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A new method in inverse scattering based on the topological derivative. (English) Zbl 1077.78010
Summary: The problem of imaging objects embedded in a transparent homogeneous medium is considered. It is assumed that the wavelength of the probing radiation is finite so that scattering effects need to be taken into consideration in the reconstruction process. This problem is commonly referred to as ’inverse scattering’ in the literature. Many algorithms, including backpropagation-based algorithms, attempt to solve this imaging problem in either of two ways: (1) by assuming linearizing approximations such as the Born, Rytov or physical optics approximations which result in closed-form expressions for the inversion formula; or (2) by solving the nonlinear inverse scattering problem using an iterative algorithm, which is computationally more expensive. In this paper, a new method for inverse scattering is proposed. This method is based on the notion of the ’optimal topology’ that solves the inverse scattering problem. To find this optimal topology, a function called the topological derivative is defined. This function quantifies the sensitivity of the scattered field to the introduction of a small scatterer at a point in the domain. Based on this definition, and the heuristic that the boundary of the objects can be considered as a group of point scatterers, we will identify high values of this function with the location of these boundaries. It is shown that the topological derivative can be calculated analytically so, as a result, the proposed reconstruction algorithm is not iterative. In addition, no approximations (such as the Born, Rytov or physical optics approximations) to the wavefield are made. The numerical examples shown in this paper demonstrate that this simple and efficient heuristic scheme can be used to accurately reconstruct the shape of scatterers.
MSC:
78A46Inverse scattering problems
35R30Inverse problems for PDE
35P25Scattering theory (PDE)
65F30Other matrix algorithms
49Q10Optimization of shapes other than minimal surfaces
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation