Regularization Properties of Mumford-Shah Type Functionals with Perimeter and Norm Constraints for Linear Ill-Posed Problems

Esther Klann*

We study the regularization properties of a Mumford-Shah type functional for linear ill-posed operator equations $Kf=g$. \newline We assume that instead of the exact data $g$, only a noisy version $g^\delta$ with known noise level $\delta$ is available. In standard regularization methods one aims for the stable reconstruction of $f$ in terms of function values only. In contrast to this we reconstruct the singularity set of $f$, i.e., the set of points where the function $f$ is discontinuous, as well as averaged function values of the function away from the singularities. To be more concrete, we assume that the solution $f$ is defined on some domain $D\subset\mathbb{R}^2$ and can be represented as a piecewise constant function, i.e., \begin{equation} \tag{$\star$} f = \sum_{i\in I} \mathrm{f}_i \chi_{\Omega_i} \end{equation} where $I$ is a finite index set, $\mathrm{f}_i\in\mathbb{R}$ and $\bigcup_{i\in I}\Omega_i = D$ with pairwise disjoint sets. \newline For the presented approach we define a solution to consist of a sequence $\boldsymbol{\Omega}$ of sets $\Omega_i\subset\mathbb{R}^2$ with $\sum_{i\in I}\chi_{\Omega_i} = \chi_D$ and a vector $\mathbf{f}$ of coefficients $\mathrm{f}_i\in\mathbb{R}$. In doing so, we achieve a segmentation of $f$ defined by $\boldsymbol{\Omega}$ and a reconstruction of $f$ as in $(\star)$. In order to find an approximation to our problem we consider the Mumford-Shah type functional \[ J_{\beta,\gamma}(\boldsymbol{\Omega}, \mathbf{f})= \|Kf-g^\delta\|^2 + \beta \|\mathbf{f}\|^2 +\gamma \sum_{i\in I}|\partial \Omega_i| \] where $\|\mathbf{f}\|^2$ denotes a norm penalty on the coefficient vector $\mathbf{f}$, and $|\partial \Omega_i|$ denotes the length of the boundary of the set $\Omega_i$. A solution to the problem of simultaneous reconstruction and segmentation is defined as minimizing argument of this functional. \newline In this talk we present an analysis of the above functional. In particular, we show existence and stability of its minimizers. As the minimizers of the Mumford-Shah type functional are pairs of sets and values, we also discuss a proper notion of convergence. Finally, we propose a rule for choosing the regularization parameters $\beta$ and $\gamma$ that ensures convergence of the sequence of minimizers $(\boldsymbol{\Omega}_{\beta,\gamma}^\delta, \mathbf{f}_{\beta,\gamma}^\delta)$ to the exact solution $(\boldsymbol{\Omega}^\dagger,\mathbf{f}^\dagger)$ as $\delta\to 0$.

Mathematics Subject Classification: 47A52

Keywords: Mumford-Shah; piecewise constant; shape sensitivity analysis; convergence

Minisymposion: Iterative Methods for Ill-Posed Problems