Qing Jun He 90121047 Math 308 Essay Iterated Functions Systems and Fractal Coding 1. Introduction Fractal coding techniques are based on the theory of Iterated Function Systems (IFS) founded by Hutchinson [24] and further developed by Barnsley in the early 1980s [1, 2, 3]. The IFS theory is based upon the Banach's Contraction Mapping Principle, which states that a contractive transformation, defined on a complete metric space, possesses a unique fixed point or attractor. For the purpose of image compression, this idea translates into finding an optimal contractive transformation whose attractor closely approximates a given target image. This problem is widely known as the inverse problem in the fractal image coding literature. The fractal-based schemes exploit the self-similarities that are inherent in many real-world images for the purpose of encoding an image as a collection of transformations. Therefore, a digitized image can be stored as a collection of IFS transformations and is easily regenerated or decoded for use or display. The storage of the IFS transformation coefficients generally requires much less memory, resulting in data compression. Iterated function systems were originally introduced to generate globally self-symmetric compact sets and natural images such as the Cantor set, the Sierpinski triangle, and the Spleenwort fern. [6]Due to these restrictions and others, the IFS scheme was initially viewed as little more than a limited scheme for representing a specific class of images, namely those that exhibit a high degree of self-similarity. Many variations of this scheme have been developed since then. These schemes have shown that the fractal-based approach provides efficient and accurate models for many real-world images, resulting in relatively high compression ratios and good reconstruction fidelity. Although fractal-based schemes are still based on exploiting self-similarities in the spatial domain of images, these self-similarities do not have to be global or highly visible. 1
In fact, most real-world images exhibit some degree of local self-similarity which can be exploited by using fractal-based image compression methods. In this section, we have seen an outline of the mathematical fractal image coding and various fractal-based schemes. Then we will show the theory of Iterated Function Systems, which represent the cornerstone of fractal image coding. 2. Iterated Function Systems An Iterated Function System (IFS) is uniquely described by a set of contractive transformations defined on a complete metric space. By the contraction mapping theorem, it possesses a unique attractor. The objective is then to construct an IFS whose attractor approximates a target image. In this part, the contraction mapping principle will be stated its use for the purpose of image compression shall be motivated.[4] The collage theorem, which is closely related to the contraction mapping principle, will also be presented. This theorem provides a method of finding a contractive transformation whose attractor or fixed point closely approximates a given target image or function. The concepts of contractivity and fixed point are first defined. that is: The Contraction Mapping Theorem guarantees that a contractive transformation defined on a complete metric space (i.e. a metric space where every Cauchy sequence converges) possesses a unique fixed point or attractor. 2
The contraction mapping theorem provides a converging algorithm for the approximation of the attractor y of a contractive transformation T. It is important to emphasize the key feature that the fixed point y _ can be closely approximated by iterating the transformation T a few times, starting with any initial seed y 0. In practice, only 10-20 iterations are needed for the estimation sequence {T (n) (y 0 )} to visibly converge within a reasonably small error tolerance. The attractor or fixed point of a contractive transformation exhibits self-tiling and symmetry characteristics, so it is also called a fractal. [6] 3. Fractals A fractal is generally a rough geometric shape that can be split into a few parts, each of which is a small size copy of the whole. A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. A fractal often has the following features [1]: It is a fine structure at arbitrarily small scales. It is self-similar. It has a Hausdorff dimension which is greater than its topological dimension It has a simple and recursive definition. 3
Since they appear at all magnification similarly, fractals are often considered to be infinitely complex. Natural objects that approximate fractals to a degree something like that mountain ranges and snow flakes. However, not all self-similar objects are fractals for example, a straight Euclidean line is formally self-similar. One of the most common and attractive fractals generated by Iterated Function Systems (IFS) is the fern leaf shown illustrated in Figure 1. The IFS corresponding to this two-dimensional fractal are described by repeatedly computing terms in two series, one series describes the x coordinate and the other series the y coordinate. The general form of the series are as follows: x n+1 = a x n + b y n + e y n+1 = c x n + d y n + f A point is drawn at each pair (x i, y i ) for the index, i, greater than some number, typically 10 to 100 [5]. Fractal coding involves finding the values of (a, b, c, d, e, f) that give the desired form. In many application it is necessary to have a number of sets of (a, b, c, d, e, f). As the series is being generated a particular set is chosen at random for each term. Such IFS systems are often known as Random Iterated Function Systems [5]. The fern, illustrated in Figure 1, can be constructed using the table of values illustrated in Table 1 [5]. The last row in the table gives the optimal probabilities. set 1 set 2 set 3 set 4 a 0.0 0.2-0.15 0.85 b 0.0-0.26 0.28 0.04 c 0.0 0.23 0.26-0.04 d 0.16 0.22 0.24 0.85 e 0.0 0.0 0.0 0.0 f 0.0 1.6 0.44 1.6 probability 0.01 0.07 0.07 0.85 Table 1 The fractal code of the fern. 4
Figure 1 The fern leaf is a typical fractal, which is self-similar at all scales. 4. Fractal Image Coding In practice, unless one carefully and appropriately chooses the transformation T, its fixed point may not have any practical use. For the purpose of fractal image representation, for the purpose of image compression and representation, one is mainly interested in the construction of an appropriate contractive transformation whose attractor closely resembles a given target image. [6]This is known as the inverse problem or the fractal image coding problem, and it can be stated in a mathematical framework as follows: Given a target image, u _, defined in a complete metric space Y, construct a contractive transformation T defined on Y, whose attractor, y _, closely approximates u _. The following theorem is a consequence of the contraction mapping theorem and is known in the IFS literature as the collage theorem. This theorem provides us with a practical and fast way to test for feasible choices of T. 5
In other words, if one can find a contractive transformation T that maps the target image, u _, close to itself, then the attractor y _ of T will closely approximate the target u _. In view of this theorem, the inverse problem for fractal image compression can be reformulated as a constrained minimization problem: Given a target image, u _, find a contractive transformation T that maps u _ closest to itself. Hence, solving for the optimal transformation T reduces to solving the following minimization problem for the parameters of T: Minimize: d Y (T( u _ ), u _ ) subject to: T is contractive. In practice, it turns out that under certain assumptions, such an optimal transformation can easily be obtained by using the least squares optimization. The formulation of this optimization problem conveys the basis of fractal image coding. IFS-type methods sought to express a target set or image as a union of shrunken copies of itself. Nevertheless, most real-world objects are rarely self-similar. Instead, self-similarity may be exhibited only locally, in the sense that sub-regions of an image may be self-similar. In the late 1980s, Jacquin developed a block-based fractal image compression scheme that exploits local self-similarities within images [6]. This fractal-based scheme is based on exploiting the inherent local self-similarities in the spatial domain of images. In fact, most real-world images exhibit some degree of local self-similarity which can be exploited by using fractal-based image compression methods. To exploit the local self-similarities within sub-regions of images, the image is subdivided into a pair of simple and uniform partitions of the image: A domain partition 6
of larger sub-blocks, also known as parent sub-blocks and a range partition of smaller sub-blocks, also known as child sub-blocks. A parent sub-block is mapped into its corresponding child sub-block using a geometric mapping, followed by a simple affine transformation, known as the gray-level map.[6] The details of this block-based fractal image coding are beyond the scope of this report. Figure 2 illustrates how iterating the generated optimal fractal code of a commonly used target image on an initial blank image converges to the target image itself after only a few iterations[5]. In practice, one may simply store the fractal code of the target image and whenever the image is needed it can be quickly generated from its fractal code. Typically, storing the fractal code of a target image requires much less bytes of storage than storing the target image itself, hence resulting in image compression. This is a basis of fractal image compression. 5. Conclusions In this report, a generated overview of Iterated Function Systems (IFS) has been presented. The theory of IFS is centered on the contraction mapping theorem. We have illustrated how the fixed point of a suitably selected IFS may represent a fractal or even a real-world image, resulting in fractal image compression. References 1. M.F. Barnsley, Fractals Everywhere. New York: Academic Press, 1988. 2. M.F. Barnsley, and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. Lond., vol. A399, pp. 243-275, 1985. 3. M. Barnsley, and A.D. Sloan, A better way to compress images, BYTE Magazine, pp. 215-223, 1998. 4. A. Jacquin, Image coding based on a fractal theory of iterated contractive image transformations, IEEE Trans. Image Processing, vol. 1, pp. 18-30, 1992 5. The University of Waterloo Fractal Coding and Analysis Group: http://links.uwaterloo.ca/ 6. Mohsen Ghazel, Adaptive Fractal and Wavelet Image Denoising, University of Waterloo, Ontario, Canada, 2004 (Pictures Resource) 7
Figure 2 Generating a fractal approximation of a target image by iterating the fractal transform T, starting with an initial blank image. 8