Two direct sampling methods for an anisotropic scatterer with a conductive boundary

TL;DR

This work tackles the inverse-shape problem for anisotropic scatterers with conductive boundaries by developing two direct sampling methods that rely on far-field or Cauchy data. A symmetric factorization F = H^*TH is established for the far-field operator, with the middle operator decomposed into a coercive plus compact part, yielding a decay estimate I(z) = O(dist(z,D)^{p(1-d)}). A reciprocity-gap-based sampling functional is introduced for Cauchy data, with a proven decay I_RG(z) = O(dist(z,D)^{p(1-d)/2}). Numerical tests in 2D confirm faithful reconstructions for circular and non-circular shapes under noise, using separation of variables and boundary-integral data generation. The methods are stable, fast, and require minimal a priori information, extending DSM techniques to anisotropic media with Robin-type boundary conditions and offering practical tools for shape recovery in scattering problems.

Abstract

In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary condition. We will assume that the corresponding far--field pattern or Cauchy data is either known or measured. The conductive boundary condition models a thin coating around the boundary of the scatterer. We will develop two direct sampling methods to solve the inverse shape problem by numerically recovering the scatterer. To this end, we study direct sampling methods by deriving that the corresponding imaging functionals decay as the sampling point moves away from the scatterer. These methods have been applied to other inverse shape problems, but this is the first time they will be applied to an anisotropic scatterer with a conductive boundary condition. These methods allow one to recover the scatterer by considering an inner--product of the far--field data or the Cauchy data. Here, we will assume that the Cauchy data is known on the boundary of a region $Ω$ that completely encloses the scatterer $D$. We present numerical reconstructions in two dimensions to validate our theoretical results for both circular and non-circular scatterers.

Figures (6)

• Figure 1: Reconstruction of circular domain with $5\%$ noise via far--field data with $p=2$ (left) and Cauchy data with $p=3$ (right).
• Figure 2: Reconstruction of circular domain with $5\%$ noise via far--field data with $p=2$ (left) and Cauchy data with $p=3$ (right).
• Figure 3: Reconstruction of the kite shape via far--field data with $5\%$ noise (left) and $10\%$ noise (right), each with $p=2$.
• Figure 4: Reconstruction of the kite shape via far--field data with $15\%$ noise (left) and $20\%$ noise (right), each with $p=2$.
• Figure 5: Reconstruction of the peanut shape via Cauchy data with $5\%$ noise (left) and $10\%$ noise (right), each with $p=3$.

Theorems & Definitions (14)

• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• Theorem 4.1
• proof
• Theorem 4.2
• proof