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A theoretical and computational framework for three dimensional inverse medium scattering using the linearized low-rank structure

Yuyuan Zhou, Lorenzo Audibert, Shixu Meng, Bo Zhang

TL;DR

This work addresses 3D inverse medium scattering by leveraging a data-driven basis formed by 3D prolate spheroidal wave functions (3D PSWFs) to induce a low-rank representation of the inverse problem. It establishes a theory with a customized Sobolev space $H_c^s$, a priori and stability estimates, and a regularization scheme using a $H_c^s$ penalty, all grounded in the dual eigenstructure of 3D PSWFs as both a restricted Fourier operator and a Sturm–Liouville operator. The computational framework computes the 3D PSWFs, their prolate eigenvalues, and reconstructs the contrast by projecting data onto the low-rank space, enabling efficient and robust inversion as well as a localized imaging capability via double orthogonality. Numerical experiments with Born-processed and full far-field data demonstrate accurate reconstructions, favorable robustness and speed relative to iterative methods, and effective localization of targeted objects within complex surroundings. Overall, the paper presents a theoretically sound and computationally practical pipeline for 3D inverse scattering that combines spectral PSWF-based low-rank modeling with novel regularization and localization techniques.

Abstract

In this work we propose a theoretical and computational framework for solving the three dimensional inverse medium scattering problem, based on a set of data-driven basis arising from the linearized problem. This set of data-driven basis consists of generalizations of prolate spheroidal wave functions to three dimensions (3D PSWFs), the main ingredients to explore a low-rank approximation of the inverse solution. We first establish the fundamentals of the inverse scattering analysis, including regularity in a customized Sobolev space and new a priori estimate. This is followed by a computational framework showcasing computing the 3D PSWFs and the low-rank approximation of the inverse solution. These results rely heavily on the fact that the 3D PSWFs are eigenfunctions of both a restricted Fourier integral operator and a Sturm-Liouville differential operator. Furthermore we propose a Tikhonov regularization method with a customized penalty norm and a localized imaging technique to image a targeting object despite the possible presence of its surroundings. Finally various numerical examples are provided to demonstrate the potential of the proposed method.

A theoretical and computational framework for three dimensional inverse medium scattering using the linearized low-rank structure

TL;DR

This work addresses 3D inverse medium scattering by leveraging a data-driven basis formed by 3D prolate spheroidal wave functions (3D PSWFs) to induce a low-rank representation of the inverse problem. It establishes a theory with a customized Sobolev space , a priori and stability estimates, and a regularization scheme using a penalty, all grounded in the dual eigenstructure of 3D PSWFs as both a restricted Fourier operator and a Sturm–Liouville operator. The computational framework computes the 3D PSWFs, their prolate eigenvalues, and reconstructs the contrast by projecting data onto the low-rank space, enabling efficient and robust inversion as well as a localized imaging capability via double orthogonality. Numerical experiments with Born-processed and full far-field data demonstrate accurate reconstructions, favorable robustness and speed relative to iterative methods, and effective localization of targeted objects within complex surroundings. Overall, the paper presents a theoretically sound and computationally practical pipeline for 3D inverse scattering that combines spectral PSWF-based low-rank modeling with novel regularization and localization techniques.

Abstract

In this work we propose a theoretical and computational framework for solving the three dimensional inverse medium scattering problem, based on a set of data-driven basis arising from the linearized problem. This set of data-driven basis consists of generalizations of prolate spheroidal wave functions to three dimensions (3D PSWFs), the main ingredients to explore a low-rank approximation of the inverse solution. We first establish the fundamentals of the inverse scattering analysis, including regularity in a customized Sobolev space and new a priori estimate. This is followed by a computational framework showcasing computing the 3D PSWFs and the low-rank approximation of the inverse solution. These results rely heavily on the fact that the 3D PSWFs are eigenfunctions of both a restricted Fourier integral operator and a Sturm-Liouville differential operator. Furthermore we propose a Tikhonov regularization method with a customized penalty norm and a localized imaging technique to image a targeting object despite the possible presence of its surroundings. Finally various numerical examples are provided to demonstrate the potential of the proposed method.
Paper Structure (22 sections, 5 theorems, 78 equations, 9 figures, 2 algorithms)

This paper contains 22 sections, 5 theorems, 78 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical model of inverse scattering in three dimensions
  3. A priori estimate and reconstruction formula
  4. Generalization of prolate spheroidal wave functions
  5. Customized Sobolev space and approximation theory
  6. Low-rank approximation and a priori estimate
  7. A versatile tool for regularization using $H^s_c$ penalty term
  8. A Localized imaging technique
  9. Computational framework
  10. Computation of 3D PSWFs and prolate eigenvalues
  11. Computation of inverse solution in the low-rank space
  12. Numerical experiments
  13. Data generation
  14. Born processed data
  15. Far field data
  16. ...and 7 more sections

Key Result

Lemma 1

Let $\epsilon>0$ be a small positive parameter, and $\pi_\epsilon u$ be the approximation given by section: theory approximation in H_c^s, then for any $u \in H_c^s(B)$, it holds that

Figures (9)

  • Figure 1: Reconstruction of three cubes with noiseless processed data, $k=15,~T=23,~M_\theta=31,~M_\phi=61$. The first row (a) ground truth, (b) (c) isosurfaces of reconstruction (real part) with level value $0.5M,~0.65M$, where $M=\hbox{max}~q^{\sigma,\delta}$; the second (resp. third) row, cross section view of the real (resp. imaginary) part.
  • Figure 2: Reconstruction of three cubes with noisy processed data, $\delta=0.2$ and all other parameters are the same as Figure \ref{['figure: Three cubes, analytic data, noise free']}.
  • Figure 3: Reconstruction of three contrasts with Born far field data with $k=15,~\delta=0.2$, and $N_1=N_2=201$. The first row: isosurface view of ground truth; the second row: isosurface of reconstruction; the third row: cross section view of the reconstruction of a ball, a cube and the oscillatory contrast, respectively.
  • Figure 4: Reconstruction of "cross3D" using full far field data with observation and incident direction pairs $N_1\times N_2=101\times 101$ and noise level $\delta=0.2$. The modeling error is ${\rm rel}(k)=1.82\%,~3.28\%,~4.75\%$, respectively for $~k=10,~15,~20$. The first column: ground truth. The second, third, and fourth columns: reconstructions using $~k=10,~15,~20$, respectively. From the top to the bottom row: isosurface and the cross section views. The width of each bar is $1/8$, and the length is $7/8$. The protruding part on the shorter side is $1/4$ in length.
  • Figure 5: Reconstruction of three cubes using full far field data with observation and incident direction pairs $N_1\times N_2=101\times 101$ and noise level $\delta=0.2$. The modeling error is ${\rm rel}(k)=1.82\%,~3.28\%,~9.15\%,~11.13\%$ for $~k=10,~15,~20,~25$, respectively.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof