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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

(



), Han-Bing Yan

2,∗

(





), and Ralph R. Martin

Department of Computer Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China

National Computer Network Emergency Response Technical Team/Coordination Center of China, Beijing 100029, China

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

Ya-Shu Liu et al.: As-Rigid-As-Possible Surface Morphing 549

Fig.1. Morphing comparison using a total of 5 frames. (a) Morphing using linear interpolation. (b) Morphing using Alexa’s method

[5]

(c) Morphing using our method. In (b), initial and ﬁnal tetrahedral meshes are shown in gray, while in (c) initial and ﬁnal triangle

meshes are shown.

The inputs to our method are a source mesh and a

target mesh, in the form of consistent triangle meshes,

and the desired number of intermediate frames. The

output is a sequence of intermediate meshes forming a

morphing sequence.

Fig.1 illustrates the results produced by linear in-

terpolation, Alexa’s original as-rigid-as-possible volume

warping method

[5]

, and our new surface-mesh-based

method. Our method produces almost identical re-

sults to Alexa’s volume-based morphing method, but at

much lower computational cost, and without the need

to produce volume meshes and make them consistent.

We note that [6] uses ideas somewhat similar to ours

to solve the deformation problem. Nevertheless, there is

a signiﬁcant diﬀerence. In our method, we incorporate

the translation vector into the error function, avoiding

the need for the user to ﬁx the location of any vertex in

the solution process. Appropriate choice of the vertex

to ﬁx is not a simple task, so our approach simpliﬁes

the interface for the user. Nevertheless, in the degen-

erate case, our error function corresponds to the form

not using a translation vector, allowing the artist to ﬁx

any vertex if desired, giving the artist the freedom to

choose how much explicit control will be used.

2 Related Work

Methods for 3D morphing typically take one of two

approaches. The ﬁrst blends simpler volumes into

which the initial and ﬁnal shapes have been embe-

dded

[7-9]

. The second directly manipulates an explicit

geometric object representation, typically a surface

mesh or volume mesh

[2,5]

. The ﬁrst class of approach

has the advantage of being able to morph objects hav-

ing diﬀerent topologies. However, mesh-based meth-

ods typically produce better results — often, shape

boundaries resulting from use of embedding methods

are not smooth enough. Thus, the focus of morphing

has shifted towards mesh-based approaches in recent

years. Our method is based on use of surface triangle

meshes.

As noted earlier, explicit surface or volume mesh

morphing typically uses two main steps: creating con-

sistent meshes, and determining vertex trajectories.We

ﬁrst brieﬂy review the former, then consider the latter

in more detail as it is the main focus of this paper.

Various work has considered how to create consis-

tent meshes for pairs of shapes of genus zero, using a

topological merging method. A frequent approach is to

ﬁrst dissect the source and target shapes into several

pieces

[10]

, then to construct a local parameterization

for each piece, and ﬁnally to perform merging

[4,11]

remeshing

[2,12]

to create the consistent mesh. Praun

[13]

gives a tracing method which can dissect source and

target shapes automatically.

Many of the above papers concentrate on the prob-

lem of creating consistent meshes, and only use simple

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