A local estimate for vectorial total variation minimization in one dimension

TL;DR

This paper establishes bounds on the vectorial total variation of minimizers and flows in one dimension, providing theoretical insights into their behavior relative to data fidelity over subintervals.

Contribution

It introduces a local estimate for vectorial total variation minimization and flow in one dimension, advancing understanding of their local properties.

Findings

Total variation of minimizer is bounded by data over subintervals.

Analogous bounds are established for the vectorial total variation flow.

Results contribute to the theoretical understanding of VTV minimization in 1D.

Abstract

Let $\boldsymbol u$ be the minimizer of vectorial total variation ($VTV$) with $L^2$ data-fidelity term on an interval $I$. We show that the total variation of $\boldsymbol u$ over any subinterval of $I$ is bounded by that of the datum over the same subinterval. We deduce analogous statement for the vectorial total variation flow on $I$.