Manifold Splines with Single Extraordinary Point
IEEE Transactions on Pattern Analysis and Machine Intelligence
Xianfeng Gu,Ying He,Miao Jin, Feng Luo,Hong Qin,Shing-Tung Yau
This paper develops a novel computational technique to define and
construct powerful manifold splines with only one singular point
by employing the rigorous mathematical theory of Ricci flow. The
central idea and new computational paradigm of manifold splines
are to systematically extend the algorithmic pipeline of spline surface
construction from any planar domain to arbitrary topology. As
a result, manifold splines can unify planar spline representations
as their special cases. Despite their earlier success, the existing
manifold spline framework is plagued by the topology-dependent,
large number of singular points (i.e., |2g?2| for any genus-g surface),
where the analysis of surface behaviors such as continuity
remains extremely difficult. The unique theoretical contribution of
this paper is that we devise new mathematical tools so that manifold
splines can now be constructed with only one singular point, reaching
their theoretic lower bound of singularity for real-world applications.
Our new algorithm is founded upon the concept of discrete
Ricci flow and associated techniques. First, Ricci flow is employed
to compute a special metric of any manifold domain (serving as a
parametric domain for manifold splines), such that the metric becomes
flat everywhere except at one point. Then, the metric naturally
induces an affine atlas covering the entire manifold except this
singular point. Finally, manifold splines are defined over this affine
atlas. The Ricci flow method is theoretically sound, and practically
simple and efficient. We conduct various shape experiments and
our new theoretical and algorithmic results alleviate the modeling
difficulty of manifold splines, and hence, promising to promote the
widespread use of manifold splines in surface and solid modeling,
geometric design, and reverse engineering.