Local estimates for vectorial Rudin–Osher–Fatemi type problems in one dimension

¹ Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
² Institute of Mathematics of the Polish Academy of Sciences, ul. ´Sniadeckich 8, 00-656 Warszawa, Poland
³ Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

^* Corresponding author: mlasica@impan.pl

Received: 23 January 2023
Accepted: 13 December 2023

Abstract

We consider the Rudin–Osher–Fatemi variational denoising model with general regularizing term and L² fidelity in one-dimensional, vector-valued setting. We obtain local estimates on the singular part of the variation measure of the minimizer in terms of the singular part of the variation measure of the datum. In the case of homogeneous regularizer, we prove local estimates on the whole variation measure of the minimizer and deduce an analogous result for the gradient flow of the regularizer. We also discuss the question of extending our results to other fidelities.