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47
A theory for multiresolution signal decomposition : the wavelet representation
 IEEE Transaction on Pattern Analysis and Machine Intelligence
, 1989
"... AbstractMultiresolution representations are very effective for analyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions ..."
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Cited by 2354 (12 self)
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AbstractMultiresolution representations are very effective for analyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions 2 ’ + ’ and 2jcan be extracted by decomposing this signal on a wavelet orthonormal basis of L*(R”). In LL(R), a wavelet orthonormal basis is a family of functions ( @ w (2’ ~n)),,,“jEZt, which is built by dilating and translating a unique function t+r (xl. This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror lilters. For images, the wavelet representation differentiates several spatial orientations. We study the application of this representation to data compression in image coding, texture discrimination and fractal analysis. Index TermsCoding, fractals, multiresolution pyramids, quadrature mirror filters, texture discrimination, wavelet transform. I I.
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
"... ..."
Framelets: MRABased Constructions of Wavelet Frames
, 2001
"... We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spl ..."
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Cited by 129 (50 self)
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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudospline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.
Wavelet theory demystified
 IEEE Trans. Signal Process
, 2003
"... Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to red ..."
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Cited by 45 (22 self)
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Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a selfcontained, accessible fashion. In particular, we prove that the Bspline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in thesense and a sharper theorem stating that smoothness implies order. Index Terms—Approximation order, Besov spaces, Hölder smoothness, multiscale differentiation, splines, vanishing moments, wavelets. I.
Multiwavelets: Theory and Applications
, 1996
"... A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gam ..."
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Cited by 35 (4 self)
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A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gamma1 V j = f0g, and translates OE(t \Gamma k) constitute a basis of V 0 . Then a basis fw jk : w jk = w(2 j t \Gamma k) j; k 2 Zg of L 2 (R) is generated by a wavelet w(t), whose translates w(t \Gamma k) form a basis of W 0 , V 1 = V 0 \Phi W 0 . A standard (scalar) MRA assumes that there is only one scaling function. We make a step forward and allow several scaling functions OE 0 (t); : : : ; OE r\Gamma1 (t) to generate a basis of V 0 . The vector OE(t) = [OE 0 (t) : : : OE r\Gamma1 (t)] T satisfies a dilation equation with matrix coefficients C k . Associated with OE(t) is a multiwavelet w(t) = [w 0 (t) : : : w r\Gamma1 (t)] T . Unlike the scalar case, construction of a multiwave...
Intertwining Multiresolution Analyses and the Construction of Piecewise Polynomial Wavelets
, 1994
"... Let (Vp ) be a local multiresolution analysis (MRA) of L 2 (R) of multiplicity r 1, i.e., V0 is generated by r compactly supported scaling functions. If the scaling functions generate an orthogonal basis of V0 then (Vp) is called an orthogonal MRA. We prove that there exists an orthogonal local M ..."
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Cited by 26 (6 self)
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Let (Vp ) be a local multiresolution analysis (MRA) of L 2 (R) of multiplicity r 1, i.e., V0 is generated by r compactly supported scaling functions. If the scaling functions generate an orthogonal basis of V0 then (Vp) is called an orthogonal MRA. We prove that there exists an orthogonal local MRA (V 0 p ) of multiplicity r 0 such that Vq ae V 0 0 ae Vq+n for some integers q 0, n 1 and r 0 ? 1. In particular, this shows that compactly supported orthogonal polynomial spline wavelets and scaling functions (of mulitplicity r 0 ? 1) of arbitrary regularity exist and we give several such examples. 1 Introduction The starting point for most wavelet constructions is a single function OE 2 L 2 (R) called a scaling function whose integer translates form a Riesz basis for a closed linear subspace V 0 ae L 2 (R). If the scaling function is compactly supported and generates an orthogonal basis of V 0 , then the associated wavelet will also be compactly supported and generate...
Moving Least Square Reproducing Kernel Method (III): Wavelet Packet Its Applications
, 1997
"... This work is a natural extension of the work done in Part II of this series. A new partition of unity  the synchronized reproducing kernel (SRK) interpolantis proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization ..."
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Cited by 25 (10 self)
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This work is a natural extension of the work done in Part II of this series. A new partition of unity  the synchronized reproducing kernel (SRK) interpolantis proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization of the reproducing kernel particle method (RKPM), which demonstrates some superior computational capability in multiple scale numerical simulations. To form such an interpolant, a class of new wavelet functions are introduced in an unconventional way, and they form an independent sequence that is referred to as the wavelet packet. By choosing different combinations in the wavelet series expansion, the desirable synchronized convergence effect in interpolation can be achieved. Based upon the builtin consistency conditions, the differential consistency conditions for the wavelet functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, w...
The undecimated wavelet decomposition and its reconstruction
 IEEE Transactions on Image Processing
, 2007
"... This paper describes the undecimated wavelet transform and its reconstruction. In the first part, we show the relation between two well known undecimated wavelet transforms, the standard undecimated wavelet transform and the isotropic undecimated wavelet transform. Then we present new filter banks s ..."
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Cited by 22 (10 self)
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This paper describes the undecimated wavelet transform and its reconstruction. In the first part, we show the relation between two well known undecimated wavelet transforms, the standard undecimated wavelet transform and the isotropic undecimated wavelet transform. Then we present new filter banks specially designed for undecimated wavelet decompositions which have some useful properties such as being robust to ringing artifacts which appear generally in waveletbased denoising methods. A range of examples illustrates the results.
Ten Good Reasons For Using Spline Wavelets
 Proc. SPIE vol. 3169, Wavelet Applications in Signal and Image Processing V
, 1997
"... The purpose of this note is to highlight some of the unique properties of spline wavelets. These wavelets can be classified in four categories: othogonal (BattleLemari), semiorthogonal (e.g., Bspline), shiftorthogonal, and biorthogonal (CohenDaubechiesFeauveau) . Unlike most other wavelet bases ..."
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Cited by 19 (5 self)
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The purpose of this note is to highlight some of the unique properties of spline wavelets. These wavelets can be classified in four categories: othogonal (BattleLemari), semiorthogonal (e.g., Bspline), shiftorthogonal, and biorthogonal (CohenDaubechiesFeauveau) . Unlike most other wavelet bases, splines have explicit formulae in both the time and frequency domain, which greatly facilitates their manipulation. They allow for a progressive transition between the two extreme cases of a multiresolution: Haar's piecewise constant representation (spline of degree zero) versus Shannon's bandlimited model (which corresponds to a spline of infinite order). Spline wavelets are extremely regular and usually symmetric or antisymmetric. They can be designed to have compact support and to achieve optimal timefrequency localization (Bspline wavelets). The underlying scaling functions are the Bsplines, which are the shortest and most regular scaling functions of order L. Finally, splines have the best approximation properties among all known wavelets of a given order L. In other words, they are the best for approximating smooth functions.