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Meshless Deformations Based on

Shape Matching

SIGGRAPH 2005

Matthias M¨uller, Bruno Heidelberger,

Matthias Teschner, Markus Gross

COMPSCI 715 S2 C - Advanced Computer Graphics

Heiko Voigt, Daniel Flower, Daniel Weisser

14 October 2005

Contents

1 Introduction 4

1.1 Problemdescription.............................. 4

1.2 Solutiondescription.............................. 4

1.3 Overview.................................... 5

2 Theoretical background 6

2.1 Physicalbackground ............................. 6

2.1.1 Motionofobjects........................... 6

Distance, velocity and acceleration . . . . . . . . . . . . . . . . . 6

Newton’s second law of motion . . . . . . . . . . . . . . . . . . . 6

2.1.2 Particlesystems............................ 7

2.1.3 Handling collisions and gravity . . . . . . . . . . . . . . . . . . . 7

Gravity ................................ 8

Collisions ............................... 8

Collisions and gravity overview . . . . . . . . . . . . . . . . . . . 10

2.2 Integrationmethods ............................. 11

2.2.1 Explicit Euler Integration . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Implicit Euler Integration . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Explicit vs. Implicit Integration . . . . . . . . . . . . . . . . . . . 12

2.3 Stability of Euler integration . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Forcecalculation ........................... 14

2.3.2 Eulerintegration ........................... 15

2.3.3 Example calculation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Stable integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 ShapeMatching............................ 16

2.4.2 Extended Euler integration . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Extended shape matching . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Rigid body dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 Linear deformations . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.3 Quadratic deformations . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Plasticity ................................... 21

2

3 Our Problems 23

3.1 Integrationscheme .............................. 23

3.2 Rotation matrix implementation . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Frame rate-independent velocity updates . . . . . . . . . . . . . . . . . . 24

3.4 Quadratic deformation of goal points . . . . . . . . . . . . . . . . . . . . 24

3.5 Plasticity ................................... 24

4 Their evaluation of the algorithm 25

4.1 Stability .................................... 25

4.2 Performance.................................. 25

5 Flaws and limitations 27

5.1 Physicalcorrectness.............................. 27

5.2 Plasticity ................................... 27

5.3 Connectivity information . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Impact of the paper 28

Bibliography 29

3

1 Introduction

1.1 Problem description

When objects in the real world have force applied to them, they change shape, i.e.

deform, and then return completely or nearly back to their original shape, even if only a

small amount. Therefore, graphical objects should also be able to change shape, because

having deformable objects in computer games and animated movies, or even scientiﬁc

simulations, would increase the realism a great deal.

However, up until now, most objects in interactive applications are not deformable. The

main reason for this is probably eﬃciency. Techniques which try to simulate physical

reality are very computationally expensive, not to mention diﬃcult to program. Further-

more, having a stable solution (for example ensuring that the volume, mass or energy

of an object stays constant) makes the technique even more computationally expensive.

Finally, the deformation needs to be controllable by the programmer, so that the object

behaves in a way that the developer intended.

The technique that we are studying [M¨uller et al., 2005] claims to solve all the above

problems by using a simple geometric model. The simplicity ensures that the technique

is eﬃcient, easy to program, stable, and controllable by varying certain parameters.

1.2 Solution description

The main point of diﬀerence with this technique compared to others is that rather than

representing the object with a mesh or surface representation, it is represented by a

particle system. The idea is that the normal shape of the object is speciﬁed by some

arrangement of particles, but when forces are applied to the particles they lose this

shape.

The main focus of the paper is on ﬁnding the best transformation to describe the change

in location of the particles from their original shape to their deformed shape, which is

called “shape matching”. Each particle is then pulled from its current position towards

its corresponding position in the shape created by the transformation (i.e. towards its

goal position), and the particles eventually return to their original positions. This is

shown graphically in ﬁgure 1.1.

4

Figure 1.1: Overview of the shape matching algorithm. A shape starts in its normal

shape, as shown in (a). After a force is applied, the particles lose their

shape, as shown in (b). The shape matching algorithm ﬁnds the best trans-

formation which will convert the original shape to the current shape with

the least error, as shown in (c). This step is what makes the goal positions

githat the particles are pulled to. Finally, in (d), the particles move towards

their goal positions. The graphic is derived from [M¨uller et al., 2005]

1.3 Overview

This report gives a detailed overview on the theoretical aspects of the paper in section

2. After that our problems with the given are described and the algorithm is evaluated.

Section 5 mentions the ﬂaws and limitations that we found, when we tried to implement

the algorithm described in the paper. The concept of clustering, as described in the

paper, is ignored in this report.

For mathematical formulae, where it is necessary to distinguish between vector and

non-vector quantities, the vector quantities are indicated in bold type.

5

2 Theoretical background

2.1 Physical background

2.1.1 Motion of objects

The paper requires a basic understanding of physics. The most important aspects to un-

derstand are the relationship between distance, velocity and acceleration, and Newton’s

second law of motion.

Distance, velocity and acceleration

Although we all know what these words mean, it is important to understand their

relationship with each other. Velocity vdescribes how distance xchanges over time,

and acceleration adescribes how velocity changes over time. In other words, velocity is

the derivative of distance over time x0, and acceleration is the derivative of velocity v0

over time. Integration goes the other way, so integrating acceleration gives velocity, and

integrating velocity gives distance. Mathematically, this can be shown as

v(t) = x0(t)

a(t) = v0(t)

and so also

a(t) = x00(t)

Figure 2.1 explains this connection graphically.

Newton’s second law of motion

Newton’s second law of motion states the relationship between the mass of an object,

its acceleration, and the force applied to achieve this acceleration, which is:

F=ma

This is an important equation because it allows us to calculate how the velocity of an

object will change when a force is applied to it. Another way to look at it is that

acceleration is equal to force divided by mass, which is important because it shows that

acceleration, and hence velocity, can only be changed when a force is applied.

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Figure 2.1: This illustrates the integration of all three items assuming a constant accel-

eration. (a(t) = 1,v(t) = t,x(t) = t2)

2.1.2 Particle systems

The solution given in this paper requires that the objects be represented as a particle

system. In graphics, objects are generally represented as meshes (i.e. vertices and

information on how the vertices are connected) or by some surface representation.

In a particle system, an object simply has a set of particles. A particle might be a single

point in space, or it may be a shape such as a small sphere. Each particle maintains its

own position, velocity, acceleration and other properties depending on the application.

The meshless deformations solution requires that a particle also keep track of its original

position, goal position, and mass.

The original position is needed to be able to calculate the displacement of a particle,

and the goal position is simply the position that the particle should move to (this is

explained in subsequent sections). The mass is required so that the object’s centre of

mass can be calculated.

At each time step, each particle is updated separately. It’s velocity is incremented by its

acceleration, and its position is updated by its velocity. External forces such as gravity

can also be added by increasing the velocity. Obviously if all particles have the same

velocity and acceleration, then the object will move as if it were a solid object. It is

only when some force disrupts this uniformity of movement, that things begin to get

interesting...

2.1.3 Handling collisions and gravity

Common sense and basic physics tells us that no object will deform without an external

force acting upon it. At ﬁrst read then, it seems most curious that a paper dealing in

deformations only mentions forces in passing. To understand why such little attention is

given to forces, it is important to understand what exactly is the concern of this paper.

As explained earlier, an object is a group of particles, where each particle has its current

position, and its goal position. When all the particles are in their goal positions, the

7

object will be in its regular shape, and the method this paper describes simply moves a

particle from its current position towards its goal position. The paper is not concerned

with how a particle became displaced from its goal position, even though without any

displacement no deformations will occur.

There are several sources of deformation, however in this section we will look at just

external, constant forces acting on all particles equally, such as gravity, and the forces

experienced through collisions by individual particles1.

Gravity

The eﬀects of gravity are introduced in the Integration section2. The term that adds

the force to a single particle is hfext(t)/mi, where his the amount of time passed since the

last update, fext(t) is the external force at that time, which is the same for all particles,

and miis the mass of the particle.

Because force equals mass multiplied by acceleration, this equation can be simply thought

of as the acceleration caused by gravity multiplied by the time step amount. Note that

this equation is completely independent of the shape matching, and so adding grav-

ity becomes the simple task of incrementing the velocity of each particle by a certain

amount.

Collisions

Dealing with collisions in a computer simulation involves two steps: collision detection

and collision response. Collision detection involves determining which, if any, objects are

colliding with each other, and where the collision takes place. Collision response applies

appropriate forces to the colliding objects so that they behave realistically. Although

response obviously comes after detection, we will look at response ﬁrst as this is more

closely related to the preceding paragraphs.

Collision response Collision response involves applying an instantaneous force to a

colliding object. An instantaneous force, as the name suggests, is a force that lasts for

only a very short time, for example when a bat hits a ball, or when you punch your best

friend in the face. Note that an instantaneous force cannot be added in the same way as

a constant force like gravity. This is because an instantaneous force should be applied

during just one time step, and because the time step length changes as the frame rate

changes, depending on the frame rate a diﬀerent amount of force would be added, and

would never be the same amount of force as was requested.

1Another source of force comes from user interaction, such as dragging a particle with the mouse. This

can be thought of as the equivalent of a collision response.

2Section 3.4, Equation 9 of the SIGGRAPH paper [M¨uller et al., 2005]

8

While the paper does not mention how to add instantaneous forces, once again because

the shape matching does not take into account any forces, nor the velocity of the particle,

we can simply set the velocity to a new value, and the next time the velocity and position

of this particle is updated, it will be moving with a new velocity. This will cause the

centre of mass of the object, and hence the goal positions, to change, and deformation

will occur.

One way to calculate the response force vector is to get the normal of the point that

the particle collided with. The response vector is simply the velocity vector reﬂected

on the normal. This can be multiplied by an elasticity value between zero and one to

simulate the energy lost in the collision, where zero means all energy was lost (and so

the particle does not rebound) and one means no energy was lost (and so the particle

will bounce around forever). The response vector can simply be added to the particle’s

velocity vector.

Collision detection Collision detection is another area which was glossed over in the

paper. Collisions with the environment and other mesh-based objects are straightfor-

ward. Each particle independently checks whether it is colliding with another object,

and numerous techniques are available for determining the point of impact (for example

there are equations to calculate the distance between a point and a plane, among many

others).

While the paper states that there are no particle-to-particle interactions within an ob-

ject, the particles of two separate objects do need to somehow react to each other. One

possibility is that particles of separate objects simply detect collisions between them-

selves, but because an object is not represented by a surface, there are many empty

spaces where collisions cannot be detected (see ﬁgure 2.2).

Figure 2.2: Because shapes are represented by particles, rather than surfaces, they can

easily pass through each other. In this example, the small, brighter ob-

ject passes right through the larger dark object without colliding with any

particles.

There are ways to mitigate this problem. One such way is to add repulsive forces between

particles so that when particles become close they push each other away. However this

can give the unrealistic eﬀect whereby two objects respond to each other without making

9

any contact, however it does make it more diﬃcult for a particle to travel “inside” another

object. Another possible solution is to simply add more particles, in eﬀect simulating a

surface (see ﬁgure 2.3). Combining these two approaches, particularly having particles

throughout the volume of the object which repulse the particles in another object, makes

it very diﬃcult for two objects to intersect each other.

Figure 2.3: An example of collision detection working only on the particles of the ob-

jects. The particles are large and bunched closely together so that when the

two objects meet, they do not intersect.

A problem with reducing the likelihood of intersection, is that it also reduces the likeli-

hood that two objects that do manage to intersect each other can untangle themselves.

Simple particle-to-particle collisions are not satisfactory when dealing with object-to-

object collisions. In the example simulations shown in the paper (for example where

hundreds of shoes successfully deform each other without intersecting each other) the

objects are loaded as meshes, and a subset of the vertices become the particles used

in the shape matching section. Conventional collision detection techniques can then be

used.

While loading objects as a mesh and using the mesh for collision detection solves the

above problems, it makes irrelevant one of the supposed advantages of meshless defor-

mations, which is that no connectivity information is needed between the points (i.e. no

mesh is needed). It would seem that this claim is only true when there is only a single

deformable object interacting with non-deformable objects3.

Collisions and gravity overview

•Shape matching involves moving a particle from its current, deformed position to

its undeformed position

•Objects only get deformed when forces are applied to them

•Forces and collision detection/response are handled independently of the shape

matching

3The usefulness of not requiring connectivity information is possibly moot anyway due to the fact that

some kind of mesh is needed in order to texture map the object.

10

•Forces are applied to single particles, rather than the object as a whole, by changing

the velocities of the particles

•Because a particles system does not represent a continuous surface, collisions be-

tween objects can only be properly handled when each object has a mesh deﬁned

2.2 Integration methods

The position x, velocity vand acceleration aare all dependent on each other by being

diﬀerent derivatives (x00(t) = v0(t) = a(t)) in time (see section 2.1.1). We need a method

to numerically calculate the integration from one to another. There is no analytical

solution due to fact that the acceleration at each time step tis only known at runtime.

The acceleration consists of diﬀerent possible forces. For example there are gravity,

shape matching and drag forces.

2.2.1 Explicit Euler Integration

The explicit Euler integration is a linear integration scheme which is based on only one

sample point f(x) plus its diﬀerence 4hf0(x).

f(t+4h) = f(x) + 4hf0(x)

Example: The integration scheme for the velocity of a spring system is based on the

acceleration force f=−k(x(t)−l0). The system is ﬁxed at the origin and the other

point with mass mis free. l0is the length of the spring and its resting state. k is the

spring constant. The acceleration a(t) is the ﬁrst derivative of the velocity v(t) with

respect to time t. This gives

v(t+4h) = v(x) + 4ha(x)

since the acceleration is a=f/m. The explicit Euler integration for the velocity is:

v(t+4h) = v(x) + 4h−k(x(t)−l0)

m

See [Weisstein, 2005] for Reference.

11

2.2.2 Implicit Euler Integration

Another approach is the implicit Euler integration. The term looks similar to the explicit

integration but takes the diﬀerence of one step ahead 4hf 0(x+4h) to integrate. This

cannot be explicitly solved and usually leads to linear equations.

f(t+4h) = f(x) + 4hf0(x+4h)

The Example for this equation is

v(t+4h) = v(x) + 4h−k(x(t+4h)−l0)

m

for the velocity. [Wikipedia, 2005]

2.2.3 Explicit vs. Implicit Integration

To compare both integration methods we take the example function f(x) = x2. The

derivative is f0(x) = 2x= 2pf(x) this leads to the following equations:

•Explicit Integration:

f(x+4h) = f(x) + 42pf(x)

•Implicit Integration:

f(x+4h) = f(x) + 42pf(x+4h)

0 = f(x+4h)2−4(4h)2f(x+4h) + f(x)2

This is then solved to

x1,2= 2(4h)2±p4(4h)4−f(x)2

The term underneath the square root can not be negative because we are integrat-

ing in the positive direction.

Figure 2.4 shows the comparison of both integration methods.

The calculation of the Implicit Integration is more complex than the Explicit scheme

but provides more accurate results in general. This evaluation scheme was described in

Kesselheim [2005]

12

Explicit Euler Integration

Implicit Euler Integration

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Figure 2.4: Comparing Euler Integration Methods

2.3 Stability of Euler integration

Stability is a key issue for 3D animation and modelling. It is not acceptable for animated

models to behave physically incorrectly if parameters of it are changed. The Euler

integration is known to be stable just under certain circumstances (see [Weisstein, 2005])

so it is not in general suitable for integrating from the applied forces to the position of

an object.

This section describes the ﬂaws of the standard Euler integration with an example where

it can be seen that this integration method is not suitable for modelling animations that

have to follow Newton’s laws.

The example that is used is based on the example of the paper [M¨uller et al., 2005]

but tries to elaborate more on the instability of the Euler integration for a spring. The

physical system that is used is a simpliﬁed mass-spring system. Figure 2.5 shows the

relevant parameters to perform the calculation. The spring is classiﬁed by a resting

length l0and a spring constant k. Physically the spring connects two points. One of the

points is ﬁxed and the other point is free and connected with the mass m.

The free point x(t) is always pulled towards the equilibrium l0of the spring.

For further calculations this is simpliﬁed and the resting length is set to 0 and the block

is initially not moving (v(0) = 0). The described example is an extended version of

[Kesselheim, 2005] and is based on the example found in [M¨uller et al., 2005, Section

3.1].

13

Figure 2.5: Overview of the mass spring system

2.3.1 Force calculation

The force F(t) that is applied to the spring is proportional to x(t). By adding the spring

constant to the equation the force can be described as

F(t) = −k·x(t)

Based on Newton’s second law of motion (see section 2.1.1) the acceleration for the

system can be determined.

a(t) = −k·(x(t))

m

With a(t) = x00(t) (see section 2.1.1) we get the diﬀerential equation

x00(t) = −k·(x(t))

m

This ordinary diﬀerential equation can be solved analytically4and leads to the following

equation for the position of x.5

x(t) = x0·cos rk

m·t!

So for this physical system it is possible to calculate the current position for an arbitrary

point in time when the initial position is given.

4The steps are described at http://www.myphysicslab.com/spring1 analytic.html

5For the solving of diﬀerential equations that are derived from Newton’s laws the paper “Ordinary Dif-

ferential Equations (ODEs)” at http://chaos.swarthmore.edu/courses/phys6 2004/QM/ODEs.pdf

provides a good overview.

14

2.3.2 Euler integration

In order to compare the exact result to the Euler integration a modiﬁed Euler integration

scheme is applied. The modiﬁcation states that for integrating the position the velocity

of the next time step is used instead of the current velocity (v(t+h) instead of v(h)).

v(t+h) = v(t) + h·a(t) = v(t) + h·−k·x(t)

m

x(t+h) = x(t) + h·v(t+h)

2.3.3 Example calculation

In this example two diﬀerent parameters for the time step are used to perform the

numerical integration. Both calculations are based on the same spring system (i.e.

spring constant k= 1, mass m= 1 and original point x(0) = 10).

The ﬁrst calculation is done with time step h= 0.1. Here it can be seen in table 2.1 and

in ﬁgure 2.6 that the Euler integration is approximating the real values quite accurately.

t0.1 0.1 0.2 0.3 0.4 0.5 0.6

correct x(t) 10.00 9.95 9.80 9.55 9.21 8.78 8.25

calculated x(t) 10.00 9.90 9.70 9.40 9.01 8.53 7.97

calculated v(t) 0.00 −1.00 −1.99 −2.96 −3.90 −4.80 −5.66

Table 2.1: Example integration with h= 0.1

7.5

8

8.5

9

9.5

10

0 0.1 0.2 0.3 0.4 0.5 0.6

Correct x(t)

Calculated x(t) with h=0.2

Figure 2.6: Diﬀerences in positions for h= 0.1

For the second calculation the time step is set to h= 2.0. The calculations (table 2.2

and ﬁgure 2.7) show that the Euler integration explodes and calculates values far away

from the correct ones.

15

t0.00 2.0 4.0 6.0 8.0 10.0 12.0

correct x(t) 10.00 −4.16 −6.54 9.60 −1.46 −8.39 8.44

calculated x(t) 10.00 −30.00 50.00 −70.00 90.00 −110.00 130.00

calculated v(t) 0.00 −20.00 40.00 −60.00 80.00 −100.00 120.00

Table 2.2: Example integration with h= 2.0

-150

-100

-50

0

50

100

150

0 2 4 6 8 10 12

Correct x(t)

Calculated x(t) with h=2.0

Figure 2.7: Diﬀerences in positions for h= 2.0

The overshooting can be especially seen for the ﬁrst calculated value of v. The value

is here bigger than the maximum that the real velocity will ever reach. The maximum

velocity is the point where x(t) = 0. This point is reached at t=π

2with a velocity

of (v(π

2) = x0(π

2) = −10). So the energy of the system increases which is physically

incorrect.

2.4 Stable integration scheme

The proposed algorithm extends the explicit numerical integration by introducing shape

matching. The shape matching step allows a more exact calculation of the actual desti-

nation for an animation step. Another advantage is that it does not overshoot like the

method shown in section 2.3.

2.4.1 Shape Matching

The idea for the calculation of the target points is based on a multiple-step calculation.

A graphical overview on the idea can be seen in ﬁgure 2.8.

•An object is divided in diﬀerent particles that are not linked to each other. Then

the physical simulation based on Newton’s laws are performed and calculated.

16

•The positions for the particles are now taken and the algorithm tries to match the

shape of the new object with the shape of the original object.

•Based on the diﬀerence to the original shape the velocity of the particle is altered

and the object tries to regain its original shape.

Figure 2.8: Idea of shape matching of particles

In order to solve the problem it is formulated mathematically. Therefore the basic idea

of shape matching is restricted to two translations and a rotation. The restrictions can

be easily explained, by translating the original object to the zero-point, rotating it and

then translating it to the destination point (see ﬁgure 2.9).

Figure 2.9: Shape matching (rotation, translations)

For the given set of original points (x0

n) and the set of deformed points (xn) of the

restricted problem we can now formulate an optimization problem. We try to ﬁnd a

rotation matrix Rand two translation vectors tand t0which minimise the following

equation:

X

i

wiRx0

i−t0+t−xi2

17

wiallows one to give the diﬀerent particles diﬀerent importance. The natural choice for

wiis to take the mass miof a particle so the formula that will be used later is reformed

to Pimi(R(x0

i−t0) + t−xi)2

In order to solve the optimisation problem we start with ﬁnding the translation vectors.

For ﬁnding the translation vectors the partial derivatives with respect to tand t0are

calculated. Due to the fact that we look for the minimum the solution for the two vectors

have to give the result 0 when they are inserted in the ﬁrst derivatives.

Therefore we get the solutions

t0=Pimix0

i

Pimi

=x0

cm and t=Pimixi

Pimi

=xcm

Physically this is the logical choice because the object will be rotated around its center

of mass.

In order to minimize Rthe solutions for tand t0are put back into the equation. This

can be seen as a rotation around the relative coordinates qiand pi, with

qi=x0

i−x0

cm

pi=xi−xcm

As a result we now try to minimize

X

i

mi(Rqi−pi)2

Finding the solution to this problem is simpliﬁed. Instead of just allowing a rotation R

an arbitrary linear transformation Ais allowed.

X

i

mi(Aqi−pi)2

For ﬁnding the solution the same approach is used that has been used for ﬁnding the

translation vectors. So the derivatives for the entries of the matrix Aare set to 0.

The solution for this is

A= X

i

mipiqT

i! X

i

miqiqT

i!−1

The rotation matrix Rcan now be found by polar decomposition. Every invertable

matrix Acan be decomposed to an orthogonal matrix Rand a symmetric positive

deﬁnite matrix Sso that A=RS. The following applies

S=√ATA

R=AS−1

18

√ATAcan be calculated with the help of diagonalization with Jacobi rotations.

After having calculated the rotation matrix Rand the translation vectors tand t0the

goal positions gican be calculated (see ﬁgure 2.9)

gi=Rx0

i−x0

cm+xcm

The matrix calculations in the implementation use the library developed at [Davies,

2005].

2.4.2 Extended Euler integration

Based on the calculated goal position the Euler integration scheme is altered. An ad-

ditional velocity is added that changes the velocity so that the shape tries to regain its

original shape (see ﬁgure 2.10).

vi(t+h) = vi(t) + α·gi(t)−xi(t)

h+h·fext

mi

xi(t+h) = xi(t) + h·vi(t+h)

Figure 2.10: Integration extension based on goal positions.

2.5 Extended shape matching

With the “basic” shape matching explained above, we can only allow for translations and

rotations. The authors give several techniques to extend the shape matching algorithm,

three of which will be explored in this section.

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2.5.1 Rigid body dynamics

Rigid bodies, or non-deformable objects, can be simulated by simply setting the value

for αto 1. This value controls how quickly a particle reaches its goal position, where

setting it to a small value will cause it to slowly return, while a larger value will speed it

up. The maximum value allowed is 1, which means each particle will instantly reach its

respective goal position, i.e. the object is never deformed, so it acts like a rigid body.

2.5.2 Linear deformations

Linear deformations allow the shape to scale and shear while they undergo deformation.

In the basic shape matching, only the rotation is taken from the best matching lin-

ear transformation, while this extension involves calculating the entire transformation.

Along with translation and rotation, a linear transformation can also express scaling

and shearing.

Because the goal positions are calculated using the transformation, the introduction of

scaling and shearing means that the shape created by the shape matching algorithm

will be deformed from the original shape. In other words, the shape will never return to

its original shape. To control the amount of inﬂuence the linear transformation has on

the shape matching, the parameter βis used, and so the equation to calculate the goal

position for a particle becomes:

gi= (βA+ (1 −β)R)x0

i−x0

cm+xcm

By using a smaller value for β, the ratio of rotation-to-pure deformation will increase,

and so the goal positions will more closely match the original, undeformed shape of the

object.

2.5.3 Quadratic deformations

The linear transformation is described using a 3 by 3 matrix. A transformation using

a matrix of this size is restricted to only translations, rotations, scaling and shearing,

however there are more kinds of deformations an object can undergo, such as twist-

ing and bending. By moving from a linear to a quadratic transformation, these extra

deformations can be simulated.

As with the linear transformation, once again the equation to calculate the goal positions

needs to be updated. This time, the transformation matrices are a lot larger (3 by 9) in

order to allow greater freedom of movement, and once again the βparameter is used to

control the amount of deformation.

20

Figure 2.11 shows the diﬀerent deformations techniques based on the same particle sys-

tem. For all pictures the β-value has been set to 0.8. In the ﬁrst picture the deformation

method (rigid body) tries to approximate the deformed shape by rotating the original

shape. For the second picture linear deformation was used. Here it can be seen that

the shape is approximated better by the goal positions but still not close enough to the

deformed object. For the last example quadratic deformation was used and here it can

be seen that with this method the deformed shape can be approximated very closely.

Figure 2.11: Comparison of the calculation of goal positions, where the spheres are par-

ticles in their current positions, and the red cubes show the goal positions

2.6 Plasticity

Imagine that you have a 30 cm metal ruler, that when bent returns to its original,

ﬂat shape. You are able to bend this ruler because it is elastic, and your bending is

equivalent to the forces as described in section 2.1.3, while the return to the original

shape corresponds to the shape matching problem.

Now imagine bending the ruler again, this time applying so much force that the ruler

does not return to its original shape. This is the idea behind plasticity: when the amount

of force exceeds some threshold, the shape of the object will be permanently changed.

An object’s propensity to undergo a permanent change in shape is referred to as that

object’s plasticity.

In a linear deformation, the linear transformation is made up of a rotational matrix

multiplied by another matrix S. In other words, Sis the deformation that takes place

before the particles are rotated, and because a rotated object is not deformed (for exam-

ple turning a book on its side does not change the book’s shape), we are only interested

in S, which represents the actual deformation.

The idea is that an object will keep track of the permanent change in shape in the matrix

Sp, this matrix transforming the original position of each particle relative to the original

centre of mass. This matrix is initialised with the identity matrix, and of course when

21

you multiply a vector by the identity matrix, you are left with the original vector, and

so to begin with there is no change in the shape due to plasticity.

The matrix Spis only updated when the squared determinant of S−Iexceeds some

threshold cyield. A small value for cyield means the object will undergo plastic defor-

mations with only a small force applied; a large value will mean a lot of force will be

required.

Once it has been determined that plastic deformation should take place, the matrix

Spneeds to be updated. If we simply set Spto be Sthen all deformations would be

permanent. However, we probably only want to update it by a small amount, hence

we multiply it by the parameter ccreep, and by multiplying it by the time step it means

we change the shape no more than to where the actual particles have moved to at the

current time. Because we only want to change it a small amount, the new value of Sp

should be near to the identity matrix, but if ccreep and the time step are small, then

the new matrix will be close to 0. Therefore, we subtract the identity matrix from S,

multiply this by ccreep and the time step, and then add the identity matrix back. Finally,

we multiply this result with the previous value of Sp, as plasticity accumulates over time.

By adjusting the values of cyield and ccreep, we are able to control the plasticity of an

object. For example, to simulate the bendy metal ruler used in the example above, we

would set cyield to be a relatively high value so that only a large force could permanently

change its shape, and ccreep to a small value, so that when the shape changed, it would

only change a small amount relative to the amount that the ruler was bent. Conversely,

to simulate a piece of wire which would stay in the shape that it was bent to, cyield would

be close to zero so that even a small amount of force altered its shape, and ccreep would

be close to one so that its shape would be permanently changed to very closely match

the shape it was bent in.

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3 Our Problems

3.1 Integration scheme

Unfortunately the integration scheme the authors present is not working in exactly the

way they describe. We had to modify our integration scheme into two parts so they

would be stable. The derived formulae term letters are related to the ones in the paper:

v(t+h) = v(t) + g(t)−¯x(t)

h+hfext(t)/mi

¯x(t) = x(t) + h¯v(t),¯v(t) = v(t) + hfext (t)/mi

x(t+h) = x(t) + hv(t+h)

¯x and ¯v are intermediate calculations which are used to calculate the goal positions gi.

The formulae for these are assimilated accordingly: pi=¯xi−xcm,xcm =imi¯xi

imi.

gi=Rqi+xcm

The authors used the same identiﬁer for calculating the goal positions and decribing the

integration scheme. This is basically only a ﬂaw in their notation and does not violate

their mathematical proves of stability.

3.2 Rotation matrix implementation

The mathematics behind the calculation for the rotation matrix to match the goal po-

sitions with the actual shape is not easily understandable. We understand roughly that

they try to ﬁnd a optimal linear transformation matrix with least square optimization.

After this step the square root of AT

pqApq is used to extract the rotational part of Apq.

This part of the calculation is not understood by our group.

23

3.3 Frame rate-independent velocity updates

The authors state that the velocity update as they presented it1depends on the time

step taken. They state that this can be solved by making α=h/τ where τ≤his a

“time constant”. Aside from the fact that this would cause αto be greater than one,

which is not desirable, this would cause the value of αto not be controllable in the

program. It is also unclear what kind of value this constant should be.

3.4 Quadratic deformation of goal points

In section 4.3 of the paper the author introduces quadratic deformation of the goal

points. This way the goal points can be matched along cubic curves. The object then

deforms along these curves. It is unclear how the 9x3 transformation Matrix along with

the 9 element vector ˜qis achieving this goal.

3.5 Plasticity

It is understandable that an object would experience plastic deformations when the

force applied exceeds a certain threshold, but it is unclear exactly why it is the squared

determinant of S−Ithat is used2. While the determinant gives information about the

volume of the object, and hence this equation is probably a measurement of the change

in volume, it is unclear to us exactly how this works. Similarly, it is unclear why dividing

Spby the cube root of its determinant conserves its volume.

1Equation 9 in their paper [M¨uller et al., 2005]

2Equation 15, Section 4.5 in the SIGGRAPH paper [M¨uller et al., 2005]

24

4 Their evaluation of the algorithm

The evaluation the authors give can be divided into stability and performance.

4.1 Stability

The unconditional stability of this technique was an important improvement over pre-

vious techniques. To illustrate the success of this aspect, a completely squashed duck

reforms to its original shape. While in the duck’s case the technique appears to be

stable, it is hardly a convincing proof of stability in all situations.

Fortunately, earlier in the paper we are shown a mathematical proof which convincingly

proves the stability.

4.2 Performance

The performance is another very important section because the ability to use this tech-

nique in interactive applications was one of its main strengths. Consequently, the authors

provide a more detailed analysis of the performance.

They supplied several diﬀerent tests, for example animating 100 objects each with 100

particles (i.e. 10,000 particles altogether) and they reported that it ran at a respectable

50 frames per second. An important ﬁnding in those tests is that the time complexity is

linear in respect to the number of particles used. More interesting is when they included

collision detection and more objects. In this example, over 55,000 particles were used,

and without collision detection the frame rate was around 25 frames per second, but with

collision detection and response the scene could not be animated in real time (they do

not give the frames per second achieved). This shows that the deformation calculations

are more eﬃcient than collision detection and response.

To illustrate that this technique is suitable for interactive applications, an example was

shown where a human head with 66 particles could be manipulated with the mouse

while handling collisions in real-time. They consider this situation “typical for games”,

however not many games consist of a single human head sitting inside a white box. They

do not give the amount of processing power used in the example which would be useful

25

to see how much computation time would be required if, for example, in a game which

took place in a complex environment the main character was deformable.

Overall though, their evaluation clearly shows that the performance achieved in this

technique is very good.

26

5 Flaws and limitations

5.1 Physical correctness

The authors speciﬁcally said that they had games in mind when developing this tech-

nique. A common mantra in games is that correctness - whether it be in graphics,

artiﬁcial intelligence, or physics etc - can be compromised as long as it appears correct.

It would seem that this technique follows that rule, with its aim to be visually pleas-

ing rather than physically correct. The beneﬁt is of course eﬃciency, so this point is a

limitation only if an exact physical representation is required.

5.2 Plasticity

Objects permanently change their shape only when the force applied to them exceed a

certain value. Real objects also change their shape when a smaller force is applied for a

long time (for example the indentations left on a sofa after sitting on it for a long time).

The method given is unable to handle this type of plasticity.

5.3 Connectivity information

One of the advantages of this technique given was that an object could be a particle sys-

tem where no connectivity information between particles is required. While technically

this is true, without connectivity information, texture mapping and collision detection,

and hence most useful applications for this technique, are at best problematic and at

worst impossible.

Indeed, in their own examples it appears that they have loaded objects as a mesh and

then used the vertices from that mesh to be the particles. This could be seen as an

advantage though, because in graphics using meshes is very common and so this could

be said to show how easy it is to extend a normal mesh to be a deformable object.

27

6 Impact of the paper

It is inevitable that in the competitive games industry, having deformable objects will

become an important aspect of many games. Until now, the diﬃculty in implementation

and the computational expense of deformations has stunted the use of these in games,

and it may just be that the technique outlined in this paper will become very popular

as a result of its ease of implementation and eﬃciency.

For these same reasons, deformable objects may be used in other applications, such as

animations, where they were not used before.

28

Bibliography

[Davies 2005]

Davies, Robert: Newmat C++ matrix library.http://www.robertnz.net/

nm intro.htm. Version:2005. – [Online; accessed 15-October-2005]. Newmat is a

powerful matrix C++ library that allows to perform the necessary operations that

the algorithm of the paper needs.

[Kesselheim 2005]

Kesselheim, Thomas: Meshless Deformations Based on Shape Matching. 2005. –

A German presentation on the paper given at a seminar at ’Lehrstuhl f¨ur Informatik

VIII, RWTH Aachen’. This presentation contains better explanations on the problems

of Euler integration in comparison of solving the diﬀerential equation.

[M¨uller et al. 2005]

M¨

uller, Matthias ; Heidelberger, Bruno ; Teschner, Matthias ; Gross,

Markus: Meshless deformations based on shape matching. In: ACM Trans. Graph.

24 (2005), Nr. 3, S. 471–478. – This is the original paper of SIGGRAPH 2005 and it

is described in this report.. – ISSN 0730–0301

[Weisstein 2005]

Weisstein, Eric W.: Euler Forward Method.http://mathworld.wolfram.com/

EulerForwardMethod.html. Version: 2005. – [Online; accessed 15-October-2005]

Mathworld contains a lot of useful mathematical background knowledge. For this

report we had a look at the diﬀerent Euler integration schemes.

[Wikipedia 2005]

Wikipedia:Implizites Euler-Verfahren.http://de.wikipedia.org/wiki/

Implizites Euler-Verfahren. Version: 2005. – [Online; accessed 15-October-2005]

29