Abstract
This article studies the denoising performance of total variation (TV) image regularization. More precisely, we study geometrical properties of the solution to the socalled RudinOsherFatemi total variation denoising method. The first contribution of this paper is a precise mathematical definition of the 'extended support' (associated to the noisefree image) of TV denoising. It is intuitively the region which is unstable and will suffer from the staircasing effect. We highlight in several practical cases, such as the indicator of convex sets, that this region can be determined explicitly. Our second and main contribution is a proof that the TV denoising method indeed restores an image which is exactly constant outside a small tube surrounding the extended support. The radius of this tube shrinks toward zero as the noise level vanishes, and we are able to determine, in some cases, an upper bound on the convergence rate. For indicators of socalled 'calibrable' sets (such as disks or properly eroded squares), this extended support matches the edges, so that discontinuities produced by TV denoising cluster tightly around the edges. In contrast, for indicators of more general shapes or for complicated images, this extended support can be larger. Beside these main results, our paper also proves several intermediate results about fine properties of TV regularization, in particular for indicators of calibrable and convex sets, which are of independent interest.
Original language  English 

Article number  015002 
Pages (fromto)  144 
Number of pages  44 
Journal  Inverse Problems 
Volume  33 
Issue number  1 
Publication status  Published  2 Dec 2016 
Profiles

Clarice Poon
Person: Research & Teaching