Coxeter Groups and Wavelet Sets

A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there is a natural notion of a dilation-reflection wavelet set. We prove that dilation-reflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilation-translation wavelet theory. There are certain measurable sets which can serve simultaneously as dilation-translation wavelet sets and dilation-reflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different

Inverse Problems and Self-similarity in Imaging

This thesis examines the concept of image self-similarity and provides solutions to various associated inverse problems such as resolution enhancement and missing fractal codes. In general, many real-world inverse problems are ill-posed, mainly because of the lack of existence of a unique solution. The procedure of providing acceptable unique solutions to such problems is known as regularization. The concept of image prior, which has been of crucial importance in image modelling and processing, has also been important in solving inverse problems since it algebraically translates to the regularization procedure. Indeed, much recent progress in imaging has been due to advances in the formulation and practice of regularization. This, coupled with progress in optimization and numerical analysis, has yielded much improvement in computational methods of solving inverse imaging problems. Historically, the idea of self-similarity was important in the development of fractal image coding. Here we show that the self-similarity properties of natural images may be used to construct image priors for the purpose of addressing certain inverse problems. Indeed, new trends in the area of non-local image processing have provided a rejuvenated appreciation of image self-similarity and opportunities to explore novel self-similarity-based priors. We first revisit the concept of fractal-based methods and address some open theoretical problems in the area. This includes formulating a necessary and sufficient condition for the contractivity of the block fractal transform operator. We shall also provide some more generalized formulations of fractal-based self-similarity constraints of an image. These formulations can be developed algebraically and also in terms of the set-based method of Projection Onto Convex Sets (POCS). We then revisit the traditional inverse problems of single frame image zooming and multi-frame resolution enhancement, also known as super-resolution. Some ideas will be borrowed from newly developed non-local denoising algorithms in order to formulate self-similarity priors. Understanding the role of scale and choice of examples/samples is also important in these proposed models. For this purpose, we perform an extensive series of numerical experiments and analyze the results. These ideas naturally lead to the method of self-examples, which relies on the regularity properties of natural images at different scales, as a means of solving the single-frame image zooming problem. Furthermore, we propose and investigate a multi-frame super-resolution counterpart which does not require explicit motion estimation among video sequences

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

Public Debt Dynamics under Ambiguity by means of Iterated Function Systems on Density Functions

We analyze a purely dynamic model of public debt stabilization under ambiguity. We assume that the debt to GDP ratio is described by a random variable, and thus it can be characterized by investigating the evolution of its density function through iteration function systems on mappings. Ambiguity is associated with parameter uncertainty which requires policymakers to respond to such an additional layer of uncertainty according to their ambiguity attitude. We describe ambiguity attitude through a simple heuristic rule in which policymakers adjust the available vague information (captured by the empirical distribution of the debt ratio) with a measure of their ignorance (captured by the uniform distribution). We show that such a model generates fractal-type objects that can be characterized as fixed-point solutions of iterated function systems on mappings. Ambiguity is a source of unpredictability in the long run outcome since it introduces some singularity features in the steady state distribution of the debt ratio. However, the presence of some ambiguity aversion removes such unpredictability by smoothing out the singularities in the steady state distribution

Iterated function systems and shape representation

We propose the use of iterated function systems as an isomorphic shape representation scheme for use in a machine vision environment. A concise description of the basic theory and salient characteristics of iterated function systems is presented and from this we develop a formal framework within which to embed a representation scheme. Concentrating on the problem of obtaining automatically generated two-dimensional encodings we describe implementations of two solutions. The first is based on a deterministic algorithm and makes simplifying assumptions which limit its range of applicability. The second employs a novel formulation of a genetic algorithm and is intended to function with general data input. Keywords: Machine Vision, Shape Representation, Iterated Function Systems, Genetic Algorithms

Gaussian heat kernel upper bounds via Phragm\'en-Lindel\"of theorem

We prove that in presence of $L^2$ Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures.Comment: 14 pages in Latex, Revtex + 4 figures in .ps attached (revised version, new title, several changes, to appear in CHAOS

Attractor image coding with low blocking effects.

by Ho, Hau Lai.Thesis (M.Phil.)--Chinese University of Hong Kong, 1997.Includes bibliographical references (leaves 97-103).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Overview of Attractor Image Coding --- p.2Chapter 1.2 --- Scope of Thesis --- p.3Chapter 2 --- Fundamentals of Attractor Coding --- p.6Chapter 2.1 --- Notations --- p.6Chapter 2.2 --- Mathematical Preliminaries --- p.7Chapter 2.3 --- Partitioned Iterated Function Systems --- p.10Chapter 2.3.1 --- Mathematical Formulation of the PIFS --- p.12Chapter 2.4 --- Attractor Coding using the PIFS --- p.16Chapter 2.4.1 --- Quadtree Partitioning --- p.18Chapter 2.4.2 --- Inclusion of an Orthogonalization Operator --- p.19Chapter 2.5 --- Coding Examples --- p.21Chapter 2.5.1 --- Evaluation Criterion --- p.22Chapter 2.5.2 --- Experimental Settings --- p.22Chapter 2.5.3 --- Results and Discussions --- p.23Chapter 2.6 --- Summary --- p.25Chapter 3 --- Attractor Coding with Adjacent Block Parameter Estimations --- p.27Chapter 3.1 --- δ-Minimum Edge Difference --- p.29Chapter 3.1.1 --- Definition --- p.29Chapter 3.1.2 --- Theoretical Analysis --- p.31Chapter 3.2 --- Adjacent Block Parameter Estimation Scheme --- p.33Chapter 3.2.1 --- Joint Optimization --- p.34Chapter 3.2.2 --- Predictive Coding --- p.36Chapter 3.3 --- Algorithmic Descriptions of the Proposed Scheme --- p.39Chapter 3.4 --- Experimental Results --- p.40Chapter 3.5 --- Summary --- p.50Chapter 4 --- Attractor Coding using Lapped Partitioned Iterated Function Sys- tems --- p.51Chapter 4.1 --- Lapped Partitioned Iterated Function Systems --- p.53Chapter 4.1.1 --- Weighting Operator --- p.54Chapter 4.1.2 --- Mathematical Formulation of the LPIFS --- p.57Chapter 4.2 --- Attractor Coding using the LPIFS --- p.62Chapter 4.2.1 --- Choice of Weighting Operator --- p.64Chapter 4.2.2 --- Range Block Preprocessing --- p.69Chapter 4.2.3 --- Decoder Convergence Analysis --- p.73Chapter 4.3 --- Local Domain Block Searching --- p.74Chapter 4.3.1 --- Theoretical Foundation --- p.75Chapter 4.3.2 --- Local Block Searching Algorithm --- p.77Chapter 4.4 --- Experimental Results --- p.79Chapter 4.5 --- Summary --- p.90Chapter 5 --- Conclusion --- p.91Chapter 5.1 --- Original Contributions --- p.91Chapter 5.2 --- Subjects for Future Research --- p.92Chapter A --- Fundamental Definitions --- p.94Chapter B --- Appendix B --- p.96Bibliography --- p.9