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