An Introduction to Fractals and their Applications in Electrical ...

An Introduction to Fractals and their . Applications in Electrical Engineenng by ARNAUD

Signal

JACQUIN

Research Department, AT&T Bell Laboratories, Avenue, P.O. Box 636, Murray Hill, NJ 0797440636,

Processing

600 Mountain

U.S.A.

Wepresent a very brief history offractals, describe their generation, their characteristics, and their relation with chaos. We point to where and why,fractals and chaotic s_wtems are commonly found in nature, which implies that they should be good candidates for modeling dijyerent types of real worldsignals. We then list a number ofdioerse research areas in electrical engineering where ,fractals and,fractal-based techniques have ,found applications. Finally, we present in some detail applications of ,fractals in signal processing, more spec~jically in the areas qf digital ima~qe modeling, synthesis, and compression. ABSTRACT :

I. Introduction

The applications of fractals in signal and image processing have been losing their boundaries more and more, as one starts to witness the merging of various fields : image synthesis and computer animation (which traditionally derive from computer graphics), with digital image and video compression (which derive from digital si~gnalprocessing, specifically information theory). To a large extent, this merging is due to the need to find ever more powerful models for real world digital signals in order to represent, store, and transmit these signals efficiently. We will give examples involving different types of signals, and correspondingly, different types of fractal models, and will try to describe the advantages and shortcomings of each of these models. The reader should be warned that our choice was to present several frameworks and examples which prevented mathematical rigor. This tutorial is mostly aimed at engineers who are interested in an overview of fractals and chaos, rather than in mathematical detail. However, references to articles and books which give the details of the various frameworks are indicated in each section. The general problem statement which underlies the use of fractals for signal modeling and compression-the main purpose of the last and largest section of this paper-is the following. Given any original object or signal, for example a discrete monochrome image specified by an array of S-bit pixel values, how can a computer construct a fractal object-the encoded object-which is both visually close to the original one, and has a digital representation which requires fewer bits than the original. Note that this latter requirement is not part of the notion of modeling peu se but is essential to the efficiency of the modeling procedure for compression or coding. For each type of object/signal to model, this problem will

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A. Jucquin be presented in the following format. First, we describe the objects under study, the class of iterated systems which can generate them, and the generation process. Second, we describe several approaches to solving what is commonly referred to as the inverse problem, which consists of devising a procedure for controlling the generation process in such a way that it produces fractal models of original real world objects. While the first part is mathematically straightforward and merely represents several instances of a very general theory of iterated contractive transformations in a metric space, the inverse problem can only be dealt with on a perframework basis and does require a significant amount of creativity on the part of the designer of a fractal modeling system.

ZZ.Fractals and Chaos 2.1. What is a jkzctal object.7 Texts on ,fractal geometry abound. We refer the reader with a strong mathematical interest in the topic to (l-7). Although it is difficult to give an allencompassing definition since fractals can be objects of different types, it is usually agreed that (deterministic) fractals arise from the repeated iteration of a transformationt. In other words, fractals are mathematical objects which can in general be written as A = ,liir ?(A,,), where A0 denotes

an initial object, and Tn =

zozo

denotes n iterations of z. They are typically a sequence of iterates

(2)

. ..05

generated

by computing

&,A,,Az,..., where A, = z(A,_,). equation :

From

Eq. (l), it is clear that fractals A = z(A),

and displaying

(3) satisfy

the invariance

(4)

which confers to them a property which we generically refer to as self-trans,formability, and which leads to the well-known properties of fractals to be rugged objects with an infinite amount of detail; objects which can be found again and again in magnified pieces of themselves, however small. This very simple formulation can lead to objects which have “pathological” mathematical propertiesI, as well as a tremendous visual complexity. It is sufficient engineers might be more likely to see tNote that where mathematicians see “iteration”, a “feedback process.” The idea is the same. ISuch as nowhere differentiability for some continuous fractal curves and surfaces, or infinite length or area. “Pathological” is meant here as opposed to the regularity of the typical curves and surfaces of Cartesian geometry and differential calculus.

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Fractals and their Applications

in Electrical

Engineering

to describe the generation of “mathematical monsters” such as the Cantor dust, Sierpinski triangle, von Koch curve and snowflake, space-filling Peano curve, Menger sponge, etc. (I, 7), which kept a number of mathematicians at the beginning of the twentieth century puzzled for a number of years, but which are by now considered to be rather tame examples of fractal objects and are all well-understood. Their status of “monsters” caused them to fall into near-oblivion until they were “rediscovered” by the mathematician Benoit Mandelbrot in the 1960s triggering a shift of attitude towards them. One can safely assume that this shift of attitude was due to the advent of the computer and graphic display devices which made these objects visual, and often strikingly so, as opposed to being characterized only by their construction and mathematical properties, or rather lack thereof. This was also due to the realization, initially by Mandelbrot but soon followed by others, that their complexity was typical of objects found in nature (cf. Section 2.2), as opposed to being aberrant. An illustration of this idea can be found in a paper by Mandelbrot entitled “How long is the coast of Britain? [. .I” (8). The argument is the following. If one set out to measure the coastline of a geographic region such as the west coast of Great Britain with yardsticks of ever decreasing size, the measurements would show that the length associated with each yardstick keeps increasing as the size of the yardstick decreases, eventually reaching infinity?. The impracticality of describing most natural objects in terms of straight lines-the basis of differential calculus and engineering-and the hint at the possibility to find accurate fractal models for them both contribute to the fascination exercised by fractals. 2.1.1. Two simple examples qf self-similar fractals : the con Koch curz’e and snoM,jake. The fractal curve known as the con Koch curre can be generated in the following way (see Fig. 1) : 1. Take as initial object a line segment of unit length S,, = [0, 11. 2. To produce the first iterate S, distort the segment by introducing a “bump” in the middle of S, in the form of two sides of an equilateral triangle with side of length f of the length of So. 3. Repeat this introduction of triangular bumps to each line segment until convergence. One can see that this construction can be equivalently formulated in the framework of Eq. (l), with a transformation r consisting of the union of four similarities which map S, to S,. The construction of the snowflake is very similar and is left as a simple exercise to the reader. The iterative construction of both objects is illustrated in Fig. 1. ?A series of such measurements was studied by L. F. Richardson, who showed that the approximate length L(q) of the west coast of Great Britain, as measured with a yardstick of size q, satisfies an empirical law of the form L(q) cc q-” (9). In (1, 8), Mandelbrot interprets the quantity D = 1+cc as the ,fractaldimension of a curve. According to the measurements, the west coast of Great Britain has a fractal dimension approximately equal to 1.25. It is easy to see that a curve with a fractal dimension strictly greater than one has infinite length. This feature is characteristic of fractal curves. Vol. 3318. No. 6, pp. 659-680, Pnnted I” Great Bntan

1994

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A. Jacquin

FIG. 1. Construction

of the von Koch curve (left) and snowflake

(right).

2.2. Fractals and chaos in the real world 2.2.1. Fractal objects in nature. Most objects found in nature such as (i) weeds, ferns, vegetables, and trees, (ii) clouds and mist, (iii) mountain ranges, coastlines, terrain, rocks, aggregates, and ice crystals, and (iv) galaxies, etc. exhibit typical fractal characteristics such as self-transformability, which manifests itself by the fact that “the object looks the same at many different scales.” This observation gives weight to the assumption that the “geometry of nature” is a,fr.actal geometry which arises from iterative processes; a notion which was popularized by Mandelbrot (1). Yet it would be naive to believe that the iterative processes at work in nature are as simple as Eq. (1). Rather, it is reasonable to assume that they are influenced by external forces and perturbations. The growth of two trees from two identical seeds, one in a protected (ideal) environment, the other in an environment plagued with acid rain, poor soil, aridity, strong winds, etc. illustrates this concept. Rather than trying to find perfect deterministic fractals in nature (the naive view) one should realize that real world objects are more likely to be the result of an iterated process of the type 4, = ~(&,)+&I, 662

(5)

Fractals and their Applications

in Electrical

Engineering

where E, would denote a random perturbation-t. The retrieval of an approximate z (and even perhaps E,) from a real-world object resulting from many iterations of Eq. (5) constitutes the basic principle of,fractal modeling which is addressed in Section IV. 2.2.2. Chaotic dynamical systems. Fractals are closely related to chaotic systems. In this section, we introduce the mathematical notion of chaos by describing and analyzing a simple chaotic dynamical system, and briefly describe the relation between fractals and chaos. For an introduction to chaos we refer the reader to (11) and to the texts (5, 7, 12-14) for a comprehensive treatment of chaotic dynamical systems and fractals. A dynamical system consists of a transformation or map s defined from a metric space of points into itself-we consider an example where the space is the unit interval I = [0, I] endowed with the Euclidean metric. The orbit of a point x,, E I is the sequence of iterates where X, = s(x,_ , ).

~O,~~I,~~2,.“,

The orbit of x0 is said to be periodic if there exists an integer VnE N,

x,,+~ = x,,,

(6) P such thata (7)

and it is then written x0x, . . . xP_ , . We consider the dyadic map s defined by

s(x)

2x

if

O