Iterated Function Systems

The method proposed by Barnsley and Demko [10, 35] for modeling natural objects does not work with geometric data, but rather with two – or higher­dimensional point sets. The structures modeled with this method therefore rep­resent only images of the objects, and no surfaces, as was the case in the pre­ceding approaches.

To define an Iterated Function System (IFS) n contractive, affine transforma­tions Mj є R2 ^ R2, M = {M1,M2, • ••,Mn} are required. Each of the affine transformations has six parameters a-f, and is defined by:

щ*,у)=(c d)(x)+(e)• (5.5)

Here point (x, y) is moved to another position. The sequential application of the transformations generates a trace of the point. While the trace is not par­ticularly interesting when applying the transformation only once, amazingly enough complex patterns evolve during the sequential application of several transformations Mj.

The attractor of the IFS is the smallest nonterminal subset A є IR2, so that Mj (p) є A for all p є A and 1 < j < n. The set A always exists and

is unambiguous (see [168]). In order to approximate A, one takes any point ps є A and combines the point sets that emerge if the transformations Mj are applied to the point. If ps is not in A, then due to the contraction characteristics of Mj it is traced back to the attractor.

Подпись:A can be generated in different ways. Deterministic procedures usually develop a tree, whose nodes consist of points A, and whose edges are applications of Mj. If for each point each function Mj is applied, a binary tree develops for j = 2; for larger values an accordingly broader tree is generated. The tree rep­resents the set of all points producible from one point, and grows exponentially with the number of function applications. Should several initial points exist, then, of course, there will be a tree for each of them.

Подпись: F ІГ ІГ г Подпись:Подпись:
In order to produce an interesting image as fast as possible out of this vast set of potential function calls, different search strategies are in place. one of them is the depth-first search, in which, outgoing from the initial point, a func­tion is applied, and following the result, is immediately succeeded by the next. Hence, it is possible to move upward in the tree very quickly. Alternatively, using breadth-first search, for a certain point all applicable transformations are applied. In this way, step by step the complete tree is constructed.

An example of this algorithm is the construction of a Sierpinski triangle. Here for the production of A the breadth-first search is applied, whereby simulta­neously an entire point set is processed. Three functions are needed, which in each case have a contraction factor of 0.5 with differing individual displace­ments of the input points:

j

a

b

c

d

e

f

1

0.5

0

0

0.5

0

0

2

0.5

0

0

0.5

0

50

3

0.5

0

0

0.5

50

50

Figure 5.14 illustrates the result of the application of the functions to a triangu­lar point set. The sides of the triangle are divided by two in each case, and the reduced triangle is moved in three different places.

Подпись: Lipschitz constantTo work with balanced trees, defined completely on each level, is a disadvan­tage if the illustrations Mj contain different contraction rates sj (the so-called Lipschitz constants). These are defined by

II Mj (pi) – Mj (P2) ||< Sj || pi – P2 II V pi, P2 є IR2. (5.6)

For such unequal Lipschitz constants, balanced scanning along the tree leads to an unequal point distribution in A. With an appropriate truncation of subtrees

this can be avoided. Figure 5.15a illustrates the effect of the uneven distribu – Section 5.11

tion and the associated visually small convergence. A tree depth of nine and Iterated Function Systems

262,144 points were used in order to assess the following IFS:

Подпись: a b c d e f 1 0 0 0 0.16 0 0 2 0.85 0.04 -0.04 0.85 0 1.60 3 0.20 -0.26 0.23 0.22 0 1.60 4 -0.15 0.28 0.26 0.24 0 0.44 Figure 5.15

Подпись: The function M1 has a strong contraction rate, M3 and M4 are medium strong contractive, while M2 hardly contracts. If subtrees are appropriately truncated, then an even accumulation of points results, as shown in Fig. 5.15b for the same number of points. (b)
Generating a fern leaf with 262,144 points each: (a) deterministic method using a balanced tree; (b) deterministic method with unbalanced tree;

(c) stochastic method

Подпись: (c)(a)

Another method introduced by Barnsley [9] consists of a stochastic depth – first search of the tree, whereby a function is selected randomly each time and is then applied. In order to improve the convergence, a similar proce­dure to that used with nonbalanced trees is applied, meaning that per func­tion Mj a probability Pj "=i Pj = 1, is defined which indicates how of­

ten Mj should be applied. If Pj is set according to the Lipschitz constant of the corresponding function, we obtain convergence characteristics as good as those obtained with the truncation method. For the above example, we apply P1 = 0.01, P2 = 0.85, P3 = 0.07, P4 = 0.07. In Fig. 5.15c the result is shown with again 262,144 points.

In his book “Fractals Everywhere” [9] Barnsley extends the method and ap­plies it to simulate images of natural scenes. In such an image many attractors are combined, and also here hierarchical IFSs are used that are not distributing points, but rather whole subimages. The method has the disadvantage that find­ing appropriate subimages can be a very cumbersome task. Often, only manual intervention yields satisfactory results. If such functions are found, then the at­tainable compression factors are, however, very high. As an extension of the procedure, if Iterated Function Systems are applied, it is even possible to con­vert arbitrary input pictures into arbitrary final results.

As an additional extension of this approach, RIFSs (recurrent IFSs) are in­troduced, in which no arbitrary transformation may follow another, but rather pairs of sequential transformations are specified. This usually is implemented using a graph, whose nodes indicate the transformations, and whose edges de­note the possible successor. The attractors of the RIFSs are therefore subsets of the attractors of common IFSs.

Prusinkiewicz and Lindenmayer in [166] describe a procedure for the transfor­mation of L-systems into RIFSs. This is successful if the L-systems describe plants with a strict self-similarity. The reverse, to generate from RIFSs a para­metric L-system, is rather simple. Nevertheless, the affine transformations in the parameters must be coded accordingly.