wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

design of regular wavelets using a three step lifting scheme we propose structural multidimensional multichannel filter banks with desirable numbers of vanishing moments for the analysis and synthesis banks for a two channel filter bank we use a three step lifting scheme as opposed to the conventional two step lifting method in order to provide more symmetry between the analysis and synthesis filters we show that the resulting filters have more regularity lower frame bounds ratio and better frequency selectivity we also extend our design to a general multichannel framework three steps of lifting scheme provides us a degree of freedom which we can benefit from toward a more flexible design

multiscale methods for data on graphs and irregular multidimensional situations for regularly spaced one dimensional data wavelet shrinkage has proven to be a compelling method for non parametric function estimation we create three new multiscale methods that provide wavelet like transforms both for data arising on graphs and for irregularly spaced spatial data in more than one dimension the concept of scale still exists within these transforms but as a continuous quantity rather than dyadic levels further we adapt recent empirical bayesian shrinkage techniques to enable us to perform multiscale shrinkage for function estimation both on graphs and for irregular spatial data we demonstrate that our methods perform very well when compared with several other methods for spatial regression for both real and simulated data although we concentrate on multiscale shrinkage regression we present our new wavelet transforms as generic tools intended to be the basis of methods that might benefit from a multiscale representation of data either on graphs or for irregular spatial data

coset sum an alternative to the tensor product in wavelet construction a multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in multiresolution analysis setup some multivariate refinement masks may be decomposed into lower dimensional refinement masks tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks we present an alternative method which we call coset sum for constructing multivariate refinement masks from univariate refinement masks the coset sum shares many essential features of the tensor product that make it attractive in practice 1 it preserves the biorthogonality of univariate refinement masks 2 it preserves the accuracy number of the univariate refinement mask and 3 the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients the coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product these features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems some experimental results using 2 d images are presented to illustrate our findings

adaptive 2 d wavelet transform based on the lifting scheme with preserved vanishing moments in this paper we propose novel adaptive wavelet filter bank structures based on the lifting scheme the filter banks are nonseparable based on quincunx sampling with their properties being pixel wise adapted according to the local image features despite being adaptive the filter banks retain a desirable number of primal and dual vanishing moments the adaptation is introduced in the predict stage of the filter bank with an adaptation region chosen independently for each pixel based on the intersection of confidence intervals ici rule the image denoising results are presented for both synthetic and real world images it is shown that the obtained wavelet decompositions perform well especially for synthetic images that contain periodic patterns for which the proposed method outperforms the state of the art in image denoising

quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions this paper deals with typical problems that arise when using wavelets in numerical analysis applications the first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions several types of quadratures are discussed and compared for different classes of wavelets since their construction using monomials is ill conditioned also a modified well conditioned construction using chebyshev polynomials is presented the second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions they are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm this is illustrated with numerical examples

on the realizability of biorthogonal m dimensional two band filter banks we show an algebraic approach for the design of ladder structures for causal biorthogonal filter banks the key ingredient of the approach is known in literature as euclid s algorithm using this algorithm we derive some strong result on the design freedom for ladder structures in particular we show that the dimensionality of the problem plays an important role we end by with some conjectures concerning the extensions to multichannel and noncausal filter banks

theory of two dimensional multirate filter banks results are presented on 2 d fir two dimensional finite impulse response filter banks for multirate applications the theory is valid for all sampling lattices conditions for alias free and perfect signal reconstruction are derived synthesis structures for paraunitary and nonparaunitary polynomial matrices are derived which yield perfect reconstruction filter banks the degrees of freedom are given for these systems linear phase conditions are posed on the polyphase form of filter banks which is used to derive a design structure for the restricted but important case of linear phase filter banks

multi d wavelet filter bank design using quillen suslin theorem for laurent polynomials in this paper we present a new approach for constructing the wavelet filter bank our approach enables constructing nonseparable multidimensional non redundant wavelet filter banks with fir filters using the quillen suslin theorem for laurent polynomials our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design first it works for any spatial dimension and for any sampling matrix second it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition third it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter

parameterization of m band affine frames with pre assigned scaling filter affine wavelet frames related with subdivision scheme provide tools for building curves and surfaces in cagd cam at first we show that scaling filters can be parameterized by decomposed elementary matrices with the form im p pz k where k is a nonnegative integer and p is a one rank constant valued idempotent matrix p2 im theorem 2 2 then we use the method to construct m band affine wavelet frames with the given scaling filter and the dual frames general solutions for affine wavelet frames and its dual affine wavelet frames are obtained with pre assigned scaling filter theorems 3 5 and 3 6 at last a design example is given

spectral properties of the transition operator associated to a multivariate refinement equation given a finitely supported sequence a on z s and an s s dilation matrix m the transition operator ta is the linear operator defined by tavoau a p b2z s aomabuvobu where a2 z s and v lies in 0oz s u the linear space of all finitely supported sequences on z s in this paper we investigate the spectral properties of the transition operator ta and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the refinement equation a p a2z s aoau omau 1999

lattice structure for generalized support multidimensional linear phase perfect reconstruction filter bank multidimensional linear phase perfect reconstruction filter bank mdlpprfb can be designed and implemented via lattice structure the lattice structure for the mdlpprfb with filter support n m has been published by muramatsu where m is the decimation matrix is a positive integer diagonal matrix and n n denotes the set of integer vectors in the fundamental parallelepiped of the matrix n obviously if is chosen to be other positive diagonal matrices instead of only positive integer ones the corresponding lattice structure would provide more choices of filter banks offering better trade off between filter support and filter performance we call such resulted filter bank as generalized support mdlpprfb gsmdlpprfb the lattice structure for gsmdlpprfb however cannot be designed by simply generalizing the process that muramatsu employed furthermore the related theories to assist the design also become different from those used by muramatsu such issues will be addressed in this paper to guide the design of gsmdlpprfb the necessary and sufficient conditions are established for a generalized support multidimensional filter bank to be linear phase to determine the cases we can find a gsmdlpprfb the necessary conditions about the existence of it are proposed to be related with filter support and symmetry polarity i e the number of symmetric filters ns and antisymmetric filters na based on a process different from the one muramatsu used that combines several polyphase matrices to construct the starting block one of the core building blocks of lattice structure the lattice structure for gsmdlpprfb is developed and shown to be minimal additionally the result in this paper includes muramatsu s as a special case

index assignment for multichannel communication under failure we consider the problem of constructing multiple description scalar quantizers and describing the achievable rate distortion tuples in that setting we model this as a combinatorial optimization problem of number arrangements in a matrix this approach gives a general technique for deriving lower bounds on the distortion at given channel rates this technique is constructive thus allowing an algorithm that gives an upper bound for the case of two communication channels with equal rates the bounds coincide thus giving the precise lowest achievable distortion at fixed rates the bounds are within a small constant for a higher number of channels to the best of our knowledge this is the first result involving systems with more than two communication channels

lipschitz continuity of refinable functions on the heisenberg group in this paper the lipschitz continuity of refinable functions related to the general acceptable dilations on the heisenberg group will be investigated in terms of the uniform joint spectral radius we also give an investigation of the refinable functions in the generalized lipschitz spaces related to a kind of special acceptable dilations

multidimensional multirate filters and filter banks derived from one dimensional filters a method by which every multidimensional m d filter with an arbitrary parallelepiped shaped passband support can be designed and implemented efficiently is presented it is shown that all such filters can be designed starting from an appropriate one dimensional prototype filter and performing a simple transformation with d denoting the number of dimensions the complexity of design and implementation of the m d filter are reduced from o n sup d to o n using the polyphase technique an implementation with complexity of only 2n is obtained in the two dimensional even though the filters designed are in general nonseparable they have separable polyphase components one special application of this method is in m d multirate signal processing where filters with parallelepiped shaped passbands are used in decimation interpolation and filter banks some generalizations and other applications of this approach including m d uniform discrete fourier transform dft quadrature mirror filter banks that achieve perfect reconstruction are studied several design example are given

construction of arbitrary dimensional biorthogonal multiwavelet using lifting scheme this paper focuses on the construction of multidimensional biorthogonal multiwavelets and the perfect reconstruction multifilter banks based on the hermite neville filter two lifting structures have been proposed and systematically investigated and a general design framework has been developed for building biorthogonal multiwavelets and hermite interpolation filter banks with any multiplicity for any lattice in any dimension with any number of primal and dual vanishing moments the construction is an important generalization of the neville based lifting scheme and inherits all of the advantages of lifting schemes such as fast transform in place computation and integer to integer transforms our multi wavelet systems preserve most of the desirable properties for applications such as interpolating short support symmetry and high vanishing moments

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

the lifting factorization of 2d 4 channel nonseparable wavelet transforms abstract one dimensional 1d wavelet transform was lifted successfully using the division with remainder of univariate laurent polynomials however division with remainder does not exist in the case of a bivariate polynomial thus it makes sense that the polyphase matrix of a two dimensional 2d nonseparable wavelet transform cannot be decomposed into the lifting format using the same solution as that of the 1d wavelet transform in this research we present a new lifting factorization method of two dimensional four channel 2d 4 channel nonseparable wavelet filter banks according to the uniform constructing format of high dimensional multivariate wavelet filter banks the general form of 2d 4 channel nonseparable wavelet filter bank is given with these filter banks the polyphase matrix of 2d 4 channel nonseparable wavelet transform is found and proven then we present the lifting factorization of the polyphase matrix and some examples are demonstrated in the factorization procedures finally the lifting performances of the proposed method are analyzed this lifting method factorizes the polyphase matrix into the product of a series of unit lower left triangular numerical matrices unit upper right triangular numerical matrices diagonal numerical matrices and diagonal polynomial matrices whose elements on the diagonal line are 1 x y and xy the original filter banks that all leading principal minors of the numerical matrices are not equal to zero can be factorized into lifting format the proposed method transforms the lifting factorization of the polyphase matrix into the decompositions of the numerical matrices without euclidian division thereby only multiplication and addition operations are performed with no fourier transformation involved when compared with the lifting method of the tensor product lifting wavelet transform and the contourlet transform the proposed lifting method can extract more edge information of images the computational complexity of the original 2d 4 channel nonseparable wavelet transform for image decomposition is n 1 times as much as that of the proposed lifting factorization method and the original wavelet transform is accelerated furthermore the proposed lifting factorization method is faster than the conventional 2d 4 channel nonseparable wavelet transform based on fourier transformation theory and convolution operation when the size of each filter in the latter is greater than 4 n 1 the proposed lifting factorization has better sparsity than that of its original 2d 4 channel nonseparable wavelet transform and other typical 2d 4 channel nonseparable wavelet transforms

convergence of nonstationary cascade algorithms a nonstationary multiresolution of l 2 mathbb r s is generated by a sequence of scaling functions phi k in l 2 mathbb r s k in mathbb z we consider phi k that is the solution of the nonstationary refinement equations phi k m sum j h k 1 j phi k 1 m cdot j k in mathbb z where h k is finitely supported for each k and m is a dilation matrix we study various forms of convergence in l 2 mathbb r s of the corresponding nonstationary cascade algorithm phi k n m sum j h k 1 j phi k 1 n 1 m cdot j as k or n tends to infty it is assumed that there is a stationary refinement equation at infty with filter sequence h and that sum k h k j h j infty the results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h

filters for distortion free two band multirate filter banks in this paper conditions are given for a two band multi rate filter bank to be alias free and to have a unity frequency response it is shown that the class of quadrature mirror filters qmf s that satisfies these conditions is quite limited a class of filters which does satisfy these conditions is given and a simple procedure for designing filters from this class is presented with an example

torsion in cohomology of compact lie groups and chow rings of reductive algebraic groups the purpose of the present note is to give a general procedure for calculating the singular cohomology h k ifp with coefficients in an arbitrary finite field ifv of a connected compact lie group k simultaneously we calculate the chow ring a g ifp a g p of the corresponding complex algebraic group g and the degrees of the basic generalized invariants of the weyl group w in arbitrary characteristic p a mysterious connection between the degrees of generators of a g ifp and the kernel of the principal nilpotent element in characteristic p is pointed out the computation of real cohomology of compact lie groups started by e cartan in 1929 was completed for classical groups by r brauer and by l pontryagin in 1935 h hopf and h samelson showed in 1941 that h k f is an exterior algebra on generators of degrees 2 d 1 1 2dr l where l r a n k k but only in 1949 c t yen managed to compute the d for all exceptional groups case by case on the other hand c chevalley and j leray showed that the d i are nothing else but the degrees of the basic invariants over of the weyl group w a simple procedure for calculating the dl was given in the late fifties by a j coleman and by b kostant nowadays this is common knowledge the problem of computing the torsion in h k z proved to be more recalcitrant this was done by several authors c e miller settled so in 1953 23 a borel settled spin ad spin and the quotients of su and sp in 1954 6 p f baum and w browder settled the remaining classical group the semispin group in 1965 4 g 2 and f 4 were settled in 1954 6 and the p 3 torsion for e 6 and p 5tors ion for ev e s in 1961 7 by a borel s araki computed the 2 torsion for e 6 e 7 1 and the 3 torsion for e 7 e s 2 and announced the 2 torsion for e 8 in a joint paper with y shikata in 1961 3 the details of the proof for the last result have appeared only recently 18 there was an error in 1 in the case of ad e 7 which has been fixed in 13

new results on edge bandwidth the edge bandwidth problem is an analog of the classical bandwidth problem in which one has to label the edges of a graph by distinct integers such that the maximum difference of labels of any two incident edges is minimized we prove tight bounds on the edge bandwidth of hypercube and butterfly graphs and complete k ary trees which extends and improves on previous known results

regularity of refinable functions with exponentially decaying masks the regularity of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing geometric and numerical solutions of elliptic partial differential equations the purpose of this paper is to characterize the regularity of refinable functions with exponentially decaying masks and a dilation matrix whose eigenvalues have the same modulus the main results of this paper are really extensions of some results in cohen et al 1999 5 jia 1999 17 and lorentz and oswald 2000 28

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

optimization of integer wavelet transforms based on difference correlation structures in this paper a novel lifting integer wavelet transform based on difference correlation structure dccs liwt is proposed first we establish a relationship between the performance of a linear predictor and the difference correlations of an image the obtained results provide a theoretical foundation for the following construction of the optimal lifting filters then the optimal prediction lifting coefficients in the sense of least square prediction error are derived dccs liwt puts heavy emphasis on image inherent dependence a distinct feature of this method is the use of the variance normalized autocorrelation function of the difference image to construct a linear predictor and adapt the predictor to varying image sources the proposed scheme also allows respective calculations of the lifting filters for the horizontal and vertical orientations experimental evaluation shows that the proposed method produces better results than the other well known integer transforms for the lossless image compression

the chow ring of the classifying space bso 2n c we compute the chow ring of the classifying space bso 2n c in the sense of totaro using the fibration gl 2n so 2n bso 2n bgl 2n and a computation of the chow ring of gl 2n so 2n in a previous paper we find this chow ring is generated by chern classes and a characteristic class defined by edidin and graham which maps to 2n 1 times the euler class under the usual class map from the chow ring to ordinary cohomology moreover we show this class represents 1 2n 1 n 1 times the nth chern class of the representation of so 2n whose highest weight vector is twice that of the half spin representation

triangular 3 subdivision schemes the regular case the paper deals with the investigation of triangular 3 subdivision schemes in the stationary shift invariant setting in section 2 we collect the available theory on refinable functions subdivision surfaces with emphasis on their sobolev and holder smoothness families of interpolatory and approximating 3 subdivision schemes are investigated in section 3 some dual 3 subdivision schemes which are related to vector valued refinable functions are also analyzed for this purpose we have developed matlab routines for numerically investigating properties of vector subdivision schemes

optimal interpolatory subdivision schemes in multidimensional spaces we analyze the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces in particular we provide a general construction of bivariate interpolatory refinement masks such that the corresponding fundamental and refinable functions attain the optimal approximation order and smoothness order in addition these interpolatory refinement masks are minimally supported and enjoy full symmetry several examples are explicitly computed

the lifting scheme for wavelet bi frames theory structure and algorithm in this paper we present the lifting scheme of wavelet bi frames along with theory analysis structure and algorithm we show how any wavelet bi frame can be decomposed into a finite sequence of simple filtering steps this decomposition corresponds to a factorization of a polyphase matrix of a wavelet bi frame based on this concept we present a new idea for constructing wavelet bi frames for the construction of symmetric bi frames we use generalized bernstein basis functions which enable us to design symmetric prediction and update filters the construction allows more efficient implementation and provides tools for custom design of wavelet bi frames by combining the different designed filters for the prediction and update steps we can devise practically unlimited forms of wavelet bi frames moreover we present an algorithm of increasing the number of vanishing moments of bi framelets to arbitrary order via the presented lifting scheme which adopts an iterative algorithm and ensures the shortest lifting scheme several construction examples are given to illustrate the results

a simple method to build oversampled filter banks and tight frames this paper presents conditions under which the sampling lattice for a filter bank can be replaced without loss of perfect reconstruction this is the generalization of common knowledge that removing up downsampling will not lose perfect reconstruction the results provide a simple way of building over sampled filter banks if the original filter banks are orthogonal these oversampled banks construct tight frames of l2 z n when iterated as an example a quincunx lattice is used to replace the rectangular one of the standard wavelet transform this replacement leads to a tight frame that has a higher sampling in both time and frequency the frame transform is nearly shift invariant and has intermediate scales an application of the transform to image fusion is also presented

uniform powell sabin spline wavelets this paper discusses how the subdivision scheme for uniform powell sabin spline surfaces makes it possible to place those surfaces in a multiresolution context we first show that the basis functions are translates and dilates of one vector of scaling functions this defines a sequence of nested spaces we then use the subdivision scheme as the prediction step in the lifting scheme and add an update step to construct wavelets that describe a sequence of complement spaces finally as an example application we use the new wavelet transform to reduce noise on a uniform powell sabin spline surface

on the edge bandwidth of graph products the edge bandwidth of a graph g is the bandwidth of the line graph of g we show asymptotically tight bounds on the edge bandwidth of two dimensional grids and tori the product of two cliques and the n dimensional hypercube

multidimensional interpolatory subdivision schemes this paper presents a general construction of multidimensional interpolatory subdivision schemes in particular we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three direction box spline br r r of equal multiplicities the resulting mask for the interpolatory subdivision exhibits all the symmetries of the three direction box spline and with this increased symmetry comes increased smoothness several examples are computed for r 2 8 regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness

characterization of smoothness of multivariate refinable functions in sobolev spaces wavelets are generated from refinable functions by using multiresolution anlalysis in this paper we investigate the srloothness properties of multivariate refinable functionis in sobolev spaces we characterize the optinlal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace several examples are provided to illustrate the general theory

convergence of cascade algorithms in sobolev spaces and integrals of wavelets the cascade algorithm with mask a and dilation m generates a sequence phi n n 1 2 ldots by the iterative process

multiresolution analysis haar bases and self similar tilings of r sup n orthonormal bases for l sup 2 r sup n are constructed that have properties that are similar to those enjoyed by the classical haar basis for l sup 2 r for example each basis consists of appropriate dilates and translates of a finite collection of piecewise constant functions the construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self similar tilings

edge bandwidth of grids and tori the edge bandwidth of a graph g is the smallest number b for which there is a bijective labeling of e g with 1 e g such that the difference between the labels at any adjacent edges is at most b here we compute the edge bandwidth for rectangular grids b pm pn 2 min m n 1 if max m n 3 where is the cartesian product and pn denotes the path on n vertices this settles a conjecture of calamoneri et al new results on edge bandwidth theoret comput sci 307 2003 503 513 we also compute the edge bandwidth of any torus a product of two cycles within an additive error of 5

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

wavelet families of increasing order in arbitrary dimensions we build discrete time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments the associated scaling functions are interpolating our construction relies on the lifting scheme and inherits all of its advantages fast transform in place calculation and integer to integer transforms we show that two lifting steps suffice predict and update the predict step can be built using multivariate polynomial interpolation while update is a multiple of the adjoint of predict while we concentrate on the discrete time case some discussion of convergence and stability issues together with examples is given

two dimensional 2 times oversampled dft modulated filter banks and critically sampled modified dft modulated filter banks this paper investigates two dimensional 2d 2 oversampled dft modulated filter banks and 2d critically sampled modified dft mdft modulated filter banks as well as their design the structure and perfect reconstruction pr condition of 2d 2 oversampled dft modulated filter banks are presented in terms of the polyphase decompositions of prototype filters pfs in the double prototype case the part solutions of the pr condition are parameterized by imposing the 2d two channel lifting structure on each pair of the polyphase components of analysis and synthesis pfs based on the parametric structure the analysis and synthesis pfs are separately designed by constrained quadratic programs the obtained filter banks are of structurally pr moreover 2d critically sampled mdft modulated filter banks are proposed it is proved that 2d critically sampled pr mdft modulated filter banks can be rebuilt from 2d 2 oversampled pr dft modulated filter banks when the decimation matrices satisfy a permissible condition and the analysis and synthesis pfs are identical and symmetric with respect to the origin a numerical algorithm is given to design 2d critically sampled pr mdft modulated filter banks and the obtained filter banks are of numerically pr

approximation theory and functional analysis part 1 research and survey articles on bernstein durrineyer polynomials with jacob weights h berens and y xu an overview of wavelets c chui a note on weak inequalities in orlicz and lorentz spaces g a edgar and l sucheston bivariate birkhoff interpolation a survey r a lorentz linear approximations of functions with several restricted derivatives y makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle k pan and e b saff box splines cardinal series and wavelets s d reimenschneider and z shen some aspects of the subspace structure of infinite dimensional banach space h rosenthal fairness and monotone curvature j roulier et al projections on 2 dimensional spaces n tomczak jacgermann real versus complex best rational approximation r s varga and a ruttan

the toeplitz theorem and its applications to approximation theory and linear pdes we take an algebraic approach to the problem of approximation by dilated shifts of basis functions given a finite collection d of compactly supported functions in lp ris 1 0 where sh is the hdilate of s we prove that sh h 0 provides lp approximation order r only if s contains all the polynomials of total degree less than r in particular in the case where d consists of a single function p with its moment f p 0 we characterize the approximation order of sh h 0 by showing that the above condition on polynomial containment is also sufficient the above results on approximation order are obtained through a careful analysis of the structure of shift invariant spaces it is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients such a system then can be reduced to an infinite system of linear equations whose solvability is characterized by an old theorem of toeplitz thus the toeplitz theorem sheds light into approximation theory it is also used to give a very simple proof for the well known ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients

separable and nonseparable multiwavelets in multiple dimensions we report on a method of constructing multidimensional biorthogonal interpolating multiwavelets the approach is based upon polynomial interpolation in rd c de boor and a ron math comput 58 198 1997 and an extension of the lifting scheme j kovacevic and w sweldens ieee trans image process 9 no 3 480 2000 the constructed wavelets have compact support are nearly isotropic and retain partial scale invariance leading to a fast and efficient multidimensional wavelet transform we demonstrate an implementation for these wavelets of variable polynomial order up to four dimensions finally we show that these wavelets have a much sparser representation of discontinuous functions as compared to tensor product wavelets which allows for a more compact and efficient representation

adaptive lifting for shape based image retrieval we propose to use adaptive wavelet lifting for image retrieval systems that are based on shape detection and multiresolution structures of objects in a database against a background of texture to measure the performance of our approach feature vectors are computed based on moment invariants of detail coefficients produced by the adaptive lifting scheme and retrieval rates are obtained by measuring distances between these vectors retrieval rates are compared with the rates obtained when using non adaptive wavelet filtering as a preprocessing step a synthetic database is created for this simulation

design of signal adapted multidimensional lifting scheme for lossy coding this paper proposes a new method for the design of lifting filters to compute a multidimensional nonseparable wavelet transform our approach is stated in the general case and is illustrated for the 2 d separable and for the quincunx images results are shown for the jpeg2000 database and for satellite images acquired on a quincunx sampling grid the design of efficient quincunx filters is a difficult challenge which has already been addressed for specific cases our approach enables the design of less expensive filters adapted to the signal statistics to enhance the compression efficiency in a more general case it is based on a two step lifting scheme and joins the lifting theory with wiener s optimization the prediction step is designed in order to minimize the variance of the signal and the update step is designed in order to minimize a reconstruction error application for lossy compression shows the performances of the method

refinement of reproducing kernels we continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel hilbert space associated with the refinement kernel contains that with the original kernel as a subspace to motivate this study we first develop a refinement kernel method for learning which gives an efficient algorithm for updating a learning predictor several characterizations of refinement kernels are then presented it is shown that a nontrivial refinement kernel for a given kernel always exists if the input space has an infinite cardinal number refinement kernels for translation invariant kernels and hilbert schmidt kernels are investigated various concrete examples are provided

lagrange wavelets for signal processing this paper deals with the design of interpolating wavelets based on a variety of lagrange functions combined with novel signal processing techniques for digital imaging halfband lagrange wavelets b spline lagrange wavelets and gaussian lagrange lagrange distributed approximating functional daf wavelets are presented as specific examples of the generalized lagrange wavelets our approach combines the perceptually dependent visual group normalization vgn technique and a softer logic masking slm method these are utilized to rescale the wavelet coefficients remove perceptual redundancy and obtain good visual performance for digital image processing

compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas in this paper for any d d dilation matrix m we demonstrate in a constructive way that we can construct compactly supported tight m wavelet frames and orthonormal m wavelet bases in l2 rd of exponential decay which are derived from compactly supported m refinable functions such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments this paper improves several results in battle comm math phys 110 1987 601 bownik j fourier anal appl 7 2001 489 grochenig and ron proc amer math soc 126 1998 1101 lemarie j math pures appl 67 1988 227 and strichartz constr approx 9 1993 327

a group theoretic approach to multidimensional filter banks theory and applications in this paper we provide a new method for analyzing multidimensional filter banks this method enables us to solve various open problems in multidimensional filter bank characterization and design the essential element in this new approach is the redefinition of polyphase components it will be shown that a rich set of mathematical tools in particular algebraic group theory will become available for use in the analysis of filter banks we demonstrate the elegance and power of the tool set by employing it for the characterization of multidimensional filter banks and applying it to two open problems the first problem is concerned with the development of a general method to design multichannel spl ges 2 multidimensional filter banks using transformations while the second problem is concerned with the derivation of general restrictions on group delays in linear phase filter banks the treatment of these problems is only an illustration of the power of the tool set of algebraic group theory employed for the first time in the context of multidimensional filter banks

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

the chow ring of the symmetric space gl 2n c so 2n c abstract we show this chow ring is z z we do this by partitioning the space into 2n subvarieties each of which is fibered over gl 2 n 2 c so 2 n 2 c

multivariate matrix refinable functions with arbitrary matrix dilation characterizations of the stability and orthonormality of a multivariate matrix refinable function d with arbitrary matrix dilation m are provided in terms of the eigenvalue and 1 eigenvector properties of the restricted transition operator under mild conditions it is shown that the approximation order of d is equivalent to the order of the vanishing moment conditions of the matrix refinement mask pa the restricted transition operator associated with the matrix refinement mask pa is represented by a finite matrix ami j i j with a idet m i 1 e p j q3 p e and p j q3 p being the kronecker product of matrices p j and p the spectral properties of the transition operator are studied the sobolev regularity estimate of a matrix refinable function 4 is given in terms of the spectral radius of the restricted transition operator to an invariant subspace this estimate is analyzed in an example

reversible fast and high quality grid conversions a new grid conversion method is proposed to resample between two 2 d periodic lattices with the same sampling density the main feature of our approach is the symmetric reversibility which means that when using the same algorithm for the converse operation then the initial data is recovered exactly to that purpose we decompose the lattice conversion process into at most three successive shear operations the translations along the shear directions are implemented by 1 d fractional delay operators which revert to simple 1 d convolutions with appropriate filters that yield the property of symmetric reversibility we show that the method is fast and provides high quality resampled images applications of our approach can be found in various settings such as grid conversion between the hexagonal and the cartesian lattice or fast implementation of affine transformations such as rotations

flexible design of multidimensional perfect reconstruction fir 2 band filters using transformations of variables an approach to designing multidimensional linear phase fir diamond subband filters having the perfect reconstruction property is presented it is based on a transformation of variables technique and is equivalent to the generalized mcclellan transformation methods for designing a whole class of transformation are given the approach consists of two parts design of the transformation and design of the 1 d filters the use of lagrange halfband filters to design the 1 d filters is discussed the modification of a particular lagrange halfband filter which gives a pair of simple 1 d filters that are almost similar to each other in their frequency characteristics but still form a perfect reconstruction pair is presented the design technique is extended to other types of two channel sampling lattice and subband shapes in particular the parallelogram and the diagonally quadrant subband cases several numerical design examples are presented to illustrate the flexibility of the design method

m band biorthogonal interpolating wavelets via lifting scheme recently the lifting scheme was generalized to the multidimensional and multiband cases and was used to design m band interpolating scaling filters and their duals based on this idea we develop a new lifting pattern namely the progressive lifting pattern this pattern allows us to pairwise generate m band interpolating filterbanks and wavelets by the order from lowpass to highpass filters a complete lifting procedure is divided into m 1 simple steps in each step a pair of filters the l th filter and its dual are generated in this way an m band biorthogonal interpolating filterbank wavelet is determined by m m 1 lifting filters the first 2 m 1 lifting filters completely characterize the two scaling filters as well as the vanishing moments of bandpass and highpass filters the residual m 1 m 2 lifting filters are used to pairwise optimize the bandpass and highpass filters in terms of the criterion of stopband energy minimization the obtained m band biorthogonal interpolating filterbanks and wavelets provide excellent frequency characteristics in particular low stopband sidelobes furthermore the pattern is also utilized to design signal adapted interpolating filterbanks and their rational coefficient counterparts in terms of subband coding gain the obtained filterbanks achieve large subband coding gains the rational coefficient filterbanks preserve the biorthogonality and allow wavelet transforms from integers to integers and a unifying lossy lossless coding framework at the cost of a slight degradation in subband coding gain

index assignment for two channel quantization this paper concerns the design of a multiple description scalar quantization mdsq system for two identical channels for an unbounded discrete information source this translates to the combinatorial problem of finding an arrangement of the integers into the infinite plane square grid so that each row and each column contains exactly n numbers such that the difference between any two numbers in the same row or column is at most d with d to be minimized for a given n the best previous lower and upper bounds on the lowest d were n sup 2 3 o n and n sup 2 2 o n we give new lower and upper bounds both of the form 3n sup 2 8 o n we also consider minimizing the maximal variance in any row or column and show that it must be at least n sup 4 60 o n sup 3 and that it does not have to be more than 3n sup 4 160 o n sup 3

approximation properties of multivariate wavelets wavelets are generated from refinable functions by using multiresolution analysis in this paper we investigate the approximation properties of multivariate refinable functions we give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask we connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators finally we demonstrate that a refinable function in w k 1 1 r s provides approximation order k

lossless image compression based on optimal prediction adaptive lifting and conditional arithmetic coding the optimal predictors of a lifting scheme in the general n dimensional case are obtained and applied for the lossless compression of still images using first quincunx sampling and then simple row column sampling in each case the efficiency of the linear predictors is enhanced nonlinearly directional postprocessing is used in the quincunx case and adaptive length postprocessing in the row column case both methods are seen to perform well the resulting nonlinear interpolation schemes achieve extremely efficient image decorrelation we further investigate context modeling and adaptive arithmetic coding of wavelet coefficients in a lossless compression framework special attention is given to the modeling contexts and the adaptation of the arithmetic coder to the actual data experimental evaluation shows that the best of the resulting coders produces better results than other known algorithms for multiresolution based lossless image coding