Constructing a map from a source domain to a target domain is an essential part of many geometry processing applications. Desirable properties of the map are smoothness, low amount of metric-or-angular distortion and the ability to handle general boundary conditions. One important property that is often required but seldom guaranteed is injectivity.
In this talk, we will explore several approaches for producing locally injective maps. Conformal maps are heavily used in geometry processing application since they are smooth, preserve angles, and are locally injective by construction. However, conformal maps do not allow for boundary positions to be prescribed. A natural extension to the space of conformal maps is the richer space of quasi-conformal maps of bounded conformal distortion.
Extremal quasi-conformal maps, that is, maps minimizing the maximal conformal distortion, have a number of appealing properties making them a suitable candidate for geometry processing tasks. Similarly to conformal maps, they are guaranteed to be locally injective; unlike conformal maps however, they have sufficient flexibility to allow for solution of boundary value problems. Moreover, these solutions are guaranteed to exist, are unique, injective and have an explicit characterization.
I will present an algorithm for computing piecewise linear approximations of extremal quasi-conformal maps for genus-zero surfaces with boundaries, based on Teichmüller’s characterization of the dilatation of extremal maps using holomorphic quadratic differentials. I will demonstrate that the algorithm closely approximates the maps when an explicit solution is available and exhibits good convergence properties for a variety of boundary conditions.