A dual self-monitored reconstruction scheme on the TV-regularized inverse conductivity problem

Abstract

Recently in Charalambopoulos et al. (2020), we presented a methodology aiming at reconstructing bounded total variation ( $TV$ ) conductivities via a technique simulating the so-called half-quadratic minimization approach, encountered in Aubert & Kornprobst (2002, Mathematical Problems in Image Processing. New York, NY: Springer). The method belongs to a duality framework, in which the auxiliary function $\omega (x)$ was introduced, offering a tool for smoothing the members of the admissible set of conductivity profiles. The dual variable $\omega (x)$ , in that approach, after every external update, served in the formation of an intermediate optimization scheme, concerning exclusively the sought conductivity $\alpha (x)$ . In this work, we develop a novel investigation stemming from the previous approach, having though two different fundamental components. First, we do not detour herein the $BV$ -assumption on the conductivity profile, which means that the functional under optimization contains the $TV$ of $\alpha (x)$ itself. Secondly, the auxiliary dual variable $\omega (x)$ and the conductivity $\alpha (x)$ acquire an equivalent role and concurrently, a parallel pacing in the minimization process. A common characteristic between these two approaches is that the function $\omega (x)$ is an indicator of the conductivity's ‘jump’ set. A fortiori, this crucial property has been ameliorated herein, since the reciprocal role of the elements of the pair $(\alpha ,\omega )$ offers a self-monitoring structure very efficient to the minimization descent.