Spectral Coarsening with Hodge Laplacians

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract / Description of output

Many computational algorithms applied to geometry operate on discrete representations of shape. It is sometimes necessary to first simplify, or coarsen, representations found in modern datasets for practicable or expedited processing. The utility of a coarsening algorithm depends on both, the choice of representation as well as the specific processing algorithm or operator. e.g. simulation using the Finite Element Method, calculating Betti numbers, etc. We propose a novel method that can coarsen triangle meshes, tetrahedral meshes and simplicial complexes. Our method allows controllable preservation of salient features from the high-resolution geometry and can therefore be customized to different applications. Salient properties are typically captured by local shape descriptors via linear differential operators ś variants of Laplacians. Eigenvectors of their discretized matrices yield a useful spectral domain for geometry processing (akin to the famous Fourier spectrum which uses eigenfunctions of the derivative operator). Existing methods for spectrum-preserving coarsening use zero-dimensional discretizations of Laplacian operators (defined on vertices). We propose a generalized spectral coarsening method that considers multiple Laplacian operators defined in different dimensionalities in tandem. Our simple algorithm greedily decides the order of contractions of simplices based on a quality function per simplex. The quality function quantifies the error due to removal of that simplex on a chosen band within the spectrum of the coarsened geometry.
Original languageEnglish
Title of host publicationSIGGRAPH '23: ACM SIGGRAPH 2023 Conference Proceedings
EditorsErik Brunvand, Alla Sheffer, Michael Wimmer
PublisherACM
Pages1-11
Number of pages11
ISBN (Electronic)9798400701597
DOIs
Publication statusPublished - 23 Jul 2023
EventSIGGRAPH 2023 - Los Angeles, United States
Duration: 6 Aug 202310 Aug 2023
https://s2023.siggraph.org/

Conference

ConferenceSIGGRAPH 2023
Country/TerritoryUnited States
CityLos Angeles
Period6/08/2310/08/23
Internet address

Keywords / Materials (for Non-textual outputs)

  • geometry processing
  • numerical coarsening
  • spectral geometry

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