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Coxeter Groups and Wavelet Sets
, 710
"... Abstract. 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/fractalsurface wavelet is obtained by defining fractal surfaces on f ..."
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Abstract. 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/fractalsurface 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 semiexpository paper. In particular, there is a natural notion of a dilationreflection wavelet set. We prove that dilationreflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilationtranslation wavelet theory. There are certain measurable sets which can serve simultaneously as dilationtranslation wavelet sets and dilationreflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different. 1.
COXETER GROUPS, WAVELETS, MULTIRESOLUTION AND SAMPLING
, 710
"... Abstract. In this short note we discuss the interplay between finite Coxeter groups and construction of wavelet sets, generalized multiresolution analysis and sampling. ..."
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Abstract. In this short note we discuss the interplay between finite Coxeter groups and construction of wavelet sets, generalized multiresolution analysis and sampling.
Multiscale Signal Processing and Shape Analysis for an Inverse SAR
"... The great challenge in signal processing is to devise computationally efficient and statistically optimal algorithms for estimating signals from noisy background and understanding their contents. This thesis treats the problem of multiscale signal processing and shape analysis for an Inverse Synthe ..."
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The great challenge in signal processing is to devise computationally efficient and statistically optimal algorithms for estimating signals from noisy background and understanding their contents. This thesis treats the problem of multiscale signal processing and shape analysis for an Inverse Synthetic Aperture Radar (ISAR) imaging system. To address some of the limitations of conventional techniques in radar image processing, an information theoretic approach for target motion estimation is first proposed. A wavelet based multiscale method for shape enhancement is subsequently derived and followed by a regression technique for shape recognition. Building on entropybased divergence measures which have shown promising results in many areas of engineering and image processing, we introduce in this thesis a new generalized divergence measure, namely the JensenRényi divergence. Upon establishing its properties such as convexity and its upper bound etc., we apply it to image registration for ISAR focusing as well as related problems in data fusion. Attempting to extend current approaches to signal estimation in a wavelet framework, which have generally relied on the assumption of normally distributed perturbations, we pro
Attractor Image Coding With Low Blocking Effects
, 1997
"... In the present work, a relatively novel method called attractor image coding (commonly known as fractal image coding) for lossy image compression is investigated. Roughly speaking, the original image is partitioned into several disjoint range blocks. For each range block, an affine transformation is ..."
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In the present work, a relatively novel method called attractor image coding (commonly known as fractal image coding) for lossy image compression is investigated. Roughly speaking, the original image is partitioned into several disjoint range blocks. For each range block, an affine transformation is constructed to approximate the range block by another part of the same image. Compression is obtained by compactly storing only the descriptions of these transformations. The decoded image is obtained by iteratively applying these transformations on any initial image. However, the attractor coding, being a blockbased algorithm, suffers from the usual blocking artifacts which is highly disturbing to human visual system (HVS). The blockiness is mainly due to the independent processing of each block in encoding. Discontinuities may occur across the block boundaries in the decoded image that are smooth in the original image. The problem is more prominent when the bit rate is reduced. In this t...
FOURIER FREQUENCIES IN AFFINE ITERATED FUNCTION SYSTEMS
, 2006
"... Abstract. We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in R d, and the “IFS” refers to such a finite system of transformations, or functions. T ..."
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Abstract. We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in R d, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X, µ) where X is a compact subset of R d, (the support of µ) and the measure µ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L 2 (X, µ); and (2) the interplay between the geometry of (X, µ) on the one
Coalescence Hidden Variable Fractal Interpolation Functions and its Smoothness Analysis
, 2005
"... ABSTRACT: We construct a coalescence hidden variable fractal interpolation function(CHFIF) through a nondiagonal iterated function system(IFS). Such a FIF may be selfaffine or nonselfaffine depending on the parameters of the defining nondiagonal IFS. The smoothness analysis of the CHFIF has been ..."
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ABSTRACT: We construct a coalescence hidden variable fractal interpolation function(CHFIF) through a nondiagonal iterated function system(IFS). Such a FIF may be selfaffine or nonselfaffine depending on the parameters of the defining nondiagonal IFS. The smoothness analysis of the CHFIF has been carried out by using the operator approximation technique. The deterministic construction of functions having order of modulus continuity O(t  δ (log t) m) (m a nonnegative integer and 0 < δ ≤ 1) is possible through our CHFIF. The bounds of fractal dimension of CHFIFs are obtained first in certain critical cases and then, using estimation of these bounds, the bounds of fractal dimension of any FIF are found.
c ○ World Scientific Publishing Company HIDDEN VARIABLE BIVARIATE FRACTAL INTERPOLATION SURFACES
, 2002
"... We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in R 4 and its projection in R 3 is taken. The extra degree of freedom coming from R 4 provides hidden variable, which is an important factor for flexibility a ..."
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We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in R 4 and its projection in R 3 is taken. The extra degree of freedom coming from R 4 provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both selfsimilar and nonselfsimilar FIS simultaneously and show that the hidden variable fractal surface may be selfsimilar under certain conditions.
FRACTAL COMPRESSION OF IMAGES WITH PROJECTED IFS
"... Standard fractal image compression, proposed by Jacquin[1], is based on IFS (Iterated Function Systems) defined in R 2. This modelization implies restrictions in the set of images being able to be compressed. These images have to be self similar in R 2. We propose a new model, the projected IFS, to ..."
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Standard fractal image compression, proposed by Jacquin[1], is based on IFS (Iterated Function Systems) defined in R 2. This modelization implies restrictions in the set of images being able to be compressed. These images have to be self similar in R 2. We propose a new model, the projected IFS, to approximate and code grey level images. This model has the ability to define affine IFS in a high dimension space, and to project it through control points, resulting in a non strictly self similar object in R 2. We proposed a method for approximating curves with such a model [2, 3]. In this paper, we extend the model capabilities to surfaces and images. This includes the combination of projected IFS in a quadtree structure and a complete coding scheme. First results show that our method gives better results than standard fractal image compression. Furthermore, in the very low bitrate context, the distortion/rate performances are equivalent to those obtained with EZW algorithm. 1.