Variational Problems and Partial Differential Equations on Implicit Surfaces: Bye Bye Triangulated Surfaces?

Abstract

A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is described in this chapter. The key idea is to implicitly represent the surface as the level set of a higher dimensional function, and solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and not-accurate computations on triangulated surfaces, as it is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, as well as image and vector field intrinsic regularization for data defined on 3D surfaces.