Appears in collection : 2019 - T1 - WS1 - Variational methods and optimization in imaging

It is folklore knowledge that the total (gradient) variation regularization tends to promote piecewise constant cartoon-like'' images. In this talk I will relate that property to the description of the extreme points of the total variation unit ball. These extreme points have been characterized by Ambrosio, Caselles, Masnou and Morel as the indicator functions ofsimple sets''. I will explain how it is possible to describe the solutions of variational problems as a sum of such functions, by using a general representation principle. This is a joint work with C. Boyer, A. Chambolle, Y. De Castro, F. de Gournay and P. Weiss.