A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images

TL;DR

This paper develops a novel second order non-smooth variational model for restoring manifold-valued images, extending techniques from real-valued image processing to manifold data using differential geometry and convex optimization.

Contribution

It introduces the first second order difference and variational models for manifold-valued data, along with an efficient algorithm for denoising in symmetric spaces.

Findings

Effective denoising on n-sphere and positive definite matrices.

Convergence of the proposed algorithms in Hadamard spaces.

Demonstrated practical performance of the method.

Abstract

We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.