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.
|Title of host publication||6th International Workshop on New Computational Methods for Inverse Problem|
|Number of pages||1|
|Publication status||Published - 2016|
|Name||Journal of Physics: Conference Series|
Chambolle, A., Duval, V., Peyré, G., & Poon, C. (2016). Total variation denoising and support localization of the gradient. In 6th International Workshop on New Computational Methods for Inverse Problem (1 ed., Vol. 756, pp. 012007). (Journal of Physics: Conference Series). IOP Publishing. https://doi.org/10.1088/1742-6596/756/1/012007/meta