Fractal Compression

We explain, in detail, the theory of Fractal Compression introduced by Barnsley and extended by Jacquin. We begin by brieey discussing metric spaces, leading to contrac-tive mappings and their xed points. Then, IFS and Local IFS are introduced. These concepts form the basic underlying theory of Fractal Compression. We then discuss our implementation of the theory and the results of our eeort. Our approach is based on Fisher's, using quadtree partitioning. We managed to achieve high delity decom-pressed images with reasonable compression ratios, as well as low quality images with high compression ratios. Our encoder is a bit slow, but our decoder works very fast. Finally, we discuss schemes which would speed up the compression time and improve the compression ratio. 1 What are Fractals and Fractal Compression? Informally, a fractal is a set that possesses self-similarity and detail at every scale. This means that an arbitrarily small portion of the set can be magniied to produce exactly the original set. A famous example of fractals is the Sierpinski triangle. In some cases, a frac-tal corresponds uniquely to a mathematical function or mapping. For example, consider a contractive map w deened as follows: w takes an image and shrinks it into an exact image of half of its size and places this new contracted image into the 2nd, 3rd, and 4th quadrants of an empty square of the same size as the original image. Then, with a pencil and paper, we can verify that after we apply w repeatedly to any image , we get a sequence of images converging to the Sierpinski triangle. So, in this case, the Sierpinski triangle can be represented by w, which turns out to be a simple mathematical formula. As a result, instead of storing the Sierpinski triangle as 256x256 (or any size) image, one can store w instead to save a lot of storage. Natural images, however, are not exactly self-similar. We don't see fractals in all everyday images; nevertheless, there is an inherent symmetry in many parts of almost all images that we encounter. For example, a person's face shows a symmetrical structure with respect to the vertical line that divides the face in half. This inherent symmetrical property of natural images gives rise to the study of fractal compression. An image is broken into several non-disjoint regions so that a good map can be found to represent each of them. These …