Total Variation Denoising and Support Localization of the Gradient

6th International Workshop on New Computational Methods for Inverse Problems

物理学;计算机科学

Chambolle, A.^1 ; Duval, V.^2 ; Peyré, G.^2,3 ; Poon, C.^2,3,4

CNRS, CMAP, Ecole Polytechnique, Palaiseau Cedex

91128, France^1

INRIA, MOKAPLAN, Domaine de Voluceau Le Chesnay, France^2

CNRS and CEREMADE, Université Paris-Dauphine, Place du Marechal De Lattre De Tassigny, Paris 16

75775, France^3

DAMTP, University of Cambridge, Wilberforce Road, Cambridge

CB3 0DZ, United Kingdom^4

关键词: Convergence rates;    Denoising methods;    Geometrical property;    Mathematical definitions;    Noise levels;    Numerical evidence;    Piece-wise constants;    Total variation;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/756/1/012007/pdf
DOI  :  10.1088/1742-6596/756/1/012007

学科分类：计算机科学（综合）

来源: IOP

【 摘 要 】

This paper describes the geometrical properties of the solutions to the total variation denoising method. A folklore statement is that this method is able to restore sharp edges, but at the same time, might introduce some staircasing (i.e. "fake" edges) in flat areas. Quite surprisingly, put aside numerical evidences, almost no theoretical result are available to backup these claims. The first contribution of this paper is a precise mathematical definition of the "extended support" (associated to the noise-free image) of TV denoising. This is intuitively the region which is unstable and will suffer from the staircasing effect. Our main result shows that the TV denoising method indeed restores a piece-wise constant image outside a small tube surrounding the extended support. Furthermore, the radius of this tube shrinks toward zero as the noise level vanishes and in some cases, an upper bound on the convergence rate is given.