Teaser: Parameterization of the Gargoyle model using (a) our As-Similar-As-Possible (ASAP) procedure, (b) As-Rigid-As-Possible (ARAP) procedure, (c) Linear ABF [ZLS07], (d) inverse curvature approach [YKL*08], and (e) curvature prescription approach [BCGB08]. The pink lines are the seams of the closed mesh when cut to a disk.
Abstract
We present a novel
approach to parameterize a mesh with disk topology to the plane in a
shape-preserving manner. Our key contribution is a local/global
algorithm, which combines a local mapping of each 3D triangle to the
plane, using transformations taken from a restricted set, with a global
"stitch" operation of all triangles, involving a sparse linear system.
The local transformations can be taken from a variety of families, e.g.
similarities or rotations, generating different types of
parameterizations. In the first case, the parameterization tries to
force each 2D triangle to be an as-similar-as-possible version of its 3D
counterpart. This is shown to yield results identical to those of the
LSCM algorithm. In the second case, the parameterization tries to force
each 2D triangle to be an as-rigid-as-possible version of its 3D
counterpart. This approach preserves shape as much as possible. It is
simple, effective, and fast, due to pre-factoring of the linear system
involved in the global phase. Experimental results show that our
approach provides almost isometric parameterizations and obtains more
shape-preserving results than other state-of-the-art approaches.
We present also a more general "hybrid" parameterization model which
provides a continuous spectrum of possibilities, controlled by a single
parameter. The two cases described above lie at the two ends of the
spectrum. We generalize our local/global algorithm to compute these
parameterizations. The local phase may also be accelerated by
parallelizing the independent computations per triangle.
Matlab source codes (0.6M)
Thanks to
Alla Sheffer for providing the models with cutting seams and thanks to Yongliang Yang and Miri Ben-Chen for providing results of the algorithms
of [YKL*08] and [BCGB08]. Thanks also to Hugues Hoppe for his many
helpful insights on this topic. This work is supported by National
Natural Science Foundation of China (grants # 60776799, 60503067). Craig
Gotsman and Steven Gortler were partially supported by United States -
Israel Binational Science Foundation grant # 2006089.
title = {A Local/Global Approach to Mesh Parameterization},
author = {Ligang Liu and Lei Zhang and Yin Xu and Craig Gotsman and Steven J. Gortler}
journal = {Computer Graphics Forum (Proceedings of Eurographics Symposium on Geometry Processing)},
volume = {27},
number = {5},
pages = {1495-1504},
year = {2008}
}