LiuYS,YanHB,MartinRRet al. As-rigid-as-possible surface morphing. JOURNAL OF COMPUTER SCIENCE AND
TECHNOLOGY 26(3): 548–557 May 2011. DOI 10.1007/s11390-011-1154-3
As-Rigid-As-Possible Surface Morphing
Ya-Shu Liu
1
(
), Han-Bing Yan
2,∗
(
), and Ralph R. Martin
3
1
Department of Computer Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China
2
National Computer Network Emergency Response Technical Team/Coordination Center of China, Beijing 100029, China
3
School of Computer Science, Cardiff University, Cardiff, U.K.
E-mail: ly
s8020@163.com; yanhb02@mails.tsinghua.edu.cn; ralph@cs.cf.ac.uk
Received October 17, 2009; revised February 25, 2011.
Abstract This paper presents a new morphing method based on the “as-rigid-as-possible” approach. Unlike the original
as-rigid-as-possible method, we avoid the need to construct a consistent tetrahedral mesh, but instead require a consistent
triangle surface mesh and from it create a tetrahedron for each surface triangle. Our new approach has several significant
advantages. It is much easier to create a consistent triangle mesh than to create a consistent tetrahedral mesh. Secondly, the
equations arising from our approach can be solved much more efficiently than the corresponding equations for a tetrahedral
mesh. Finally, by incorporating the translation vector in the energy functional controlling interpolation, our new method
does not need the user to arbitrarily fix any vertex to obtain a solution, allowing artists automatic control of interpolated
mesh positions.
Keywords morphing, simplex, transformation, interpolation
1 Introduction
Morphing, also called metamorphosis or shape
blending, is a technique used to smoothly transform
one graphical shape into another. Many 2D morphing
methods have been developed to assist 2D animation
making
[1]
. In recent years, with the introduction of 3D
cartooning techniques, 3D morphing has increased in
importance in games and animation production.
The most popular type of model used in 3D graphics
is a surface triangle mesh. In this paper, we propose a
new morphing method that works well even when two
input 3D triangle meshes have very different shapes.
Many successful approaches to morphing use a frame-
work based on firstly creating consistent meshes for the
original source and target models, i.e., source and tar-
get meshes having one to one correspondences between
their vertices, edges and faces. Given these, a path may
then be determined for each vertex to follow during the
morphing process. Much work has considered the first
issue
[2-4]
; this paper focuses on the trajectory problem.
The approach used in this paper is based on the
as-rigid-as-possible warping method
[5]
for generating
a smooth morph between two quite different shapes.
However, that method requires 3D objects to be rep-
resented as volume, tetrahedral meshes. Unfortunately,
as is well known, meshing complex solid is not easy, and
creating consistent tetrahedral meshes is notoriously
difficult. Indeed, we are unaware of any satisfactory
general solution to that problem.
Our new method has the following merits compared
to the original as-rigid-as-possible method, while still
providing very good 3D morphing results.
• Our method works directly with surface triangle
meshes instead of requiring tetrahedral meshes that
represent the interior of each object.
• Creating consistent triangle meshes is a much ea-
sier problem to solve than creating consistent tetrahe-
dral meshes.
• The number of vertices in consistent triangle
meshes is far fewer than that in corresponding tetrahe-
dral meshes of the same objects at the same resolution,
resulting in greatly reduced computing time.
• By incorporating the translation vector into our
energy function, we do not need to fix some arbi-
trary vertex when computing the solution; the original
method needs user selection of at least one fixed vertex.
Regular Paper
This work was supported by the National Natural Science Foundation of China under Grant No. 61003132, the EPSRC Travel
Grant, the Technology Project of MOUHURD of China under Grant No. 2010-K9-25, and the Development Project of BMCE under
Grant No. KM200710016001.
∗
Corresponding Author
2011 Springer Science + Business Media, LLC & Science Press, China
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