Adjoint Method in PDE-based Image Compression

TL;DR

This paper introduces an adjoint-based shape optimization method for PDE-driven image compression, effectively reconstructing missing image regions by optimizing shape functionals with topological derivatives.

Contribution

It develops a novel adjoint method for computing topological derivatives in PDE-based image inpainting, enabling efficient shape optimization for image compression.

Findings

The method accurately reconstructs missing image regions.

Numerical experiments demonstrate the efficiency of the approach.

Theoretical results improve understanding of shape derivatives in PDE inpainting.

Abstract

We consider a shape optimization based method 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 an $L^p$-norm between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing, the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.