Ofir Weber, Ph.D. Thesis Seminar
Wednesday, 30.6.2010, 11:00
Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data values. Traditionally, the coordinates are required to satisfy a number of key properties, the first being that they are real. In this work we relax this requirement, allowing the barycentric coordinates to be complex numbers. This allows us to generate new families of barycentric coordinates, which have some powerful advantages over traditional ones and are especially useful for creating detail-preserving planar shape deformations. We first use Cauchy's theorem to construct three types of complex barycentric coordinates that can be used to generate holomorphic functions. Such functions can be interpreted as conformal planar mappings as long as the derivative of the function doesn't vanish.
We then construct another type of complex barycentric coordinates based on the Hilbert transform. Combined with a novel 2D shape deformation system, we show how to generate "foldovers free" pure conformal planar deformations. Our system provides the user a large degree of control over the result. For example, it allows discontinuities at user-specified boundary points, so true "bends" can be introduced into the deformation. It also allows the prescription of angular constraints at corners of the target image. Beyond deforming a given shape into a new one at each key frame, our method also provides the ability to interpolate between shapes in a very natural way, such that also the intermediate deformations are conformal.
Finally, we show that non-holomorphic complex barycentric coordinates also exists and may be used to construct planar deformations with exact boundary behavior.