# The regularized monotonicity method: detecting irregular indefinite inclusions

## TL;DR

This paper extends the monotonicity method for electrical impedance tomography to irregular indefinite inclusions, introducing a regularization scheme and a peeling algorithm, with numerical validation demonstrating robustness and accuracy.

## Contribution

It formulates a regularized monotonicity method for irregular indefinite inclusions without regularity assumptions, including a new peeling-type reconstruction algorithm and numerical examples.

## Key findings

- The method can reconstruct the outer support of indefinite inclusions independently.
- The regularization scheme ensures robustness against noise and modeling errors.
- Numerical examples demonstrate the effectiveness of the proposed approach.

## Abstract

In inclusion detection in electrical impedance tomography, the support of perturbations (inclusion) from a known background conductivity is typically reconstructed from idealized continuum data modelled by a Neumann-to-Dirichlet map. Only few reconstruction methods apply when detecting indefinite inclusions, where the conductivity distribution has both more and less conductive parts relative to the background conductivity; one such method is the monotonicity method of Harrach, Seo, and Ullrich. We formulate the method for irregular indefinite inclusions, meaning that we make no regularity assumptions on the conductivity perturbations nor on the inclusion boundaries. We show, provided that the perturbations are bounded away from zero, that the outer support of the positive and negative parts of the inclusions can be reconstructed independently. Moreover, we formulate a regularization scheme that applies to a class of approximative measurement models, including the Complete Electrode Model, hence making the method robust against modelling error and noise. In particular, we demonstrate that for a convergent family of approximative models there exists a sequence of regularization parameters such that the outer shape of the inclusions is asymptotically exactly characterized. Finally, a peeling-type reconstruction algorithm is presented and, for the first time in literature, numerical examples of monotonicity reconstructions for indefinite inclusions are presented.