Diffusion generated methods for denoising target-valued images
Provides a principled and efficient denoising method for images with non-standard pixel values, addressing a niche but important problem in image processing.
This paper tackles denoising of target-valued images, where each pixel belongs to a constrained set (e.g., convex sets or spheres). They propose a diffusion-generated method that efficiently minimizes a relaxed energy, proving stability and convergence, and demonstrate it on synthetic and real problems.
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.