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A novel direct imaging method for passive inverse obstacle scattering problem

Yunwen Yin, Liang Yan

TL;DR

This work tackles passive inverse obstacle scattering where incident sources are random and uncontrolled, which undermines traditional direct sampling. It introduces the Doubly Cross-Correlating Method (DCM), which forms a cross-correlation from two passive measurements and uses the Helmholtz-Kirchhoff identity to connect to active scattering, yielding a correlation-based imaging functional akin to reverse-time migration. The paper provides rigorous resolution and stability analyses and demonstrates through extensive numerical experiments that DCM is computationally efficient, robust to noise, and capable of resolving single, multiple, and multiscale obstacles, including closely spaced features at higher wavenumbers. The method offers a practical, fast alternative to forward solvers and has potential extensions to three-dimensional settings and other obstacle types.

Abstract

This paper investigates the inverse scattering problem of recovering a sound-soft obstacle using passive measurements taken from randomly distributed point sources. The randomness introduced by these sources poses significant challenges, leading to the failure of classical direct sampling methods that rely on scattered field measurements. To address this issue, we introduce the Doubly Cross-Correlating Method (DCM), a novel direct imaging scheme that consists of two major steps. Initially, DCM creates a cross-correlation between two passive measurements. This specially designed cross-correlation effectively handles the uncontrollability of incident sources and connects to the active scattering model via the Helmholtz-Kirchhoff identity. Subsequently, this cross-correlation is used to create a correlation-based imaging function that can qualitatively identify the obstacle. The stability and resolution of DCM are theoretically analyzed. Extensive numerical examples, including scenarios with two closely positioned obstacles and multiscale obstacles, demonstrate that DCM is computationally efficient, stable, and fast.

A novel direct imaging method for passive inverse obstacle scattering problem

TL;DR

This work tackles passive inverse obstacle scattering where incident sources are random and uncontrolled, which undermines traditional direct sampling. It introduces the Doubly Cross-Correlating Method (DCM), which forms a cross-correlation from two passive measurements and uses the Helmholtz-Kirchhoff identity to connect to active scattering, yielding a correlation-based imaging functional akin to reverse-time migration. The paper provides rigorous resolution and stability analyses and demonstrates through extensive numerical experiments that DCM is computationally efficient, robust to noise, and capable of resolving single, multiple, and multiscale obstacles, including closely spaced features at higher wavenumbers. The method offers a practical, fast alternative to forward solvers and has potential extensions to three-dimensional settings and other obstacle types.

Abstract

This paper investigates the inverse scattering problem of recovering a sound-soft obstacle using passive measurements taken from randomly distributed point sources. The randomness introduced by these sources poses significant challenges, leading to the failure of classical direct sampling methods that rely on scattered field measurements. To address this issue, we introduce the Doubly Cross-Correlating Method (DCM), a novel direct imaging scheme that consists of two major steps. Initially, DCM creates a cross-correlation between two passive measurements. This specially designed cross-correlation effectively handles the uncontrollability of incident sources and connects to the active scattering model via the Helmholtz-Kirchhoff identity. Subsequently, this cross-correlation is used to create a correlation-based imaging function that can qualitatively identify the obstacle. The stability and resolution of DCM are theoretically analyzed. Extensive numerical examples, including scenarios with two closely positioned obstacles and multiscale obstacles, demonstrate that DCM is computationally efficient, stable, and fast.
Paper Structure (16 sections, 4 theorems, 50 equations, 13 figures)

This paper contains 16 sections, 4 theorems, 50 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Direct scattering problem
  4. Passive inverse obstacle scattering problem
  5. The doubly cross-correlating method
  6. DCM for the inverse problem
  7. The resolution analysis and stability analysis
  8. Numerical experiments
  9. Example 1: Single obstacle case
  10. The influence of the perturbation $\xi$ and the number $L$ of incident random sources
  11. The influence of the wavenumber $k$
  12. The influence of the noise level $\delta$
  13. Example 2: Multiple obstacles case
  14. Tow obstacles close to each other
  15. Multiscale
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $\widetilde{\omega}\in H^{1/2}(\partial D)$, the forward obstacle scattering problem: admits a unique solution $\omega\in H^{1}_{loc}(\mathbb{R}^2 \backslash \overline{D})$. Furthermore, there exists a constant $C\in\mathbb{R}_{+}$ such that where $\nu$ denotes the unit outer normal to $\partial D$.

Figures (13)

  • Figure 1: In the passive imaging (Left), the incident sources are random and uncontrolled, but in the active imaging (Right), the incident sources are controlled and fixed.
  • Figure 2: The imaginary parts of $C_{jm}$ and $N^{s}_{jm}$ with $\xi=0.4$ and $\xi=2$ in the case of $L=64$ for the kite (Top) and the peanut (Bottom). The first column displays the imaginary parts of $C_{jm}$ with $\xi=0.4$, the second column displays the imaginary parts of $C_{jm}$ with $\xi=2$ and the third column displays the imaginary parts of $N^{s}_{jm}$.
  • Figure 3: Recoveries of $\partial D$ by DCM with $\xi=0.4$ and $\xi=2$ in the case of $L=64$ for the kite (Top) and the peanut (Bottom). The first column shows the exact boundary $\partial D$ and $L=64$ measurement points, the second column shows the reconstructed $\partial D$ with $\xi=0.4$ and the third column shows the reconstructed $\partial D$ with $\xi=2$.
  • Figure 4: The imaginary parts of $C_{jm}$ and $N^{s}_{jm}$ with $\xi=0.4$ and $\xi=2$ in the case of $L=256$ for the kite (Top) and the peanut (Bottom). The first column displays the imaginary parts of $C_{jm}$ with $\xi=0.4$, the second column displays the imaginary parts of $C_{jm}$ with $\xi=2$ and the third column displays the imaginary parts of $N^{s}_{jm}$.
  • Figure 5: Recoveries of $\partial D$ by DCM with $\xi=0.4$ and $\xi=2$ in the case of $L=256$ for the kite (Top) and the peanut (Bottom). The first column shows the exact boundary $\partial D$ and $L=256$ measurement points, the second column shows the reconstructed $\partial D$ with $\xi=0.4$ and the third column shows the reconstructed $\partial D$ with $\xi=2$.
  • ...and 8 more figures

Theorems & Definitions (8)

  • Lemma 3.1: DRIAMcLean1
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof