Harmonic Maps and Their Applications in Surface Matching

Abstract

The surface-matching problem is investigated in this paper using a mathematical tool called harmonic maps. The theory of harmonic maps studies the mapping between different metric manifolds from the energy-minimization point of view. With the application of harmonic maps, a surface representation called harmonic shape images is generated to represent and match 3D freeform surfaces. The basic idea of harmonic shape images is to map a 3D surface patch with disc topology to a 2D domain and encode the shape information of the surface patch into the 2D image. This simplifies the surface-matching problem to a 2D image-matching problem. Due to the application of harmonic maps in generating harmonic shape images, harmonic shape images have the following advantages: they have sound mathematical background; they preserve both the shape and continuity of the underlying surfaces; and they are robust to occlusion and independent of any specific surface sampling scheme. The performance of surface matching using harmonic maps is evaluated using real data. Preliminary results are presented in the paper.