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.
Total Generalized Variation for Manifold-valued Data
University of Graz（格拉茨大学）；Heidelberg Collaboratory for Image Processing, Universität Heidelberg（海德堡大学图像分析协作中心）；Institute of Computational Biology, Helmholtz Zentrum München（慕尼黑亥姆霍兹中心计算生物学研究所）；Department of Mathematics and Natural Sciences, Hochschule Darmstadt（达姆施塔特应用技术大学数学与自然科学系）
A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数学研究所）；Technische Universität Kaiserslautern（凯泽斯劳滕工业大学）；Technische Universität München（慕尼黑工业大学）；Helmholtz-Zentrum München（赫尔姆霍兹慕尼黑中心）
Variational Regularization of Inverse Problems for Manifold-Valued Data
Image Analysis and Learning Group, Interdisciplinary Center for Scientific Computing, Universität Heidelberg, Germany（图像分析与学习组，跨学科科学计算中心，海德堡大学，德国）；Department of Mathematics and Natural Sciences, Hochschule Darmstadt, and Institute of Computational Biology, Helmholtz Zentrum München, Germany（数学与自然科学系，达姆施塔特应用科学大学，以及计算生物学研究所，海德堡研究中心，德国）
