Inverse Scale Space Iterations for Non-Convex Variational Problems: The Continuous and Discrete Case

Abstract

Non-linear filtering approaches allow to obtain decompositions of images with respect to a nonclassical notion of scale, induced by the choice of a convex, absolutely one-homogeneous regularizer. The associated inverse scale space flow can be obtained using the classical Bregman iteration with quadratic data term. We apply the Bregman iteration to lifted, i.e. higherdimensional and convex, functionals in order to extend the scope of these approaches to functionals with arbitrary data term. We provide conditions for the subgradients of the regularizer -in the continuous and discrete setting-under which this lifted iteration reduces to the standard Bregman iteration. We show experimental results for the convex and non-convex case. Motivation and IntroductionIn modern image processing tasks, variational problems constitute an important tool [4,40]. They are used in a variety of applications such as denoising [38], segmentation [17], and depth estimation [41,43]. In this work, we consider variational image processing problems withThe authors acknowledge support through DFG grant LE 4064/1-1 "Functional Lifting 2.0: Efficient Convexifications for Imaging and Vision" and NVIDIA Corporation.