Year: 2018
Type: article
Abstract: Graph-based methods have been proposed as a unified framework for discrete calculus of local and nonlocal image processing methods in 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 and light detection and ranging data.
Source: SIAM Journal on Imaging Sciences
DOI: 10.1137/17m1118567
Authors: Ronny Bergmann, Daniel Tenbrinck
Language: English
Subfield: Computer Vision and Pattern Recognition
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G1680711539, G4891414081
OpenAlex ID: W2760923457