On Variational and PDE-Based Distance Function Approximations

• Published in: Computer graphics forum Vol. 34; no. 8; pp. 104 - 118
• Main Authors: Belyaev, Alexander G., Fayolle, Pierre-Alain
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.12.2015
• Subjects: Analysis; and systems; and systems; G.1.8 [Numerical Analysis]: Partial Differential Equations—Iterative solution techniques; Approximation; Computation; Computer graphics; distance function approximations; G.1.8 [Numerical Analysis]: Partial Differential Equations-Iterative solution techniques; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms; iterative optimization; languages; Mathematical analysis; Mathematical models; Partial differential equations; Studies; Two dimensional; variational methods
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.12611

In this paper, we deal with the problem of computing the distance to a surface (a curve in two dimensional) and consider several distance function approximation methods which are based on solving partial differential equations (PDEs) and finding solutions to variational problems. In particular, we deal with distance function estimation methods related to the Poisson‐like equations and generalized double‐layer potentials. Our numerical experiments are backed by novel theoretical results and demonstrate efficiency of the considered PDE‐based distance function approximations. In this paper, we deal with the problem of computing the distance to a surface (a curve in two dimensional) and consider several distance function approximation methods which are based on solving partial differential equations (PDEs) and finding solutions to variational problems. In particular, we deal with distance function estimation methods related to the Poisson‐like equations and generalized double‐layer potentials. Our numerical experiments are backed by novel theoretical results and demonstrate efficiency of the considered PDE‐based distance function approximations.