Pub Date : 2024-07-25DOI: 10.1088/1361-6420/ad5e18
Lorenzo Audibert and Shixu Meng
In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.
{"title":"Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator","authors":"Lorenzo Audibert and Shixu Meng","doi":"10.1088/1361-6420/ad5e18","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5e18","url":null,"abstract":"In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-24DOI: 10.1088/1361-6420/ad5fb1
Lionel Tondji, Idriss Tondji and Dirk Lorenz
We consider the block Bregman–Kaczmarz method for finite dimensional linear inverse problems. The block Bregman–Kaczmarz method uses blocks of the linear system and performs iterative steps with these blocks only. We assume a noise model that we call independent noise, i.e. each time the method performs a step for some block, one obtains a noisy sample of the respective part of the right-hand side which is contaminated with new noise that is independent of all previous steps of the method. One can view these noise models as making a fresh noisy measurement of the respective block each time it is used. In this framework, we are able to show that a well-chosen adaptive stepsize of the block Bregman–Kaczmarz method is able to converge to the exact solution of the linear inverse problem. The plain form of this adaptive stepsize relies on unknown quantities (like the Bregman distance to the solution), but we show a way how these quantities can be estimated purely from given data. We illustrate the finding in numerical experiments and confirm that these heuristic estimates lead to effective stepsizes.
{"title":"Adaptive Bregman–Kaczmarz: an approach to solve linear inverse problems with independent noise exactly","authors":"Lionel Tondji, Idriss Tondji and Dirk Lorenz","doi":"10.1088/1361-6420/ad5fb1","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5fb1","url":null,"abstract":"We consider the block Bregman–Kaczmarz method for finite dimensional linear inverse problems. The block Bregman–Kaczmarz method uses blocks of the linear system and performs iterative steps with these blocks only. We assume a noise model that we call independent noise, i.e. each time the method performs a step for some block, one obtains a noisy sample of the respective part of the right-hand side which is contaminated with new noise that is independent of all previous steps of the method. One can view these noise models as making a fresh noisy measurement of the respective block each time it is used. In this framework, we are able to show that a well-chosen adaptive stepsize of the block Bregman–Kaczmarz method is able to converge to the exact solution of the linear inverse problem. The plain form of this adaptive stepsize relies on unknown quantities (like the Bregman distance to the solution), but we show a way how these quantities can be estimated purely from given data. We illustrate the finding in numerical experiments and confirm that these heuristic estimates lead to effective stepsizes.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-23DOI: 10.1088/1361-6420/ad6284
Travis Askham and Carlos Borges
In inverse scattering problems, a model that allows for the simultaneous recovery of both the domain shape and an impedance boundary condition covers a wide range of problems with impenetrable domains, including recovering the shape of sound-hard and sound-soft obstacles and obstacles with thin coatings. This work develops an optimization framework for recovering the shape and material parameters of a penetrable, dissipative obstacle in the multifrequency setting, using a constrained class of curvature-dependent impedance function models proposed by Antoine et al (2001 Asymptotic Anal.26 257–83). We find that in certain regimes this constrained model improves the robustness of the recovery problem, compared to more general models, and provides meaningfully better obstacle recovery than simpler models. We explore the effectiveness of the model for varying levels of dissipation, for noise-corrupted data, and for limited aperture data in the numerical examples.
在反向散射问题中，允许同时恢复领域形状和阻抗边界条件的模型涵盖了具有不可穿透领域的各种问题，包括恢复声硬和声软障碍物以及具有薄涂层的障碍物的形状。本研究利用 Antoine 等人提出的一类受约束的曲率依赖阻抗函数模型（2001 Asymptotic Anal.26，257-83），建立了一个优化框架，用于在多频环境下恢复可穿透耗散障碍物的形状和材料参数。我们发现，在某些情况下，与更一般的模型相比，这种约束模型提高了恢复问题的鲁棒性，与更简单的模型相比，障碍物恢复效果更好。我们在数值示例中探讨了该模型对不同程度的耗散、噪声干扰数据和有限孔径数据的有效性。
{"title":"Reconstructing the shape and material parameters of dissipative obstacles using an impedance model","authors":"Travis Askham and Carlos Borges","doi":"10.1088/1361-6420/ad6284","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6284","url":null,"abstract":"In inverse scattering problems, a model that allows for the simultaneous recovery of both the domain shape and an impedance boundary condition covers a wide range of problems with impenetrable domains, including recovering the shape of sound-hard and sound-soft obstacles and obstacles with thin coatings. This work develops an optimization framework for recovering the shape and material parameters of a penetrable, dissipative obstacle in the multifrequency setting, using a constrained class of curvature-dependent impedance function models proposed by Antoine et al (2001 Asymptotic Anal.26 257–83). We find that in certain regimes this constrained model improves the robustness of the recovery problem, compared to more general models, and provides meaningfully better obstacle recovery than simpler models. We explore the effectiveness of the model for varying levels of dissipation, for noise-corrupted data, and for limited aperture data in the numerical examples.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-17DOI: 10.1088/1361-6420/ad602e
Alen Alexanderian, Ruanui Nicholson and Noemi Petra
We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.
我们考虑了在模型不确定的情况下，由偏微分方程（PDEs）控制的贝叶斯非线性逆问题的最优实验设计（OED）。具体来说，我们考虑的反演问题中，除了反演参数外，支配偏微分方程的还包括次要不确定参数。我们将重点放在具有无限维反演和次要参数的问题上，并为此类问题的优化设计提出了一个可扩展的计算框架。所提出的方法可在统一框架内实现不确定条件下的贝叶斯反演和 OED。我们以贝叶斯近似误差（BAE）方法为基础，将建模不确定性纳入贝叶斯反演问题，并提出了无穷维贝叶斯非线性反演问题的 A 优化设计方法。具体来说，最大后验概率点的后验高斯近似用于定义不确定性感知 OED 目标，该目标易于评估和优化。特别是，OED 目标的计算成本（PDE 求解次数）不会随着离散反演和次要参数维度的增加而增加。OED 问题被表述为一个二元双级 PDE 受限优化问题，而贪婪算法提供了一种务实的方法，用于寻找最优设计。我们在三维域上演示了由椭圆 PDE 控制的模型逆向问题的拟议方法的有效性。我们的计算结果还强调了在 OED 和/或推理阶段忽略建模不确定性的缺陷。
{"title":"Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty","authors":"Alen Alexanderian, Ruanui Nicholson and Noemi Petra","doi":"10.1088/1361-6420/ad602e","DOIUrl":"https://doi.org/10.1088/1361-6420/ad602e","url":null,"abstract":"We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141739684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-10DOI: 10.1088/1361-6420/ad5e19
Haie Long, Ye Zhang and Guangyu Gao
With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.
{"title":"An accelerated inexact Newton regularization scheme with a learned feature-selection rule for non-linear inverse problems","authors":"Haie Long, Ye Zhang and Guangyu Gao","doi":"10.1088/1361-6420/ad5e19","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5e19","url":null,"abstract":"With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-09DOI: 10.1088/1361-6420/ad5d0d
Yijie Yang, Qifeng Gao and Yuping Duan
The unrolling method has been investigated for learning variational models in x-ray computed tomography. However, for incomplete data reconstruction, such as sparse-view and limited-angle problems, the unrolling method of gradient descent of the energy minimization problem cannot yield satisfactory results. In this paper, we present an effective CT reconstruction model, where the low-resolution image is introduced as a regularization for incomplete data problems. In what follows, we utilize the deep equilibrium approach to unfolding of the gradient descent algorithm, thereby constructing the backbone network architecture for solving the minimization model. We theoretically discuss the convergence of the proposed low-resolution prior equilibrium (LRPE) model and provide the necessary conditions to guarantee its convergence. Experimental results on both sparse-view and limited-angle reconstruction problems are provided, demonstrating that our end-to-end LRPE model outperforms other state-of-the-art methods in terms of noise reduction, contrast-to-noise ratio, and preservation of edge details.
在 X 射线计算机断层扫描中，已经研究了用于学习变分模型的展开方法。然而，对于不完整数据重建，如稀疏视图和有限角度问题，能量最小化问题梯度下降的展开法无法获得令人满意的结果。本文提出了一种有效的 CT 重建模型，其中引入了低分辨率图像作为不完整数据问题的正则化。接下来，我们利用深度均衡方法来展开梯度下降算法，从而构建出求解最小化模型的骨干网络架构。我们从理论上讨论了所提出的低分辨率先验均衡（LRPE）模型的收敛性，并提供了保证其收敛性的必要条件。我们提供了稀疏视图和有限角度重建问题的实验结果，证明我们的端到端 LRPE 模型在降噪、对比度-噪声比和边缘细节保留方面优于其他最先进的方法。
{"title":"Low-resolution prior equilibrium network for CT reconstruction","authors":"Yijie Yang, Qifeng Gao and Yuping Duan","doi":"10.1088/1361-6420/ad5d0d","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5d0d","url":null,"abstract":"The unrolling method has been investigated for learning variational models in x-ray computed tomography. However, for incomplete data reconstruction, such as sparse-view and limited-angle problems, the unrolling method of gradient descent of the energy minimization problem cannot yield satisfactory results. In this paper, we present an effective CT reconstruction model, where the low-resolution image is introduced as a regularization for incomplete data problems. In what follows, we utilize the deep equilibrium approach to unfolding of the gradient descent algorithm, thereby constructing the backbone network architecture for solving the minimization model. We theoretically discuss the convergence of the proposed low-resolution prior equilibrium (LRPE) model and provide the necessary conditions to guarantee its convergence. Experimental results on both sparse-view and limited-angle reconstruction problems are provided, demonstrating that our end-to-end LRPE model outperforms other state-of-the-art methods in terms of noise reduction, contrast-to-noise ratio, and preservation of edge details.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-09DOI: 10.1088/1361-6420/ad5d0e
Alfred K Louis
Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems38 065008) containing the results from Louis (2022 Inverse Problems38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.
大多数 X 射线或拉顿变换反演公式的推导都是基于投影定理，即在固定方向上计算 X 射线或拉顿变换的傅里叶变换，并将其与搜索函数的傅里叶变换进行比较。与此相反，我们在这里从搜索到的场开始，计算其固定方向的傅里叶变换，现在它是一个矢量或张量场，然后我们在一个合适的与方向相关的基础上对其进行扩展。展开系数分别被视为纵向、横向或混合射线变换的傅里叶变换或矢量拉顿变换。搜索场的反傅里叶变换可直接得出这些变换的反演公式，并应用与问题相适应的反推。在考虑场的亥姆霍兹分解时，我们可以立即找到这些横向或纵向变换的反演公式。Natterer-Wübbeling 于 2001 年给出了纵向射线变换的反演公式，Natterer 和 Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM))也给出了与 Natterer（1986 The Mathematics of Computerized Tomography (Teubner and Wiley)）类似的 X 射线断层摄影反演公式，但其他作者并未继续研究。在本文中，我们介绍了上述方法，并以统一的方式推导出路易斯（2022 逆问题 38 065008）中处理的射线变换的反演公式，其中包含路易斯（2022 逆问题 38 065008）中作为特例的结果。此外，我们还提出了矢量拉顿变换的新反演公式。因此，反演公式直接给出了矢量或张量变换的 Plancherel 公式。通过与射线或拉顿变换无关的纳特勒不等式，我们提出了这些矢量和张量变换的纳特勒稳定性。
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Pub Date : 2024-07-09DOI: 10.1088/1361-6420/ad5a35
Martin Burger, Thomas Schuster, Anne Wald
We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.
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Pub Date : 2024-07-09DOI: 10.1088/1361-6420/ad5b83
F M Watson, D Andre, W R B Lionheart
Through-wall synthetic aperture radar (SAR) imaging is of significant interest for security purposes, in particular when using multi-static SAR systems consisting of multiple distributed radar transmitters and receivers to improve resolution and the ability to recognise objects. Yet there is a significant challenge in forming focused, useful images due to multiple scattering effects through walls, whereas standard SAR imaging has an inherent single scattering assumption. This may be exacerbated with multi-static collections, since different scattering events will be observed from each angle and the data may not coherently combine well in a naive manner. To overcome this, we propose an image formation method which resolves full-wave effects through an approximately known wall or other arbitrary obstacle, which itself has some unknown ‘nuisance’ parameters that are determined as part of the reconstruction to provide well focused images. The method is more flexible and realistic than existing methods which treat a single wall as a flat layered medium, whilst being significantly computationally cheaper than full-wave methods, strongly motivated by practical considerations for through-wall SAR.
穿墙合成孔径雷达（SAR）成像在安全领域具有重要意义，尤其是在使用由多个分布式雷达发射器和接收器组成的多静态 SAR 系统来提高分辨率和识别物体的能力时。然而，由于穿墙的多重散射效应，而标准合成孔径雷达成像具有固有的单散射假设，因此在形成聚焦、有用的图像方面存在巨大挑战。由于从每个角度都会观察到不同的散射事件，数据可能无法以天真的方式连贯组合，因此多静态采集可能会加剧这一问题。为了克服这一问题，我们提出了一种图像形成方法，通过近似已知的墙壁或其他任意障碍物来解决全波效应，这些障碍物本身有一些未知的 "干扰 "参数，这些参数将作为重建的一部分来确定，以提供聚焦良好的图像。这种方法比现有的将单面墙壁视为平面分层介质的方法更灵活、更逼真，同时计算成本也比全波方法低得多，这主要是出于对穿墙合成孔径雷达的实际考虑。
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Pub Date : 2024-07-03DOI: 10.1088/1361-6420/ad5b81
Tian Niu, Junliang Lv and Jiahui Gao
In this paper, we establish the uniqueness of identifying a smooth grating profile with a mixed boundary condition (MBC) or transmission boundary conditions (TBCs) from phaseless data. The existing uniqueness result requires the measured data to be in a bounded domain. To break this restriction, we design an incident system consisting of the superposition of point sources to reduce the measurement data from a bounded domain to a line above the grating profile. We derive reciprocity relations for point sources, diffracted fields, and total fields, respectively. Based on Rayleigh’s expansion and reciprocity relation of the total field, a grating profile with a MBC or TBCs can be uniquely determined from the phaseless total field data. An iterative algorithm is proposed to recover the Fourier modes of grating profiles at a fixed wavenumber. To implement this algorithm, we derive the Fréchet derivative of the total field operator and its adjoint operator. Some numerical examples are presented to verify the correctness of theoretical results and to show the effectiveness of our numerical algorithm.
在本文中，我们通过无相数据确定了具有混合边界条件（MBC）或传输边界条件（TBC）的光栅轮廓的唯一性。现有的唯一性结果要求测量数据位于有界域中。为了打破这一限制，我们设计了一个由点源叠加组成的入射系统，将测量数据从有界域减少到光栅轮廓上方的一条线上。我们分别推导出了点源、衍射场和总场的互易关系。根据雷利展开和总场的互易关系，可以从无相总场数据唯一确定具有 MBC 或 TBC 的光栅轮廓。我们提出了一种迭代算法来恢复光栅轮廓在固定波长下的傅里叶模式。为了实现这一算法，我们推导出了总场算子的弗雷谢特导数及其邻接算子。为了验证理论结果的正确性，并显示我们的数值算法的有效性，我们给出了一些数值示例。
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