Construction of fractals using formal languages and matrices of attractors

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Construction of fractals using formal languages and matrices of attractors

Joëlle Thollot, Eric Tosan

To cite this version:

Joëlle Thollot, Eric Tosan. Construction of fractals using formal languages and matrices of attractors.

Conference on Computational Graphics and Visualization Techniques (Compugraphics), 1993, Alvor,

Portugal. pp.74–81. �inria-00510143�

matrices of attractors

J. THOLLOT and E. TOSAN

LIGIA-LI SPI, b^at710

43,b oulevard du11Novembre1918

69622 VILLEURBANNE Cedex{ France

jthollot@ligia.univ-lyon1.fr

ABSTRACT

LRIFS's (Languag e-Restricted Iterated Function Systems) generalize the originaldeni-

tion of IFS's(Iterated FunctionSystems) by providingto ols for restrictingthe sequences

of applicable transformations.In this pap er, we study an approach of LRIFS's based on

matrices and graph theory. This enables us to generate a matrix which elements are

attractors.

Key Words: formal language, fractal, iterated function system, language-restricted it-

erated functionsystem,graph, automaton, matrix,diod.

INTRODUCTION

Fractal gemetry was rst intro duced by

Mandelbrot

[Mandelbrot83].

However, he

didn't give any precise denition of what a

fractal is. Thus, several dierent denitions

(notnecessarilyequivalent)and severalconc-

tructionmetho ds havesince b een tested.

The IFSmo del [

Barnsley88 ]

isparticularly

interesting due to it's rigourous formalism

and it's simplicity : a fractal is enco ded by

a nite numb er of contractive transforma-

tions. Several authors have tried to gen-

eralize this mo del. Equations systems [

Cu-

lik I I-Dub e93 ] [

Hart92 ]

and matrices [

Peilgen

et al.92]

are away to deneattractor vectors

(i.e. which elementsare non-empty compact

sets). Languag es acceptedby nite-state au-

tomatonareawaytodeneasubsetofanIFS

attractor [

Prusinkiewicz-Hammel92 ][

Berstel-

Morcrette89][Berstel-NaitAb dallah89].

We prop ose an approach based on matri-

cesofthe LRIFSmo del thatwillenableusto

constructamatrixofattractors. Wewillrst

to LRIFS and nite-state automaton. Then

we will show how to asso ciate a matrix to a

LRIFS. Finally, we will intro duce the deni-

tion of our matrix of attractors as well as a

constructionmetho d.

DEFINITIONS

LRIFS's intro duced by Prusinkiewicz and

Hammel

[Prusinkiewicz-Hammel92]

general-

ize

IFS'sintro ducedbyBarnsley

[Barnsley88].

Thus we will give the classical denitions of

anIFSandaLRIFS.Thenwewillintro ducea

new denitionofaLRIFS usinganite-state

automaton insteadof a language.

Iterated function systems

Let (X;d) b e a complete metric space. De-

note by H(X) the set of all non-empty com-

pacts of X. With the Hausdorff distance

d

H

, (H(X);d

H

) isa completemetric space.

An IFS isa set T = fT

1

;:::;T

N

g of con-

F : P(X) 0! P(X)

f 70! S

N

i=1 T

i

(f)=T f

isacontraction onH(X). Thusthisop erator

has a unique xedp oint :

f =F(f)=T

1

(f)[T

2

(f)[:::[T

N (f)

f is called the attractor of the IFS T and is

denoted by A(T).

Language-restricted iterated function

systems

Barnsleydenedan IFSasasetofN func-

tions. Prusinkiewicz denes an IFS as a

tuple of functions inorder to use formal lan-

guages. Thisenableshimtodenethealpha-

b etof contraction lab elsand alanguage over

this alphab et.

ALRIFSisaquadrupletI

L

= (T;6;h;L)

where:

T = (T

1

;:::;T

N

) is a tuple of contrac-

tions inX.

6 = f1;:::;Ng is an alphab et of con-

traction lab els.

h isa lab eling function,dened as :

h: 6 0! T

i 70! T

i

L6 3

is a language over6.

The function h is generalized to languages

over6 using the equations :

h()=T

1 T

2

:::T

k

if =

1

2 :::

k

h(L)=fh()=2Lg

h(6 3

)=fh()=26 3

g=T 3

L enables usto construct subsets of trans-

formationsandsubsetsofcompactsoftheIFS

attractor :

h(L)h(6 3

)

A

L

(p) =adh (h(L)(p))

A (p) A(T)=adh (h(6 3

)(p))

L

choice of the starting p oint p. However,

if the language L is p ostx extensible (i.e.

La L, the smallest setA

L

do es existand

can b e found as :

A

L

=adh (h(L)(p

0 ))

wherep

0

isthe xedp ointofthe transforma-

tion h(a).

Example : We use ane transformations of

R 2

2R 2

. The following notations are used :

T(a;b) isa translation by vector (a;b).

R(a)isarotationbyangleawithresp ect

to the originof the co ordinate system.

H(a)isa scalingwith resp ectto the ori-

gin of the co ordinate system.

The followinggure shows the attractor of

the IFS

T =fT

1

;T

2

;T

3

;T

4 g

where :

T

1

= H(0:5)

T

2

= T(0;0:5)H(0:5)

T

3

= T(0;1)R(=4)H(0:5)

T

4

= T(0;1)R(0=4)H(0:5)

The following gureshows the corresp ond-

ing attractor of the LRIFS

I

L

=(T;6;h;L)

where :

T = (T

1

;T

2

;T

3

;T

4 )

6 = f1;2;3;4g

L = f3;4g 3

f1;2g 3

The branchingstructure oftheattractor of

the LRIFS is clearly a subset of the original

attractor of the IFS.

Finiteautomaton

Inthe following, wewillonlyuse regularlan-

guages as generally admitted in the littera-

ture. We have chosen to work on the graph

of the automaton as these languages are ac-

cepted by nite automaton. Thus, we will

rst intro duce the denition of a nite au-

tomaton and of the graph of an automaton.

A nite automaton is aquintuplet

M=(Q;6;;Q

I

;Q

F )

where:

Qis a nite setof states.

6is an alphab et.

: Q26 ! P(Q) is a state transition

relation.

Q

I

Qis the set of initial states.

Q

F

Q is the setof nal states.

The language accepted by M is:

L(M)=f! 26 3

=9q2Q

I

:(q;!)Q

F g

where (q;a!) = ((q ;a);!) if a 2 6 and

! 2 6 3

.

Wecan representM as the directedgraph

G(M) = (Q;E) with no des representing

statesandarcsrepresentingtransitions. (The

initialandnalstateswillb edistinguishedby

short arrows ingures.)

So we will intro duce a novel denition of

a LRIFS : a LRIFS is given by a quadruplet

I =(T;6;h;M).

LRIFS

We prop ose a formalism based on matrices.

This is p ossible b ecause of the prop erties of

thespacesweworkon. Thus,wewillseewhat

are these prop erties and how to construct a

matrixasso ciatedwithaLRIFS.Moredetails

are givenin

[Thollot89].

Diods

We use the prop erties of diods in order to

construct matrices [

Gondran-Minoux86 ]

.

A diod isa triplet(D ;+;2) where :

D isasetasso ciated withtwoop erations

+ and 2.

+ iscommutativeand asso ciative.

2 is distributiveover +.

GivenadiodD onecanconstructandma-

nipulatevectorsandmatriceswhichelements

are in D . The following tripletsare diods:

(P(6 3

);[;3): the setoflanguages asso-

ciated with union and concatenation.

(P(h(6 3

));[;) : the set of transforma-

tionssetsasso ciatedwithunionandcom-

p osition.

Moreover, we have the following prop er-

ties:

h(L

1 [L

2

)=h(L

1

)[h(L

2 )

h(L

1 3L

2

)=h(L

1

)h(L

2 )

Thus, formulae over languages will b e the

same over sets of transformations.

Matrix associated with a graph of

automaton

In(P(6 3

);[;3),wecandenethe matrix as-

so ciated with agraph by:

A(M)=(A

ij )

A

ij

=fa=(q

i

;a;q

j

)2Eg

Example : Let M b e the automaton that

acceptsthe language L=f3;4g f1;2g . The

graphof Mis shown inFigure1.

0 q ₁

q

1,2 3,4

1,2

Figure1

The matrix associated with Mis :

A(M)=

f3;4g f1;2g

; f1;2g

!

This matrix do es not give the initial and

nalstates. That'swhywedenetwovectors

I and F that are resp ectively the initial and

nal vectors.

I =(I

j )=

(

fg if q

j 2Q

I

; if q

j 62Q

I

F =(F

j )=

(

fg if q

j 2Q

F

; if q

j 62Q

F

Proposition : The set of words which

lenght is n acceptedbyM is:

L

n

=I t

A(M) n

F

Example : Let A b e the matrix :

A =

f3;4g n

S

i+j=n;j1 f3;4g

i

f1;2g j

; f1;2g

n

!

The language accepted by the automaton

shown inFigure 1is :

L

n

=

fg ;

A fg

fg

!

Matrix associated with a LRIFS

Usingthe matrixassociatedwith anautoma-

ton, wecan now construct the matrix asso ci-

ated witha LRIFS I

M

=(T;6;h;M) as :

H =h(A(M)) =(h(A ))

The elements of this matrix are the sets of

transformations asso ciated with the words of

A(M).

Example : The automaton shown in Fig-

ure 1 givesthe matrix :

H

M

=h

f3;4g f1;2g

; f1;2g

!

H

M

= fT

3

;T

4 g fT

1

;T

2 g

; fT

1

;T

2 g

!

MATRIX OF ATTRACTORS

We have shown in the previous section how

to asso ciate a matrix with a graph. We will

nowseehowtoasso ciateamatrixofattractor

with thismatrix. We willintro duce an appli-

cation relating a transformation to a p oint.

Then we willgeneralizeit to sets of transfor-

mations and to matrices which elements are

sets of transformations.

Case of one transformation

We use aconsequence of the xedp oint the-

orem :

Proposition : Let (X;d) b e a complete

metric space. Let S b e a set of contractive

transformations, stable for . Let T 2 S b e

acontractivetransformation. T has aunique

xedp oint c and wehave :

8p2X lim

n!1 T

n

(p) =c

Thus,we can dene an application by :

: S 0! X

T 70! c

We can also dene the following op erator :

T 1

= lim

n!1 T

n

=cst(c)

where cst(c)is the constant function.

This is a uniform convergence [

Gentil92 ]

.

Case of a set of transformations

Let H(S) b e the set of all non-empty com-

pacts of S. With the Hausdorff distance,

H

we have:

(T 0

) 1

= lim

n!1 (T

0

) n

=cst(A(T 0

))

(T 0

)=A(T 0

)

where T 0

= fT

!1

;:::;T

!

k

g 2 H(S) and

A(T 0

) isthe attractor ofthe IFST 0

.

Example : Consider the LRIFS

I

M

=(T;6;h;M)

where

T =fT

1

;:::T

8 g=I

1 [I

2

where :

I

1

=fT

1

;T

2

;T

3

;T

4 g

I

2

=fT

5

;T

6

;T

7

;T

8 g

T

1

= H(0:5)

T

2

= T(0;0:5)H(0:5)

T

3

= T(0;1)R(=4)H(0:5)

T

4

= T(0;1)R(0=4)H(0:5)

T

5

= H(1=3)

T

6

= T(1;0)R(=3)H(1=3)

T

7

= T(1;0)R(=3)T(1;0)

R(02=3)H(1=3)

T

8

= T(2;0)H(1=3)

6=f1;:::;8g

Mis givenby :

1 S ₂

{ } S

q 1

0 q

where

S

1

=f1;2;3;4g

S

2

=f5;6;7;8g

Then we have:

(I

1 )=A

1

whereA

1

isthe attractor of I

1 .

(I

2 )=A

2

whereA isthe attractor of I .

sets of transformations

LetM(H(S))b ethe setofallmatriceswhich

elementsarecompacts of S. Let d

m

b e adis-

tance denedby :

d

m

(A;B)= max

1iN;1jN d

H (A

ij

;B

ij )

Then (M(H(S));d

m

) is a completemetric

space, and we have:

(H

M )

1

= lim

n!1 (H

M )

n

=cst((A

ij ))

(H

M

)=(A

ij )

Example :

h(A(M))=H

M

= I

1 fidg

; I

2

!

H k

M

= I

k

1 S

i+j=k 0 1 I

i

1 I

j

2

; I

k

2

!

H 1

M

= I

1

1 I

1

1 [I

3

1 I

1

2

; I

1

2

!

where I 3

1

= S

k I

k

1 .

And

(H

M )=

A

11 A

12

A

21 A

22

!

(H

M )=

A

1 A

1 [I

3

1 (A

2 )

; A

2

!

The elementsof (H

M

)will b e :

A

A

12

A

21

=;

A

22

VISUALIZATION

We use the Deterministic Algorithm in or-

der to visualize each element of the matrix

[

Barnsley88 ]

. Wewillnowgiveaconstruction

ofanattractorandavisualizationmetho dus-

ing the automaton.

Attractor associated with an

automaton

Prusinkiewicz has given a denition of an

attractorbasedonalanguage. Wegiveadef-

inition of an attractor based on an automa-

ton. This denition is a consequence of the

constructionof the matrixof attractors.

Denition : The attractor asso ciated with

I

M

=(T;6;h;M) is:

A

M

=

[

q

i 2Q

I

;q

j 2Q

F A

ij

A(T)

Thesetofalltransformationsasso ciatedwith

the words ofL(M) whichlenght is n is :

h(L

n

)=h(I t

A(M) n

F)=h(I) t

H n

M h(F)

Thenwehave :

lim

n!1 h(L

n

) = h(I) t

(lim

n!1 H

n

M )h(F)

= h(I) t

H 1

M h(F)

= h(I) t

(cst(A

ij

))h(F)

= S

q

i 2Q

I

;q

j 2Q

F cst(A

ij )

= cst(A

M )

Andso we get :

lim

n!1 h(L

n

)=cst(A

M )

Visualization method

Prusinkiewicz used the escap e-

timemetho dtovisualizetheattractorasso ci-

ated with a regular language [

Prusinkiewicz-

Hammel92].

Culik used the Chaos Game

Algorithmtovisualizetheattractorvectoras-

so ciated with an equations system

[Culik

I I-

Dub e93].

We use the Deterministic Algo-

rithm to visualize the elements of our ma-

trix of attractors (or an attractor asso ciated

withanautomaton). Thisalgorithmwas also

prop osed by Culikand used by Mocrette

and Berstel

[Berstel-Morcrette89][Berstel-

Nait

Ab dallah89].

Let f b e a compact (generally the unit

ball). Thenthesequence(f

n )

n2N

denedby:

f

n

=h(L

n )f

tends to the attractor

A

M

=

[

q

i 2Q

I

;q

j 2Q

F A

ij

WithQ

I

=figand Q

F

=fjg wehave :

f

n

=h(A n

M )

ij f

Thissequencetendstothe elementA

ij ofthe

Consider the LRIFS

I

M

=(T;6;h;M)

where

T =fT

1

;T

2

;T

3

;T

4

g where :

T

1

= H(0:5)

T

2

= T(0:5;0)H(0:5)

T

3

= T(0;0:5)H(0:5)

T

4

= T(0:5;0:5)R()H(0:5)

6=f1;:::;4g

We havechosen the matrix :

H

M

= fT

1

;T

4 g fT

2

;T

3 g

fT

2

;T

3

g fT

4 g

!

The matrix ofattractors corresp ondingis:

(H

M )=

A

11 A

12

A

21 A

22

!

The following gures give the elements of

(H

M ) :

A =A

A

21

=A

22

CONCLUSION

One can dene a matrix of attractors as a

limitofaserieofmatriceswhichelementsare

setsoftransformations. Givenamatrixofat-

tractors, one can dene subsets of attractors

asso ciated witha nite automaton. This is a

way to combine attractors, by comp osing el-

ementsof the matrix,yielding an interesting

approach.

However, further studies are needed in or-

der to :

Establish relationsb etweenthe dierent

attractors.

Developgeneralvisualization algorithms

of these attractors.

Classify the attractors using matricesof

languages.

Establishsome\rules"forcomp osingat-

tractors.

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