Generalized Barycentric Interpolation
Interpolating data with functions is fundamental to many fields and applications in computer science, mathematics, and engineering. Barycentric coordinates, introduced by Möbius in 1827, provide a very convenient way to linearly interpolate data given at the vertices of a d-dimensional simplex. This kind of barycentric interpolation is widely used in computer graphics for interpolating vertex attributes such as colour, normals, or texture coordinates over the individual triangles of a triangle mesh, but it also plays a key role in other disciplines. For example, in the finite element method, the related piecewise linear, cardinal basis functions over planar triangulations can be adopted as trial and test approximations. Starting with the seminal work of Wachspress in 1975, the idea of barycentric coordinates and barycentric interpolation have been extended in recent years to arbitrary polygons in the plane, general polytopes in higher dimensions, and smooth domains, which in turn has led to novel solutions in graphics applications (e.g., mesh parameterization, image warping, and mesh deformation) and in mechanics (e.g., fracture and topology optimization). The aim of this project is to further develop the theoretical foundations of generalized barycentric interpolation and to provide a better understanding of the underlying principles. We plan to attack several open problems, to investigate novel interesting research questions, and to considerably advance the state-of-the-art in this field. We expect our results to also have an impact on related applications and to possibly lead to new scenarios where barycentric interpolation can provide elegant and favourable solutions. In particular, we will address the following four problems: First, we will explore the possibility of deriving generalized barycentric coordinates which are smooth, positive at every point inside an arbitrary polygon, and have a simple closed-form expression. So far, none of the known constructions have all properties. Second, we will consider the problem of transfinite barycentric interpolation and investigate the particular case where the data is given along the limit curve of a linear subdivision scheme. We expect that the local and recursive nature of the subdivision scheme leads to efficient algorithms for evaluating the resulting transfinite interpolant, which outperform state-of-the-art methods that rely on numerical integration. Third, we will extend our work on complex barycentric mappings and develop strategies for guaranteeing the bijectivity of these mappings. This is important in the context of image warping and shape deformation and can be achieved with current methods only for some special cases that are too restrictive for real-world applications. Finally, we will study a novel idea for interpolating scattered data with bivariate functions. Our approach is motivated by our results on univariate rational barycentric interpolation and can be seen as a generalization of Shepherd's interpolation method.