Barnsley and Hurd invite the reader to enter a mathematical wonderland, the land of fractals. This time, however, the fractals are doing down-to-earth work, besides being beautiful. They are used to compress such disk space fillers as images. Image compression is not new, but the approach revealed in this book is. The field of computer and electronic engineering has some great cornerstone ideas (with a much broader impact, of course), including the fast Fourier transform (FFT), back propagation in neural nets, and object-oriented programming, and I think of fractal approximation as a real candidate for this list. The field of fractal image compression is being pioneered by the authors of this book. This work is the first time the key ideas have been collected in a book. The techniques were first mentioned in the October 1991 issue of SunWorld by Louisa Anson (illustrator of the book) and Barnsley, the inventor of the methods. These methods are now applied in a variety of fields requiring high image compression ratios with high to acceptable quality, thanks to software and hardware tools from Barnsley’s company, Iterated Systems Incorporated.

The book has six chapters and one appendix. Chapter 1 is merely a brief introduction and outline of the rest of the book. Chapter 2 lays the mathematical foundations for image models. It is a strange mix of philosophical, mathematical, and artistic viewpoints. The groundwork is laid down in chapter 3, where we learn about affine transforms, metric spaces, and contraction mapping. The key chapter is chapter 4. Here we are introduced to iterated function systems (IFSs), the collage theorem, and the photocopy algorithm for binary and gray-level images. We learn how to compute attractors of an IFS, one of the bases of fractal compression. C source code is included here to illustrate these definitions. The strength of fractal compression for binary images is given in a few words at the bottom of page 100: “to find an IFS whose attractor is close to or looks like a given set (of points in a region of the binary image), one must try to find a set of transformations, contraction mappings on a suitable space within which the given set lies, such that union, or collage, of the images of the given set under the transformations is close to or looks like the given set.…” A similar assertion can be found at the top of page 113 for gray-scale images. Although well written and superbly illustrated, these two chapters are far easier to comprehend if one consults Barnsley’s previous book [1]. Most theorem proofs and other mathematical details are found there.

Chapter 5 treats classical topics of information theory in a pleasant and intuitive manner. Here the reader is introduced to Huffman coding and arithmetic coding, with a nice parallel between the latter and IFS. Illustrative source code is provided. Chapter 6 gives what the reader probably expects from the beginning, theorems and algorithms for performing gray-level fractal image compression and decompression. To get a deeper understanding, however, most readers would probably have to refer to the previous book [1]. C source code for fractal image compression and decompression is also provided, and although the results obtained are slightly worse than a JPEG compression (an ISO/CCITT standard for image compression, named for the Joint Photographic Experts Group) with an approaching ratio, the code has the merit that it really works and is the only publicly available code. Small changes have to be made to work with images of size at least 256×256, including compiling with a huge model and replacing alloc and free functions with their far versions. The programs also takes several hours to complete for images of this size, but this algorithm is more than enough from an inventor whose patented methods run an entire company. The appendix is devoted to the JPEG alternative, well illustrated by C code once more. Although a deep treatment is not given, this material is enough for comparisons, and many of the good references in the field are given at the end of the appendix.

This work is a good and so far unique book on fractal compression of images. Even people not professionally involved in this field can benefit from the ideas it presents.