Complex Transfinite Barycentric Mappings with Similarity Kernels

Abstract

Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define inter-polants of values given on a smooth planar contour. When the data is two-dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some of the basic complex-valued algebra needed to describe these planar maps, and use it to define similarity kernels, a natural alternative to the usual barycentric kernels. We develop the theory behind similarity kernels, explore their properties, and show that the transfinite versions of the popular three-point barycentric coordinates (Laplace, mean value and Wachspress) have surprisingly simple similarity kernels. We furthermore show how similarity kernels may be used to invert injective transfinite barycentric mappings using an iterative algorithm which converges quite rapidly. This is useful for rendering images deformed by planar barycentric mappings.