An algorithm for the construction of intrinsic delaunay triangulations with applications to digital geometry processing

• Published in: Computing Vol. 81; no. 2-3; pp. 199 - 213
• Main Authors: Fisher, M., Springborn, B., Schröder, P., Bobenko, A. I.
• Format: Journal Article
• Language: English
• Published: Wien Springer 01.11.2007; Springer Nature B.V
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Combinatorics; Computer science; control theory; systems; Construction; Exact sciences and technology; Geometry; Laplace transforms; Mathematical analysis; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations; Sciences and techniques of general use; Studies; Theoretical computing; Laplace operator; Triangulation; Grid pattern; Numerical linear algebra; Triangle; Maximum principle; Parameterization; Algorithm; Discretization method; Surface; Denoising; Geometry; Direct method; Numerical analysis; Linear system; Scientific computation; Matrix inversion
• ISSN: 0010-485X; 1436-5057
• DOI: 10.1007/s00607-007-0249-8

The discrete Laplace-Beltrami operator plays a prominent role in many digital geometry processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate the use of the intrinsic Laplace-Beltrami operator. It satisfies a local maximum principle, guaranteeing, e.g., that no flipped triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace-Beltrami operator is based on an intrinsic Delaunay triangulation of the surface. We detail an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic Laplace-Beltrami operator. [PUBLICATION ABSTRACT]