A nonlocal graph-PDE and higher-order geometric integration for image labeling
This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in \textit{J.~Math.~Imaging \& Vision} 58(2), 2017. Due to this parameterization,...
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| Main Authors: | , , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
4 Oct 2022
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| Edition: | Version v2 |
| In: |
Arxiv
Year: 2022, Pages: 1-60 |
| DOI: | 10.48550/arXiv.2205.03991 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2205.03991 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2205.03991 
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| Author Notes: | Dmitrij Sitenko, Bastian Boll, Christoph Schnörr |
| Summary: | This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in \textit{J.~Math.~Imaging \& Vision} 58(2), 2017. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions (DC) decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments. |
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| Item Description: | Version 1 vom 9 Mai 2022, Version 2 vom 4 Oktober 2022 Gesehen am 14.10.2022 |
| Physical Description: | Online Resource |
| DOI: | 10.48550/arXiv.2205.03991 |