# Calder\'on cavities inverse problem as a shape-from-moments problem

## TL;DR

This paper presents a non-iterative method to recover the shape and location of cavities in a 2D domain using the Dirichlet-to-Neumann map, by transforming the inverse problem into a shape-from-moments problem.

## Contribution

It introduces a novel approach combining generalized Pólia-Szegö tensors with shape-from-moments techniques for cavity reconstruction in inverse conductivity problems.

## Key findings

- Method effectively reconstructs cavity shapes from DtN data.
- Numerical results demonstrate efficiency for arbitrary geometries.
- The approach simplifies the inverse problem into a shape-from-moments framework.

## Abstract

In this paper, we address a particular case of Calder\'on's (or conductivity) inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e. heterogeneities of infinitely high conductivities). We aim to recover the location and the shape of the cavities from the knowledge of the Dirichlet-to-Neumann (DtN) map of the problem. The proposed reconstruction method is non iterative and uses two main ingredients. First, we show how to compute the so-called generalized P\'olia-Szeg\"o tensors (GPST) of the cavities from the DtN of the cavities. Secondly, we show that the obtained shape from GPST inverse problem can be transformed into a shape from moments problem, for some particular configurations. However, numerical results suggest that the reconstruction method is efficient for arbitrary geometries.