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50
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...
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
"... ..."
Fast Multiresolution Surface Meshing
, 1995
"... We are presenting a new method for adaptive surface meshing and triangulation which controls the local levelofdetail of the surface approximation by local spectral estimates. These estimates are figured out by a wavelet representation of the surface data. The basic idea is to decompose the initial ..."
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Cited by 70 (3 self)
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We are presenting a new method for adaptive surface meshing and triangulation which controls the local levelofdetail of the surface approximation by local spectral estimates. These estimates are figured out by a wavelet representation of the surface data. The basic idea is to decompose the initial data set by means of an orthogonal or semiorthogonal tensor product wavelet transform (WT) and to analyze the resulting coefficients. In surface regions, where the partial energy of the resulting coefficients is low, the polygonial approximation of the surface can be performed with larger triangles without loosing too much fine grain details. However, since the localization of the WT is bound by the Heisenberg principle the meshing method has to be controlled by the detail signals rather than directly by the coefficients. The dyadic scaling of the WT stimulated us to build an hierachical meshing algorithm which transforms the initially regular data grid into a quadtree representation by...
Quantitative Fourier Analysis of Approximation Techniques: Part II  Wavelets
 IEEE Trans. Signal Processing
, 1999
"... In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic ..."
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Cited by 63 (28 self)
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In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter. This is, in particular, true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asymptotically optimal upper bounds for the leastsquares approximation error. Another contribution is the application of these results to Bsplines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify ...
Fast parametric elastic image registration
 IEEE Transactions on Image Processing
, 2003
"... Abstract—We present an algorithm for fast elastic multidimensional intensitybased image registration with a parametric model of the deformation. It is fully automatic in its default mode of operation. In the case of hard realworld problems, it is capable of accepting expert hints in the form of so ..."
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Cited by 60 (4 self)
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Abstract—We present an algorithm for fast elastic multidimensional intensitybased image registration with a parametric model of the deformation. It is fully automatic in its default mode of operation. In the case of hard realworld problems, it is capable of accepting expert hints in the form of soft landmark constraints. Much fewer landmarks are needed and the results are far superior compared to pure landmark registration. Particular attention has been paid to the factors influencing the speed of this algorithm. The Bspline deformation model is shown to be computationally more efficient than other alternatives. The algorithm has been successfully used for several twodimensional (2D) and threedimensional (3D) registration tasks in the medical domain, involving MRI, SPECT, CT, and ultrasound image modalities. We also present experiments in a controlled environment, permitting an exact evaluation of the registration accuracy. Test deformations are generated automatically using a random hierarchical fractional waveletbased generator. Index Terms—Elastic registration, image registration, landmarks, splines. I.
Efficient triangular surface approximation using wavelets and quadtree data structures
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 1996
"... We present a new method for adaptive surface meshing and triangulation which controls the local levelofdetail of the surface approximation by local spectral estimates. These estimates are determined by a wavelet representation of the surface data. The basic idea is to decompose the initial data se ..."
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Cited by 51 (4 self)
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We present a new method for adaptive surface meshing and triangulation which controls the local levelofdetail of the surface approximation by local spectral estimates. These estimates are determined by a wavelet representation of the surface data. The basic idea is to decompose the initial data set by means of an orthogonal or semiorthogonal tensor product wavelet transform (WT) and to analyze the resulting coefficients. In surface regions, where the partial energy of the resulting coefficients is low, the polygonal approximation of the surface can be performed with larger triangles without loosing too much fine grain details. However, since the localization of the WT is bound by the Heisenberg principle the meshing method has to be controlled by the detail signals rather than directly by the coefficients. The dyadic scaling of the WT stimulated us to build an hierarchical meshing algorithm which transforms the initially regular data grid into a quadtree representation by rejection of unimportant mesh vertices. The optimum triangulation of the resulting quadtree cells is carried out by selection from a lookup table. The tree grows recursively as controlled by detail signals which are computed from a modified inverse WT. In order to control the local levelofdetail, we introduce a new class of wavelet space filters acting as “magnifying glasses ” on the data. We show that our algorithm performs a low algorithmic complexity, so that surface meshing can be achieved at interactive rates, such as required by flight simulators. However, other applications are possible as well, such as mesh reduction in complex data, FEM or radiosity meshing. The method is applied on different types of data comprising both digital terrain models and laser range scans. In addition, quantitative investigations on error analysis are carried out.
LongRange Dependence: revisiting Aggregation with Wavelets.
 Journal of Time Series Analysis
, 1998
"... The aggregation procedure is a natural way to analyse signals which exhibit longrange dependent features and has been used as a basis for estimation of the Hurst parameter, H. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly ..."
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Cited by 40 (12 self)
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The aggregation procedure is a natural way to analyse signals which exhibit longrange dependent features and has been used as a basis for estimation of the Hurst parameter, H. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly related to approximations of the signal in the sense of a Haarmultiresolution analysis. A natural wavelet based generalisation to traditional aggregation is then proposed: "aaggregation". It is shown that aaggregation cannot lead to good estimators of H, and so a new kind of aggregation, "daggregation", is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of H based on daggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data.
Statistical Texture Characterization From Discrete Wavelet Representations
 IEEE Transactions on Image Processing
, 1999
"... We conjecture that texture can be characterized by the statistics of the wavelet detail coefficients and therefore introduce 2 feature sets: 1) the wavelet histogram signatures which capture all first order statistics using a model based approach; 2) the wavelet cooccurrence signatures, which reflec ..."
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Cited by 32 (0 self)
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We conjecture that texture can be characterized by the statistics of the wavelet detail coefficients and therefore introduce 2 feature sets: 1) the wavelet histogram signatures which capture all first order statistics using a model based approach; 2) the wavelet cooccurrence signatures, which reflect the coefficients' second order statistics. The introduced feature sets outperform the traditionally used energy. Best performance is achieved by combining histogram and cooccurrence signatures.