Optimal Surface Parameterization Using Inverse Curvature Map
IEEE TVCG 2008
Yongliang Yang, Junho Kim, Feng Luo, Shimin Hu and Xianfeng Gu
Mesh parameterization is a fundamental technique
in computer graphics. The major goals during mesh parameterization
are to minimize both the angle distortion and the
area distortion. Angle distortion can be eliminated by the use of
conformal mapping, in principle. Our paper focuses on solving
the problem of finding the best discrete conformal mapping that
also minimizes area distortion.
Firstly, we deduce an exact analytical differential formula to
represent area distortion by curvature change in the discrete
conformal mapping, giving a dynamic Poisson equation. On a
mesh, the vertex curvature is related to edge lengths by the
curvature map. Our result shows the map is invertible, i.e.
the edge lengths can be computed from the curvature (by
integration). Furthermore, we give the explicit Jacobi matrix of
the inverse curvature map.
Secondly, we formulate the task of computing conformal parameterizations
with least area distortions as a constrained nonlinear
optimization problem in curvature space. We deduce explicit
conditions for the optima.
Thirdly, we give an energy form to measure the area distortions,
and show that it has a unique global minimum. We use this
to design an efficient algorithm, called free boundary curvature
diffusion, which is guaranteed to converge to the global minimum;
it has a natural physical interpretation.
This result proves the common belief that optimal parameterization
with least area distortion has a unique solution and can
be achieved by free boundary conformal mapping.
Major theoretical results and practical algorithms are presented
for optimal parameterization based on the inverse curvature map.
Comparisons are conducted with existing methods and using
different energies. Novel parameterization applications are also
introduced. The theoretical framework of the inverse curvature
map can be applied to further study discrete conformal mappings.