Modular proximal optimization for multidimensional total-variation\n regularization

Abstract

We study \\emph{TV regularization}, a widely used technique for eliciting\nstructured sparsity. In particular, we propose efficient algorithms for\ncomputing prox-operators for $\\ell_p$-norm TV. The most important among these\nis $\\ell_1$-norm TV, for whose prox-operator we present a new geometric\nanalysis which unveils a hitherto unknown connection to taut-string methods.\nThis connection turns out to be remarkably useful as it shows how our geometry\nguided implementation results in efficient weighted and unweighted 1D-TV\nsolvers, surpassing state-of-the-art methods. Our 1D-TV solvers provide the\nbackbone for building more complex (two or higher-dimensional) TV solvers\nwithin a modular proximal optimization approach. We review the literature for\nan array of methods exploiting this strategy, and illustrate the benefits of\nour modular design through extensive suite of experiments on (i) image\ndenoising, (ii) image deconvolution, (iii) four variants of fused-lasso, and\n(iv) video denoising. To underscore our claims and permit easy reproducibility,\nwe provide all the reviewed and our new TV solvers in an easy to use\nmulti-threaded C++, Matlab and Python library.\n