Closed-Form Approximation of the Total Variation Proximal Operator

TL;DR

This paper introduces a theoretically grounded closed-form approximation for the total variation proximal operator, enabling more efficient TV-regularized image reconstruction with controlled error.

Contribution

It provides a novel theoretical analysis proving the approximation's convexity, equivalence to gradient descent on a smoothed TV, and error control via a scaling parameter.

Findings

The approximation corresponds to a gradient step on a smoothed TV.

Error bounds for the approximation are derived and controllable.

Experimental validation on denoising and CT reconstruction confirms theoretical insights.

Abstract

Total variation (TV) is a widely used function for regularizing imaging inverse problems that is particularly appropriate for images whose underlying structure is piecewise constant. TV regularized optimization problems are typically solved using proximal methods, but the way in which they are applied is constrained by the absence of a closed-form expression for the proximal operator of the TV function. A closed-form approximation of the TV proximal operator has previously been proposed, but its accuracy was not theoretically explored in detail. We address this gap by making several new theoretical contributions, proving that the approximation leads to a proximal operator of some convex function, it is equivalent to a gradient descent step on a smoothed version of TV, and that its error can be fully characterized and controlled with its scaling parameter. We experimentally validate our theoretical results on image denoising and sparse-view computed tomography (CT) image reconstruction.