Fractal image compression and the self-affinity assumption : a stochastic signal modelling perspective

Bibliography: p. 208-225.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

The development of audio data compression methods and facilities with the use of fractal approach

The development of modern telecommunication networks is characterized by an increase in the share of multimedia traffic. An important component of multimedia traffic is audio information, especially voice information. This problem appeared during the working process on workstations and personal computers. The problem is associated with a large volume for their transfer and storage. There are many different algorithms for audio data archiving, but they either provide insufficient compression ratios, or lead to a significant loss of data, what is connected with deterioration of consumer information quality. According to this, there is a relevant task to develop new approaches to audio data compression

The development of audio data compression methods and facilities with the use of fractal approach

The development of modern telecommunication networks is characterized by an increase in the share of multimedia traffic. An important component of multimedia traffic is audio information, especially voice information. This problem appeared during the working process on workstations and personal computers. The problem is associated with a large volume for their transfer and storage. There are many different algorithms for audio data archiving, but they either provide insufficient compression ratios, or lead to a significant loss of data, what is connected with deterioration of consumer information quality. According to this, there is a relevant task to develop new approaches to audio data compression

Giving eyes to ICT!, or How does a computer recognize a cow?

Het door Schouten en andere onderzoekers op het CWI ontwikkelde systeem berust op het beschrijven van beelden met behulp van fractale meetkunde. De menselijke waarneming blijkt mede daardoor zo efficiënt omdat zij sterk werkt met gelijkenissen. Het ligt dus voor de hand het te zoeken in wiskundige methoden die dat ook doen. Schouten heeft daarom beeldcodering met behulp van 'fractals' onderzocht. Fractals zijn zelfgelijkende meetkundige figuren, opgebouwd door herhaalde transformatie (iteratie) van een eenvoudig basispatroon, dat zich daardoor op steeds kleinere schalen vertakt. Op elk niveau van detaillering lijkt een fractal op zichzelf (Droste-effect). Met fractals kan men vrij eenvoudig bedrieglijk echte natuurvoorstellingen maken. Fractale beeldcodering gaat ervan uit dat het omgekeerde ook geldt: een beeld effectief opslaan in de vorm van de basispatronen van een klein aantal fractals, samen met het voorschrift hoe het oorspronkelijke beeld daaruit te reconstrueren. Het op het CWI in samenwerking met onderzoekers uit Leuven ontwikkelde systeem is mede gebaseerd op deze methode. ISBN 906196502

Multiscale Methods in Image Modelling and Image Processing

The field of modelling and processing of 'images' has fairly recently become important, even crucial, to areas of science, medicine, and engineering. The inevitable explosion of imaging modalities and approaches stemming from this fact has become a rich source of mathematical applications. 'Imaging' is quite broad, and suffers somewhat from this broadness. The general question of 'what is an image?' or perhaps 'what is a natural image?' turns out to be difficult to address. To make real headway one may need to strongly constrain the class of images being considered, as will be done in part of this thesis. On the other hand there are general principles that can guide research in many areas. One such principle considered is the assertion that (classes of) images have multiscale relationships, whether at a pixel level, between features, or other variants. There are both practical (in terms of computational complexity) and more philosophical reasons (mimicking the human visual system, for example) that suggest looking at such methods. Looking at scaling relationships may also have the advantage of opening a problem up to many mathematical tools. This thesis will detail two investigations into multiscale relationships, in quite different areas. One will involve Iterated Function Systems (IFS), and the other a stochastic approach to reconstruction of binary images (binary phase descriptions of porous media). The use of IFS in this context, which has often been called 'fractal image coding', has been primarily viewed as an image compression technique. We will re-visit this approach, proposing it as a more general tool. Some study of the implications of that idea will be presented, along with applications inferred by the results. In the area of reconstruction of binary porous media, a novel, multiscale, hierarchical annealing approach is proposed and investigated

Fractal methods in image analysis and coding

In this thesis we present an overview of image processing techniques which use fractal methods in some way. We show how these fields relate to each other, and examine various aspects of fractal methods in each area. The three principal fields of image processing and analysis th a t we examine are texture classification, image segmentation and image coding. In the area of texture classification, we examine fractal dimension estimators, comparing these methods to other methods in use, and to each other. We attempt to explain why differences arise between various estimators of the same quantity. We also examine texture generation methods which use fractal dimension to generate textures of varying complexity. We examine how fractal dimension can contribute to image segmentation methods. We also present an in-depth analysis of a novel segmentation scheme based on fractal coding. Finally, we present an overview of fractal and wavelet image coding, and the links between the two. We examine a possible scheme involving both fractal and wavelet methods