on the asymptotic convergence of b spline wavelets to gabor functions a family of nonorthogonal polynomial spline wavelet transforms is considered these transforms are fully reversible and can be implemented efficiently the corresponding wavelet functions have a compact support it is proven that these b spline wavelets converge to gabor functions modulated gaussian pointwise and in all l sub p norms with 1

shift orthogonal wavelet bases shift orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation or shifts within one level but not with respect to dilations across scales we characterize these wavelets and investigate their main properties by considering two general construction methods in the first approach we start by specifying the analysis and synthesis function spaces and obtain the corresponding shift orthogonal basis functions by suitable orthogonalization in the second approach we take the complementary view and start from the digital filterbank we present several illustrative examples including a hybrid version of the battle lemarie 1987 1988 spline wavelets we also provide filterbank formulas for the fast wavelet algorithm a shift orthogonal wavelet transform is closely related to an orthogonal transform that uses the same primary scaling function both transforms have essentially the same approximation properties one experimentally confirmed benefit of relaxing the interscale orthogonality requirement is that we can design wavelets that decay faster than their orthogonal counterpart

wavelet decomposition of harmonizable random processes the discrete wavelet decomposition of second order harmonizable random processes is considered the deterministic wavelet decomposition of a complex exponential function is examined where its pointwise and bounded convergence to the function is proved this result is then used for establishing the stochastic wavelet decomposition of harmonizable processes the similarities and differences between the wavelet decompositions of general harmonizable processes and a subclass of processes having no spectral mass at zero frequency e g those that are wide sense stationary and have continuous power spectral densities are also investigated the relationships between the harmonization of a process and that of its wavelet decomposition are examined finally certain linear operations such as addition differentiation and linear filtering on stochastic wavelet decompositions are considered it is shown that certain linear operations can be performed term by term with the decomposition

analytical footprints compact representation of elementary singularities in wavelet bases we introduce a family of elementary singularities that are point holder regular these singularities are self similar and are the green functions of fractional derivative operators i e by suitable fractional differentiation one retrieves a dirac function at the exact location of the singularity we propose to use fractional operator like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain we show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed form expression termed the analytical footprint practically this means that the dictionary of wavelet footprints is embodied in a single analytical form we show that the wavelet coefficients of the nonredundant decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity we propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

on the shiftability of dual tree complex wavelet transforms the dual tree complex wavelet transform dt bbcwt is known to exhibit better shift invariance than the conventional discrete wavelet transform we propose an amplitude phase representation of the dt ibbc wt which among other things offers a direct explanation for the improvement in the shift invariance the representation is based on the shifting action of the group of fractional hilbert transform fht operators which extends the notion of arbitrary phase shifts from sinusoids to finite energy signals wavelets in particular in particular we characterize the shift ability of the dt bbcwt in terms of the shifting property of the fhts at the heart of the representation are certain fundamental invariances of the fht group namely that of translation dilation and norm which play a decisive role in establishing the key properties of the transform it turns out that these fundamental invariances are exclusive to this group next by introducing a generalization of the bedrosian theorem for the fht operator we derive an explicitly understanding of the shifting action of the fht for the particular family of wavelets obtained through the modulation of lowpass functions e g the shannon and gabor wavelet this in effect links the corresponding dual tree transform with the framework of windowed fourier analysis finally we extend these ideas to the multidimensional setting by introducing a directional extension of the fht the fractional directional hilbert transform in particular we derive a signal representation involving the superposition of direction selective wavelets with appropriate phase shifts which helps explain the improved shift invariance of the transform along certain preferential directions

a computational framework for a class of fully coupled models for elastoplastic damage at finite strains with reference to the linearization aspects abstract a fully coupled model for finite strain elastoplastic damage based on the multiplicative decomposition of the deformation gradient which incorporates non linear isotropic and kinematic hardening is described the numerical integration of its constitutive equations is performed by an algorithm which generalizes the standard return mapping procedures of the infinitesimal theory the exact consistent linearization of the model described is carried out in detail it provides the basis of the full newton raphson procedure for iterative solution of the associated equilibrium problem by means of finite element simulations the performance of the exact tangent modulus derived is compared with an approximate version and some practical implications are discussed

non smooth multisurface plasticity and viscoplasticity loading unloading conditions and numerical algorithms rate independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non smooth fashion are considered in detail it is shown that the standard kuhn tucker optimality conditions lead to the only computationally useful characterization of plastic loading on the computational side an unconditionally convergent return mapping algorithm is developed which places no restrictions aside from convexity on the functional forms of the yield condition flow rule and hardening law the proposed general purpose procedure is amenable to exact linearization leading to a closed form expression of the so called consistent algorithmic tangent moduli for viscoplasticity a closed form algorithm is developed based on the rate independent solution the methodology is applied to structural elements in which the elastic domain possesses a non smooth boundary numerical simulations are presented that illustrate the excellent performance of the algorithm

construction of coarse grained order parameters in nonequilibrium systems we develop a renormalization group rg procedure that includes important system specific features the key ingredient is to systematize the coarse graining procedure that generates the rg flow the coarse graining technology comes from the control and operator theoretic model reduction the resulting generalized rg is a consistent generalization of the wilsonian rg we apply the procedure to a deterministic nonlinear wave equation nlwe with probabilistic initial conditions we derive the form of the projection operator from the dynamics of the nlwe and then use it to generate the rg flow for the distribution of initial conditions the probability density of the initial conditions is chosen to be a boltzmann weight that is quartic in the field variables in our calculation we find that in contrast to conventional implementations of the rg naive power counting breaks down we also show that the resulting rg equations are different from those derived from the conventional rg

design of simple low order finite elements for large strain analysis of nearly incompressible solids a simple four node quadrilateral and an eight node hexahedron for large strain analysis of nearly incompressible solids are proposed based on the concept of deviatoric volumetric split and the replacement of the compatible deformation gradient with an assumed modified counterpart the formulation developed is applicable to arbitrary material models the closed form of the corresponding exact tangent stiffnesses which have a particularly simple structure is derived it ensures asymptotically quadratic rates of convergence of the newton raphson scheme employed in the solution of the implicit finite element equilibrium equations from a practical point of view the incorporation of the proposed elements into existing codes is straightforward it requires only small changes in the routines of the standard displacement based 4 node quadrilateral and 8 node brick a comprehensive set of numerical examples involving hyperelasticity as well as multiplicative elasto plasticity is provided it illustrates the performance of the proposed elements over a wide range of applications including strain localisation problems metal forming simulation and adaptive analysis

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

biorthogonal butterworth wavelets derived from discrete interpolatory splines we present a new family of biorthogonal wavelet transforms and a related library of biorthogonal periodic symmetric waveforms for the construction we used the interpolatory discrete splines which enabled us to design a library of perfect reconstruction filterbanks these filterbanks are related to butterworth filters the construction is performed in a lifting manner the difference from the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast fourier transform fft two ways to choose the control filters are suggested the proposed scheme is based on interpolation and as such it involves only samples of signals and it does not require any use of quadrature formulas these filters have a linear phase property and the basic waveforms are symmetric in addition these filters yield refined frequency resolution

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

aspects of the formulation and finite element implementation of large strain isotropic elasticity the paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation on the theoretical side an eulerian setting of isotropic elasticity is discussed exclusively in terms of the finger tensor as a strain measure noval aspects are a direct representation of the eulerian elastic moduli in terms of the finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts the isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the finger tensor a constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity on the computational side the implementation of the constitutive model in three possible finite element formulations is discussed after pointing out algorithmic techniques for the treatment of incompressible elasticity several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity

a phase cell cluster expansion for a hierarchical o 3 4 model the formalism developed in a previous paper is applied to yield a phase cell cluster expansion for a hierarchical b model the field is expanded into modes with specific renormatization group scaling properties the present cluster expansion for a vacuum expectation value is formally the natural factorization of each term in the perturbation expansion into the contribution of modes connected to the variables in the expectation via interactions and that of the complementary set the expectation value is thus realized as a sum of contributions due to finite subsets of the modes we emphasize the following additional features 1 partitions of unity are not used 2 there are essentially no cut offs 3 the expansion is developed directly without an initial need to prove an ultraviolet stability bound the most difficult part of the traditional approach our main interest in the present phase cell cluster expansion is founded in the belief that it may be the right vehicle for proving the existence of a non trivial four dimensional field theory

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

a hyperelastic based large strain elasto plastic constitutive formulation with combined isotropic kinematic hardening using the logarithmic stress and strain measures summary this paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity the combined isotropic kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions i the formulation is hyperelastic based ii the stress strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the hencky strain tensor and its elastic work conjugate stress tensor and iii the multiplicative decomposition of the deformation gradient is adopted since no stress rates are present the formulation is of course numerically objective in the time integration it is shown that the model gives adequate physical behaviour and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor

wavelets and recursive filter banks it is shown that infinite impulse response iir filters lead to more general wavelets of infinite support than finite impulse response fir filters a complete constructive method that yields all orthogonal two channel filter banks where the filters have rational transfer functions is given and it is shown how these can be used to generate orthonormal wavelet bases a family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband butterworth filters when there is an odd number of zeros at pi it is shown that closed forms for the filters are available without need for factorization a still larger class of orthonormal wavelet bases having the same moment properties and containing the daubechies and butterworth filters as the limiting cases is presented it is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case and a constructive method is given it is also shown how compactly supported bases may be orthogonalized and bases for the spline function spaces are constructed

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline 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 n a self contained accessible fashion in particular we prove that the b spline 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 the l sub p sense and a sharper theorem stating that smoothness implies order

on the computation of battle lemarie s wavelets we propose a matrix approach to the computation of battle lemarie s wavelets the fourier transform of the scaling function is the prod uct of the inverse f x of a square root of a positive trigonometric polynomial and the fourier transform of a b spline of order m the polynomial is the symbol of a bi infinite matrix b associated with a b spline of order 2m we approximate this bi infinite matrix b2m by its finite section as a square ma trix of finite order we use as to compute an approximation s of x whose discrete fourier transform is f x we show that xs converges pointwise to x exponentially fast this gives a feasible method to compute the scaling func tion for any given tolerance similarly this method can be used to compute the wavelets

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

wavelets in wandering subspaces mallat s construction via a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions the connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in hilbert space is exploited in the general setting an example of multiresolution approximation generated by cardinal hermite b splines is constructed

pointwise convergence of wavelet expansions in this note we announce that under general hypotheses wavelet type ex pansions of functions in lp p in one or more dimensions converge pointwise almost everywhere and identify the lebesgue set of a function as a set of full measure on which they converge it is shown that unlike the fourier sum mation kernel wavelet summation kernels pj are bounded by radial decreasing l convolution kernels as a corollary it follows that best l spline approxi mations on uniform meshes converge pointwise almost everywhere moreover summation of wavelet expansions is partially insensitive to order of summation we also give necessary and su cient conditions for given rates of conver gence of wavelet expansions in the sup norm such expansions have order of convergence s if and only if the basic wavelet is in the homogeneous sobolev space h s d h we also present equivalent necessary and su cient conditions on the scaling function the above results hold in one and in multiple dimen sions

block renormalization group for euclidean fermions block renormalization group transformations rgt for lattice and continuum euclidean fermions in d dimensions are developed using fermionic integrals with exponential and function weight functions for the free field the sequence of actionsdk generated by the rgt from d the dirac operator are shown to have exponential decay uniform ink after rescaling to the unit lattice it is shown that the two point functiond 1 admits a simple telescopic sum decomposition into fluctuation two point functions which for the exponential weight rgt have exponential decay contrary to rg intuition the sequence of rescaled actions corresponding to the function rgt do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight rgt also

noise covariance properties in dual tree wavelet decompositions dual tree wavelet decompositions have recently gained much popularity mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy when the decomposition of a random process is performed which occurs in particular when an additive noise is corrupting the signal to be analyzed it is useful to characterize the statistical properties of the dual tree wavelet coefficients of this process as dual tree decompositions constitute over complete frame expansions correlation structures are introduced among the coefficients even when a white noise is analyzed in this paper we show that it is possible to provide an accurate description of the covariance properties of the dual tree coefficients of a wide sense stationary process the expressions of the cross covariance sequences of the coefficients are derived in the one and two dimensional cases asymptotic results are also provided allowing to predict the behavior of the second order moments for large lag values or at coarse resolution in addition the cross correlations between the primal and dual wavelets which play a primary role in our theoretical analysis are calculated for a number of classical wavelet families simulation results are finally provided to validate these results

rational filter wavelets abstract in this paper we introduce two types of rational filters for the construction of wavelets the rational filters are a natural development of the polynomial filters we use rational filters to derive a large family of the wavelets which can include daubechies wavelets and battle lemarie wavelets especially the ii type rational filter wavelets among the family have linear phases furthermore we analyze the regularity of the rational filter wavelets and estimate their regularity indices some examples are also given

wavelet packets of fractional brownian motion asymptotic analysis and spectrum estimation this paper provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional brownian motion it also discusses the convergence speed to the limit autocorrelation function when the input random process is either a fractional brownian motion or a wide sense stationary second order random process the analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the shannon paraunitary filters from this analysis we derive wavelet packet based spectrum estimation for fractional brownian motions and wide sense stationary random processes experimental tests show good results for estimating the spectrum of 1 f processes

periodic quasi orthogonal spline bases and applications to least squares curve fitting of digital images presents a new covariant basis dubbed the quasi orthogonal q spline basis for the space of n degree periodic uniform splines with k knots this basis is obtained analogously to the b spline basis by scaling and periodically translating a single spline function of bounded support the construction hinges on an important theorem involving the asymptotic behavior in the dimension of the inverse of banded toeplitz matrices the authors show that the gram matrix for this basis is nearly diagonal hence the name quasi orthogonal the new basis is applied to the problem of approximating closed digital curves in 2d images by least squares fitting since the new spline basis is almost orthogonal the least squares solution can be approximated by decimating a convolution between a resolution dependent kernel and the given data the approximating curve is expressed as a linear combination of the new spline functions and new control points another convolution maps these control points to the classical b spline control points a generalization of the result has relevance to the solution of regularized fitting problems

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

lattice and continuum wavelets and the block renormalization group the authors obtain a resolution of the identity operator for functions on a lattice var epsilon z sup d which is derived from the block renormalization group the authors use eigenfunctions of the terms of the decomposition to form a basis for l sub 2 var epsilon z sup d and show how the basis is generated from lattice wavelets the lattice spacing var epsilon is taken to zero and continuum wavelets are obtained 12 refs

introduction to wavelets in engineering the aim of this paper is to provide an introduction to the subject of wavelet analysis for engineering applications the paper selects from the recent mathematical literature on wavelets the results necessary to develop wavelet based numerical algorithms in particular we provide extensive details of the derivation of mallat s transform and daubechies wavelet coefficients since these are fundamental to gaining an insight into the properties of wavelets the potential benefits of using wavelets are highlighted by presenting results of our research in one and two dimensional data analysis and in wavelet solutions of partial differential equations

wavelet packet division multiplexing and wavelet packet design under timing error effects wavelet packet division multiplexing wpdm is a multiple signal transmission technique in which the message signals are waveform coded onto wavelet packet basis functions for transmission the overlapping nature of such waveforms in time and frequency provides a capacity improvement over the commonly used frequency division multiplexing fdm and time division multiplexing tdm schemes while their orthogonality properties ensure that the overlapping message signals can be separated by a simple correlator receiver the interference caused by timing offset in transmission is examined a design procedure that exploits the inherent degrees of freedom in the wpdm structure to mitigate the effects of timing error is introduced and a waveform that minimizes the energy of the timing error interference is designed an expression for the probability of error due to the presence of gaussian noise and timing error for the transmission of binary data is derived the performance advantages of the designed waveform over standard wavelet packet basis functions are demonstrated by both analytical and simulation methods the capacity improvement of wpdm its simple implementation and the possibility of having optimum waveform designs indicate that wpdm holds considerable promise as a multiple signal transmission technique

from differential equations to the construction of new wavelet like bases in this paper an approach is introduced based on differential operators to construct wavelet like basis functions given a differential operator l with rational transfer function elementary building blocks are obtained that are shifted replicates of the green s function of l it is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential b spline representation the corresponding b splines are entirely specified by their poles and zeros they are compactly supported have an explicit analytical form and generate multiresolution riesz bases moreover they satisfy generalized refinement equations with a scale dependent filter and lead to a representation that is dense in l sub 2 this allows us to specify a corresponding family of semi orthogonal exponential spline wavelets which provides a major extension of earlier polynomial spline constructions these wavelets are completely characterized and it is proven that they satisfy the following remarkable properties 1 they are orthogonal across scales and generate riesz bases at each resolution level 2 they yield unconditional bases of l sub 2 either compactly supported b spline type or with exponential decay orthogonal or dual type 3 they have n vanishing exponential moments where n is the order of the differential operator 4 they behave like multiresolution versions of the operator l from which they are derived and 5 their order of approximation is n m where n and m give the number of poles and zeros respectively last but not least the new wavelet like decompositions are as computationally efficient as the classical ones they are computed using an adapted version of mallat s filter bank algorithm where the filters depend on the decomposition level

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

wspm wavelet based statistical parametric mapping recently we have introduced an integrated framework that combines wavelet based processing with statistical testing in the spatial domain in this paper we propose two important enhancements of the framework first we revisit the underlying paradigm i e that the effect of the wavelet processing can be considered as an adaptive denoising step to improve the parameter map followed by a statistical detection procedure that takes into account the non linear processing of the data with an appropriate modification of the framework we show that it is possible to reduce the bias of the method with respect to the best linear estimate providing conservative results that are closer to the original data second we propose an extension of our earlier technique that compensates for the lack of shift invariance of the wavelet transform we demonstrate experimentally that both enhancements have a positive effect on performance in particular we present a reproducibility study for multi session data that compares wspm against spm with different amounts of smoothing the full approach is available as a toolbox named wspm for the spm2 software it takes advantage of multiple options and features of spm such as the general linear model

coherent structures in random media and wavelets abstract we construct low dimensional dynamical models for the motion of coherent structures such as those frequently observed in extended turbulent flows these models are derived from the wavelet based galerkin projection of the pde describing the flow and account for the local interaction of a small number of coherent structures we show that under rather general assumptions the wavelet projection is close to the proper orthogonal decomposition in the average energy sense in the specific case of the 1d kuramoto sivashinsky equation we show numerically that the dynamics of our model agrees qualitatively with that of the original ks equation

ondelettes and phase cell cluster expansions a vindication y meyer has recently developed a particularly useful o n basis forl2 r d expansions using this basis i e expansions into ondelettes or wavelets have yielded important new results in soft and hard analysis the expansion into ondelettes of a boson scalar field naturally leads to phase cell cluster expansions a formalism already developed by the authors using other related bases adoption of ondelettes expansions into the phase cell program gives improvements of some extant results and excises an early error

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

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

a fourier domain formula for the least squares projection of a function onto a repetitive basis in n dimensional space a theorem concerning the least squares projection of an arbitrary function onto an infinite basis of translated function is given the theorem provides an explicit formula for the fourier transform of the projected function the formula has the advantage of being valid for least squares projections in any n dimensional space the expression for the projected function can be approximately inverted using the discrete fourier transform to find the actual basis coefficients

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

a linear wavelet filter for parametric imaging with dynamic pet describes a new filter for parametric images obtained from dynamic positron emission tomography pet studies the filter is based on the wavelet transform following the heuristics of a previously published method that are here developed into a rigorous theoretical framework it is shown that the space time problem of modeling a dynamic pet sequence reduces to the classical one of estimation of a normal multivariate vector of independent wavelet coefficients that under least squares risk can be solved by straightforward application of well established theory from the study of the distribution of wavelet coefficients of pet images it is inferred that a james stein linear estimator is more suitable for the problem than traditional nonlinear procedures that are incorporated in standard wavelet filters this is confirmed by the superior performance of the james stein filter in simulation studies compared to a state of the art nonlinear wavelet filter and a nonstationary filter selected from literature finally the formal framework is interpreted for the practitioner s point of view and advantages and limitations of the method are discussed

determining significant connectivity by 4d spatiotemporal wavelet packet resampling of functional neuroimaging data an active area of neuroimaging research involves examining functional relationships between spatially remote brain regions when determining whether two brain regions exhibit significant correlation due to true functional connectivity one must account for the background spatial correlation inherent in neuroimaging data we define background correlation as spatiotemporal correlation in the data caused by factors other than neurophysiologically based functional associations such as scanner induced correlations and image preprocessing we develop a 4d spatiotemporal wavelet packet resampling method which generates surrogate data that preserves only the average background spatial correlation within an axial slice across axial slices and through each voxel time series while excluding the specific correlations due to true functional relationships we also extend an amplitude adjustment algorithm which adjusts our surrogate data to closely match the amplitude distribution of the original data our method improves upon existing wavelet based methods and extends them to 4d we apply our resampling technique to determine significant functional connectivity from resting state and motor task fmri datasets

phase space localization theorem for ondelettes it is proven that an orthonormal basis of ondelettes can never have exponential localization in both position space and momentum space

pointwise error bounds for orthogonal cardinal spline approximation for orthogonal cardinal spline approximation closed form expressions of the reproducing kernel and the peano kernels in terms of exponential splines are proved concrete and sharp pointwise error bounds are deduced for low degree splines

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

klein gordon propagation of daubechies wavelets scale commensurate decay estimates inside the light cone for the solution of the cauchy problem for the klein gordon equation with a daubechies wavelet as initial data were obtained similar estimates on the solution of the radiation problem with a space time daubechies wavelet as the source were derived

quantitative fourier analysis of approximation techniques ii wavelets for pt i see ibid vol 47 no 10 p 2783 95 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 two scale 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 least squares approximation error another contribution is the application of these results to b splines and daubechies 1988 1992 scaling functions which yields explicit asymptotic developments and upper bounds thanks to these explicit expressions we can quantify the improvement that can be obtained by using b splines instead of daubechies wavelets in other words we can use a coarser spline sampling and achieve the same reconstruction accuracy as daubechies specifically we show that this sampling gain converges to spl pi as the order tends to infinity

local reproducible smoothing without shrinkage a simple local smoothing filter is defined for curves or surfaces combining the advantages of gaussian smoothing and fourier curve description unlike gaussian filters the filter described has no shrinkage problem repeated application of the filter does not yield a curve or surface smaller than the original but simply reproduces the approximate result that would have been obtained by a single application at the largest scale unlike fourier description the filter is local in space the wavelet transform of y meyer 1989 is also shown to have these properties

denoising methods for thermomechanical decomposition for quasi equilibrium molecular dynamics simulations in this work we applied robust denoising methods well established in the signal processing field for the thermomechanical decomposition of velocity data obtained from molecular dynamics md simulations in the decomposition the atomic velocity was assumed to be the sum of the mechanical velocity and the thermal velocity which can be linked to the stress and temperature field at the continuum scale respectively for the quasi equilibrium process with the thermal velocity treated as the gaussian distributed stationary noise with zero mean and a positive variance that is linearly proportional to temperature and mechanical velocity as the clean signal the velocity decomposition can be recasted into a denoising problem for which powerful denoising methods have been developed to estimate a clean signal from noisy data we investigated the widely used linear parametric real domain linear nonparametric fourier based and nonlinear nonparametric wavelet based denoising methods first on their theoretical properties and then made comparsion among them for denosing some synthetic noisy 1 dimensional 1d data generated from md simulations the nonlinear wavelet based thresholding estimator possessed better optimality properties than the other estimators and also outperformed the other estimators in the synthetic data test a further test comparing the various denoising methods for an adiabatic shear crack nucleation and propagation process simulated using md simulations showed better performance by the wavelet based denoisng method results from this work reveal good potential of applying wavelet based denoising method to the study of thermomechanical processes simulated using md simulations

a new family of spline based biorthogonal wavelet transforms and their application to image compression in this paper we design a new family of biorthogonal wavelet transforms and describe their applications to still image compression the wavelet transforms are constructed from various types of interpolatory and quasiinterpolatory splines the transforms use finite impulse response and infinite impulse response filters that are implemented in a fast lifting mode

localisation property of battle lemarie wavelets sums abstract explicit formulae are given for a type of battle lemarie scaling functions and related wavelets compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to nikolskii besov spaces on r

a new computational model for tresca plasticity at finite strains with an optimal parametrization in the principal space abstract a new computational model for the rate independent elasto plastic solids characterized by yield surfaces containing singularities and general nonlinear isotropic hardening is presented within the context of fully implicit return mapping algorithms a numerical scheme for integration of the constitutive equations is formulated in the space of principal stresses as a direct consequence of the principal stress approach the representation of a yield surface is cast in terms of optimal parameterization which for the tresca yield criterion takes a simple linear form the associated return mapping equations then reduce to a remarkably simple format in addition due to assumed isotropy of the models the associated algorithmic incremental constitutive functionals can be identified as particular members of a class of isotropic tensor functions of one tensor in which the function eigenvalues are expressed in terms of the eigenvalues of the argument this observation leads to a simple closed form derivation of the consistent tangent moduli associated with the described integration algorithms the extension of the present model to finite strains is carried out following standard multiplicative plasticity described in terms of logarithmic stretches and exponential approximation to the flow rule the efficiency and robustness of the computational model are illustrated on a range of numerical examples

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed