Applied Mathematics and Plasma Physics

Brendt Wohlberg, *Fractal Image Compression and the Self-Affinity Assumption: A Stochastic Signal Modelling Perspective*, PhD thesis, University of Cape Town, 1996

Fractal image compression is a comparatively new technique which has gained considerable attention in the popular technical press, and more recently in the research literature. The most significant advantages claimed are high reconstruction quality at low coding rates, rapid decoding, and "resolution independence" in the sense that an encoded image may be decoded at a higher resolution than the original. While many of the claims published in the popular technical press are clearly extravagant, it appears from the rapidly growing body of published research that fractal image compression is capable of performance comparable with that of other techniques enjoying the benefit of a considerably more robust theoretical foundation.

So called because of the similarities between the form of image representation and a mechanism widely used in generating deterministic fractal images, fractal compression represents an image by the parameters of a set of affine transforms on image blocks under which the image is approximately invariant. Although the conditions imposed on these transforms may be shown to be sufficient to guarantee that an approximation of the original image can be reconstructed, there is no obvious theoretical reason to expect this to represent an efficient representation for image coding purposes. The usual analogy with vector quantisation, in which each image is considered to be represented in terms of code vectors extracted from the image itself is instructive, but transforms the fundamental problem into one of understanding why this construction results in an efficient codebook.

The signal property required for such a codebook to be effective, termed "self-affinity", is poorly understood. A stochastic signal model based examination of this property is the primary contribution of this dissertation. The most significant findings (subject to some important restrictions) are that "self-affinity" is not a natural consequence of common statistical assumptions but requires particular conditions which are inadequately characterised by second order statistics, and that "natural" images are only marginally "self-affine", to the extent that fractal image compression is effective, but not more so than comparable standard vector quantisation techniques.

@phdthesis{wohlberg-1996-fractal,

author = {Brendt Wohlberg},

title = {Fractal Image Compression and the Self-Affinity Assumption: A Stochastic Signal Modelling Perspective},

year = {1996},

urlpdf = {http://math.lanl.gov/~brendt/Publications/Docs/wohlberg-1996-fractal.pdf},

urlps = {http://math.lanl.gov/~brendt/Publications/Docs/wohlberg-1996-fractal.ps.gz},

school = {University of Cape Town},

address = {Cape Town, South Africa}

}