\Chaos Games" for Iterated Function Systems with
Grey Level Maps
B. Forte1;2, F. Mendivil2 and E.R. Vrscay2
1
Facolta di Scienze MM. FF. e NN. a Ca Vignal
Universita Degli Studi di Verona
Strada Le Grazie
37134 Verona, Italy
e-mail:
[email protected]
Department of Applied Mathematics
Faculty of Mathematics
University of Waterloo
Waterloo, Ontario, Canada N2L 3G1
2
e-mail:
[email protected] [email protected]
(January 21, 1997)
Abstract: Two random iteration algorithms or \chaos games" for Iterated Function Systems
on function spaces, namely IFS with grey level maps (IFSM), are described. The rst algorithm
can be interpreted as a \chaos game in code space" and works only for the case of nonoverlapping IFS maps. In the second algorithm, applicable to IFSM with overlapping IFS maps
but ane grey level maps, the (normalized) IFSM attractor function u serves as the density
for an invariant measure of an IFS with probabilities with condensation measure. As such,
approximations to the attractor function of the IFSM are yielded by visitation histograms, as
in the case of IFS with probabilities on measure spaces.
1
1 Introduction
In this paper we formulate two random iteration algorithms or \chaos games" for Iterated
Function Systems with Grey Level Maps (IFSM). As in the case of Iterated Function Systems
with probabilities (IFSP) on probability measure spaces, the chaos game is a kind of \bin
counting" algorithm which can be used to generate approximations to IFSM attractor functions.
For both the IFSM and IFSP, such a random iteration algorithm represents an alternative to
a deterministic algorithm for constructing such approximations.
In the remainder of this section, the basic denitions for IFSM are presented. In Section II,
we outline a rst kind of chaos game for IFSM, motivated by the chaos game for IFSP. (The
important features of the latter are given in the Appendix.) Its applicability in analyzing
the IFSM attractor function u(x) is restricted to the case where the IFS contraction maps
are \nonoverlapping". In this case, as expected, a sampling of function values by a random
\chaos game" walk produces converging estimates of the average value of the attractor over a
subset/pixel. The breakdown of the algorithm in the overlapping case can be understood by
reformulating it as a chaos game over code space. A convergence result for this chaos game is
provided by Elton's ergodic theorem. However, it can provide information about u(x) on X only
in the nonoverlapping case. In Section III, we introduce another chaos game for ane IFSM in
L1 in the general overlapping case. The (normalized) IFSM attractor function u is considered
as the density of the invariant measure for an IFSP with condensation. The oset terms i in
the ane grey level maps i(t) = it + i will play the role of the condensation measure while
the scaling terms i will contribute to mixing probabilities for these condensation measures.
1.1 Basics of IFSM
Let (X; d) denote a complete metric space, the \base space" which is typically [0; 1] or [0; 1]2
with Euclidean metric. Let w = fw1; w2; : : : ; wN g be a set of one-to-one contraction maps
on (X; d) with contraction factors ci 2 [0; 1). For simplicity, we assume that the IFS maps
are ane. Associated with the IFS maps wi is a set of grey level maps = f1; 2; : : :; N g,
i : R ! R, assumed to be Lipschitz on R with Lipschitz constants Ki. The set of IFS maps
w and associated grey level maps comprises an IFSM on (X; d).
Associated with the N -map IFSM (w; ) is a fractal transform operator T : Lp(X ) ! Lp(X ),
p 2 [1; 1) [7, 8]. For u 2 Lp(X ),
(Tu)(x) =
N
X
k=1
fk (x);
x 2 X;
(1)
where the fractal components fk (x) are given by
(
);
fk (x) = 0;k (u(wk (x))); xx 22= wwk ((X
k X ):
1
(2)
In other words, the kth fractal component fk (x) is a modication of the grey level value of u
at the preimage wk 1 (x) (provided this preimage exists).
2
For u; v 2 Lp(X ), p 2 [1; 1),
k Tu Tv kp Cp k u v kp; Cp =
N
X
jJk j =pKk ;
(3)
1
k=1
where jJk j is the Jacobian associated with the transformation x = wk (y). These bounds may
be improved in the case of -nonoverlapping, i.e. where the sets Xi = wi(X ) overlap only on
sets of zero Lebesgue measure on X (a common assumption in the literature).
k Tu Tv kp Cp k u v kp; Cp =
"X
N
k=1
jJk jKkp
# =p
1
:
(4)
If Cp < 1 then T is contractive in (Lp; dp). From the Banach Contraction Mapping Theorem,
there exists a unique xed point u 2 Lp(X ), i.e. T u = u. Furthermore, for any u 2 Lp(X ),
dp(T nu; u) ! 0 as n ! 1. This is the basis for the deterministic algorithm to generate
approximations to u.
In what follows it will be useful to consider the code space associated with the N -map IFS
w. Recall that there exists a unique compact set A X , the attractor of the IFS, such that
A = [Ni=1 wi(A). Dene
= f = (1; 2; : : :)ji 2 f1; 2; : : : ; N g 8 i 1g:
(5)
Then for any x 2 A, there exists at least one code 2 such that
x = nlim
!1 w1 w2 : : : w (y )
n
(6)
for any y 2 X [2, 1]. As well, for any 2 , there exists a unique point x 2 X . We let
a : ! X denote the address map so that a() = x. If the sets wi(A) are disjoint, then a is
one-to-one.
Finally, each IFS map wi on X induces an equivalent action on . The code space map corresponding to wi is !i : (1; 2; : : :; ) 7! (i; 1; 2; : : :; ). These actions are depicted schematically
below.
!i
- a
a
?
X
wi
3
?
- X
2 A Simple Chaos Game
2.1 Nonoverlapping Sets X
i
=
w (X )
i
Let PK = fBk gKk=1 denote a partition of X into Borel subsets Bk . Associated with each set Bk
is a \cumulative sum" Sk which will initially be set to 0. Let (w; ) be an N -map IFSM on X
satisfying the following properties:
1. The sets Xi = wi(X ) cover X, i.e. SNk=1 Xi = X .
2. As well, m(Xi \ Xj ) = 0 for i 6= j , i.e. \nonoverlapping" sets,
3. The grey level maps i are contractive on R, i.e. Ki 2 [0; 1). (This implies that T is
contractive on L1 (X ) [3, 7].)
From the above assumptions, almost every point x 2 X (Lebesgue measure) has a unique
code = (1; 2; : : :) 2 . If u is the attractor of the N -map IFSM (w; ), then u(x) =
1 u(w11(x)), which may be iterated to obtain (using the contractivity of the i)
u(x) = nlim
!1 1 2 : : : (t );
n
0
(7)
where t0 2 R.
Let p = fp1; p2; : : : png, pk > 0, PNk=1 pk = 1, be a set of probabilities to be associated with
each IFS-grey level map pair (wk ; k ). The choice of the pk - a crucial point - will be specied
below. We now outline the rst algorithm:
1. Initialize x0 as the xed point of w1 (merely for convenience).
2. Initialize u0 to be \close" to u(x0) by setting u0 = 1m(1), where m is suciently large
to obtain the desired accuracy. (Convergence is guaranteed by the contractiveness of the
k maps and the nonoverlapping property of the Xi.)
3. Initialize the sum Sj0 = u0, where x0 2 Bj0 .
4. Choose a pair fw0 ; 0 g, 0 2 f1; 2; : : : ; N g according to the probabilities pi.
5. Set x1 = w0 (x0) and u1 = 0 (u0).
6. Increment the sum Sj1 by u1, where x1 2 Bj1 .
7. Continue in this way by returning to 4 above, i.e.
xn = w (xn); un = (un);
+1
n
+1
n
n 2 f1; 2; : : : ; N g;
(8)
where the k are chosen according to the probabilities pi, and then updating the appropriate Sj +1 .
n
4
Proposition 1 For each k 2 f1; 2; : : : ; K g,
Sk ! Z u(x)d(x) as n ! 1;
(9)
n
B
where is the invariant measure of the IFSP (w; p) (see Appendix for the denition of the
k
latter).
Proof: From the assumptions that (1) the i maps are contractive and (2) the sets wi(X )
nonoverlapping, it follows that un u(xn ). Let Ik denote the characteristic function of Bk .
Then at the nth stage of this chaos game,
n
Sk 1 X
(10)
n n m=1 Ik (xm)u(xm):
From Elton's
R Theorem [5], in the limit n ! 1 the right hand side of the above expression
becomes B u(x)d(x).
Corollary 1 Dene the probabilities to be pi = m(Xi)= Pk m(Xk ). Then
1 Sk =
1 Z u(x)d(x)
lim
n!1 n m(Bk )
m(Bk ) B
= uav (Bk );
(11)
the average value of u over Bk .
Proof: An easy calculation shows that the invariant measure for the IFSP (w; p) is = m.
From Eq. (10) the desired result follows.
Proposition 2 Let Pn be a nested sequence of Borel partitions whose \sizes" go to zero as
n ! 1. Let un be the average value function of u associated with Pn. Suppose that T is
contractive in Lp (X; m) (1 p < 1) so that its xed point u 2 Lp. Then un converges to u in
Lp .
Proof: Notice that un is the conditional expectation of u given Pn. Thus, un forms a martingale sequence which is Lp bounded. The desired convergence then follows by the Martingale
Convergence Theorem [10].
k
k
Remarks:
1. The above implies that we may obtain an approximation to any accuracy (in the Lp sense)
by using a suciently ne partition of X . In the particular case that u 2 L1 a stronger
result follows from the Martingale Convergence Theorem, namely, that un ! u pointwise
a:e:.
2. In the general case of probabilities pi, PNi pi = 1, the limit in Corollary 1 becomes
uav (Bk ; ), the -average value of u over Bk . Then Proposition 2 is generalized to convergence of un to u in Lp(). The specic results of Corollary 1 and Proposition 2, i.e.
convergence with respect to Lebesgue measure, are more relevant to computer approximations using the chaos game.
5
2.2 Overlapping w (X ) and a \Chaos Game in Code Space"
i
In the case that the sets Xi = wi(X ) overlap, i.e. m(Xi \ Xj ) 6= 0 for some pair (i; j ), i 6= j ,
the chaos game of the previous section fails. One immediate consequence of overlapping is
that in Step 2 of the algorithm, it is not guaranteed that u0 may be made \close" to u(x0),
since x0 may have several preimages wi 1 (x0). In order to further understand this problem, we
formulate a chaos game algorithm on the code space rather than on the base space X . This
is possible from the equivalence of actions in both spaces as shown in the diagram at the end
of Section 1.1. Associated with the partition PK of X into Borel subsets Bk is a partition of into subsets dened by Tk = a 1(Bk ) for all k where a : ! X is the address map dened in
Section 1. To each Tk we now associate a cumulative sum Sk , initialized to zero.
Instead of considering functions u : X ! R as was done in the previous section, we consider
the function f : ! R dened as follows: For 2 , dene
f () = nlim
!1 1 2 : : : (t );
n
0
(12)
where t0 2 R. The limit exists and is independent of t0 by the assumption that the i are
contractive on R. This is the \code space analogy" of the base space attractor function u as
dened in Eq. (7).
We now modify the algorithm of the previous section to produce a chaos game on instead of
X . This is simply done by replacing Bj by Tj , i.e. the \bins" are now in the code space instead of the base space X . Let 1; 2; : : :; n; : : : be the indices of the (wi; i) pairs chosen.
Elton's ergodic theorem guarantees the following result.
Proposition 3 For each k 2 f1; 2; : : : ; K g,
Sk X 0f ( )I ! Z f ()dP (); as n ! 1:
(13)
n T
n
T
The prime indicates summation over codes n = (n l ; n l+1 ; : : : ; n ), where l is suciently
large so that f (n ) f ( ). As well, P denotes the invariant measure of the IFS with probabilities (w; p) on .
In order to obtain an approximation to the IFSM attractor u(x) on X , it is necessary to interpret
the above algorithm as acting on X . One may try to accomplish this by using the address map
a to \push" the process onto X . From Eq. (13),
Sk ! Z f ()dP ():
(14)
n
T
This integral may involve a summation over dierent regions Bk which are mapped to Tk . In
order to obtain a true approximation to u on X , however, we require the following quantity:
k
k
j
k
k
i
1
Z 0 X
@
f ()A dP (a (x)):
B
1
k
2a 1 (x)
(15)
The quantities in Eqs. (14) and (15) are not necessarily identical. Equality is guaranteed only
in the case that a is injective, i.e. the sets Xi = wi(X ) are nonoverlapping. We illustrate the
6
problem with the overlapping case by means of a simple example.
Example: X = [0; 1] with
w (x) = w (x) = 12 x; w (x) = 21 x + 21 ;
(16)
(t) = (t) = 21 ; (t) = 1:
Then u(x) = 1. Let B = [0; 1=2] = w (X ) and B = [1=2; 1] = w (X ). Consider the case
k = 1 in Eqs. (14) and (15). Then f = 1=2 on T because = = 1=2. The integral in Eq.
(14) becomes
Z 1
Z 12 1
Z 1
dP () +
dP () =
2 dP (a (x))
1 2
1 2
= 12 (p + p ):
(17)
1
1
1
2
3
3
2
1
2
3
1
1
2
1
=1
=2
0
1
2
The corresponding integral in Eq. (15) is
Z
Z 1 1
Z1
( 21 + 21 )dP () +
( 2 + 2 )dP () = 2 ( 12 + 21 )dP (a 1(x))
0
1 =1
1 =2
= (p1 + p2 ):
(18)
The two integrals are clearly not equal. The problem is due to the existence of \cross terms"
in the integral of Eq. (15) which are not present in Eq. (14).
In the case that the address map a is injective, the integrals in Eqs. (14) and (15) are identical
and Proposition 1 of the previous section follows. The following generalization of Corollary 1
also follows:
Z
1
1
S
k
nlim
!1 n (Bk ) = (Bk ) B u(x)d(x)
= uav (Bk ; );
(19)
the -average of u over Bk . Approximations to this average value may be obtained by running
the standard chaos game for the invariant measure of the IFSP (w; p) simultaneously: Simply
include another set of accumulation variables mk , 1 k K , and increment the appropriate
mj by 1 at each step. Then mk =n ! (Bk ) as n ! 1 so that Sk =mk ! uav (Bk ; ).
In summary, the simple \chaos game" algorithms outlined above - one in the base space X
and the other in the code space - are guaranteed to work only in in the special case of
nonoverlapping wi(X ). We are unable to construct comparable algorithms for the more general
case of overlapping wi(X ).
This is not to say that the algorithms never work in the overlapping case. They may work
in \nongeneric" situations, for example, when grey level maps i corresponding to overlapping
IFS maps are identically zero. These are special cases, however. Our simple example clearly
illustrates that the algorithms are not universally applicable. This serves as a motivation for
the work outlined in the next section, in which a chaos game based on IFS with probabilities
and condensation measures is devised.
k
7
3 Chaos Game Using IFSP with Condensation
In what follows we assume, for simplicity of notation, that X = [0; 1]. Let (w; ) denote an
N -map ane IFSM, i.e. both IFS and grey level maps are ane:
wi(x) = six + ai; i(t) = i t + i; 1 i N:
(20)
Note that the sets wi(X ) are not assumed to be nonoverlapping. The associated fractal transform operator T has the form
x ak N
N
X
X
I
k IX (x)
(21)
(Tu)(x) = k u
X (x) +
sk
k
k
k
k
=1
=1
where Xk = wk (X ). We shall write the above operation symbolically as
T (u) = A(u) + b;
(22)
where b(x) is dened by the second sum in Eq. (21). We also assume that i; i 0 and that
C =
1
N
X
k=1
ci i < 1;
(23)
where ci = jsij, i.e. T is contractive in L1. Then the xed point u = T u may be written as
follows:
u = b + A(u)
1
X
An(b):
=
(24)
n=0
The iterated application of A on the function b mimics the operation of \condensation" in IFSP
with condensation measures (reviewed in the Appendix). The nature of this condensation is
claried if we consider the (normalized) attractor u as the density function of a probability
measure on X .
From the relation T u = u, one may easily compute the following integral:
P c
Z
k k k :
hui = X u(x)dx = 1 P
(25)
k ck k
(Note that the denominator does not vanish.) As well,
hbi =
Z
X
b(x)dx =
X
k
ck k:
(26)
Also from the relation T u = u, we have, for any Borel set S X ,
Z
S
u(x)dx =
N
X
k=1
k ck
Z
w k 1 (S )
8
u(x)dx +
Z
S
b(x)dx:
(27)
Dene the normalized functions u1(x) = u(x)=hui and b1(x) = b(x)=hbi so that hu1i = hb1i = 1.
Rewrite Eq. (27) in terms of these normalized functions to obtain:
N
X
N
X
k ck (wk (S )) + [1
k c k ] (S );
(28)
k
k
R
R
where (S ) = S u (x)dx and (S ) = S b (x)dx. Thus, the measure 2 M(X ), with density
u , is the invariant measure of an IFSP with condensation measure 2 M(X ) with density b ,
cf. Eq. (44) in the Appendix. PLet pi = ici, 1 i N be the probabilities associated with
N c to be the probability associated with the condensation
the IFS maps wi and p = 1
i i i
measure . Our chaos game for ane IFSM will now be based on a chaos game for IFSP with
condensation.
As in the previous section, we assume that X is partitioned into Borel subsets Bk . The cumulative sums Sk associated with the sets Bk are again initialized to zero. Here, however, they
will supply information only on the visitation of the sets Bk . The algorithm is as follows:
1. Initialize x as the xed point of w (merely for convenience).
2. Set Sj0 = 1, where xj0 2 Bj .
3. Choose a = i 2 f0; 1; 2; : : : ; N g according to the probabilities pi. If
(a) 1, then dene x = wi(x ). Increment the sum Sj1 by one, where x 2 Bj1 ,
(b) = 0, then choose x according to the distribution with b as its density. Increment
the sum Sj1 by one, where x 2 Bj1 ,
4. Continue in this way, choosing the next n according to the probabilities pi, either (a)
setting xn = w (xn) or (b) sampling from b . Then increment the appropriate Sj +1
accordingly.
At the nth stage, the approximation to u on Bk yielded by the above algorithm will be given
by
! P
!
1
S
j
k ck
k
uav (Bk ) n m(B ) 1 P c :
(29)
j
k i i
Proposition 4 The above approximations converge to the average value of u over the set Bk
as n ! 1.
Proof: By Proposition 4 in [4], we have Sk =n ! (Bk ), where is the invariant measure for
the IFSP with condensation (w; p). We write the IFSP Markov operator as
N
X
pi (wi (B )) + p (B )
(M )(B ) =
i
)(B ) + p (B ):
= (A
(30)
(S ) =
1
=1
=1
1
1
1
1
0
0
1
1
1
1
1
1
1
0
1
1
+1
1
n
n
1
0
=1
0
Thus,
=
1
X
An():
n
9
(31)
This shows that is absolutely continuous with respect to Lebesgue measure since each term
An() is. Since is invariant with respect to M , its Radon-Nikodym derivative (i.e. its density)
must also be invariant with respect to M . By scaling, we obtain the desired result.
In order for the connection between IFSM and IFSP with condensation to be possible, the
IFSM must be ane so that the operator T may be written as in Eq. (22). However, some
generalizations may be made:
1. The shift or \oset" terms i in the grey level maps may be generalized to nonconstant
functions of x 2 X . In other words, the function b(x) in Eq. (22) need not be a piecewise
constant function. In order to dene a density function for the condensation measure ,
cf. Eqs. (27),(28), it is sucient that the i 2 L1(X ). The nonnegativity condition on
the i may also be relaxed, as shown below.
2. Consider the ane IFSM operator T in Eq. (22), with b(x) negative but bounded from
below on X . Now let c be a positive constant (or a nonnegative L1 function) such that
b + c 0 on X . Now dene the ane IFSM operator T 0, where
T 0(u) = A(u) + (b + c); u 2 L1(X ):
(32)
Then T 0 is contractive with xed point u0 2 L1(X ) given by
u0 =
1
X
n=0
1
X
An(b + c)
1
X
An(c)
An(b) +
n
n
= u + v;
(33)
where v is the xed point for the IFSM operator Tc, where Tc(u) = A(u) + c. Therefore
u = u0 v. \IFSP with condensation" chaos games may now be run separately for
the two operators T 0 and Tc since they both have nonnegative condensation functions.
Two accumulation sums are now computed in the algorithm and approximations to the
attractor u may then be obtained by subtracting these sums.
Example: Consider the following 3-map ane IFSM on [0,1] with overlapping IFS maps:
w (x) = 0:5x;
(t) = 0:6t + 0:2;
w (x) = 0:4x + 0:3; (t) = 0:25t + 0:25;
(34)
w (x) = 0:6x + 0:4; (t) = 0:4t + 0:6:
The IFSM operator T is contractive in L (X ). A histogram approximation of the attractor u
of this IFSM is shown in Figure 1. This approximation was obtained by using the deterministic
algorithm on an equipartition of [0,1] using 2000 subintervals. A discrete version of the IFSM
operator was then iterated until satisfactory convergence was obtained. (25 iterations required
about 0.03 CPU sec. All calculations were performed on an IBM RISC 6000 Model 43P-100.)
Figures 2, 3 and 4 show histogram approximations to u yielded by the chaos game after 250,000,
2,000,000 and 10,000,000 iterations, respectively. (Again, 2000 \bins" Bk were employed.) In
all three cases, less than 0.01 CPU sec. was required. It is evident that the approximations
yielded by the chaos game are converging.
=
=0
=0
1
1
2
2
3
3
1
10
4
3.5
3
u(x)
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 1: The attractor u to the 3-map ane IFSM in Eq. (34).
4
3.5
3
u(x)
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 2: Chaos game approximation to u after 250,000 iterations.
11
4
3.5
3
u(x)
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 3: Chaos game approximation to u after 2,000,000 iterations.
4
3.5
3
u(x)
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 4: Chaos game approximation to u after 10,000,000 iterations.
12
Acknowledgments
This research has been supported by the following grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) which are gratefully acknowledged: an Operating
Grant (ERV) as well as an NSERC Collaborative Projects Grant (BF and ERV, along with C.
Tricot, E cole Polytechnique, U. de Montreal and J. Levy-Vehel, INRIA, Rocquencourt, France).
References
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[3] M.F. Barnsley and L.P. Hurd, Fractal Image Compression, A.K. Peters, Wellesley, MA,
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[5] J. Elton, An ergodic theorem for iterated maps, J. Erg. Th. Dyn. Sys. 7, 481-488 (1987).
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[7] B. Forte and E.R. Vrscay, Solving the inverse problem for functions and image approximation using iterated function systems, Dyn. Cont. Impul. Sys. 1 177-231 (1995).
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13
Appendix: IFSP and the \Chaos Game"
Let (w; p) be an N -map IFS with probabilities, i.e. p = fp1; p2; :::; pN g, pi 0. Let B(X )
denote the -algebra of Borel subsets of X and M(X ) the set of all probability measures on
B(X ). Associated with the N -map IFSP (w; p) is a (Markov) operator M : M(X ) ! M(X )
such that for a 2 M(X ) and any S 2 B(X ),
(M)(S ) =
N
X
pi (wi (S )):
(35)
1
i=1
M is contractive on M(X ) [9]:
dH (M; M ) cdH (; ); 8; 2 M(X );
(36)
where c = max iN fcig and dH is the Monge-Kantorovich metric, referred to in the IFS
literature as the \Hutchinson metric" due to its use in [9]. Thus, there exists a unique 2 H(X )
such that (1) M = and (2) dH (M n ; ) ! 0 as n ! 1 for any 2 M(X ). Moreover,
supp() A, with the equality when all pi > 0.
Given an f 2 C (X ) and a 2 M(X ), then
Z
Z
f (x)d(M)(x) = (M yf )d;
(37)
1
X
X
where the adjoint operator M y : C (X ) ! C (X ) (referred to as T in [2]) is given by
(M y f )(x) =
X
k=1
pk (f wk )(x):
(38)
The above procedure may be iterated to obtain, for n = 1; 2; : : :,
Z
X
f (x)d(M n )(x) =
N
X
i1 ;:::;in
pi1 : : :pi
Z
n
X
(f wi1 : : : wi )(x)d(x):
n
(39)
Now set = x0 , the Dirac unit mass at x0 2 X and f (x) = IS (x), where S X , to give
(S ) = nlim
!1
N
X
i1 ;:::;in
pi1 : : : pi IS (wi1 : : : wi (x )):
n
n
(40)
0
The term involving IS indicates whether or not the point wi1 : : : wi (x0) lies in S . The
quantity pi1 pi2 : : :pi represents the probability of choosing the nite sequence fi1 ; i2 ; : : :; i g.
Therefore for each n > 0, the sum is equal to the probability that the point xn lies in S .
There is a connection between Eq. (40) and the Random Iteration Algorithm or \Chaos Game"
[1], dened as follows: Pick an x0 2 X and dene the iteration sequence
n
n
n
xn = w (xn); n = 0; 1; 2; : : : ;
(41)
where the n are chosen randomly and independently from the set f1; 2; : : : ; N g with probabilities P (n = i) = pi . A straightforward coding argument shows that for almost every code
+1
n
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sequence = f1; 2; : : :) the orbit fxng is dense on the attractor A of the IFS w. As such, the
Chaos Game can be used to generate computer approximations of A. However, it also provides
approximations to the invariant measure as a consequence of the following ergodic theorem
for IFS [5]: For almost all code sequences = (1; 2; : : :),
Z
n
1 X
f (x)(x)
f
(
x
)
=
lim
k
n!1 n + 1
(42)
X
k=0
for all continuous (and simple) functions f : X ! R. By setting f (x) = IS (x) in Eq. (42) for
an S X , we obtain
n
X
1
(S ) = nlim
IS (xk ):
(43)
!1 n + 1
k=0
In other words, (S ) is the limit of the relative visitation frequency of S during the chaos game.
IFSP with Condensation
Consider an N -map IFSP on (X; d) with a condensation measure 2 M(X ), i.e.
1. wi : X ! X , 1 i N , contractive IFS maps on (X; d), with associated probabilities
pi , 1 i N ,
2. a condensation measure 2 M(X ) with support L 2 B(X ) such that (L) = 1 and
(B ) = (B \ L) for B 2 B(X ).
The Markov operator M : M(X ) ! M(X ) is given by
(M)(S ) =
Thus
Z
X
N
X
i=1
f (x)d(M)(x) =
pi (wi (S )) + p (S ); S 2 H(X ):
1
0
Z
N Z
X
pi (f wi)(x)d(x) + p f (x)d(x):
X
X
0
i=1
(44)
(45)
Following [2], dene the IFSP (w1; : : :; wN ; p^1; : : :; p^N g, where p^i = pi=(1 p0), with associated
Markov operator M^ : M(X ) ! M(X ). Dene the ane map M c : M(X ) ! M(X ), where
^ + p0 ; 2 M(X ):
Mc = (1 p0 )M
(46)
Mc is contractive in the dH metric. Hence, there exists a unique c 2 M(X ) such that
Mc c = c .
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