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Numerous applications in statistics, signal processing, and machine learning regularize using Total Variation (TV) penalties. We study anisotropic (l1-based) TV and also a related l2-norm variant. We consider for both variants associated (1D) proximity operators, which lead to challenging optimization problems. We solve these problems by developing Newton-type methods that outperform the state-of-the-art algorithms. More importantly, our 1D-TV algorithms serve as building blocks for solving the harder task of computing 2- (and higher)-dimensional TV proximity. We illustrate the computational benefits of our methods by applying them to several applications: (i) image denoising; (ii) image deconvolution (by plugging in our TV solvers into publicly available software); and (iii) four variants of fused-lasso. The results show large speedups--and to support our claims, we provide software accompanying this paper.

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