An iterated `1 Algorithm for Non-smooth Non-convex Optimization in Computer Vision

Natural image statistics indicate that we should use nonconvex norms for most regularization tasks in image processing and computer vision. Still, they are rarely used in practice due to the challenge to optimize them. Recently, iteratively reweighed `1 minimization has been proposed as a way to tackle a class of non-convex functions by solving a sequence of convex `2-`1 problems. Here we extend the problem class to linearly constrained optimization of a Lipschitz continuous function, which is the sum of a convex function and a function being concave and increasing on the non-negative orthant (possibly non-convex and nonconcave on the whole space). This allows to apply the algorithm to many computer vision tasks. We show the effect of non-convex regularizers on image denoising, deconvolution, optical flow, and depth map fusion. Non-convexity is particularly interesting in combination with total generalized variation and learned image priors. Efficient optimization is made possible by some important properties that are shown to hold.