Variational Methods in Surface Parameterization

A surface parameterization is a function that maps coordinates in a 2-dimensional parameter space to points on a surface. This thesis investigates two kinds of parameterizations for surfaces that are disc-like in shape. The first is a map from a region of the plane to the surface. The second is a mapping from one surface to another, which defines a correspondence between them. The main challenge in both cases is the construction of a smooth map with low distortion. In this thesis we present a variational approach to surface parameterization that addresses these challenges. The first contribution of this thesis is the development of a variational framework for parameterizations. This framework encompasses the mapping of a region of the plane to a surface that is isomorphic to a disc, and the mapping between such surfaces. It is based on the rich mathematical theory built up over decades in the study of rational mechanics. Because of its roots in mechanics, our parameterizations are guaranteed to be smooth and locally bijective, and optimal parameterizations which minimize a variational energy are known to exist. A proof of existence is given for the case of optimal parameterizations in the plane. Our second contribution is a set of algorithms to construct parameterizations for surface triangulations. We describe in detail free-boundary methods that use standard numerical optimization algorithms for the computation of optimal parameterizations. A flexible set of parameters is offered to the user to formulate preferences for the trade-off between angle, area and length distortion in parameterizations in the plane. In the specification of a correspondence between surfaces, we provide user control through feature lines which are mapped as sets onto corresponding feature lines. Additionally we allow for a partial correspondence of the surfaces which is particularly important for correlating surfaces with boundaries. Our third contribution is an analysis of the performance of the algorithms based on our implementations. Our testing focuses on parameterizations of physically-acquired triangle mesh data. The efficiency of our methods is measured by analyzing the rate of convergence of the energy minimization, and execution times are shown to be quite reasonable. Robustness is established in the vi presence of large deformations in the parameterizations and the stability of our parameterizations is demonstrated under different discretizations of the surfaces. Our fourth contribution is a set of concrete, compelling applications of surface parameterization. Non-trivial examples which draw from texture mapping, morphing and facial animation provide further evidence and insight into the versatility of our parameterization framework.