In this work we revisit the Mumford-Shah functional, one
of the most studied variational approaches to image segmentation.
The contribution of this paper is to propose an
algorithm which allows to minimize a convex relaxation of
the Mumford-Shah functional obtained by functional lifting.
The algorithm is an efficient primal-dual projection algorithm
for which we prove convergence. In contrast to existing
algorithms for minimizing the full Mumford-Shah this
is the first one which is based on a convex relaxation. As
a consequence the computed solutions are independent of
the initialization. Experimental results confirm that the proposed
algorithm determines smooth approximations while
preserving discontinuities of the underlying signal.