Numerical Methods for Digital Geometry Processing extended abstract

Digital Geometry Processing recently appeared (in the middle of the 90’s) as a promising avenue to solve the geometric modeling problems encountered when manipulating surfaces represented by discrete elements (i.e. meshes). Since a mesh may be considered to be a sampling of a surface in other words a signal the DGP (digital signal processing) formalism was a natural theoretic background for this discipline (see e.g. [20]). In this discipline, discrete fairing [13] and mesh parameterization [9] have been two active research topics these last few years. In parallel with the evolution of this discipline, acquisition techniques have made huge advances, and todays meshes acquired from real objects by range-laser scanners are larger and larger (30 million triangles is now common). This causes difficulties when trying to apply DGP tools to these meshes. The kernel of a DGP algorithm is a numerical method, used either to solve a linear system, or to minimize a multivariate function. The Gauss-Seidel iteration and gradient descent methods used at the early ages of DGP do not scale-up when applied to huge meshes. In this presentation, our goal is to give a survey of classic and more recent numerical methods, to show how they can be applied to DGP problems, from a theoretic point of view down to implementation. We will focus on two different classes of DGP problems (mesh fairing and mesh parameterization), show solutions for linear problems, quadratic problems, and general non-linear problems, with and without constraint. In particular, we give a general formulation of quadratic problems with reduced degrees of freedom that can be used as a general framework to solve a wide class of DGP problems. Our method is implemented in the OpenNL library, freely available on the web. The presentation will be illustrated with live demos of the methods.