Minimizing Isotropic Total Variation without Subiterations

Total variation (TV) is one of the most popular regularizers in the context of ill-posed image reconstruction problems. Due to its particular structure, minimization of a TV-regularized function with a fast iterative shrinkage/thresholding algorithm (FISTA) requires additional sub-iterations, which may lead to a prohibitively slow reconstruction when dealing with very large scale imaging problems. In this work, we introduce a novel variant of FISTA for isotropic TV that circumvents the need for subiterations. Specifically, our algorithm replaces the exact TV proximal with a componentwise thresholding of the image gradient in a way that ensures the convergence of the algorithm to the true TV solution with arbitrarily high precision.