  We consider the inverse problem of denoising an image where each point
(pixel) is an element of a target set, which we refer to as a target-valued
image. The target sets considered are either (i) a closed convex set of
Euclidean space or (ii) a closed subset of the sphere such that the closest
point mapping is defined almost everywhere. The energy for the denoising
problem consists of an $L^2$-fidelity term which is regularized by the
Dirichlet energy. A relaxation of this energy, based on the heat kernel, is
introduced and the associated minimization problem is proven to be well-posed.
We introduce a diffusion generated method which can be used to efficiently find
minimizers of this energy. We prove results for the stability and convergence
of the method for both types of target sets. The method is demonstrated on a
variety of synthetic and test problems, with associated target sets given by
the semi-positive definite matrices, the cube, spheres, the orthogonal
matrices, and the real projective line.
