Robust Computation of Bounded-Distortion Maps of Surfaces
Abstract: Computing a surface-to-surface or a surface-to-plane map is one of the most fundamental tasks in geometry processing and is a core component in countless number of applications. Applications range from standard and ubiquitous tools in computer graphics and visualization (texture mapping, image-and-shape-deformation and animation) to engineering analysis and scientific simulation (quadrangulation, remeshing and solving equations on surfaces) as well as geometric data analysis for biological and medical applications in particular (shape comparison, feature analysis and shape compression).
Given a curved or flat surface, the goal is to find a map to a target domain with bounded (and low) amount of distortion while satisfying different types of user constraints (boundary position or orientation, feature point positions etc.). Despite extensive efforts and considerable progress on the mapping problem, a fully robust and efficient parameterization method for surfaces with general constraints still has not been found.
In this talk we will explore several approaches for computing maps that have bounded minimal amount of conformal (angular) distortion. First, we will consider using the subspace of complex holomorphic functions. These maps have zero amount of conformal distortion everywhere except at finite number of points where the map is singular. Then, we will learn how to avoid getting singularities by parameterizing the space of conformal (angle preserving) maps based on the complex logarithm derivative function. Conformal maps have zero angular distortion throughout their domain and are consider ideal for many tasks. However, they do not have the flexibility to support arbitrary position constraints.
In order to support arbitrary position constraints we relax the conformality restriction by allowing the map to be quasiconformal. That is a map that have bounded amount of conformal distortion. Specifically, we will be looking for a quasiconformal map which minimizes the maximal distortion among all possible maps while satisfying the boundary constraints. Such a map is called extremal. The appealing properties of extremal quasiconformal maps fascinated many pure mathematicians who studied them deeply in the last several decades. Yet, they were never computed or visualized before. I will present the first algorithm for computing these maps followed by results.