Restoration of Manifold-Valued Images by Half-Quadratic Minimization

TL;DR

This paper extends the half-quadratic minimization technique to restore images with values on Riemannian manifolds, proving convergence and demonstrating effectiveness on various manifold-valued images including spheres, SO(3), and positive definite matrices.

Contribution

It adapts and proves convergence of the half-quadratic minimization method for manifold-valued images, with extensive numerical validation on diverse manifolds.

Findings

Effective restoration of manifold-valued images demonstrated.

Convergence proven for Hadamard spaces.

Promising results for material science applications.

Abstract

The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3) and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.