Dual Constrained TV-based Regularization on Graphs

Abstract

Abstract. Algorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa-Osher's projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. On denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TV-based approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering.

Epigraphical projection and proximal tools for solving constrained convex optimization problems

“…Consequently, primal-dual methods are often easier to implement than primal ones, but their convergence may be slower [71,72].…”

A variational model for thin structure segmentation based on a directional regularization

“…Peyré et al [15] proposed a gradient computed on a weighted graph, generalized in [16]. Both the graph and its weights are computed, based on the initial image at each step of the optimization algorithm, resulting in an algorithm which may be time-consuming.…”

A random block-coordinate primal-dual proximal algorithm with application to 3D mesh denoising

“…A quite general formulation of optimization problems arising in many application areas such as machine learning, computer vision or inverse problems [1][2][3] is as follows: minimize x 1 ∈H 1 ,...,xp∈Hp p j=1 fj(xj) + hj(xj)…”

“…. , p}, Hj is a separable real Hilbert space, fj and hj are proper lower-semicontinuous convex functions from Hj to ]−∞, +∞], hj being assumed to be Lipschitz differentiable, 1 • for every k ∈ {1, . .…”

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Primal-dual proximal optimization methods have recently gained much interest for dealing with very large-scale data sets encoutered in many application fields such as machine learning, computer vision and inverse problems [1][2][3]. In this work, we propose a novel random block-coordinate version of such algorithms allowing us to solve a wide array of convex variational problems.

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