Iterative thresholding meets free discontinuity problems

TL;DR

This paper explores the use of iterative thresholding algorithms to solve discrete free-discontinuity problems, linking sparse recovery techniques with fracture mechanics and image segmentation applications.

Contribution

It introduces an iterative thresholding algorithm tailored for discrete free-discontinuity problems, offering new insights into solution properties and potential computational methods.

Findings

Established a connection between sparse recovery and free-discontinuity problems.

Developed an iterative thresholding algorithm for these problems.

Provided analysis of solution properties and convergence.

Abstract

Free-discontinuity problems describe situations where the solution of interest is defined by a function and a lower dimensional set consisting of the discontinuities of the function. Hence, the derivative of the solution is assumed to be a `small' function almost everywhere except on sets where it concentrates as a singular measure. This is the case, for instance, in crack detection from fracture mechanics or in certain digital image segmentation problems. If we discretize such situations for numerical purposes, the free-discontinuity problem in the discrete setting can be re-formulated as that of finding a derivative vector with small components at all but a few entries that exceed a certain threshold. This problem is similar to those encountered in the field of `sparse recovery', where vectors with a small number of dominating components in absolute value are recovered from a few given linear measurements via the minimization of related energy functionals. Several iterative thresholding algorithms that intertwine gradient-type iterations with thresholding steps have been designed to recover sparse solutions in this setting. It is natural to wonder if and/or how such algorithms can be used towards solving discrete free-discontinuity problems. The current paper explores this connection, and, by establishing an iterative thresholding algorithm for discrete free-discontinuity problems, provides new insights on properties of minimizing solutions thereof.