Identifying convex obstacles from backscattering far field data

TL;DR

The paper proves that, for strictly convex obstacles, both the boundary geometry and the boundary condition can be uniquely determined from multi-frequency backscattering far-field data, via a high-frequency asymptotic expansion due to Majda. It then develops a fast, non-iterative reconstruction method that first identifies the boundary condition (or the impedance) using data from specially chosen backscattering configurations, and then images the obstacle boundary with direct sampling indicators, optionally using a secondary indicator that works without boundary information. Numerical experiments on egg- and kite-shaped domains demonstrate accurate reconstruction of both shape and impedance, including when the boundary is not known a priori, and show robustness to noise and limited data. The approach leverages the asymptotic link between far-field patterns and boundary data, enabling simultaneous recovery of geometric and physical properties from backscattering measurements. The work thus advances the theoretical and computational understanding of inverse obstacle problems in backscattering settings and provides practical tools for applications requiring joint shape and boundary-property identification.

Abstract

The recovery of anomalies from backscattering far field data is a long-standing open problem in inverse scattering theory. We make a first step in this direction by establishing the unique identifiability of convex impenetrable obstacles from backscattering far field measurements. Specifically, we prove that both the boundary and the boundary conditions of the convex obstacle are uniquely determined by the far field pattern measured in backscattering directions for all frequencies. The key tool is Majda's asymptotic estimate of the far field patterns in the high-frequency regime. Furthermore, we introduce a fast and stable numerical algorithm for reconstructing the boundary and computing the boundary condition. A key feature of the algorithm is that the boundary condition can be computed even if the boundary is not known, and vice versa. Numerical experiments demonstrate the validity and robustness of the proposed algorithm.

Figures (15)

• Figure 1: Physical optics diagram for incident direction $\theta$ and observation direction $\hat{x}$.
• Figure 2: As $M$ increases, the peak of finite $M$-term Fourier series gradually approaches discontinuous points.
• Figure 3: The values of $\mathcal{L}(\hat{x}, \alpha_1)$ for four different boundary conditions with frequency band $[20, 50]$.
• Figure 4: True and reconstructed $\lambda_1$ and $\lambda_2$ using \ref{['reconstruct-lambda']} with frequency band $[20, 50]$.
• Figure 5: Reconstructions of $\partial D$ with $\mathcal{I}(z)$ in different case using frequency band $[20, 50]$. Top: Plot of $\mathcal{I}(z)$. Bottom: Comparison with true $\partial D$ (red dotted curve).

Theorems & Definitions (15)

• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• proof : Proof of Lemma \ref{['lem-impe']}
• Theorem 3.5
• proof
• Remark 3.6
• Corollary 3.7