Spectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace-Beltrami operator, which has the often-cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator.
A unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds was introduced. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. Showcased are various applications of our operators, with improved numerics over prior work.
Zi Ye, Olga Diamanti, Chengcheng Tang, Leonidas Guibas, and Tim Hoffmann.
A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing.
Computer Graphics Forum, 37(5):93–106, August 2018.
Prof. Dr. Olga Diamanti +
University: TU Graz, Department of Mathematics, NT04012
Address: Kopernikusgasse 24/IV, 8010 Graz, AUSTRIA
Tel: +43 316 8738442
Prof. Leonidas J. Guibas
Prof. Dr. Tim Hoffmann +
University: TU München, Department of Mathematics, 02.06.021
Address: Boltzmannstr. 3, 85748 Garching, GERMANY
Tel: +49 89 28918384
Zi Ye +