Topology-based Surface Mapping with Exact Feature Alignment
Christopher Carner, Miao Jin, Xianfeng Gu, and Hong Qin
Proceedings of IEEE Visualization 2005.
Topological concepts and techniques have been broadly applied
in computer graphics and geometric modeling. However,
the homotopy type of a mapping between two surfaces
has not been addressed before. In this paper, we present a
novel solution to the problem of computing continuous maps
with different homotopy types between two arbitrary triangle
meshes with the same topology. Inspired by the rich
theory of topology as well as the existing body of work on
surface mapping, our newly-developed mapping techniques
are both fundamental and unique, offering many attractive
advantages. First, our method allows the user to change
the homotopy type or global structure of the mapping with
minimal intervention. Moreover, to locally affect shape correspondence,
we articulate a new technique that robustly
satisfies hard feature constraints, without the use of heuristics
to ensure validity. In addition to acting as a useful tool
for computer graphics applications, our method can be used
as a rigorous and practical mechanism for the visualization
of abstract topological concepts such as homotopy type of
surface mappings, homology basis, fundamental domain, and
universal covering space. At the core of our algorithm is a
procedure for computing the canonical homology basis and
using it as a common cut graph for any surface with the
same topology. We demonstrate our results by applying our
algorithm to shape morphing in this paper.