Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012)
P. Kry and J. Lee (Editors)

Real-Time Example-Based Elastic Deformation
Yuki Koyama1

Kenshi Takayama1,2

1 The University of Tokyo, Japan

Rest shape!

Nobuyuki Umetani1

2 ETH Zürich, Switzerland

Takeo Igarashi1,3
3 JST ERATO, Japan

Example pose!
(a)!

(b)!

(c)!

Figure 1: (a) User-specified input data comprising the rest shape of the model and its preferred deformed pose(s). (b) A result
animation of the example-based elastic material generated from the input data. (c) A slope scene containing two types ofexample-based elastic cuboids (generated from twist-shaped and S-shaped example poses) represented by 2250 particles intotal; this scene is simulated at 49 frames per second on a desktop with Core i7 2.67GHz CPU and Radeon HD 6630M.
Abstract
We present an example-based elastic deformation method that runs in real time. Example-based elastic deformation was originally presented by Martin et al. [MTGG11], where an artist can intuitively control elastic material
behaviors by simply giving example poses. Their FEM-based approach is, however, computationally expensive
requiring nonlinear optimization, which hinders its use in real-time applications such as games. Our contribution is to formulate an analogous concept using the shape matching framework, which is fast, robust, and easy
to implement. The key observation is that each overlapping local region’s right stretch tensor obtained by polar
decomposition is a natural choice for a deformation descriptor. This descriptor allows us to represent the posespace as a linear blending of examples. At each time step, the current deformation descriptor is linearly projected
onto the example manifold, and then used to modify the rest shape of each local region when computing goal positions. Our approach is two orders of magnitude faster than Martin et al.’s approach while producing comparable
example-based elastic deformations.
Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-Dimensional
Graphics and Realism—Animation

1. Introduction
Achieving intuitive artist control in physically based animations has been a longstanding goal in computer graphics.
Many approaches have been proposed for achieving various
phenomena such as fluids [TKPR06], rigid bodies [TJ08],
cloths [BMWG07], and deformable solids [KKA05].
Recently, Martin et al. [MTGG11] proposed an approachof example-based elastic materials, where an artist can conThe definitive version is available at http://diglib.eg.org/ .

trol the elastic behavior of a deformable object by simply
giving examples of poses, while other data-driven elastic
simulations [SZT∗ 08,JF03] mainly focus on physical plausibility rather than artistic control. Martin et al.’s system computes an additional force that pulls the current pose toward
the space of preferable poses defined by examples (Fig. 1 (a),(b)). Compared to the previous techniques for artistic controlof physically-based animations that are constrained by somefixed keyframe or trajectory, example-based elastic materi-Y. Koyama, K. Takayama, N. Umetani, and T. Igarashi / Real-Time Example-Based Elastic Deformation

als can be used in a free environment without a predefined
scenario or an animation trajectory. Martin et al.’s approach
is more desirable for artist control than the approach of setting actual physical material parameters [BBO∗ 09], because
the artist only needs to specify examples of desired poses.
Their FEM-based approach is, however, computationallydemanding which limits its use in offline simulations only.
This is unfortunate, as artistic expressions are of high importance also for many real-time applications such as interactive virtual environments. The fundamental cause of thehigh computational cost of Martin et al.’s approach is thatthey use an additional elastic potential to attract the current
deformation toward the space of preferable deformations defined by example poses. This approach requires a projectionof the current deformation to the space of preferable deformations, called an example manifold. Since this manifold isnonlinear with respect to their deformation descriptor, theprojection is done by nonlinear optimization which is thebottleneck of their approach.
We achieve example-based elastic deformations in realtime by basing our approach on shape matching [MHTG05].
With shape matching, we can attract the current deformation
toward the space of preferable deformations without relying
on an elastic potential; instead, we can simply modify therest shape of each overlapping local region. Since the example manifold is linear in our approach, we only need toperform a linear projection at each time step, achieving realtime performance. Our approach is two orders of magnitude
faster than Martin et al.’s approach while producing comparable example-based elastic deformations.
Similar to Martin et al., we can extend our approach toallow local examples by dividing the entire set of local regions into separate groups. In addition, by regarding each
local region as a separate group (i.e., defining an independent example manifold for each local region), we can cause
the effect of example-based elastic deformations to appear
non-uniformly according to the stress applied to each local
region. Because shape matching is a purely geometric technique, our approach does not provide as much physical accuracy as Martin et al.’s approach; however, we believe thatphysical accuracy would be of secondary importance in thecontext of art-directed animations.

1.1. Related Work
Shape matching [MHTG05] inspired a number of subsequent extensions. FastLSM [RJ07] achieves large deformations at fast rates by defining particles on a cubic latticeand performing shape matching on many overlapping local regions using a fast summation operator. Steinemannet al. [SOG08] extended this approach to achieve levelsof details for deformations using an octree, while Rungjiratananon et al. [RKN10] modified FastLSM to simulate
complex hairstyles. Müller and Chentanez [MC11] proposed

Current pose!

Example manifold!
Example pose 1!

Projection!

Rest pose!
Goal pose!

Example pose 2!

Figure 2: Overview of run-time operations. Here, the elastic
cuboid model has its rest pose, current pose, and two example poses. In each time step, the current pose is projected
onto the example manifold E; then, the projected point is assumed to be the goal pose, and the current pose is pulled
toward it.
oriented particles to achieve stability with much fewer particles by augmenting each particle with orientation information. Diziol et al. [DBB11] took a different approach, where
surface meshes are used for simulation. To our knowledge,
we are the first that extend shape matching to example-basedelastic deformations. Note that our approach is orthogonal tothese previous extensions and thus can benefit from them.
Ijiri et al. [ITYI09] used shape matching to procedurally
generate active deformations of unarticulated objects such as
jellyfish by expanding and contracting local regions. They
actively change the rest shapes of local regions to generate amotion while we passively change rest shapes responding to
deformations caused by external forces.
2. Overview of Run-time Operations
Before describing our method in detail, we first provide an
overview of run-time operations. In the proposed method,
the concept of example-based elastic materials is realized byprocessing the following two operations in each time step:
1. Linearly projecting the current deformed pose toward theexample manifold E to obtain a goal pose. (§4.2)
2. Pulling the current deformed pose toward the goal pose.
(§3)
The key difference between Martin et al.’s and our method
is that we define the example manifold E linearly; that is,
we define it as a convex hull of the linear blending of therest and example poses (§4.1). Thus, the projection can be
efficiently computed without any nonlinear optimizations.
Fig. 2 presents a schematic representation of these operations.
3. Modification in Shape Matching Dynamics
We closely follow the approach of convolving shape matching operators on overlapping local regions to drive deforma-

Y. Koyama, K. Takayama, N. Umetani, and T. Igarashi / Real-Time Example-Based Elastic Deformation

tions [RJ07, ITYI09]. The input to our system consists ofa volumetric tetrahedral mesh (rest shape) and n example
poses. A triangular surface mesh embedded into the tetrahedral mesh is used to render the model with detailed geometry. Each tetrahedral mesh vertex is treated as a particle,
and a local region Ni is defined for each particle i by collecting the particle itself and its 1-ring neighbors [ITYI09].
Note that our approach does not assume any specific way ofconstructing particles and local regions, and works well withother extensions of shape matching such as FastLSM [RJ07]
and oriented particles [MC11].
The rest position, current position, and mass of particle iare denoted as x0i , xi , and mi , respectively. The system first
performs shape matching on each local region r independently. That is, a moment matrix is computed as
−1



  X
 X



T
T



m̃i qi qi  ∈ R3×3
(1)m̃i pi qi  
Ar = 
i∈Nr

x 04

x 03

x4

x10

x 02

x3

g r,4

g r,1

x1

g r,3

x4

x2

g r,2

x3

g r,4

g r,3
x2

g r,1

g r,2

x1

Figure 3: In order to obtain the goal position gr,i of particle
i in local region r, the rest shape of the region is appropriately deformed by the strain S̃r that is obtained by projection (§4.2) in our method, while the standard shape matching uses the undeformed rest shape as the goal shape.

E(x)

( )

E x1

S(x)

E ( x )

( )

S x1

( )

S = ∑ wk S x k
k

i∈Nr

P
m̃ x0
Pi∈Nr i i and cr =i∈Nr m̃iP
m̃ x
Pi∈Nr i i the centers of mass, and pi = xi − cr and qi = x0 − c0rii∈Nr m̃imiwhere m̃i = |Nis the effective mass, c0r =i|the relative positions with respect to the centers of mass. The
moment matrix is then decomposed into its rotation part Rr
andq stretch/shear part Sr using polar decomposition as Sr =ATr Ar and Rr = Ar S−1
r . The goal position of particle i withrespect to local region r is computed as
gr,i = Rr x0i − c0r + cr
(2)
in conventional shape matching, while we modify it as
gr,i = Rr S̃r x0i − c0r + cr

(3)

to achieve example-based elastic deformations. Here, S̃r is
obtained by performing projection onto the example manifold as explained in the next section. This effectively deforms the rest shape of the local region r when computing
goal positions (Fig. 3). The goal position gi of particle i iscomputed as an average of goal positions estimated by its
overlapping local regions as gi = hgr,i ir∈Ni . Finally, the position xi and velocity vi are updated as
fext (t)gi (t) − xi (t)+hhmixi (t + h) = xi (t) + hvi (t + h)vi (t + h) = vi (t) + α

(4)
(5)

where h is the time step, fext is the external force, and α ∈[0 . . . 1] is a parameter of stiffness.
4. Example-Based Elastic Deformation
Similar to Martin et al., we define a space of preferable deformations implied by example poses. We first define a descriptor of a deformation and an example manifold spanned
by descriptors of the example deformations. We then explain how the current deformation descriptor is projected
onto the example manifold. Finally, we show an extension

( )

E x0

Martin et al.!

( )

S x0

Our method!

Figure 4: Comparison between Martin et al.’s (left) and our(right) approach. In Martin et al.’s approach, example manifold E and the projection are nonlinearly defined. In contrast, in our approach, example manifold E can be defined
as a simple linear interpolation, thus the projection can behandled easily.
that achieves locally independent example-based elastic deformations by dividing the entire set of local regions into
separate groups.
4.1. Deformation Descriptor and Example Manifold
We provide a brief review of Martin et al.’s approach forcomparison with ours. Further details for Martin et al.’s approach can be found in [MTGG11]. In their approach, in order to compute an additional elastic potential, it is necessary
to find the deformed pose from a space of preferable posesthat is closest to the current pose. To measure the similarity between deformations, they define a unique descriptorof a deformation x by concatenating the Green strain tensorEi ∈ R6 of each tetrahedral element i as E(x)T = ET1 · · · ETm ∈R6m (3 × 3 symmetric matrices are represented as 6D vectors hereafter). An important fact is that the image of a map
x 7→ E(x), called realizable manifold F ⊂ R6m , is a nonlinear manifold which means that a linear combination of
two descriptors w1 E(x1 ) + w2 E(x2 ) is in general not reconstructible; i.e., it does not correspond to any deformed pose.
Isolated tetrahedra can satisfy the prescribed strain but theirassembly cannot (Fig. 4 left). A continuous space of preferable deformations, called an example manifold E, is there-

Y. Koyama, K. Takayama, N. Umetani, and T. Igarashi / Real-Time Example-Based Elastic Deformation

fore defined by projecting
nP the convex
o hull of example deformation descriptors nk=0 wk E(xk ) onto F . At each time
step, the current deformation is projected onto E using nonlinear optimization, which is the bottleneck of Martin et al.’s
approach.
In our approach based on shape matching, the systempulls the current deformation toward the space of preferable
deformations by modifying the rest shape of each local region (Fig. 3), as opposed to using an additional elastic potential in Martin et al.’s FEM-based approach. We observethat the moment matrix Ar in (1) can be regarded as an approximation of the deformation gradient of local region r as
detailed in Appendix, and thus its strain component Sr as
the right stretch tensor. Analogous to Martin et al., we define a deformation descriptor
by concatenating
Sr of each
local region r as S(x)T = ST1 · · · STm ∈ R6m . Unlike Martin etal., we can compute a valid deformation from an arbitrary
deformation descriptor, because inconsistent goal positions
estimated by overlapping local regions are blended by averaging. In other words, the issue of reconstructibility of deformation descriptors is nonexistent in our case, allowing us
to define an example manifold
nP as a convex
o hull of example
n w S(xk ) (Fig. 4 right). We
deformation descriptorskk=0
linearly project S onto the example manifold to yield S̃ as
explained below, and use it when computing goal positionsin (3).
4.2. Projection to Example Manifold
Denoting the rest pose’s descriptor (i.e., concatenation ofidentity matrices) and the k-th example pose’s descriptor as
S0 = S(x0 ) and Sk = S(xk ), respectively, our goal is to findP
a convex combination S̃ = nk=0 wk Sk that is closest to thecurrent pose descriptor S = S(x). We first compute weights
w1 , . . . , wn by minimizing a quadratic energy
n
X

(6)

k=1

whose solution wT = (w1 , . . . , wn )T is obtained as−1
w = LT L LT (S − S0 )0.995!

0.95!

0.8!

0.5!

Figure 5: Varying the magnitude of the example-based elastic deformation effect by changing β.
Rest shape! Example pose!

: Control handle!

Figure 6: Local regions of the teddy bear are divided into
two separate groups at its middle. The teddy bear’s left andright halves can be manipulated independently.
of each local region when computing goal positions (Fig.
3). This is in contrast to Martin et al.’s approach where theconventional potential pulls the deformation back to the rest
pose. To avoid this problem, we modify the weights such thatthe rest pose always gets a slightly higher weight than others.
We introduce a parameter β ∈ [0, 1) that modifies the weightsP
as w00 = w0 + (1 − β) nk=1 wk and w0k = βwk (k = 1, . . . , n). This
modification has a large enough impact even when β is only
slightly smaller than 1 (e.g., β = 0.995), and the effect ofexample-based elastic deformation decreases as β becomes
smaller (Fig. 5).
4.3. Extension to Local Examples

2

wk (Sk − S0 ) − (S − S0 )

1.0!

(7)

where L = (S1 − S0 · · · Sn − S0 ) ∈ R6m×n which is constantP
during simulation. We then obtain w0 = 1 − nk=1 wk .
This may yield negative weights that will result in extrapolation beyond example poses, which would be undesirable
for artist control. We eliminate negative weights by iteratively performing the following simple procedure: the smallest negative weight is chosen and set to zero, and its absolute value is divided by n and subtracted from all the otherweights.
Another concern is that the deformation may become
stuck somewhere in the space of preferable poses instead ofgoing back to the rest pose, because we change the rest shape

We can achieve an effect of local examples similar to thosedemonstrated by Martin et al (Fig. 6). We divide the entire
set of local regions into separate groups and define an example manifold for each group independently. Furthermore, bytreating each local region as a separate group (i.e., each local
region’s example manifold lies in 6D space), we can achieve

Global!

Local!

Figure 7: Each local region has its own example manifold
independently. The difference from the case of using a globally defined example manifold is evident.

Y. Koyama, K. Takayama, N. Umetani, and T. Igarashi / Real-Time Example-Based Elastic Deformation

1!

2!

3!

4!

model
cuboid (global)
cuboid (local)
cylinderteddycar

Figure 8: Four cylinders undergoing very different elastic
deformations.

(No example)!

Arch!

Twist!

Wave!

#particles
225
225
2025
1280
192

an effect where each local region deforms non-uniformly according to the stress applied to it (Fig. 7). This is closer tothe physical law than the case of using a globally definedexample manifold, because in the real world an object’s deformation occurs as a result of stress applied to each local
region.
5. Results and Discussion
We can simulate many objects that exhibit example-basedelastic deformations in real time (Fig. 1, 8, 9). Table 1 shows
timings for the models shown in this paper. Note that thecomputational cost of projecting the current pose onto theexample manifold is much smaller than those of the otherprocesses. Our approach is two orders of magnitude fasterthan Martin et al.’s approach which typically took a few seconds per time step for models with about thousand DOFs.
This real-time performance is already achieved with ourunoptimized implementation; combining our approach withother techniques for accelerating shape matching such as
FastLSM [RJ07] and oriented particles [MC11] would further improve the performance.
Comparison. Fig. 10 shows a rough comparison between
Martin et al.’s example-based elastic materials and ours. We
believe that our method successfully reproduces practicallythe same behavior as Martin et al.’s approach, while achieving real-time performance. Note that our 3D model, example pose, and simulation and rendering conditions are not

tproj
0.091
0.17
0.86
0.72
0.078

ttot
0.33
0.40
3.1
2.2
0.27

Table 1: Timings per time step in milliseconds measured
using a single core of Intel Core i7 M620 2.67 GHz CPU.
tsm , tproj , and ttot denote timings for shape matching, projection onto the example manifold, and total simulation, respectively.

Martin et al.!

Figure 9: Cars colliding against wall, without example
shape and with three kinds of example shapes.

tsm
0.21
0.21
2.0
1.3
0.17

Our method!

Figure 10: Comparison between Martin et al.’s twistingelastic cuboid and ours under gravity. Note that the simulation and rendering conditions are not exactly identical. The
left image is from [MTGG11], c 2011 Association for Computing Machinery, Inc. Reprinted by permission.
exactly the same as Martin et al.’s. For a comparison of theanimations, please see the supplementary material.
Limitation. Although physical accuracy would be of secondary importance in the context of art-directed animations,
it is desirable that the other elastic behaviors orthogonal toexample poses are as physically accurate as possible to ensure naturalness. Our approach based on shape matchingis inherently limited in this respect because of its purely
geometrically-based nature, as opposed to Martin et al.’s
FEM-based approach.
Future work. It would be possible to apply our approachto elastic thin shells and rods by using other extensions ofshape matching such as oriented particles [MC11] and chain
shape matching [RKN10]. Another possibility of further investigation is different way of projecting the current pose
onto the example manifold. It would be interesting to take
into account other contextual information such as velocity,
forces, and relative positions among characters in proximity.
Acknowledgement
We would like to thank Sebastian Martin and the coauthorsof [MTGG11] for permitting the reuse of their video clipin our supplemental material. We also thank the anonymous
reviewers for their helpful comments. The car model is fromthe Princeton Shape Benchmark. The teddy model is from
Watertight Models Track of SHREC 2007.

Y. Koyama, K. Takayama, N. Umetani, and T. Igarashi / Real-Time Example-Based Elastic Deformation

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Appendix: Moment Matrix as Approximate
Deformation Gradient Tensor
Here we show that the moment matrix Ar in (1) approximates the deformation gradient tensor with respect to the
local region r. As shown by Müller et al. [MHTG05], themoment matrix is the minimizer of a quadratic energyXm̃i kAqi − pi k2(8)iwhere qi = x0i − c0 and pi = xi − c are relative particle positions with respect to the center of mass for the rest andcurrent configurations, respectively (subscript r is omitted
hereafter). In continuum mechanics [BW08], the deformation gradient is defined as follows: considering a material
point inside an object as a pivot whose rest and current positions are denoted as X and x, respectively, another material
point infinitesimally close to the pivot whose rest and current
positions are denoted as X0 and x0 , respectively, is related tothe pivot asx0 − x = F(X0 − X)

(9)where F is called the deformation gradient tensor. By setting
the pivot to the center of mass and assuming that F approximates the deformations of particles within the region, F is
the minimizer of a quadratic energyXwi kFqi − pi k2(10)iwhere wi are weights. By setting weights wi to m̃i , (10) is exactly the same with (8). Note that Gerszewski et al. [GBB09]
used a similar formulation for the purpose of point-based
simulation of elastoplastic solids.