Polygon Laplacian Made Robust
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Date
2024
Journal Title
Journal ISSN
Volume Title
Publisher
The Eurographics Association and John Wiley & Sons Ltd.
Abstract
Discrete Laplacians are the basis for various tasks in geometry processing. While the most desirable properties of the discretization invariably lead to the so-called cotangent Laplacian for triangle meshes, applying the same principles to polygon Laplacians leaves degrees of freedom in their construction. From linear finite elements it is well-known how the shape of triangles affects both the error and the operator's condition. We notice that shape quality can be encapsulated as the trace of the Laplacian and suggest that trace minimization is a helpful tool to improve numerical behavior. We apply this observation to the polygon Laplacian constructed from a virtual triangulation [BHKB20] to derive optimal parameters per polygon. Moreover, we devise a smoothing approach for the vertices of a polygon mesh to minimize the trace. We analyze the properties of the optimized discrete operators and show their superiority over generic parameter selection in theory and through various experiments.
Description
CCS Concepts: Computing methodologies -> Mesh geometry models; Mathematics of computing -> Mesh generation; Discretization
@article{10.1111:cgf.15025,
journal = {Computer Graphics Forum},
title = {{Polygon Laplacian Made Robust}},
author = {Bunge, Astrid and Bukenberger, Dennis R. and Wagner, Sven Dominik and Alexa, Marc and Botsch, Mario},
year = {2024},
publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
ISSN = {1467-8659},
DOI = {10.1111/cgf.15025}
}