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142
The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
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Cited by 541 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 225 (39 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
Biorthogonal SplineWavelets on the Interval  Stability and Moment Conditions
 Appl. Comp. Harm. Anal
, 1997
"... This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and co ..."
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Cited by 99 (48 self)
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This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly...
Stability of Multiscale Transformations
 J. Fourier Anal. Appl
, 1996
"... After briefly reviewing the interrelation between Rieszbases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It i ..."
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Cited by 98 (22 self)
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After briefly reviewing the interrelation between Rieszbases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It is based on direct and inverse estimates for a certain scale of spaces induced by the underlying multiresolution sequence. Furthermore, we highlight some properties of these spaces pertaining to duality, interpolation, and applications to norm equivalences for Sobolev spaces. AMS Subject Classification: 41A17, 41A65, 46A35, 46B70, 46E35 Key Words: Riesz bases, biorthogonality, stability, projectors, interpolation theory, Kmethod, duality, Jackson, Bernstein inequalities 1 Background and Motivation A standard framework for approximately recapturing a function v in some infinite dimensional separable Hilbert space V , say, either from explicitly given data or as a solution of an operator equ...
Directional Multiscale Modeling of Images using the Contourlet Transform
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2004
"... The contourlet transform is a new extension to the wavelet transform in two dimensions using nonseparable and directional filter banks. The contourlet expansion is composed of basis images oriented at varying directions in multiple scales, with flexible aspect ratios. With this rich set of basis ima ..."
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Cited by 89 (5 self)
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The contourlet transform is a new extension to the wavelet transform in two dimensions using nonseparable and directional filter banks. The contourlet expansion is composed of basis images oriented at varying directions in multiple scales, with flexible aspect ratios. With this rich set of basis images, the contourlet transform can effectively capture the smooth contours that are the dominant features in natural images with only a small number of coefficients. We begin with a detailed study on the statistics of the contourlet coefficients of natural images, using histogram estimates of the marginal and joint distributions, and mutual information measurements to characterize the dependencies between coefficients. The study reveals the nonGaussian marginal statistics and strong intrasubband, crossscale, and crossorientation dependencies of contourlet coefficients. It is also found that conditioned on the magnitudes of their generalized neighborhood coefficients, contourlet coefficients can approximately be modeled as Gaussian variables. Based on these statistics, we model contourlet coefficients using a hidden Markov tree (HMT) model that can capture all of their interscale, interorientation, and intrasubband dependencies. We experiment this model in the image denoising and texture retrieval applications where the results are very promising. In denoising, contourlet HMT outperforms wavelet HMT and other classical methods in terms of visual quality. In particular, it preserves edges and oriented features better than other existing methods. In texture retrieval, it shows improvements in performance over wavelet methods for various oriented textures.
A new class of twochannel biorthogonal filter banks and wavelet bases
 IEEE Trans. Signal Processing
, 1995
"... Abstract: We propose a novel framework for a new class of twochannel biorthogonal filter banks. The framework covers two useful subclasses: (i) causal stable 11R filter banks; (ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for su ..."
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Cited by 85 (1 self)
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Abstract: We propose a novel framework for a new class of twochannel biorthogonal filter banks. The framework covers two useful subclasses: (i) causal stable 11R filter banks; (ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. We also provide a novel mapping of the proposed one dimensional (1D) framework into two dimensional (2D). The mapping preserves: (i) perfect reconstruction; (ii) stability in the IIR case; (iii) linear phase in the FIR case; (iv) zeros at aliasing frequency; (v) frequency characteristic of the filters.
Multivariate refinement equations and convergence of subdivision schemes
 SIAM J. Math. Anal
, 1998
"... Abstract. Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lpconvergence of a subdivision scheme in te ..."
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Cited by 72 (47 self)
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Abstract. Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lpconvergence of a subdivision scheme in terms of the pnorm joint spectral radius of a collection of matrices associated with the refinement mask. In particular, the 2norm joint spectral radius can be easily computed by calculating the eigenvalues of a certain linear operator on a finite dimensional linear space. Examples are provided to illustrate the general theory.
A stability criterion for biorthogonal wavelet bases and their related subband coding scheme
 Duke Math. J
, 1992
"... biorthogonal bases of compactly supported wavelets, i.e., pairs of dual Riesz bases generated from two single compactly supported functions q and by means of dilations and translations ..."
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Cited by 61 (1 self)
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biorthogonal bases of compactly supported wavelets, i.e., pairs of dual Riesz bases generated from two single compactly supported functions q and by means of dilations and translations
Biorthogonal Wavelet Expansions
 Constr. Approx
"... This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connectio ..."
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Cited by 60 (7 self)
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This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connection of this issue with stationary subdivision schemes. Key Words: Finiteley generated shiftinvariant spaces, stationary subdivision schemes, matrix refinement relations, biorthogonal wavelets. AMS Subject Classification: 39B62, 41A63 1 Introduction During the past few years the construction of multivariate wavelets has received considerable attention. It is quite apparent that multivariate wavelets with good localazition properties in frequency and spatial domains which constitute an orthonormal basis of L 2 (IR s ) are hard to realize. On the other hand, it turns out that in many applications orthogonality is not really important whereas locality, in particular, compact support is very...