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Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 434 (7 self)
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ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
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 377 (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...
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 84 (46 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...
A Multiresolution Framework for Variational Subdivision
, 1998
"... Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is ..."
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Cited by 42 (0 self)
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Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a given set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces nonuniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consequently we show how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.
Local Decomposition of Refinable Spaces and Wavelets
, 1996
"... A convenient setting for studying multiscale techniques in various areas of applications is usually based on a sequence of nested closed subspaces of some function space F which is often referred to as multiresolution analysis. The concept of wavelets is a prominent example where the practical use o ..."
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Cited by 33 (8 self)
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A convenient setting for studying multiscale techniques in various areas of applications is usually based on a sequence of nested closed subspaces of some function space F which is often referred to as multiresolution analysis. The concept of wavelets is a prominent example where the practical use of such a multiresolution analysis relies on explicit representations of the orthogonal difference between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of IR s , say. It is then usually difficult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements, in particular, those corresponding to biorthogonal wavelets. The main objective of this paper is therefore to study possibly nonorthogonal but in a certain sense stable and even local decompositions of nested spaces and to develop tools which are not necessarily confined to ...
Reversing Subdivision Rules: Local Linear Conditions and Observations on Inner Products
 Journal of Computational and Applied Mathematics
, 1999
"... In a previous work [32] we investigated how to reverse subdivision rules using global least squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finitedimensional Cartesian vector space. We produced simple an ..."
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Cited by 26 (17 self)
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In a previous work [32] we investigated how to reverse subdivision rules using global least squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finitedimensional Cartesian vector space. We produced simple and sparse reconstruction filters, but had to appeal to matrix factorization to obtain an efficient, exact decomposition. We also made some observations on how the inner product that defines the semiorthogonality inuences the sparsity of the reconstruction filters. In this work we carry the investigation further by studying biorthogonal systems based upon subdivision rules and local least squares fitting problems that reverse the subdivision. We are able to produce multiresolution structures for some common univariate subdivision rules that have both sparse reconstruction and decomposition filters. Three will be presented here  for quadratic and cubic Bspline subdivision and for the 4point interpola...
Multiresolution Curve and Surface Representation: Reversing Subdivision Rules by LeastSquares Data Fitting
, 1998
"... This work explores how three techniques for defining and representing curves and surfaces can be related efficiently. The techniques are subdivision, leastsquares data fitting, and wavelets. We show how leastsquares data fitting can be used to "reverse" a subdivision rule, how this reversal is rela ..."
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Cited by 25 (13 self)
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This work explores how three techniques for defining and representing curves and surfaces can be related efficiently. The techniques are subdivision, leastsquares data fitting, and wavelets. We show how leastsquares data fitting can be used to "reverse" a subdivision rule, how this reversal is related to wavelets, how this relationship can provide a multilevel representation, and how the decomposition/reconstruction process can be carried out in linear time and space through the use of a matrix factorization. Some insights that this work brings forth are that the inner product used in a multiresolution analysis influences the support of a wavelet, that wavelets can be constructed by straightforward matrix observations, and that matrix partitioning and factorization can provide alternatives to inverses or duals for building efficient decomposition and reconstruction processes. We illustrate our findings using an example curve, greyscale image, and tensorproduct surface. Keywords: Su...
Commutation For Irregular Subdivision
, 1998
"... We present a generalization of Lemarié's commutation formula to irregular subdivision schemes and wavelets. We show how in the noninterpolating case the divided differences need to be adapted to the subdivision scheme. As an example we include the construction of an entire family of biorthogonal co ..."
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Cited by 7 (1 self)
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We present a generalization of Lemarié's commutation formula to irregular subdivision schemes and wavelets. We show how in the noninterpolating case the divided differences need to be adapted to the subdivision scheme. As an example we include the construction of an entire family of biorthogonal compactly supported irregular knot Bspline wavelets starting from Lagrangian interpolation.
Biorthogonal wavelets for subdivision volumes
 In: Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications
, 2002
"... Figure 1: Volume subdivision, manipulation, and fitting. A lattice (top left) is recursively subdivided and reshaped at the fourth subdivision level. This shape is lowpass filtered by removing fineresolution wavelet coefficients (bottom right). We present a biorthogonal wavelet construction based ..."
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Cited by 5 (0 self)
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Figure 1: Volume subdivision, manipulation, and fitting. A lattice (top left) is recursively subdivided and reshaped at the fourth subdivision level. This shape is lowpass filtered by removing fineresolution wavelet coefficients (bottom right). We present a biorthogonal wavelet construction based on CatmullClarkstyle subdivision volumes, like multilinear cell averaging (MLCA). Our wavelet transform is the threedimensional extension of a previously developed construction of subdivisionsurface wavelets that was used for multiresolution modeling of largescale isosurfaces. Subdivision surfaces provide a flexible modeling tool for geometries of arbitrary topology and for functions defined thereon. Wavelet representations add the ability to compactly represent largescale geometries at multiple levels of detail. Our wavelet construction based on subdivision volumes extends these concepts to trivariate geometries, such as timevarying surfaces, freeform deformations, and solid models with nonuniform material properties.