# Topological derivative-based technique for imaging thin inhomogeneities with few incident directions

## TL;DR

This paper provides a rigorous mathematical analysis of topological derivative imaging for thin inhomogeneities, demonstrating that few incident directions can be sufficient and identifying optimal configurations through Bessel function series representation.

## Contribution

It offers a theoretical foundation for topological derivative imaging with limited incident directions and explores optimal incident direction configurations.

## Key findings

- Few incident directions are sufficient for accurate imaging.
- The imaging function can be represented as a series of Bessel functions.
- Numerical simulations support the theoretical analysis.

## Abstract

Many non-iterative imaging algorithms require a large number of incident directions. Topological derivative-based imaging techniques can alleviate this problem, but lacks a theoretical background and a definite means of selecting the optimal incident directions. In this paper, we rigorously analyze the mathematical structure of a topological derivative imaging function, confirm why a small number of incident directions is sufficient, and explore the optimal configuration of these directions. To this end, we represent the topological derivative based imaging function as an infinite series of Bessel functions of integer order of the first kind. Our analysis is supported by the results of numerical simulations.