# A Graph Framework for Manifold-valued Data

## TL;DR

This paper extends graph-based variational and PDE methods to manifold-valued data, enabling advanced processing of complex data types like DTI and LiDAR through novel operators and numerical algorithms.

## Contribution

It introduces a generalized graph framework for manifold-valued data, including calculus and operators for variational models and PDEs, expanding beyond Euclidean domains.

## Key findings

- Successfully applied to synthetic and real-world data
- Enabled total variation and Tikhonov regularization on manifolds
- Demonstrated effectiveness on DTI and LiDAR data

## Abstract

Graph-based methods have been proposed as a unified framework for discrete calculus of local and nonlocal image processing methods in the recent years. In order to translate variational models and partial differential equations to a graph, certain operators have been investigated and successfully applied to real-world applications involving graph models. So far the graph framework has been limited to real- and vector-valued functions on Euclidean domains. In this paper we generalize this model to the case of manifold-valued data. We introduce the basic calculus needed to formulate variational models and partial differential equations for manifold-valued functions and discuss the proposed graph framework for two particular families of operators, namely, the isotropic and anisotropic graph~$p$-Laplacian operators, $p\geq1$. Based on the choice of $p$ we are in particular able to solve optimization problems on manifold-valued functions involving total variation ($p=1$) and Tikhonov ($p=2$) regularization. Finally, we present numerical results from processing both synthetic as well as real-world manifold-valued data, e.g., from diffusion tensor imaging (DTI) and light detection and ranging (LiDAR) data.