Image Processing Variations with Analytic Kernels

TL;DR

This paper explores variations of the Rudin-Osher-Fatemi image processing model using analytic kernels, characterizing minimizers and their smoothness, and showing radial symmetry results for certain cases.

Contribution

It introduces and analyzes new functional variations of the R-O-F model with analytic kernels, providing detailed properties and symmetry results of minimizers.

Findings

Minimizers exhibit real analytic surface level sets under certain conditions.

If both the image and kernel are radial, the minimizer is a radial step function.

Minimizers have specific smoothness and structural properties depending on the kernel and data.

Abstract

Let $f\in L^1(\R^d)$ be real. The Rudin-Osher-Fatemi model is to minimize $\|u\|_{\dot{BV}}+\lambda\|f-u\|_{L^2}^2$, in which one thinks of $f$ as a given image, $\lambda > 0$ as a "tuning parameter", $u$ as an optimal "cartoon" approximation to $f$, and $f-u$ as "noise" or "texture". Here we study variations of the R-O-F model having the form $\inf_u\{\|u\|_{\dot{BV}}+\lambda \|K*(f-u)\|_{L^p}^q\}$ where $K$ is a real analytic kernel such as a Gaussian. For these functionals we characterize the minimizers $u$ and establish several of their properties, including especially their smoothness properties. In particular we prove that on any open set on which $u \in W^{1,1}$ and $\nabla u \neq 0$ almost every level set $\{u =c\}$ is a real analytic surface. We also prove that if $f$ and $K$ are radial functions then every minimizer $u$ is a radial step function.