Modal Shape Analysis beyond Laplacian

Please always quote using this URN: urn:nbn:de:0296-matheon-9678
  • In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies or deformation energies. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces.

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Author:Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, Konrad Polthier
Referee:Konrad Polthier
Date of first Publication:2012/01/24
Institute:Freie Universit├Ąt Berlin
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Dxx Numerical approximation and computational geometry (primarily algorithms) (For theory, see 41-XX and 68Uxx) / 65D18 Computer graphics, image analysis, and computational geometry [See also 51N05, 68U05]
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section {04 in that areag 68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) / 68Uxx Computing methodologies and applications / 68U05 Computer graphics; computational geometry [See also 65D18]
Preprint Number:867