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The inverse obstacle scattering with incident tapered waves

Deyue Zhang, Mengjiao Bai, Yan Chang, Yukun Guo

TL;DR

The paper addresses the 2D acoustic obstacle reconstruction problem by employing Thorsos tapered waves with width controlled by $\lambda$ to achieve local boundary illumination. A direct imaging scheme uses an indicator function $I(z; d)$ derived under the physical optics approximation to extract boundary points from near-field data $u^s(x; d)$, with local maxima corresponding to the illuminated boundary patch $\Gamma_d$. The authors provide a stability analysis and demonstrate robust, high-resolution reconstructions for diverse shapes and apertures, including noisy and limited-view scenarios, highlighting the method's speed and ease of implementation. The approach holds promise for rapid boundary recovery and can be extended to 3D and to other wave types such as electromagnetic or elastic waves.

Abstract

This paper is concerned with the reconstruction of the shape of an acoustic obstacle. Based on the use of the tapered waves with very narrow widths illuminating the obstacle, the boundary of the obstacle is reconstructed by a direct imaging algorithm. The stability of the imaging scheme is mathematically analyzed. We emphasize that different from the incident plane waves or point sources, the tapered waves with narrow widths bring several benefits in the inverse scattering: 1. local property. A tapered wave can illuminate only on a local part of the boundary of the obstacle, which generates the scattered field; 2. high resolution. We need only reconstruct the boundary near the beam, which improves the quality of some well-known algorithms; 3. fast and easy to implement. Numerical examples are included to demonstrate the effectiveness of the tapered waves.

The inverse obstacle scattering with incident tapered waves

TL;DR

The paper addresses the 2D acoustic obstacle reconstruction problem by employing Thorsos tapered waves with width controlled by to achieve local boundary illumination. A direct imaging scheme uses an indicator function derived under the physical optics approximation to extract boundary points from near-field data , with local maxima corresponding to the illuminated boundary patch . The authors provide a stability analysis and demonstrate robust, high-resolution reconstructions for diverse shapes and apertures, including noisy and limited-view scenarios, highlighting the method's speed and ease of implementation. The approach holds promise for rapid boundary recovery and can be extended to 3D and to other wave types such as electromagnetic or elastic waves.

Abstract

This paper is concerned with the reconstruction of the shape of an acoustic obstacle. Based on the use of the tapered waves with very narrow widths illuminating the obstacle, the boundary of the obstacle is reconstructed by a direct imaging algorithm. The stability of the imaging scheme is mathematically analyzed. We emphasize that different from the incident plane waves or point sources, the tapered waves with narrow widths bring several benefits in the inverse scattering: 1. local property. A tapered wave can illuminate only on a local part of the boundary of the obstacle, which generates the scattered field; 2. high resolution. We need only reconstruct the boundary near the beam, which improves the quality of some well-known algorithms; 3. fast and easy to implement. Numerical examples are included to demonstrate the effectiveness of the tapered waves.
Paper Structure (5 sections, 3 theorems, 29 equations, 11 figures, 1 table)

This paper contains 5 sections, 3 theorems, 29 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. The inverse scattering and physical optics approximation
  3. Physical optics approximation and the indicator functions
  4. Numerical examples
  5. Conclusion

Key Result

Lemma 2.1

For the scattering of a tapered wave incidence $u^i$ from an obstacle $D$, we have

Figures (11)

  • Figure 1: An illustration of the tapered wave $(k=25,\theta =\frac{3}{4}\pi)$.
  • Figure 2: An illustration of the inverse obstacle scattering problem.
  • Figure 3: Reconstruction of the circle-shaped obstacle with $k=25.$
  • Figure 4: The sampling process for a single incident direction. (The first row: $j=5$; the second row: $j=100$, the third row: $j=237,$ the last row: $j=430$).
  • Figure 5: Reconstruction of the kite-shaped obstacle for $k=20.$
  • ...and 6 more figures

Theorems & Definitions (9)

  • Lemma 2.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4