A Graph Framework for Manifold-valued 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, $p\ge 1$. 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.

References in corpus (5)

• Consistency of spectral clustering
• Image Labeling by Assignment
• A Parallel Douglas Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds
• Graph clustering, variational image segmentation methods and Hough transform scale detection for object measurement in images
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• Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems
• First- and Second-Order Analysis for Optimization Problems with Manifold-Valued Constraints
• Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds
• p-Laplacians for Manifold-valued Hypergraphs