Sensing with inverse multiple scattering with no phase measurement

Abstract

A permittivity sensor for determining a map of a distribution of a permittivity of a material of an object in a scene is provided, the permittivity sensor comprising an input interface, a hardware processor, and an output interface. The input interface is configured to accept an apodized measurement of a propagation of a known incident field through the scene and scattered by a material of an object in the scene. The hardware processor is configured to solve a multivariate minimization problem for an unknown phase of the apodized measurement and an unknown map of a permittivity of the material of the object by minimizing a difference between a nonlinear function of the known incident field and the unknown map and a product of a known magnitude and the unknown phase of the apodized measurement. Further, the output interface is configured to present the permittivity of the material of the object provided by a solution of the multivariate minimization problem.

Description

Technical Field

[0001] The present invention relates to sensing systems and methods, and more particularly, to systems and methods for determining an image of the dielectric constant distribution of a material based on a phaseless (intensity only) measurement of the scattered electromagnetic field of the material. Background Technology

[0002] The composition or internal structure of an object can be visualized by numerically generating an image representing the distribution of the dielectric constant of the material within the object. A transmitter emits a signal in some mode, such as electromagnetic (EM), light, or ultrasound or pulse, which propagates through the object, reflects off various structures within the object, and propagates to a receiver sensor array. Specifically, due to variations in the object's material composition and structure, the non-uniform distribution of the dielectric constant within the object forces waves or pulses to deviate from straight trajectories and scatter along different paths. Part of the scattered signal is measured by sensors arranged inside or around the object. Inverse scattering is the problem of reconstructing the distribution of the dielectric constant within the object from the measured scattered waves or pulses. The composition of an object can also be visualized by numerically generating an image representing this distribution of the dielectric constant of the material within the object.

[0003] Depending on the material composition of the object, the received signal is often generated by multiple reflections of the propagating pulse due to multiple scattering from structures within the object, leading to artifacts that scramble the reconstructed image. Most traditional methods for addressing this problem consider linear forward models, which achieve an efficient convex formulation of the inverse problem by neglecting multiple scattering. However, in high-contrast scenarios, i.e., when the dielectric constant changes significantly between different structures in the object and background, these linear models tend to be highly inaccurate. This effect is particularly pronounced when the size of the structure is large relative to the wavelength of the incident wave.

[0004] To address this limitation, some methods employ nonlinear formulas for image acquisition, which provide a more accurate representation of the physical setup by modeling multiple scattering and facilitate imaging in high-contrast scenarios. These nonlinear inverse problems are generally more challenging than solving their linear counterparts. However, several methods have recently been developed to directly invert the nonlinear forward model, enabling imaging in high-dielectric-constant contrast scenarios, provided that the amplitude and phase of the scattered wave are available.

[0005] However, in many applications, measuring the phase of the scattered wave can be impractical or expensive. In other cases, phase measurements may contain a great deal of noise or be unreliable. In these applications, the problem becomes phase-less image recovery, i.e., recovery from measurements lacking phase information of the measured scattered wave.

[0006] A common approach to solving the phase-free image restoration problem is alternating minimization, which involves alternating between phase estimation (using phase restoration techniques) and inversion (based on the estimated phase). However, alternating minimization is sensitive to the choice of optimization parameters. Some other methods not based on alternating minimization are less effective for practical imaging problems. Some methods utilize good initialization or initial guesses to determine the solution. However, providing good initialization for the phase-free image restoration problem is difficult. Other methods elevate the phase-free image restoration problem to a different domain (i.e., a higher-dimensional domain) to solve the problem in that domain. However, solving the problem in an elevated higher-dimensional space is computationally forbidden and may make merging image priors impractical.

[0007] Therefore, a method is needed to reconstruct objects from phase-free measurements in the context of inverse multiple scattering. Summary of the Invention

[0008] Technical issues

[0009] Some implementations are based on an understanding of the difficulties in solving the phase-free image reconstruction problem. In solving this problem, the dielectric constant distribution of the material in the measured object is the unknown image to be reconstructed. In the presence of phase information, typical image reconstruction methods attempt to interpret the acquired data as a function of the measured image. This function is formulated based on the physical principles of wave propagation suitable for the application domain and modes, and is known as a forward model. Therefore, the image reconstruction method determines the unknown image of the scene through this forward model, which interprets the measurement data. Naturally, when the measured data lacks phase, phase-free image reconstruction methods tend to modify the existing formulas for the image reconstruction problem by inserting unknown phases to correct the image's function, allowing the existing formulas to be reused. However, this approach leads to multiplicative coupling between the unknown phase and the unknown image.

[0010] Some implementations are based on the understanding that the multiplicative coupling between the unknown phase and the unknown image exacerbates the nonconvexity of the image reconstruction problem, especially in the case of nonlinearity due to multiple scattering and high contrast. To address this, there is a tendency to use alternating minimization, which is very sensitive to the choice of initialization and easily converges to undesirable local minima, failing to provide high-quality reconstruction. In the case of a linear forward model, it is possible to elevate the image reconstruction problem to a higher-dimensional space, making the resulting problem convex and eliminating the unknown phase. However, this leads to a very large optimization problem. Furthermore, in the case of nonlinearity due to multiple scattering, the problem remains nonconvex.

[0011] Therefore, some implementations aim to decouple the variables representing the unknown image and the unknown phase in the forward model of image acquisition. This decoupling allows the implementation to solve the reconstruction problem simultaneously with respect to both unknowns, for example, with appropriate regularization for each unknown.

[0012] Specifically, some implementations are based on the understanding that the unknown phase can be incorporated into the image reconstruction problem by multiplying and correcting known phase-free measurements, allowing the phase-incorporating measurements to be interpreted from the image using a forward model. To this end, phase-free measurements are obtained through a measurement system that can be modeled as being acquired by obtaining the amplitude values ​​of complex nonlinear measurements of the unknown image (i.e., known phase-free measurements). Using this implementation, the unknown phase is used to multiplicatively correct known parameters, i.e., the phase-free measurements. This contrasts with previous ideas that attempted to recover the phase by multiplicatively modifying the forward model to interpret the phase-free measurements. Thus, in the previous ideas, the unknown phase is multiplied by a function of the unknown image; i.e., there is a multiplicative coupling of unknown quantities. In contrast, the implementations decouple the unknown phase from the unknown image by coupling the unknown phase with known measurements.

[0013] However, due to the nonlinearity of the forward model under multiple scattering, the resulting problem remains nonlinear and nonconvex. This nonlinearity stems from the interdependence between the unknown picture and the total field generated by the propagation of waves through the object and the interaction between the waves and the object's material structure.

[0014] Some implementations are based on an alternative approach where the nonlinearity of image acquisition can be modeled by representing the total field of wave propagation using a known incident field generated by the transmitter and a scattered field from the object in the unknown image. This implementation allows the product of a phase-free measurement and an unknown phase to be represented as a nonlinear function of the known incident field of the object and the unknown image.

[0015] This concept remains nonlinear and nonconvex. However, due to the decoupling of the multiplicative relationship between the unknown image and the unknown phase, the nature of this concept leads to a better-performing nonconvex objective function (a function of the unknown image and the unknown phase), thus allowing various nonconvex and even convex solvers to obtain high-quality reconstructed images. For example, some implementations solve the phase-free image restoration problem by making an objective function that includes the sum of a data fidelity term and a regularization term, where the data fidelity term is the difference between a nonlinear function of the unknown image with known incident field and scene and the product of the phase-free measurement and the unknown phase, and the regularization term includes the sum of a penalty for the total variation of the unknown image and an amplitude constraint on the unknown phase, such as an effective phase variable having an amplitude equal to 1. In various implementations, implementations use convex and / or nonconvex solvers to minimize the objective function using an alternating minimization method or a simultaneous multivariate minimization method, which simultaneously optimizes with respect to two unknowns by updating two unknowns for each iteration. Examples of simultaneous multivariate minimization methods include proximal gradient methods, such as the Fast Iterative Shrink Thresholding Algorithm (FISTA) and its variations. In particular, some implementations use FISTA to minimize the aforementioned objective function, and the resulting algorithm is called the Phase-Free Iterative Shrinkage Thresholding Algorithm (PISTA). In some cases, simultaneous multivariate minimization methods and alternating minimization methods show better results.

[0016] In some embodiments, the measurement system is implemented within a civil infrastructure system that solves a phase-free image reconstruction problem using convex and / or non-convex solvers to detect at least one of the following: material defects, cavities, invisible objects within the civil infrastructure (e.g., underground objects, leaking or cavitary pipes under roads, or defects within bridge structures), or any three-dimensional object in three-dimensional space. In some embodiments, the civil infrastructure system is a mobile platform that moves within the civil infrastructure to detect at least one of the following: material defects, cavities, or defects in invisible objects within the civil infrastructure. For example, the system may include a vehicle that travels around monitoring the infrastructure.

[0017] Technical solutions to the problem

[0018] Therefore, one embodiment discloses a dielectric constant sensor for determining the distribution of the dielectric constant of a material of an object in a scene. The dielectric constant sensor includes: an input interface configured to receive a phase-free measurement of the propagation of a known incident field through the scene and scattered by the material of the object in the scene; a hardware processor configured to solve a multivariate minimization problem of an unknown image of the unknown image with respect to the unknown phase of the phase-free measurement and the dielectric constant of the object's material by minimizing the difference between a nonlinear function of the known incident field and the unknown image and the product of the known amplitude and unknown phase of the phase-free measurement; and an output interface configured to present the dielectric constant of the object's material provided by the solution of the multivariate minimization problem.

[0019] Some implementations also use regularization terms for unknown images and unknown phases to improve convergence.

[0020] Another embodiment discloses a method for determining the distribution of the dielectric constant of a material of an object in a scene, the method comprising the steps of: receiving a phaseless measurement of the propagation of a known incident field through the scene and scattered by the material of the object in the scene; solving a multivariate minimization problem of an unknown image of the unknown phase of the phaseless measurement and the dielectric constant of the material of the object by minimizing the difference between a nonlinear function of the known incident field and the unknown image and the product of the known amplitude and the unknown phase of the phaseless measurement; and presenting the dielectric constant of the material of the object as provided by the solution of the multivariate minimization problem.

[0021] The currently disclosed embodiments will be further explained with reference to the accompanying drawings. The drawings shown are not necessarily drawn to scale, but generally focus on illustrating the principles of the currently disclosed embodiments. Attached Figure Description

[0022] [ Figure 1 ]

[0023] Figure 1 A schematic diagram of an apparatus for backscattering according to some embodiments is shown.

[0024] [ Figure 2A ]

[0025] Figure 2A A block diagram of a dielectric constant sensor, according to some embodiments, is shown for determining the distribution of the dielectric constant of a material in an object in a scene.

[0026] [ Figure 2B ]

[0027] Figure 2B The determination of an optimization algorithm for propagation-based phase-free measurement according to some implementations is shown.

[0028] [ Figure 3 ]

[0029] Figure 3 This diagram illustrates a workflow for determining the distribution of the dielectric constant of the material of an object in a scene according to some implementation methods.

[0030] [ Figure 4A ]

[0031] Figure 4A A schematic diagram of a civil infrastructure system for generating a distribution map of invisible objects in civil infrastructure is shown, according to some embodiments.

[0032] [ Figure 4B ]

[0033] Figure 4B An exemplary output of a distribution map of invisible objects according to some implementations is shown.

[0034] [ Figure 4C ]

[0035] Figure 4C Another exemplary output of a distribution map of invisible objects according to some implementations is shown.

[0036] [ Figure 5A ]

[0037] Figure 5A A schematic diagram of a civil infrastructure system for generating a distribution map of cavities of invisible objects in civil infrastructure is shown, according to some embodiments.

[0038] [ Figure 5B ]

[0039] Figure 5B An exemplary output of a distribution map of cavities of invisible objects in a civil infrastructure according to some implementations is shown.

[0040] [ Figure 6 ]

[0041] Figure 6 A schematic diagram showing a comparison of experimental results of the dielectric constant distribution of materials of objects using the PISTA algorithm and prior art algorithms in high-contrast settings according to some embodiments is illustrated.

[0042] [ Figure 7 ]

[0043] Figure 7 A schematic diagram is shown comparing experimental results of the dielectric constant distribution of the material of the object at different frequencies using the PISTA algorithm and existing algorithms according to some embodiments. Detailed Implementation

[0044] In the following description, numerous specific details are set forth for purposes of explanation in order to provide a thorough understanding of this disclosure. However, it will be apparent to those skilled in the art that this disclosure may be practiced without these specific details. In other instances, apparatuses and methods are shown only as block diagrams to avoid obscuring this disclosure.

[0045] As used in this specification and claims, the terms “for example,” “like,” and “such as,” as well as the verbs “comprising,” “having,” “including,” and other forms thereof, when used in conjunction with a list of one or more components or other items, are to be understood as open-ended, meaning that such list should not be construed as excluding other additional components or items. The term “based on” means at least partially based on. Furthermore, it should be understood that the wording and terminology used herein are for descriptive purposes and should not be considered restrictive. Any headings used in this specification are for convenience only and have no legal or limiting effect.

[0046] Technical Solution Overview

[0047] Figure 1 A schematic diagram of an apparatus 100 for inverse scattering image reconstruction according to some embodiments is shown. Figure 1 The image shows objects 106 and 108 in a scene (i.e., a bounded domain Ω104) and a sensor 112 located outside the bounded domain Ω104. Objects 106 and 108 may collectively correspond to a single two-dimensional or three-dimensional entity having multiple sub-parts (e.g., a first part 106 and a second part 108). Alternatively, objects 106 and 108 may correspond to a single part as a complete entity. An input wave 102 is also shown incident on and further scattered from objects 106 and 108 (as scattered wave 110). Scattered wave 110 is also shown being received by sensor 112. Wave 102 is emitted by transmitter 111. In this exemplary embodiment, transmitter 111 and sensor 112 are located on opposite sides of objects 106 and 108. In an alternative embodiment, transmitter 111 and sensor 112 are located on the same side of objects 106 and 108. In some implementations, the transmitter 111 and the sensor 112 may use the same physical device that operates in either transmit or receive mode.

[0048] Based on the reconstruction of inverse scattering images according to some embodiments, the spatial permittivity distribution ε(x) in the bounded domain Ω104 is estimated, where, This represents the spatial coordinates within the bounded domain Ω104. For example, one or more transmitters 111 emit an incident wave 102 in some mode, such as electromagnetic (EM), light, ultrasound, or pulse, which propagates through and illuminates objects 106 and 108. The input wave 102 incident on objects 106 and 108 is denoted as the input field u. in The scattered field, which is usually known and determined by system design and wave propagation laws, is formed by the scattering of incident wave 102 inside and outside the domain Ω104. Therefore, the scattered field caused by the scattered wave 110 can be established and measured. Specifically, the non-uniform distribution of the dielectric constant inside objects 106 and 108 due to variations in material composition and structure forces the incident wave 102 to deviate from a straight trajectory and scatter along different paths. The scattered field is measured by sensor 112 as u. sc The scattered field u measured from a single sensor or multiple sensors (called a sensor array) sc Images used to reconstruct objects 106 and 108.

[0049] From the scattering field u sc The reconstructed images of objects 106 and 108 are called inverse scattering. Therefore, a system including such a sensor (i.e., sensor 112 or sensor array) can also be called a measurement and / or sensing system.

[0050] In this way, the total field u in the imaging device 100 is two fields (i.e., the input field u). in and scattering field u sc The sum of ) makes

[0051] u = u in +u sc .

[0052] Furthermore, the scalar Lippmann-Schwinger relation is used in the total field u and the input field u in The relationship between and the scattering objects 106 and 108 can be expressed by the following equation:

[0053]

[0054] Where, f(x) = k 2 (ε(x)-ε b ) represents the scattering potential, ε b The dielectric constant representing the background is given by k = 2π / λ, which is the wave number in vacuum. It is a Green's function in 2D free space, where Denotes the zeroth-order Hankel function of the first kind. represents the background wave number, and ||·|| represents the Euclidean norm.

[0055] Furthermore, the discretized system of the imaging device 100 is represented as follows:

[0056]

[0057] u = u in +Gdiag(f)u, (1)

[0058] in, and They are f(x), u(x), and u, respectively. in (x) represents the samples obtained at N points in the domain Ω104. diag(f) denotes a diagonal matrix, where f lies on its main diagonal. This represents the scattered wave observed in the sensor domain Γ. In some implementations, using appropriately fine discretization, N points in the domain are chosen to lie on a regular grid. In different implementations, the discretization accuracy is a design parameter and / or guided by the wavelength of the emitted pulse to ensure that the object or characteristics of the object are taken into account (if they are large enough to interact with the incident or scattered wave).

[0059] Furthermore, the matrix in formula (1) This corresponds to the mapping from the image domain to the sensor domain Γ, as defined by discretizing the continuous Green's function g(xx′) for x∈Γ and x′∈Ω. Similarly, the matrix... It is a mapping in the image domain, as defined by discretizing the Green's function g(xx′) for x,x′∈Ω.

[0060] Some implementations are based on the understanding that measuring the phase of a scattered wave can be impractical or expensive in many applications. Furthermore, phase measurements can be highly noisy or unreliable. In these applications, the problem becomes phase-free image recovery—recovering the phase from a measurement lacking information about the phase of the scattered wave. For example, the phase-free image recovery problem is applicable to optical sensing, such as Fourier overlay microscopy and optical diffraction tomography, due to the difficulty of nonlinear coherent measurements. It is also applicable to motion-related applications due to the nonlinearity of motion-induced distortion, and to terahertz (THz) sensing due to the difficulty of performing phase measurements at sufficient resolution.

[0061] Therefore, in some implementations, the measurement system is phase-free, that is, it only records the scattered wave u. sc The amplitude. Therefore, the obtained data (in the absence of noise) can be expressed by formula (1):

[0062] y = |Hdiag(u)f|

[0063] u = uin +Gdiag(f)u sc (2)

[0064] Formula (2) corresponds to nonlinear and phase-free image reconstruction, where, Represents the scattered wave observed in the sensor domain Γ. The range.

[0065] Some implementations are based on the understanding that image reconstruction methods determine an unknown image of a scene, which interprets the measurement data through a forward model. Naturally, when the measured data lacks phase, phase-free image reconstruction methods tend to modify the existing formulation of the image restoration problem by inserting a function of the unknown phase to correct the image, allowing the existing formulation to be reused. However, this approach leads to multiplicative coupling between the unknown phase and the unknown image. This multiplicative coupling exacerbates the nonconvexity of the image restoration problem, especially in the case of nonlinearity due to multiple scattering and high contrast.

[0066] Therefore, some implementations aim to decouple the variables representing the unknown image and the unknown phase in the forward model of captured image acquisition. This decoupling allows the implementation to solve the reconstruction problem simultaneously with respect to both unknowns using appropriate regularization for each unknown.

[0067] Specifically, some implementations are based on the understanding that the unknown phase can be incorporated into the image reconstruction problem by multiplying and correcting known phase-free measurements, allowing the phase-incorporating measurements to be interpreted from the image using a forward model. To this end, phase-free measurements are obtained through a measurement system that can be modeled as being obtained by acquiring the amplitude values ​​of complex nonlinear measurements of the unknown image (i.e., known phase-free measurements). Using this implementation, the unknown phase is used to multiplicatively correct known parameters, i.e., the phase-free measurements. This contrasts with previous ideas that attempted to recover the phase by multiplicatively modifying the forward model to interpret the phase-free measurements. Thus, in the previous ideas, the unknown phase is multiplied by a function of the unknown image, i.e., there is a multiplicative coupling of unknown quantities. In contrast, the implementations decouple the unknown phase from the unknown image by coupling the unknown phase with known measurements.

[0068] For example, the unknown phase of the scattered wave 110 is obtained through the complex phase vector. To model, that is Where ⊙ represents element-wise product. Therefore, phase-free observations satisfy y⊙p=H diag(u)f. In the presence of noise, equation (2) can be expressed as:

[0069] diag(y)p=Hdiag(u)f+e

[0070] u = u in +Gdiag(f)u (3)

[0071] in, Indicates noise.

[0072] Some implementations are based on the following objective: under the constraints that f is a piecewise constant and the unknown phase p is only a phase vector, through phase-free observation and incident field u in Estimate the unknown object f and the unknown phase p.

[0073] However, due to the nonlinearity of the forward model under multiple scattering, the resulting problem remains nonlinear and nonconvex. This nonlinearity stems from the interdependence between the unknown picture and the total field generated by the propagation of the wave through the object and the interaction between the wave and the material structure of the object.

[0074] Some implementations are based on an alternative approach where the nonlinearity of image acquisition can be modeled by representing the total field of wave propagation using a known incident field generated by the transmitter and a scattered field from objects in the unknown image. This implementation allows the product of a phase-free measurement and an unknown phase to be represented as a nonlinear function of the known incident field of the scene and the unknown image.

[0075] Therefore, some implementations use a known incident field u in The multivariate minimization problem of the unknown image of objects 106 and 108 with respect to the unknown phase and the dielectric constant of the materials is solved by minimizing the difference between the nonlinear function of the unknown image and the product of the known amplitude and unknown phase of the phase-free measurement. Some implementations also include regularization terms for the unknown image and the unknown phase.

[0076] This concept remains nonlinear and nonconvex. However, due to the decoupling of the multiplicative relationship between the unknown image and the unknown phase, the nature of this concept leads to a better-performing nonconvex objective function (a function of the unknown image and the unknown phase), thus allowing various nonconvex and even convex solvers to obtain high-quality reconstructed images. For example, if the total field u is known, Equation (3) becomes linear because the formula decouples the unknown phase p from the unknown image f. Therefore, for example, some implementations solve the phase-free image restoration problem by making the objective function include the sum of a data fidelity term and a regularization term, where the data fidelity term includes the difference between the known incident field of the scene and the nonlinear function of the unknown image and the product of the phase-free measurement and the unknown phase, and the regularization term includes the sum of the total variation penalty of the unknown image and the amplitude constraint of the unknown phase (the effective phase variable has an amplitude equal to 1).

[0077] In various implementations, the methods employ convex and / or non-convex solvers to minimize the objective function using alternating minimization or simultaneous multivariate minimization methods. Simultaneous multivariate minimization methods optimize with respect to two unknowns simultaneously by updating both unknowns for each iteration. Examples of simultaneous multivariate minimization methods include proximal gradient methods, such as the Fast Iterative Shrinking Thresholding Algorithm (FISTA) and its variants. Specifically, some implementations use FISTA to minimize the aforementioned objective function, and the resulting algorithm is called the Phase-Free Iterative Shrinking Thresholding Algorithm (PISTA). Based on the solution to the multivariate minimization problem, images of the dielectric constants of the materials of objects 106 and 108 are presented.

[0078] System Overview

[0079] Figure 2A A block diagram of a dielectric constant sensor 200, according to some embodiments, for determining an image of the dielectric constant distribution of the materials of objects 106, 108 in a scene is shown. The dielectric constant sensor 200 includes an input interface 202 configured to receive a propagated phase-free measurement 216 from a device, such as from sensor 112, for determining an image of the dielectric constant distribution of the materials of objects 106, 108.

[0080] Furthermore, the dielectric constant sensor 200 includes a network interface controller (NIC) 212 adapted to connect the dielectric constant sensor 200 to a network 214 via a bus 210. The dielectric constant sensor 200 can receive propagated phase-free measurements 216 wirelessly or via a line through the network 214. The system 200 stores image acquisition settings 209 used by multivariate minimization 208 to reconstruct an image of the scene. The image acquisition settings may include settings used by a forward operator to capture details of the image acquisition, such as the type and relative arrangement of imaging devices that transmit and receive reflected signals from one or more objects in the scene, the type and frequency of the signals, the optics of the imaging devices, and the position and motion of the imaging devices.

[0081] Image acquisition setting 209 allows you to define the input field u in Alternatively, the phase-free measurement 216 of the propagation may include an input field u corresponding to the incident wave 102. in Information and characteristics of the scattered wave 110, such as known parameters of the scattered wave 110, such as the amplitude of the scattered wave 110.

[0082] The dielectric constant sensor 200 includes a memory 206 storing instructions executable by a processor 204. The processor 204 may be a hardware processor configured to execute the stored instructions to control the operation of the dielectric constant sensor 200. The processor 204 may be a single-core processor, a multi-core processor, a graphics processing unit (GPU), a computing cluster, or any other configuration. The memory 206 may include random access memory (RAM), read-only memory (ROM), flash memory, or any other suitable memory system.

[0083] In some implementations, memory 206 may store programming instructions to perform multivariate minimization 208 with decoupled unknowns. The solution of multivariate minimization 208 performs non-convex phase-free image reconstruction by decoupling the unknown phase and the unknown image f. More specifically, based on the execution of such programming instructions, processor 204 is configured to perform reconstruction by making the known incident field u in The problem of minimizing the difference between the nonlinear function of the unknown image f and the product of the known amplitude and unknown phase of the phase-free measurement 216 (which may include regularization terms for the unknown image and unknown phase) is solved to solve a multivariate minimization problem of the unknown image f with respect to the unknown phase of the phase-free measurement 216 and the dielectric constant of the materials of objects 106 and 108.

[0084] In some implementations, processor 204 is configured to solve multivariate minimization problem 208 by simultaneously minimizing the two unknowns with respect to both unknowns for each iteration. Examples of simultaneous multivariate minimization include multivariate gradient descent that finds the minimum of a function, such that the optimization algorithm provides a solution to the inverse multiscattering image reconstruction problem.

[0085] Therefore, the memory can store a set of instructions for solving the multivariate minimization problem 208. Examples of such instructions include proximal gradient methods such as the Phase-Free Iterative Shrinking Thresholding Algorithm (PISTA) to simultaneously update the unknown phase of the unknown objects (i.e., images of objects 106, 108) and the measurement (i.e., the scattering field 110) at each iteration. Alternatively, the memory 206 can store a set of program instructions corresponding to the alternating minimization algorithm and other algorithms capable of solving non-convex optimization problems.

[0086] In some embodiments, the dielectric constant sensor 200 is connected via a bus 210 to an output interface 222 adapted to connect the dielectric constant sensor 200 to an external device 224 configured to present the dielectric constant distribution of the materials of objects 106, 108 output by the dielectric constant sensor 200. In some other embodiments, the dielectric constant sensor 200 is communicatively coupled to multiple external devices.

[0087] Alternatively, in one embodiment, the dielectric constant sensor 200 is connected via bus 210 to a display interface 218 adapted to connect the dielectric constant sensor 200 to a display device 220. The display device 220 is configured to present an image inversion result corresponding to an image of the dielectric constant distribution of the materials of the objects 106, 108 output by the dielectric constant sensor 200.

[0088] Some implementations are based on the understanding that objects 106, 108 are irradiated by incident waves 102 emitted from one or more emitters, and that the incident waves 102 are further scattered by the materials of objects 106, 108. The incident waves 102 are selected before irradiating objects 106, 108 based on various combinations of frequencies. Therefore, the incident waves 102 have known parameters, and the incident field u... in It is known.

[0089] Furthermore, the processor 204 is configured to receive, via the input interface 202, a known incident field u that passes through the scene and is scattered by the material of objects 106, 108 present in the scene. in Phase-free measurement 216 of propagation. Phase-free measurement 216 of propagation includes the input field u corresponding to the incident wave 102. in Information and characteristics of the scattered wave 110, such as known parameters of the scattered wave, such as the amplitude of the scattered wave 110.

[0090] Based on the acceptance of the propagating phaseless measurement 216, the processor 204 is also configured to acquire a set of program instructions stored in the memory 206 in order to solve a multivariate minimization problem of an unknown phase of the phaseless measurement 216 and an unknown image f of the dielectric constant of the materials of objects 106, 108.

[0091] To this end, processor 204 executes the acquired set of program instructions to minimize the objective function, which includes the known incident field u. in The difference between the nonlinear function of the unknown image f and the product of the known amplitude and unknown phase of the phase-free measurement 216 may include regularization terms for the unknown image and the unknown phase. More specifically, the processor 204 executes a set of program instructions based on the propagated phase-free measurement 216 to minimize the objective function, thereby solving a multivariate minimization problem.

[0092] In some implementations, processor 204 executes a set of program instructions corresponding to alternating minimization. Alternatively, processor 204 executes a set of instructions corresponding to a multivariate proximal gradient method. The multivariate proximal gradient method is implemented using the execution of an algorithm such as PISTA.

[0093] By solving a multivariate minimization problem, processor 204 is configured to determine the dielectric constant distribution of the materials of objects 106 and 108. More specifically, processor 204 determines an image of the dielectric constant distribution of the materials of objects 106 and 108. Furthermore, the dielectric constants of the materials of objects 106 and 108 are presented on external device 224 via output interface 222. Alternatively, the dielectric constants of the materials of objects 106 and 108 are presented on display device 220 via display interface 218.

[0094] By solving a multivariate minimization problem concerning the unknown phase and the unknown image f of the phaseless measurement 216, the processor 204, given the scattering field u, sc The unknown parameters can be optimized under the given amplitude. Therefore, the dielectric constant sensor 200 is convenient for optimizing unknown parameters based on a given scattering field u. sc The unknown parameters of the amplitude are optimized to accurately determine the dielectric constant distribution of the materials of objects 106 and 108.

[0095] Exemplary Solution

[0096] Figure 2B The determination of an optimization algorithm for propagation-based phase-free measurement 216 according to some embodiments is shown.

[0097] One implementation aims to decouple the variables representing the unknown image and the unknown phase in the forward model acquired from the captured image, so as to achieve a given scattering field u sc The unknown parameters are optimized under the condition of amplitude, so that the reconstruction problem with respect to the unknown image and the unknown phase can be solved simultaneously, and the multiplicative coupling between the unknown image and the unknown phase can be eliminated.

[0098] Reference Figure 1 The discretized system of the imaging device 100 is formulated as Equation (1) 228, which is further rearranged as Equation (2) based on the phase-free measurement system in the absence of noise. Furthermore, in the presence of noise, Equation (2) is expressed as Equation (3) 230.

[0099] Equation (3) removes the multiplicative coupling between the unknown phase and the unknown image f in the discretized system of the imaging device 100. Therefore, Equation (3) defines a multivariate minimization problem of the unknown image with respect to the unknown phase without phase measurement and the dielectric constant of the object material, which minimizes the difference between the nonlinear function of the known incident field 243 and the unknown image 244 and the product of the known amplitude 242 and the unknown phase 242 without phase measurement;

[0100] By rearranging formula (3), we obtain the following expression 232:

[0101]

[0102] [I-Gdiag(f)]u=u in (5),

[0103] In formula (4), p and f are located in the null space of matrix [diag(y),-Hdiag(u)] in the absence of noise, and they may not be unique.

[0104] More specifically, p and f in equation (4) in the null space include trivial all-zero solutions that lead to fuzzy results. This effect is called a multivariate minimization problem. Furthermore, the constraint (5) on the total field u reduces the size of the solution space and introduces a set of constraints to further solve the problem.

[0105] Therefore, some implementations are based on the understanding that the object dielectric constant contrast f is generally smooth in spatial coordinates and that the object dielectric constant contrast f is only associated with non-negative values. Additionally, p is a phase-only complex vector with a unit quartile magnitude, such that for i = 1, ..., M, |p i |=1. Utilizing these additional priors and constraints, some implementations employ total variational (TV)-based regularization, with a non-negativity constraint on the object's dielectric constant contrast f and a unit size constraint on only the phase vector p. Therefore, phase-free inverse scattering requires solving the following optimization 234.

[0106]

[0107]

[0108] in,

[0109] This indicates TV regularization.

[0110] D i Let represent the discrete difference operator in the i-th spatial dimension, and

[0111] λ is the optimization parameter.

[0112] According to (5), u is written as a function f and inserted into the first term of (6), where the first term of (6) is a differentiable function of f and its gradient can be efficiently computed using the conjugate gradient method. Therefore, processor 204 can be configured to solve a multivariate minimization problem by using the proximal gradient method to perform simultaneous multivariate minimization with respect to two unknowns by updating the two unknowns for each iteration. The gradients of f and p are recursively estimated up to a fixed number of iterations or until a termination condition is met. The termination condition can be based on the simultaneous estimation of the gradients of f and p until a unique value of the gradient is achieved.

[0113] Therefore, formula (6) can be derived by using a known phase-free measurement (e.g., a known incident field u). in The solution is obtained using a method (hereinafter referred to as "Algorithm 1") that utilizes the known amplitude of phase-free measurement 216. This algorithm is called PISTA, and it uses FISTA as the solver for the optimization problem (6). In some implementations, equation (6) is solved based on FISTA. In some alternative implementations, equation (6) is solved based on other algorithms capable of solving non-convex and non-smooth optimization problems.

[0114] To this end, processor 204 can be configured to execute a set of instructions corresponding to the PISTA algorithm to simultaneously recover the phase of the unknown object and the measurement from a nonlinear and nonconvex inverse scattering problem.

[0115] Algorithm 1: PISTA

[0116]

[0117] In Algorithm 1,

[0118] α represents the step size.

[0119] P TV,λ,|·|≥0 This represents the proximal mapping with non-negativity constraints and TV regularization.

[0120] P |·|＝1 This represents a non-convex proximal mapping to the surface of an M-dimensional complex sphere, which is obtained by scaling each complex entry of the vector to a unit size.

[0121] and These represent the gradients of the data fidelity term for smoothing. The f and p related components.

[0122] More specifically, some implementations are based on the realization of the overall gradient of the data fidelity term in Equation (6), as described in the following proposal:

[0123] proposal:

[0124] In some implementations, the p and f components of the gradient of the data fidelity term in formula (6) can be expressed as follows: as well as

[0125]

[0126] in,

[0127] r = [Hdiag(u)f - diag(y)p] is the residual vector.

[0128] A = I - Gdiag(f), and

[0129] u and w correspond to the solutions of the following linear system:

[0130] Au = u in A H w = diag(f)H H r, (9)

[0131] Linearize the current estimate of f.

[0132] Proof. The gradient of a data term with respect to p, It can be determined based on the matrix derivative. Furthermore, referring to formula (6), the derivative with respect to f, It can be represented as:

[0133]

[0134] Among them, J f The Jacobian determinant for f can be further expressed as:

[0135]

[0136] Furthermore, the derivative in formula (11) can be solved using differentials as follows:

[0137] d(Hdiag(u)f)=Hdiag(u)df+Hd(diag(u))f

[0138] =Hdiag(u)df+Hdiag(f)du, (12)

[0139] Some implementations are based on the understanding that u is a function of f, and that when differentiating only the total field u, the following relationship is established:

[0140] d(diag(u))f=d(diag(u)f)=d(diag(f)u)=diag(f)du.

[0141] Furthermore, in u = A -1 u in In this case, the differential of u is expressed as:

[0142] du=-A -1 dAA -1 u in =-A -1 dA u. (13)

[0143] Furthermore, the differential of A is expressed as:

[0144] dA=d(I-Gdiag(f))=-Gd(diag(f)), (14)

[0145] It can be used to obtain dAu=-Gd(diag(f))u=-Gdiag(u)df.

[0146] By using all differentials, the Jacobian determinant can be expressed as:

[0147] J f =Hdiag(u)+Hdiag(f)A -1 Gdiag(u). (15)

[0148] Finally, combining (15) and (10) yields (8).

[0149] Furthermore, based on the gradient expressions (7) and (8), the processor 204 is configured to calculate the gradient required to determine the dielectric constant distribution of the materials of objects 106 and 108.

[0150] According to the execution of the PISTA algorithm, the processor 204 solves a multivariate minimization problem of unknown phase and unknown image x of the dielectric constant of the materials of objects 106 and 108 with respect to the unknown phase of phase-free measurement 216.

[0151] Therefore, processor 204 determines the dielectric constant distribution of the materials of objects 106 and 108 with high resolution and high accuracy based on the solution of the multivariate minimization problem. Furthermore, processor 204 presents the determined dielectric constants of the materials of objects 106 and 108 via output interface 222.

[0152] Figure 3 A schematic diagram of the workflow is shown, illustrating the distribution of dielectric constants of the materials of objects 106, 108 in a defined scenario according to some embodiments.

[0153] This workflow is executed by processor 204 based on propagating phaseless measurement 216.

[0154] Using the 302 instruction set, the known incident field u is obtained by passing through the scene and being scattered by the materials of objects 106 and 108 in the scene. in The phase-free measurement 216 of the propagation. Furthermore, the processor 204 executes a set of instructions based on the phase-free measurement 216 of the propagation, so as to make the known incident field u... in The difference between the nonlinear function of the unknown image x and the product of the known amplitude and unknown phase of the phaseless measurement 216 of the propagation is minimized 304, optionally with additional constraints and priors on the unknown image and unknown phase (as described in (6)).

[0155] Furthermore, processor 204 executes a set of program instructions to solve a multivariate minimization problem 306 for an unknown phase of the propagating phaseless measurement 216 and an unknown image f of the dielectric constant of the materials of objects 106 and 108 (as in...). Figure 2B (As described in the description). By solving the multivariate minimization problem, the processor 204 presents the dielectric constant 308 of the materials of objects 106 and 108 via the output interface 222 or the display interface 218. Thus, a high-precision image of the materials of objects 106 and 108 is obtained.

[0156] Figure 4A A schematic diagram of a civil infrastructure system for generating a distribution map of invisible objects in civil infrastructure is shown, according to some embodiments.

[0157] exist Figure 4A The diagram illustrates a civil infrastructure system 400 within a civil infrastructure 404. The civil infrastructure system 400 includes at least a dielectric constant sensor 402, one or more transmitters configured to emit one or more electromagnetic signals, and one or more receivers configured to receive one or more scattered waves. The dielectric constant sensor 402 corresponds to... Figure 2A The dielectric constant sensor 200 is described. Figure 4A The diagram also shows an underground pipe 406 located in mud 408 beneath civil infrastructure 404, wherein the particles (e.g., rock, soil, water, etc.) of the underground pipe 406 and mud 408 may correspond to invisible objects of civil infrastructure 404. Alternatively, civil infrastructure 404 may include multiple underground pipes as invisible objects.

[0158] In some embodiments, the civil infrastructure system 400 is a mobile platform that travels on the ground of the civil infrastructure 404. The civil infrastructure system 400 generates electromagnetic signals (such as those from a known incident field u) on the ground of the civil infrastructure 404. in When the electromagnetic signal propagates through the ground, it is reflected from invisible objects (i.e., particles of underground pipe 406 and mud 408).

[0159] Based on reflection (i.e., one or more scattered waves), the civil infrastructure system 400 detects the presence of an invisible object located underground in the civil infrastructure 404.

[0160] For this purpose, a dielectric constant sensor 402, arranged on a moving platform, receives reflected electromagnetic signals from particles from underground pipe 406 and from mud 408. The dielectric constant sensor 402 also receives phaseless measurements of propagating electromagnetic signals with unknown phase (e.g., propagating phaseless measurement 218).

[0161] Furthermore, the dielectric constant sensor 402 solves a multivariate minimization problem concerning the unknown phase of the unknown phase of the phase-free measurement and the unknown image of the dielectric constant of the material of the particles in the underground pipe 406 and mud 408 by minimizing the difference between the nonlinear function of the known incident field and the unknown image and the product of the known amplitude and unknown phase of the phase-free measurement. This may include, for example, the following: Figure 2A and Figure 2B The additional constraints on the unknown image and unknown phase described in the description.

[0162] Based on the solution to the multivariate minimization problem, the dielectric constant sensor 402 determines the dielectric constant distribution of the materials of the underground pipe 406 and the mud 408. The dielectric constant sensor 402 also generates a distribution map of the particles of the underground pipe 406 and the mud 408 based on the dielectric constant of the underground pipe 406, and presents the distribution map via an output interface (e.g., output interface 222). Figure 4B The output shows the distribution map of underground pipe 406 and mud particles, which shows the distribution of the dielectric constant of the material of the invisible objects (i.e., the particles of underground pipe 406 and mud 408).

[0163] Figure 4B An exemplary output of a distribution map of invisible objects according to some implementations is shown. Figure 4B The diagram 410 shows the dielectric constant distribution of the materials corresponding to the particles of underground pipe 406 and mud 408. The diagram 410 includes an image 412 corresponding to the particles of mud 408 and an image 414 corresponding to underground pipe 406. Therefore, invisible objects in civil infrastructure 404 are detected based on phase-free measurements using a dielectric constant sensor 402 of the civil infrastructure system 400.

[0164] Figure 4C Another exemplary output of a distribution map of invisible objects according to some implementations is shown. Figure 4C The diagram shows a two-dimensional distribution map 410C, which corresponds to the dielectric constant distribution of the material of particles in the underground pipe 406 and slurry 408 for a pipe cross-section. Distribution map 410C includes an image 412C corresponding to the particles of slurry 408 and an image 414C corresponding to the underground pipe 406. Therefore, invisible objects in the civil infrastructure 404 are detected based on phase-free measurements by means of the dielectric constant sensor 402 of the civil infrastructure system 400.

[0165] Figure 5A A schematic diagram of a civil infrastructure system for generating a distribution map of cavities of invisible objects in civil infrastructure is shown, according to some embodiments.

[0166] exist Figure 5AThe image illustrates a civil infrastructure comprising objects such as pipes (e.g., pipe 500). Pipe 500 includes a set of cavities (e.g., cavity 502). In some embodiments, cavity 502 may be a defect in pipe 500. A civil infrastructure system 504 is also shown, which includes at least a dielectric constant sensor 506, one or more transmitters configured to emit one or more electromagnetic signals, and one or more receivers configured to receive one or more scattered electromagnetic waves. The dielectric constant sensor 506 corresponds to... Figure 2A The dielectric constant sensor 200 and as described above Figure 4A The dielectric constant sensor 402 is described above.

[0167] like Figure 5A As shown, the civil infrastructure system 504 is a mobile platform traveling within a conduit (i.e., conduit 500). As the civil infrastructure system 504 moves within the conduit 500, it generates electromagnetic signals (e.g., signals with a known incident field u) on the internal structure of the conduit 500 via one or more transmitters. in ).

[0168] When an electromagnetic signal propagates within the pipe 500, it is reflected from the internal structure of the pipe 500 and received by the civil infrastructure system 504 via one or more receivers. However, because the incident electromagnetic wave propagates through the cavity 502 and further into the space outside the pipe 500, the portion of the pipe 500 containing the cavity 502 does not reflect the electromagnetic wave back. Therefore, the civil infrastructure system 504 does not receive electromagnetic waves from the cavity 502.

[0169] Based on the reflection of electromagnetic waves from pipe 500, civil infrastructure system 504 generates a distribution map 508 of the dielectric constant distribution of the internal structure of pipe 500 using dielectric constant sensor 506 (e.g., Figure 2B and Figure 4A (As described in the description). Such a distribution diagram is also known as a cavity distribution diagram of the pipe. Figure 5B The output of the cavity distribution diagram (i.e., distribution diagram 508) is shown, which shows the distribution of dielectric constant of the internal structure of the pipe 500.

[0170] Figure 5B An exemplary output of a distribution map of cavities of invisible objects in a civil infrastructure according to some embodiments is shown. Figure 5B The diagram 508 is shown, where a portion 510 of the diagram 508 represents a cavity 502. Therefore, the civil infrastructure system 504 detects defects in objects of the civil infrastructure by using a dielectric constant sensor 506.

[0171] Figure 6A schematic diagram is shown comparing experimental results of the dielectric constant distribution of an object material using the PISTA algorithm and a prior art algorithm with a high contrast setting, according to some embodiments.

[0172] For example, an object with physical dimensions of 15cm × 15cm has two cylinders with dielectric constant contrast values ​​of 2 and 0.45, respectively, and is illuminated by one or more light signals containing multiple wavelengths (e.g., 6cm, 7.5cm, 10cm, 15cm, and 30cm) through multiple emitters. Furthermore, to obtain experimental results, one or more scattered light signals are received from the object.

[0173] For this purpose, the pixel size was set to 0.4688 cm, and the number of emitters was set to 24, with 36 receivers measuring the scattered light signals received from the object. The emitters and receivers were uniformly placed within a circle of radius 1.67 m around the object. Furthermore, one or more scattered light signals were processed using existing algorithmic techniques (e.g., Alternating Direction Method with Multipliers (ADMM)) and the PISTA algorithm to obtain experimental results. More specifically, the phase-free measurement of the propagation of one or more light signals was obtained by a processor (e.g., processor 204), wherein the propagation measurement included information on at least one or more scattered light signals, one or more light signals incident on the object, or the amplitude of one or more light signals. Furthermore, the processor executed a set of program instructions as follows: Figure 2A and Figure 2A The description describes solving the nonlinear image inversion problem based on propagation measurements.

[0174] Therefore, based on the solutions to the nonlinear image inversion problem using the ADMM algorithm and the PISTA algorithm under different contrast settings, the image of the object is reconstructed into output distribution map 602 and output distribution map 604.

[0175] exist Figure 6 In the diagram, a set of experimental results with high contrast settings is shown in the top and bottom rows, where the high contrast settings are based on the maximum dielectric constant contrast setting. More specifically, the top row contains experimental results for an object with a maximum dielectric constant contrast of 2, and the image is reconstructed using phase-free measurements of scattered waves at multiple wavelengths. The bottom row contains experimental results for an object with a maximum dielectric constant contrast of 10, using phase-free measurements of scattered waves at only one wavelength (i.e., a wavelength of 30 cm).

[0176] Simulated images 600 of the object are also shown in the top and bottom rows. Furthermore, output distribution plots 602, which are experimental results generated based on the PISTA algorithm, are shown in the top and bottom rows. Additionally, output distribution plot 604, which is experimental results generated based on the existing ADMM algorithm, is shown in the top and bottom rows.

[0177] In addition, such as Figure 6 As shown, the reconstruction performance of the two methods (i.e., ADMM algorithm and PISTA algorithm) is reasonable for medium contrast; however, for high contrast settings with only one frequency, the PISTA algorithm is significantly better than the ADMM algorithm.

[0178] Figure 7 A schematic diagram comparing experimental results of the dielectric constant distribution of the material of the object using the PISTA algorithm and existing algorithms at different frequencies according to some embodiments is shown.

[0179] exist Figure 7 The image shows experimental measurements of the FoamDielExtTM object from the Fresnel Research Institute's public dataset. For example, consider a 15cm × 15cm area around the origin where the FoamDielExtTM object was placed. Furthermore, eight transmitters and 360 receivers were evenly placed within a circle with a radius of 1.67m around the origin. One transmitter was activated at a time, and data from only 241 receivers was used. The remaining 119 receivers closest to the transmitters remained inactive.

[0180] In addition, Figure 7 In the diagram, the top and bottom rows show reconstructed images (i.e., images 702-708) of the object FoamDielExtTM (based on the basic fact 700) and the object based on the PISTA and ADMM algorithms (i.e., dielectric constant distribution maps). Images 702 and 706 are based on the PISTA algorithm, and images 704 and 708 are based on the ADMM algorithm. Furthermore, images 702 and 704 (in the top row) are reconstructed for a single frequency at a wavelength of 10 cm, and images 706 and 708 (in the bottom row) are reconstructed for multiple frequencies at wavelengths of 5 cm, 6 cm, 7.5 cm, 10 cm, and 15 cm with a pixel size of 0.4688 cm. The reconstructions of images 702-708 of the object are consistent with those in the diagram. Figure 2B and Figure 6 The description is the same as that in the description.

[0181] In addition, according to Figure 7 It is clear that, in some cases, PISTA achieves better reconstruction than ADMM.

[0182] The above description provides exemplary embodiments only and is not intended to limit the scope, applicability, or configuration of this disclosure. Rather, the above description of the exemplary embodiments will provide those skilled in the art with a description enabling the implementation of one or more exemplary embodiments. Various changes to the function and arrangement of the elements are contemplated without departing from the spirit and scope of the subject matter set forth in the appended claims.

[0183] Specific details are set forth in the foregoing description to provide a full understanding of the embodiments. However, as will be understood by those skilled in the art, embodiments may be practiced without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in block diagram form so as not to obscure the embodiments with unnecessary details. In other instances, well-known processes, structures, and techniques may be shown without unnecessary details to avoid obscuring the embodiments. Furthermore, the same reference numerals and designations in the various figures indicate the same elements.

[0184] Furthermore, the various implementations can be described as processes depicted as flowcharts, flow diagrams, data flow diagrams, structural diagrams, or block diagrams. Although flowcharts may describe operations as sequential processes, many operations can be performed in parallel or simultaneously. Additionally, the order of operations can be rearranged. A process may terminate upon completion of its operations, but may have additional steps not discussed or included in the diagram. Moreover, not all operations within any specifically described process will occur in all implementations. A process may correspond to a method, function, process, subroutine, subroutine, etc. When a process corresponds to a function, the termination of the function may correspond to the function returning to the calling function or the main function.

[0185] Furthermore, implementations of the disclosed subject matter can be carried out, at least partially, manually or automatically. Manual or automatic implementations can be performed or at least assisted by using machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, program code or code segments for performing the necessary tasks can be stored in a machine-readable medium. The processor can then perform the necessary tasks.

[0186] The various methods or processes outlined herein can be encoded as software that can be executed on one or more processors employing any of a variety of operating systems or platforms. Furthermore, such software can be written using any of a number of suitable programming languages ​​and / or programming or scripting tools, and can also be compiled into executable machine language code or intermediate code that executes on a framework or virtual machine. Typically, in various implementations, the functionality of program modules can be combined or distributed as needed.

[0187] Implementations of this disclosure can be carried out as a method, examples of which have been provided. The actions performed as part of the method can be ordered in any suitable manner. Therefore, implementations in which actions are performed in a different order than those shown can be constructed, which may include performing some actions simultaneously, although shown as sequential actions in the illustrative embodiments.

[0188] While this disclosure has been described with reference to certain preferred embodiments, it should be understood that various other adaptations and modifications can be made within the spirit and scope of this disclosure. Therefore, the aspects of the appended claims cover all such variations and modifications that fall within the true spirit and scope of this disclosure.

Claims

1. A permittivity sensor for determining a map of a distribution of a permittivity of a material of an object in a scene, the permittivity sensor comprising: an input interface configured to accept an apn-phase measurement of a propagation of a known incident field through the scene and scattered by the material of the object in the scene; a hardware processor configured to solve a multivariate minimization problem with respect to an unknown phase and an unknown map of a distribution of a permittivity of the material of the object by minimizing a difference between the known incident field and a nonlinear function of the unknown map of the permittivity of the material of the object and a product of a known amplitude of the apn-phase measurement and an unknown phase of the apn-phase measurement; and an output interface configured to present a map of a distribution of the permittivity of the material of the object provided by a solution of the multivariate minimization problem.

2. The permittivity sensor of claim 1, wherein, The processor is configured to solve the multivariate minimization problem using simultaneous multivariate minimization with respect to two unknowns by updating the two unknowns for each iteration.

3. The permittivity sensor of claim 1, wherein, The multivariate minimization problem further includes additional constraints on the unknown map and the unknown phase.

4. The dielectric constant sensor of claim 2, wherein, The simultaneous multivariate minimization uses a multivariate proximal gradient method.

5. The dielectric constant sensor of claim 4, wherein, The multivariate proximal gradient method is implemented using a fast iterative shrinkage thresholding algorithm (FISTA).

6. The permittivity sensor of claim 1, wherein, The processor is configured to solve the multivariate minimization problem using alternating minimization.

7. The permittivity sensor of claim 1, wherein, The output interface presents the map of a distribution of the permittivity of the material of the object on a display device.

8. The permittivity sensor of claim 1, wherein, The output interface presents the permittivity of the material of the object to a civil infrastructure system configured to detect at least one of a defect of the material of the object, a cavity of a civil infrastructure, or an invisible object.

9. The permittivity sensor of claim 8, wherein, The civil infrastructure system is configured to detect the invisible object including a pipe located underground, and wherein the permittivity sensor is arranged on a mobile platform traveling on a ground surface above the underground pipe, and wherein the civil infrastructure system is configured to generate a map of a distribution of the underground pipe.

10. The dielectric constant sensor of claim 8, wherein, The civil infrastructure system is configured to detect the cavity of the object of the civil infrastructure including a pipe, and wherein the permittivity sensor is arranged on a mobile platform traveling through the pipe, and wherein the civil infrastructure system is configured to generate a map of a distribution of the cavity of the pipe.

11. The permittivity sensor of claim 1, wherein, The permittivity of the material of the object corresponds to at least one of an internal structure of the object or an external structure of the object.

12. A method for determining a map of a distribution of a permittivity of a material of an object in a scene, the method comprising the steps of: accepting an apn-phase measurement of a propagation of a known incident field through the scene and scattered by the material of the object in the scene; solving a multivariate minimization problem with respect to an unknown phase and an unknown map of a distribution of a permittivity of the material of the object by minimizing a difference between the known incident field and a nonlinear function of the unknown map of the permittivity of the material of the object and a product of a known amplitude of the apn-phase measurement and an unknown phase of the apn-phase measurement; and presenting an image of a distribution of a dielectric constant of a material of the object provided by a solution of the multivariate minimization problem.

13. The method of claim 12, wherein, the multivariate minimization problem further comprises additional constraints on the unknown image and the unknown phase.

Images