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

TL;DR

This paper extends the Rudin-Osher-Fatemi denoising model to one-dimensional vector-valued data, providing local estimates on the variation measure of minimizers and their gradient flows, enhancing understanding of their regularity properties.

Contribution

It introduces local estimates for the variation measure of minimizers in vector-valued Rudin-Osher-Fatemi models in one dimension, including the case of homogeneous regularizers.

Findings

Local estimates on the singular part of the variation measure of minimizers.

Global estimates for the variation measure in the case of homogeneous regularizers.

Results applicable to the gradient flow of the regularizer.

Abstract

We consider the Rudin-Osher-Fatemi variational denoising model with general regularizing term 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.