Characterization of minimizers of an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term

TL;DR

This paper characterizes the minimizers of an anisotropic Rudin-Osher-Fatemi functional with an L1 fidelity term, linking them to the Wulff shape and dual anisotropy, with applications to 2D barcode denoising.

Contribution

It provides a detailed characterization of minimizers for the anisotropic ROF functional, including the subdifferential, and applies it to specific cases relevant to image denoising.

Findings

Minimizers are characterized via the Wulff shape and dual anisotropy.

Explicit shape of minimizers for circular characteristic functions in 2D.

Application to denoising of 2D bar codes.

Abstract

In this paper we study an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term of the form \[ E(u) = \int_{\mathbb{R}^n} \phi(\nabla u) + \lambda \| u -f \|_{L^1(\mathbb{R}^n)}. \] We will characterize the minimizers of $E$ in terms of the Wulff shape of $\phi$ and the dual anisotropy. In particular we will calculate the subdifferential of $E$. We will apply this characterization to the special case $\phi = |\cdot|_1$ and $n=2$, which has been used in the denoising of 2D bar codes. In this case, we determine the shape of a minimizer $u$ when $f$ is the characteristic function of a circle.