Obstacle and Boundary Determination from Scattering Data

Abstract

In this paper, we are concerned with the identification of complex obstacles from the scattering data for the acoustic problem. The complex obstacle is characterized by its shape and the boundary values of the impedance coefficient. We establish point-wise formulas which can be used to reconstruct the shape of the obstacle and give explicitly the values of the surface impedance as a function of the far field. In addition, these formulas enable us to distinguish and recognize the coated and the non-coated parts of the obstacle.

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

“…More explicitly, in the original version of the probe method, the indicator function is constructed by firstly computing the D-to-N map (1.5) for unknown D from the far-field data {w ∞ (θ, d) : θ, d ∈ S}, with suitable choices of and f related to some detection point z outside D. We then get the profile of the scatterer by capturing the blow-up behavior of the indicator function as z → ∂ D. If the scatterer is an obstacle with the boundary of impedance type, then we can also recover the surface impedance from the Dto-N map by moment method as suggested in [1]. Hence, the reconstruction of the D-to-N map from the far-field measurements is very interesting and important in inverse scattering problems, although the indicator function can be constructed directly from the far-field data as done in [12][13][14]. Using Green's representation theorem and properties of layer potentials, this problem can be reduced to solving an integral equation of the second kind, where the kernels are singular and involve the derivatives of the scattered waves for point sources.…”

Reconstruction of shapes and impedance functions using few far-field measurements

“…This is a rather weak assumption and it can be given naturally in some real cases that the location of the obstacle is known. It can also be obtained by other direct and non-iterative imaging methods, such as [15] which uses full far-field data and multiple frequencies to obtain accurate shape reconstructions, or [10] which uses topological derivatives to obtain rough shape reconstructions from full or partial far-field data, or [23,20] which use full far-field data to reconstruct an approximation of the complex obstacle ð@D; rÞ. Particularly, we can further extend our work by using the reconstructions of the complex obstacle ð@D; rÞ from [23,20] as the initial guess for the shape and the impedance function.…”

On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates

“…In the original version of this scheme, the indicator is constructed by firstly computing the D-to-N map (5) using the far-field data for suitably chosen Ω and boundary value f . Here, we would like to point out that the indicator in the probe method can also be constructed directly from the far-field pattern in one step [15], which is firstly noticed in [17] and then is developed in [16]. For other reconstruction schemes for the obstacle with impedance boundary, we refer to [3] and the references therein.…”

“…These numerical implementations did not involve the process of reconstructing Dto-N map from the far-field data, which is the first step in the initial version of probe method using the physical measurement data. Although the probe method can use an indicator constructed from the far-field data directly as done in [15][16][17], the original version based on the D-to-N map is still of great importance for the thorough understanding on the probe method, i.e., the probe method essentially extracts the information about the Green function of the boundary value problem from the far field data to reconstruct the obstacle boundary. Moreover, to the authors' knowledge, the numerical implementations of D-to-N map from the far-field pattern have not yet been realized efficiently up to now.…”