Inverse obstacle scattering problems with a single incident wave and the logarithmic differential of the indicator function in the Enclosure Method

TL;DR

This paper introduces a new approach using the logarithmic differential of the indicator function in the Enclosure Method, improving the accuracy and providing direct vertex coordinates of unknown polygonal obstacles from a single wave.

Contribution

It proposes a novel method employing the logarithmic differential of the indicator function, enhancing convergence and enabling direct extraction of obstacle vertices in inverse scattering.

Findings

Improved convergence rate over previous indicator functions

Direct determination of obstacle vertices from a single incident wave

Applicability to polygonal sound-hard obstacles and thin obstacles

Abstract

This paper gives a remark on the Enclosure Method by considering inverse obstacle scattering problems with a single incident wave whose governing equation is given by the Helmholtz equation in two dimensions. It is concerned with the indicator function in the Enclosure Method. The previous indicator function is essentially real-valued since only its absolute value is used. In this paper, another method for the use of the indicator function is introduced. The method employs the logarithmic differential with respect to the independent variable of the indicator function and yields directly the coordinates of the vertices of the convex hull of unknown polygonal sound-hard obstacles or thin ones. The convergence rate of the formulae is better than that of the previous indicator function. Some other applications of this method are also given.