# Recent Advances in Denoising of Manifold-Valued Images

## TL;DR

This paper reviews recent variational model-based methods and algorithms for denoising manifold-valued images, emphasizing practical implementations and software tools for real-world applications.

## Contribution

It provides a comprehensive overview of recent advances in variational denoising techniques for manifold-valued data, including algorithm generalizations and software resources.

## Key findings

- Algorithms like subgradient, half-quadratic, cyclic proximal point, and Douglas-Rachford are adapted for manifolds.
- Software packages such as Manopt and MVIRT facilitate practical implementation.
- Recent methods improve denoising quality for manifold-valued signals and images.

## Abstract

Modern signal and image acquisition systems are able to capture data that is no longer real-valued, but may take values on a manifold. However, whenever measurements are taken, no matter whether manifold-valued or not, there occur tiny inaccuracies, which result in noisy data. In this chapter, we review recent advances in denoising of manifold-valued signals and images, where we restrict our attention to variational models and appropriate minimization algorithms. The algorithms are either classical as the subgradient algorithm or generalizations of the half-quadratic minimization method, the cyclic proximal point algorithm, and the Douglas-Rachford algorithm to manifolds. An important aspect when dealing with real-world data is the practical implementation. Here several groups provide software and toolboxes as the Manifold Optimization (Manopt) package and the manifold-valued image restoration toolbox (MVIRT).