Continuous map with constraints for homeomorphic surfaces

Abstract

This thesis investigates the problem of establishing a continuous map between two homeomorphic surfaces, given some correspondence specified over parts of their surfaces. This is the foundation of many morphing techniques with user-specified correspondences. In general, with specified correspondence, it is not always possible to have a continuous map between the two surfaces. This thesis presents an approach that re-expresses each surface as a re-tessellated mesh and systematically analyzes and processes the constraints to get the correspondence over the surface. In particular, it identifies situations where the specified correspondence makes a continuous map between the two surfaces impossible, and computes the necessary relaxation required on the specified correspondence so that a continuous map becomes possible. To stay as close to the user-specified constraints, these modifications are to be kept minimal, and restricted to only the boundary of the regions specified in the constraints.