or a manually driven vehicle. The vehicle 1004 may be driven on a road under hazardous weather conditions such as fog, storm, rain, etc. The vehicle 1004 may have difficulty navigating a road when the hazardous weather conditions may affect the visibility of the road. In such a scenario, the vehicle 1004 may be more likely to be placed in an undesirable situation. As an example, the vehicle 1004 may collide with another vehicle or fall into a pothole. Therefore, the vehicle 1004 may be assisted in road navigation during hazardous weather conditions by the tomography system 900 for detecting obstacles such as other vehicles on the same road, potholes, etc. In some example embodiments, the tomography system 900 may be realized within the vehicle 1004. In some other example embodiments, the tomography system 900 may be remotely connected to the vehicle 1004 via the network 1006. The vehicle 1004 may transmit waves using sensors in the vehicle 1006. The sensors receive scattered waves from the road and obstacles. The transmitted and scattered waves are processed by the tomography system 900 to render a reconstructed image corresponding to the obstacle. The reconstructed image may be transmitted to an object detection device in the vehicle 1004. The object detection device in the vehicle 1004 may process this image and return a command to alert an operator of the vehicle 1004. As an example, the command may be a voice or text command such as "hole detected" to alert a driver or passenger in the vehicle 1004.

[0103] The tomography system 900 may also be implemented in medical diagnostic applications, cell research applications, etc. One use case for such an application is now described in FIG.

[0104] FIG. 11 illustrates a use case 1100 for implementation of the tomography system 900, according to some embodiments of the present disclosure. In the use case 1100, the tomography system 900 is used to reconstruct an image of cells submerged in a fluid contained in a cell culture dish 1102. The tomography system 900 may be implemented in an electron microscope 1104. Alternatively, the tomography system 900 may be connected to the electron microscope 1104 via a network, such as the network 920. The electronic device 1104 may emit electromagnetic waves or pulses into the cell culture dish 1102. The fluid and submerged cells in the cell culture dish 1102 may scatter the emitted electromagnetic waves. The scattered and emitted electromagnetic waves are processed by the tomography system 900 to render a reconstructed image of the cells. The reconstructed image may be displayed on the device 1106.

[0105] The above-described embodiments of the present disclosure may be implemented in any of numerous ways. For example, the 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 across multiple computers. Such a processor may be implemented as an integrated circuit, with one or more processors being components of the integrated circuit. However, the processor may be implemented using circuits in any suitable format.

[0106] Also, embodiments of the present disclosure may be implemented as a method, an example of which is provided. The order of operations performed as part of the method may be determined in any suitable manner. Thus, embodiments may be configured to perform operations in an order different from that illustrated, which may include performing some operations simultaneously, although in the illustrated embodiment they are shown as a sequence of operations.

[0107] Although the disclosure has been described with examples of preferred embodiments, it will be understood that various other adaptations and modifications can be made within the spirit and scope of the disclosure. Therefore, it is the object of the appended claims to cover all such variations and modifications that come within the true spirit and scope of the disclosure.

Claims

1. 1. A tomography system for reconstructing an image of an internal structure of an object, the tomography system comprising: a transmitter configured to transmit an incident wave field to an object embedded in a heterogeneous reflectance background medium having a known internal structure, the object residing in a background domain within the heterogeneous reflectance background medium, the tomography system further comprising: a receiver configured to receive a measurement of a total wave field, the total wave field being a combination of the incident wave field and a wave field scattered by an unknown internal structure of the object and the known internal structure of the background domain, the object existing in a computational domain smaller than the background domain within the heterogeneous reflectivity background medium, the tomography system further comprising: a memory configured to store a cross-domain operator and a residual measurement operator, the cross-domain operator defining a propagation of the total wave field from the computational domain of the object having the unknown internal structure of the object into a residual domain and back to the computational domain, the residual domain being in a portion of the background domain located outside the computational domain, the residual measurement operator defining a propagation of the total wave field from the computational domain into the residual domain and back to the receiver, the tomography system further comprising: a processor configured to reconstruct an image of the unknown internal structure of the object from the measurements by solving an optimization problem corresponding to the computational domain of the object subject to stored boundary conditions of the cross-domain operator and the residual measurement operator, and iteratively solving the optimization problem to minimize a difference between the measurements of the total wave field and a wave field synthesized from the reconstructed image using a measurement operator and a Green's function operator, and the tomography system further comprises: A tomography system comprising an output interface configured to render the reconstructed image.

2. 2. The tomography system of claim 1, wherein the background domain is configured to attenuate a wave field propagating between the transmitter and the receiver through the background domain such that measurements of the total wave field are unaffected from propagation of the total wave field outside of the background domain.

3. The cross-domain operator comprises a mapping function for the entire wave field in the computational domain, the mapping function comprising a mapping from each grid point in the computational domain to each corresponding grid point in the residual domain, and performing a series of integration operations, the series of integration operations comprising: a first operation of performing, for each grid point in the residual domain, an integration over each grid point in the computational domain using a Green's function centered at the grid point in the residual domain; a second operation of calculating an updated total wave field resulting from the interaction of the total wave field in the residual domain with the background medium in the residual domain; and a third operation of performing, for each grid point in the computational domain, an integration over each grid point in the residual domain using a Green's function centered at the grid point in the computational domain.

4. The residual measurement operator comprises a mapping function for the total wave field in the computational domain, the mapping function comprising a mapping from each grid point in the computational domain to each position of a corresponding array of antennas of the receiver, and performs a series of integration operations, the series of integration operations comprising: a first operation of performing, for each grid point in the residual domain, an integration over each grid point in the computational domain using a Green's function centered at the grid point in the residual domain; a second operation of calculating an updated total wave field resulting from the interaction of the total wave field in the residual domain with the background medium in the residual domain; and a third operation of performing, for each corresponding antenna array position of the receiver, an integration over each grid point in the residual domain using a Green's function centered at the corresponding antenna array position.

5. The optimization problem for the computational domain subject to the boundary conditions governed by the stored cross-domain operator and the residual measurement operator is transformed into another optimization problem having free-space boundary conditions that is solved to reconstruct an image of the unknown internal structure of the object, the transformation further comprising: combining a background wave field at each grid point in the background domain with a measurement of the background wave field at the receiver resulting from propagation of the incident wave field into the background domain having the known internal structure of the inhomogeneous reflectivity background medium; 2. The tomography system of claim 1, further comprising: forming a differential partial differential equation (PDE) operator that discretizes a PDE that describes the interaction of the cross-domain operator, the residual measurement operator, the measurement operator, the Green's function operator, the incident wave field, the background wave field, the differential wave field, and the unknown internal structure of the object, thereby simultaneously calculating the differential wave field along with the unknown internal structure of the object.

6. The optimization problem to be solved to reconstruct an image of the unknown internal structure of the object further comprises: an inversion operation to determine a Jacobian of the scattered wave field for the current image of the internal structure of the object by inverting the PDE operator for the initialized current image of the internal structure of the object; an update operation, updating the current image of the internal structure of the object by minimizing the cost function between the measurements of the total wave field and a total wave field synthesized by combining the Jacobian of the scattered wave field and a quasi-Newton descent direction of the cost function for the image of the internal structure of the object; and projecting the current image of the internal structure of the object onto a constrained total change penalty function.

7. The cost function between the total wave field measurements and the combined total wave field is: the Euclidean distance between all measurements of the total wave field and the combined total wave field; the 1-norm distance between the total wave field measurement and the combined total wave field; and 7. The tomography system of claim 6, further comprising one or a combination of the Euclidean distance and a barrier function, the barrier function being a sum of exponential functions that penalizes updating the current image of the internal structure of the object.

8. 7. The tomography system of claim 6, wherein the inversion operation of inverting the PDE operator is performed using a preconditioned conjugate gradient method, and a preconditioner for the PDE operator is computed using a deep learning network, and the deep learning network is trained to determine an estimate of the wave field scattered from the incident wave field for the reconstructed image.

9. The tomography system of claim 1 , wherein the reconstructed image is an image of the refractive index of a material within the object.

10. The tomography system of claim 1 , wherein the reconstructed image comprises a distribution of dielectric constants of materials within the object.

11. 1. A tomography method for reconstructing an image of an internal structure of an object, the method using a processor in combination with stored instructions for implementing the method, the instructions performing steps of the method when executed by the processor, the method comprising: transmitting an incident wave field to an object, the object being embedded in a heterogeneous reflectance background medium having a known internal structure, the object residing in a background domain within the heterogeneous reflectance background medium, the method further comprising: receiving a measurement of a total wave field, the total wave field being a combination of the incident wave field and a wave field scattered by an unknown internal structure of the object and the known internal structure of the background domain, the object being within the heterogeneous reflectance background medium in a computational domain smaller than the background domain, the method steps further comprising: reconstructing an image of the unknown internal structure of the object from the measurements by solving an optimization problem corresponding to the computational domain of the object subject to boundary conditions of a stored cross-domain operator and a residual measurement operator, minimizing a difference between measurements of the total wave field and a wave field synthesized from the reconstructed image using a measurement operator and a Green's function operator by iteratively solving the optimization problem, the method further comprising: generating the reconstructed image via an output interface.

12. The tomography method of claim 11 , wherein the background domain is configured to attenuate a wave field propagating between a transmitter and a receiver through the background domain such that the measurements are unaffected from propagation of the wave field outside the background domain.

13. and executing a mapping function that maps the total wave field in the computational domain from each grid point in the computational domain to a corresponding grid point in a residual domain using a cross-domain operator, the mapping function performing a series of integration operations, the series of integration operations comprising: a first operation of performing, for each grid point in the residual domain, an integration over each grid point in the computational domain using a Green's function centered at the grid point in the residual domain; a second operation of calculating an updated total wave field resulting from the interaction of the total wave field in the residual domain with the background medium in the residual domain; and a third operation of performing, for each grid point in the computational domain, an integration over each grid point in the residual domain using a Green's function centered at the grid point in the computational domain.

14. and executing a mapping function using a residual measurement operator to map the total wave field in the computational domain from each grid point in the computational domain to each location of an array of corresponding antennas of a receiver, the mapping function performing a series of integration operations, the series of integration operations comprising: a first operation of performing, for each grid point in a residual domain, an integration over each grid point in the computational domain using a Green's function centered at the grid point in the residual domain; a second operation of calculating an updated total wave field resulting from the interaction of the total wave field in the residual domain with the background medium in the residual domain; and a third operation of performing, for each corresponding antenna array position of the receiver, an integration over each grid point in the residual domain using a Green's function centered at the corresponding antenna array position.

15. The optimization problem for the computational domain subject to the boundary conditions governed by the stored cross-domain operator and the residual measurement operator is transformed into another optimization problem having free-space boundary conditions that is solved to reconstruct an image of the unknown internal structure of the object, the transformation further comprising: combining a background wave field at each grid point in the background domain with a measurement of the background wave field at a receiver resulting from propagation of the incident wave field into the background domain having the known internal structure of the inhomogeneous reflectivity background medium; 12. The tomography method of claim 11, further comprising: forming a differential partial differential equation (PDE) operator that discretizes a PDE that describes the interaction of the cross-domain operator, the residual measurement operator, the measurement operator, the Green's function operator, the incident wave field, the background wave field, the differential wave field, and the unknown internal structure of the object, thereby simultaneously calculating the differential wave field along with the unknown internal structure of the object.

16. Solving the optimization problem further comprises: determining a Jacobian of the scattered wave field for the current image of the internal structure of the object by inverting the PDE operator for the initialized current image of the internal structure of the object; updating the current image of the internal structure of the object by minimizing the cost function between the measurements of the total wave field and a total wave field synthesized by combining the Jacobian of the scattered wave field and a quasi-Newton descent direction of the cost function for the image of the internal structure of the object; and projecting the current image of the internal structure of the object onto a constrained total variation penalty function.

17. The cost function between the measured wavefield and the combined total wavefield is: the Euclidean distance between all measurements of the total wave field and the combined total wave field; the 1-norm distance between the total wave field measurement and the combined total wave field; and 17. The method of claim 16, further comprising one or a combination of the Euclidean distance and a barrier function, the barrier function being a sum of exponential functions that penalizes updating the current image of the internal structure of the object.

18. 17. The tomography method of claim 16, wherein inverting the PDE operator is performed using a preconditioned conjugate gradient method, and a preconditioner for the PDE operator is computed using a deep learning network, the deep learning network being trained to determine an estimate of the wave field scattered from an input wave field for the reconstructed image corresponding to a probe pulse.

19. 1. A non-transitory computer readable storage medium containing a program executable by a processor to reconstruct an image of an internal structure of an object, the method comprising: transmitting an incident wave field to an object, the object being embedded in a non-homogeneous reflectance background medium having a known internal structure, the object residing in a background domain within the non-homogeneous reflectance background medium; and receiving a measurement of a total wave field, the total wave field being a combination of the incident wave field and a wave field scattered by an unknown internal structure of the object and the known internal structure of the background domain, the object being within the heterogeneous reflectance background medium in a computational domain smaller than the background domain; and reconstructing an image of the unknown internal structure of the object from the measurements by solving an optimization problem corresponding to the computational domain of the object subject to boundary conditions of a stored cross-domain operator and a residual measurement operator, minimizing a difference between measurements of the total wave field and a wave field synthesized from the reconstructed image using a measurement operator and a Green's function operator by solving the optimization problem iteratively; and and generating the reconstructed image via an output interface.

20. 20. The non-transitory computer-readable storage medium of claim 19, wherein the background domain is configured to attenuate a wave field propagating between a transmitter and a receiver through the background domain such that the measurements are unaffected from propagation of the wave field outside of the background domain.

Images