# Reconstruction of piecewise constant images via total (gradient) variation regularization

**Romain Petit**(University of Genoa, Italy)

November 16, 2023 — 11:00 — "new L2S location (IBM building), room to be announced" (and Teams)

###### Abstract

In this talk, I will consider the reconstruction of some unknown image from possibly noisy linear measurements. I will focus on a variational reconstruction method that uses a specific regularizer: the total (gradient) variation. It is well known that minimizing this functional produces piecewise constant reconstructions. It is therefore natural to study the case where the unknown image has precisely this structure. I will present two works on this topic, which are collaborations with Yohann De Castro and Vincent Duval. The first concerns a noise robustness result, stating that, in a low noise regime, the reconstruction is also piecewise constant, and one exactly recovers the number of shapes in the unknown image. The second is about introducing a new numerical method for solving the inverse problem. Its main feature is that it does not rely on the introduction of a fixed spatial discretization (e.g. a pixel grid), and builds a sequence of iterates that are linear combinations of indicator functions.

###### Biography

Romain Petit is a postdoctoral researcher at the University of Genoa (Italy), hosted by Giovanni S. Alberti. He received the PhD degree in mathematics from Université Paris Dauphine, under the supervision of Yohann De Castro and Vincent Duval. His main research interests are inverse problems in imaging and (grid-free) sparse estimation.