SIERPINSKI JELLY

Iterated Function Systems as Elastic Bodies

Pawel Filipczuk

, Slawomir Nikiel

and Korneliusz Warszawski

Institute of Control & Computation Engineering, University of Zielona Gora, Podgorna 50, Zielona Gora, Poland

Faculty of Electrical Engineering, Computer Science and Telecommunication, University of Zielona Gora

Podgorna 50, Zielona Gora, Poland

Keywords: Fractals, Iterated Function System, Real-time Simulation, Animation.

Abstract: Relatively simple ideas of fractal geometry result in an infinite number of complex images and objects.

Fractals are used in computer graphics to increase visual fidelity of the vector models. Although derived

from dynamical systems, fractals are usually presented as static objects. The paper presents a new concept

of embedding physical description in the model of IFS (Iterated Function System). Dynamically changing

fractal structures offer better sense of ‘material’ than static images or key-framed animations. The model

can augment IFS attractors with the illusion of softness, weight and other material-related features. The

proposed model is flexible, deterministic and offers high rendering performance.

1 INTRODUCTION

We can observe tremendous upgrade in computer

graphics. Computing power of currently available

PCs and game consoles support real-time rendering

of high-resolution images. Image quality is still a

result of trade–off between geometry and

performance. Vector graphics is a core of 3D

computer simulation models (Hearn and Baker,

1997); (Heckbert, 1994). Shading algorithms add

visual details to the rendered scene. Fractals has

always been a source for procedural shaders, greatly

improving quality of images. There are fractal

models that perfectly describe some organic

structures (Prusinkiewicz and Lindemeyer, 1990).

Genetic Programming and fractals are also used in

shape grammar-based rendering (Glassner, 1989;

1992); (Mignonneau and Somerrer, 2000); (Sims,

1991). Most fractal models, however, are usually

presented as static structures. This is probably due to

their geometric complexity, easily consuming

available computation power. IFS proposed by

Barnsley were originally focused on application of

affine transformations to image analysis and

synthesis (Barnsley, 1993). They proved to be

enough flexible to be rendered even on limited-

resource mobile devices (Nikiel, 2007). IFS are part

of dynamic systems (Clempner and Poznyak, 2011);

(Di Trapani and Inanc, 2010), but their graphic

representations are rather static. Recent

developments of Super-IFS and IFS

homeomorphism open path to dynamical morphing

of flat images and textures (Barnsley, 2006).

Extension of classical IFS with a vector model along

with the purely deterministic rendering algorithm

enabled real-time IFS shape modelling (Nikiel,

2005). Adding non-scaling parameters such as a type

of vector object to the affine maps enhances the

process of fractal shape construction. Key-framed

and parameter-driven animations of IFS attractors

have also been discussed in literature (Barnsley,

2006). Adding physical description to IFS model

offers quite new interaction and simulation

properties. Fractals behave ‘naturally’ reacting to

gravity and deformations in realistic manner. IFS

attractors can be elastic bodies. User can interact

with them the same way as with their real-life

counterparts.

The paper is organized as follows: First Section

describes theoretical background of the model

including necessary precautions to determine the set

of IFS functions. Then the IFS with embedded

physics model is described. It is followed by

presentation and discussion on real-time rendering

and simulation. Concluding remarks sum up

advantages of the proposed method. Directions for

further developments are indicated at the end of the

paper.

361

Filipczuk P., Nikiel S. and Warszawski K..

SIERPINSKI JELLY - Iterated Function Systems as Elastic Bodies.

DOI: 10.5220/0003829803610364

In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2012), pages 361-364

ISBN: 978-989-8565-02-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 ITERATED FUNCTION

SYSTEMS

2.1 Background

The Iterated Function Systems theory defines

mathematically some concepts of chaos and

irregularity in geometry. Research done mainly by

Barnsley led to significant new methods for image

understanding (Barnsley, 1993; 2006). A basic set of

tools for image construction is created through a set

of simple geometric transformations. IFS are based

on mathematical foundations laid by Hutchinson. An

IFS fractal is constructed from a collage of

transformed copies of itself. The transformation is

performed by a set of affine maps. An affine

mapping on a plane is usually a combination of

rotation, scaling and translation in R

. There are no

particular conditions imposed on the maps except

their contraction (Falconer, 1990).

A set of affine transformations is accompanied

by respective contraction factors. They are relatively

easy to estimate for classical linear two-dimensional

IFS. When all contraction factors are less than 1 the

IFS are called to be hyperbolic IFS. If each mapping

has a specific measure assigned a probability IFS are

called IFS with probabilities. Probabilities can be

proportional according to Jacobians of

transformation matrices. Proper adjustment of

probability according to a given contraction factor

enhances the rendering process. The average

contraction suggests that even non-hyperbolic and

non-linear IFS can posses their constant points,

called the attractors. Estimation of the contraction

factor for non-linear IFS is not trivial (Skarbek,

2006). It is often a matter of trial and error process

to check the existence of the IFS attractor. If we

consider an IFS to be a hyperbolic and linear, then

the situation is quite safe. We will always obtain the

image of the IFS attractor. With the help of the

Collage Theorem it is possible to design

interactively nature-like images and shapes.

2.2 IFS with Physics

First implementations of the IFS used binary images

and were rather inefficient. There have been many

improvements made over the IFS models and

alternative fractal rendering methods. IFS

(Partitioned IFS) and VRIFS (Vector Recurrent IFS)

are incorporated in fractal image compression and

decompression schemes (Barnsley, 1993). Polar IFS

along with Genetic Programming is a very

interesting alternative targeted at inverse problem:

how to find IFS set for a given image (Collet et al.,

1999). Adding color-space to the IFS codes

enhanced the process of colorization of fractal

models (Nikiel, 1998). In the field of rendering

methods, Dubuc and Elquortobi recognized that only

new points of the IFS attractor are necessary to

visualize the attractor (Dubuc, 1990). Monroe and

Dudbridge developed an optimized version for on-

screen display of IFS images, called the Minimal

Plotting Algorithm (Monroe and Dudbridge, 1995).

Bell proposed a recursive rendering scheme called

Tesseral Synecdoche Algorithm (Bell, 1995); (Bell

and Holroyd, 1991). Vector Recursive Rendering is

another algorithm that can be used to transform an

arbitrary set of points in n-dimensional spaces

(Nikiel, 2005). Considering above-mentioned

developments and purely deterministic character of

some IFS models it is possible to render their

attractors in real time. This enables interaction and

animation of IFS fractals. There have been many

attempts to use key-framed or parameter-driven

animations (Barnsley, 2006). Adding physical

description to IFS model opens quite new ways of

interaction and simulation. Fractals behavior might

be sensitive to deformations, gravity or other kind of

forces.

Let (F, d) be a complete metric space. Let F→F

be a collection of mappings (ω

; i=1,2,…,L)

operating on points (p) in F, then the

Ω=(F, (ω

); i=1,2,…,N) (1)

is called the Iterated Function System. An affine

transformation ω

scaling and translating points (p)

in R

has the form, and can be treated as a

transformation matrix M





























11000

PMP

(2)

where (t

, t

) determine translations and (s

, s

)

determine scaling. A 2D example of such IFS model

might be Sierpinski triangle built with three

transformations. The IFS described above can be

treated then as an elastic body by using classical

physical formulas to define its behavior. Fractal

objects can be influenced by external forces of

gravity and wind. They can also collide with each

other as well as with other objects or with the

ground.

GRAPP 2012 - International Conference on Computer Graphics Theory and Applications

362

Figure 1: The model of IFS representing rigid bodies

connected with springs.

Fig. 1 depicts an example, in which transformations

are treated as rigid bodies connected by springs (for

the model defined by Eq. 2). Spring forces are

calculated using following formula:

bvstretchkFFF

dampingspring









(3)

where k is the spring stiffness coefficient, stretch is

the difference between neutral length of the spring

and its current length, b is the dumping coefficient

and v is a relative velocity between both ends of the

spring. The resultant force F

resultant

acting on objects

is a sum of forces of connected springs and other

forces like gravity, wind, etc. In the example

presented in this paper, simple Euler integration

method have been used.

Let P=[t

, t

] represent a position of the fractal

object. Then, following formulas describe new value

of P after time:

resultant



vtavv





resultant

vPtvPsPP

resultant

















 ''

(4)

where v is velocity and m is mass of the object. Final

transformation used in the IFS to generate the object

in current frame of simulation depends on forces

acting on it, time between the current and the

previous frame, on its mass and on previous position

and velocity. Final transformation matrix based on

Eq. 2 can be described by:

Figure 2: A few screens from the simulation.





































































1000

100

010

001

(5)

It is possible to apply this method to other

transformations, like scaling, rotation or skew but

their physical interpretation is less obvious and

could not be explained directly as for translation.

Final transformation set used in our prototype

application consists of all these transformations and

is calculated as follows:

TRCRRCS

MMMMMM 

1

(6)

where M

is scaling, M

is rotation, M

is translation

and M

is the center of rotation matrix.

2.3 Rendering Algorithm

The physical IFS objects are rendered with vector

recursive rendering (VRR) (Nikiel, 2005). It

provides fast and deterministic way to calculate

structure of the object. According to the VRR

algorithm set transformations are used N times on

initial point to estimate fractal attractor. The final set

of points might be drawn directly on the screen or

rendered as coordinates of billboards or CSG sets.

The amount of physical calculations depends on the

number of transformation, not on N or the final

complexity of the object.

A deterministic rendering scheme is necessary to

construct physics-based three-dimensional fractals.

Either Tesseral Synecdoche Algorithm or Vector

Recursive Rendering Algorithm can be adapted to

handle the model described in the previous Section.

SIERPINSKI JELLY - Iterated Function Systems as Elastic Bodies

363

3 IMPLEMENTATION

The simulation and rendering procedure was

implemented using C++ and DirectX API. The

application delivers full structure in real-time. The

overall performance is very fast and some of the

operations may be performed at the GPU. The scene

was composed of 256 geometric objects, which

contained 10,752 vertices and 20,480 faces all

together. Average efficiency was about 165 FPS.

Without rendering it was about 780 FPS. The

prototype application was running on AMD Athlon

64 3000+ (1.8GHz) with ATI Radeon X700Pro

graphics card. Fig. 2. presents sample screens from

the simulation.

4 CONCLUSIONS

The developments described in the paper broaden

the application area of fractal modeling in three

dimensional vector graphics with the idea of

physical behavior of fractals. It opens up even more

possibilities to create artificial objects. When

simulated or interactively manipulated, physics-

based IFS attractors acts ‘naturally’ showing their

mass (inertia) or elasticity (they behave like jelly,

strings, plants or fur). The model presented in the

paper is relatively simple and provides real-time

interaction. It changes appearance of fractals from

complex static images to dynamically changing

structures. It is possible to describe IFS physics in

more advanced way, including collision detection.

Transformations might influence each other with

gravity or electromagnetic field instead of springs.

That would simulate objects like galaxies or atoms.

Considering plants, the hierarchical structure might

be used.

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