  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.
