Total variation projection with first order schemes

Abstract

This paper proposes a new class of algorithms to compute the projection onto the set of images with a total variation bounded by a constant. The projection is computed on a dual formulation of the problem that is minimized using either a one-step gradient descent method or a multi-step Nesterov scheme. This yields iterative algorithms that compute soft thresholding of the dual vector fields. We show the convergence of the method with a convergence rate of O(1/k) for the one step method and O(1/k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) for the multi-step one, where k is the iteration number. The projection algorithm can be used as a building block in several applications, and we illusrtate it by solving linear inverse problems under total variation constraint. Numerical results show that our algorithm competes favorably with state-of-the-art TV projection methods to solve denoising, inpainting and deblurring problems.