Iterative Thresholding Meets Free-Discontinuity Problems

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 iterati...

• Type: Journal
• Year: 2010
• Where: FOCM
• Authors: Massimo Fornasier, Rachel Ward
• Keywords: Discrete Free-discontinuity Problems, FOCM 2010, Free-discontinuity Problems, Iterative Thresholding