# Reconstruction of piecewise constant layered conductivities in electrical impedance tomography

## TL;DR

This paper introduces a new constructive proof and a practical reconstruction method for identifying piecewise constant layered conductivities in electrical impedance tomography, leveraging monotonicity principles for efficient numerical implementation.

## Contribution

It provides a novel constructive uniqueness proof and a modified monotonicity-based reconstruction method for PCLC conductivities, overcoming limitations of existing inclusion detection techniques.

## Key findings

- The method can reconstruct layered conductivities using local Cauchy data.
- It decomposes conductivities into nested layers of perturbations for layer-by-layer determination.
- The approach is efficient and adaptable to electrode models.

## Abstract

This work presents a new constructive uniqueness proof for Calder\'on's inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call "piecewise constant layered conductivities" (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that "holes in inclusions" or "inclusions-within-inclusions" cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.