Optimal Interpolation Data for PDE-based Compression of Images with Noise

TL;DR

This paper develops shape-based models for optimal interpolation data in PDE-based image compression with noise, analyzing their theoretical properties and confirming practical effectiveness through numerical experiments.

Contribution

It introduces a novel framework combining shape models and $ ext{Gamma}$-convergence analysis for PDE-based image compression with noise.

Findings

Theoretical insights into pixel relevance via topological asymptotics.

Asymptotic analysis of 'fat pixels' models as radius vanishes.

Numerical results validate the theoretical models for image compression.

Abstract

We introduce and discuss shape-based models for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in the $L^2$-norm between the images and their reconstructed counterparts using time-dependent PDE inpainting. We analyze the proposed models in the framework of the $\Gamma$-convergence from two different points of view. First, we consider a continuous stationary PDE model, obtained by focusing on the first iteration of the discretized time-dependent PDE, and get pointwise information on the "relevance" of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting of the continuous model based on "fat pixels" (balls with positive radius), and we study by $\Gamma$-convergence the asymptotics when the radius vanishes. Numerical computations are presented that confirm the usefulness of our theoretical findings for non-stationary PDE-based image compression.