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

uniform positivity improving property sobolev inequalities and spectral gaps abstract we introduce a certain kind of strong ergodicity condition to study the existence of spectral gap for markov generator we can estimate the spectral gap using the ergodicity condition and a sobolev type inequality we apply our results to the dirichlet form on wiener spaces riemannian manifolds and loop spaces

on the approximation power of convolution based least squares versus interpolation there are many signal processing tasks for which convolution based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine based methods the coefficients of the corresponding signal approximations are typically obtained by direct sampling interpolation or quasi interpolation or by using least squares techniques that apply a prefilter prior to sampling we compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h specifically we review several results in approximation theory with a special emphasis on the strang fix 1971 conditions which relate the general o h sup l behavior of the error to the ability of the representation to reproduce polynomials of degree n l 1 we use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the l sub 2 error as h tends to zero we also propose a new improved l sub 2 error bound for the least squares case in the process we provide all the relevant bound constants for polynomial splines some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order we present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

mathematical properties of the jpeg2000 wavelet filters the legall 5 3 and daubechies 9 7 filters have risen to special prominence because they were selected for inclusion in the jpeg2000 standard we determine their key mathematical features riesz bounds order of approximation and regularity holder and sobolev we give approximation theoretic quantities such as the asymptotic constant for the l sup 2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection we also derive new asymptotic error formulae that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet the daubechies 9 7 stands out because it is very close to orthonormal but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments

3 d shape estimation of dna molecules from stereo cryo electron micro graphs using a projection steerable snake we introduce a three dimensional 3 d parametric active contour algorithm for the shape estimation of dna molecules from stereo cryo electron micrographs we estimate the shape by matching the projections of a 3 d global shape model with the micrographs we choose the global model as a 3 d filament with a b spline skeleton and a specified radial profile the active contour algorithm iteratively updates the b spline coefficients which requires us to evaluate the projections and match them with the micrographs at every iteration since the evaluation of the projections of the global model is computationally expensive we propose a fast algorithm based on locally approximating it by elongated blob like templates we introduce the concept of projection steerability and derive a projection steerable elongated template since the two dimensional projections of such a blob at any 3 d orientation can be expressed as a linear combination of a few basis functions matching the projections of such a 3 d template involves evaluating a weighted sum of inner products between the basis functions and the micrographs the weights are simple functions of the 3 d orientation and the inner products are evaluated efficiently by separable filtering we choose an internal energy term that penalizes the average curvature magnitude since the exact length of the dna molecule is known a priori we introduce a constraint energy term that forces the curve to have this specified length the sum of these energies along with the image energy derived from the matching process is minimized using the conjugate gradients algorithm we validate the algorithm using real as well as simulated data and show that it performs well

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

coercive inequalities for gibbs measures abstract we prove the generalized nash and logarithmic nash inequalities for gibbs measures with dirichlet form associated to the kawasaki dynamics

robust dimension free isoperimetry in gaussian space we prove the first robust dimension free isoperimetric result for the standard gaussian measure n and the corresponding boundary measure n in rn the main result in the theory of gaussian isoperimetry proven in the 1970s by sudakov and tsirelson and independently by borell states that if n a 1 2 then the surface area of a is bounded by the surface area of a half space with the same measure n a 2 1 2 our results imply in particular that if a rn satisfies n a 1 2 and n a 2 1 2 then there exists a half space b rn such that n a b clog 1 2 1 for an absolute constant c since the gaussian isoperimetric result was established only recently a robust version of the gaussian isoperimetric result was obtained by cianchi et al who showed that n a b c n for some function c n with no effective bounds compared to the results of cianchi et al our results have optimal i e no dependence on the dimension but worse dependence on

fresnelets new multiresolution wavelet bases for digital holography we propose a construction of new wavelet like bases that are well suited for the reconstruction and processing of optically generated fresnel holograms recorded on ccd arrays the starting point is a wavelet basis of l sub 2 to which we apply a unitary fresnel transform the transformed basis functions are shift invariant on a level by level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the fresnel domain we derive a heisenberg like uncertainty relation that relates the localization of fresnelets with that of their associated wavelet basis according to this criterion the optimal functions for digital hologram processing turn out to be gabor 1948 functions bringing together two separate aspects of the holography inventor s work we give the explicit expression of orthogonal and semi orthogonal fresnelet bases corresponding to polynomial spline wavelets this special choice of fresnelets is motivated by their near optimal localization properties and their approximation characteristics we then present an efficient multiresolution fresnel transform algorithm the fresnelet transform this algorithm allows for the reconstruction backpropagation of complex scalar waves at several user defined wavelength independent resolutions furthermore when reconstructing numerical holograms the subband decomposition of the fresnelet transform naturally separates the image to reconstruct from the unwanted zero order and twin image terms this greatly facilitates their suppression we show results of experiments carried out on both synthetic simulated data sets as well as on digitally acquired holograms

embedding of tori and grids into twisted cubes the hypercube is one of the most popular interconnection networks since it has a simple structure and is easy to implement an n dimensional twisted cube tq n is an important variation of hypercube q n and preserves many of its desirable properties the problem of how to embed a family of disjoint meshes or tori into a host graph has attracted great attention in recent years however there is no systematic method proposed to generate the desired meshes and tori in tq n in this paper we develop two systematic linear time algorithms for embedding disjoint multi dimensional tori into tq n n 7 as follows 1 for a positive integer m with email protected email protected email protected n 4 a family of 2 m disjoint k dimensional tori of size 2 s 1x2 s 2x x2 s k each can be embedded with unit dilation where k 2 and i 1 ks email protected n m and 2 for a positive integer m with email protected email protected n 5 a family of 2 m disjoint k dimensional tori of size 2 s 1x2 s 2x x2 s k each can be embedded with unit dilation where k 2 s i 2 i 1 ks email protected n m and max 1 i k s i n 2m moreover we also provide similar embedding results for meshes and hypercubes our results mean that a family of torus structured mesh structured or hypercube structured parallel algorithms can be executed on the same twisted cube efficiently and in parallel

asymptotic error expansion of wavelet approximations of smooth functions ii we generalize earlier results concerning an asymptotic error expansion of wavelet approximations the properties of the monowavelets which are the building blocks for the error expansion are studied in more detail and connections between spline wavelets and euler and bernoulli polynomials are pointed out the expansion is used to compare the error for different wavelet families we prove that the leading terms of the expansion only depend on the multiresolution subspaces v j and not on how the complementary subspaces w j are chosen consequently for a fixed set of subspaces v j the leading terms do not depend on the fact whether the wavelets are orthogonal or not we also show that daubechies orthogonal wavelets need in general one level more than spline wavelets to obtain an approximation with a prescribed accuracy these results are illustrated with numerical examples

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

analysis and geometry on configuration spaces the gibbsian case using a natural riemannian geometry like structure on the configuration space over rd we prove that for a large class of potentials the corresponding canonical gibbs measures on 1 can be completely characterized by an integration by parts formula that is if is the gradient of the riemannian structure on one can define a corresponding divergence div such that the canonical gibbs measures are exactly those measures for which and div are dual operators onl2 one consequence is that for such the corresponding dirichlet forms e are defined in addition each of them is shown to be associated with a conservative diffusion process on with invariant measure the corresponding generators are extensions of the operator div the diffusions can be characterized in terms of a martingale problem and they can be considered as a brownian motion on perturbed by a singular drift another main result of this paper is the following if is a canonical gibbs measure then it is extreme or a pure phase if and only if the corresponding weak sobolev spacew1 2 on is irreducible as a consequence we prove that for extreme canonical gibbs measures the above mentioned diffusions are time ergodic in particular this holds for tempered grand canonical gibbs measures ruelle measures provided that the activity constant is small enough we also include a complete discussion of the free case i e 0 where the underlying space rdis even replaced by a riemannian manifoldx

extremal properties of half spaces for log concave distributions n w x n where m s m m is the product measure in r d s y1 1 is the n n dimensional cube in r n and the infimum is then over all borel measurable n n sets a r of measure m a g p 0 p 1 h 0 we are searching for necessary and sufficent conditions under which the n 4 infimum in 1 1 is attained at the half spaces a s x g r x f c of 1 measure p for all p and h the half spaces of this form will be called standard in these notes we give such conditions in the case where m is log concave there exists the following probabilistic consequence of the above property if z z are independent random variables on some probability space 1 n v i p with common distribution m consider a family of random variables s t indexed by a set t and formed by linear combinations n s t s a t z i i is1 where the coefficients a are arbitrary functions on t such that i n s s sup a t f 1 1 i t is1 set m s sup s t then the extremal property of the standard half spaces in t

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

least squares image resizing using finite differences we present an optimal spline based algorithm for the enlargement or reduction of digital images with arbitrary noninteger scaling factors this projection based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are b splines of any degree n a noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a for a given choice of basis functions the results of our method are consistently better than those of the standard interpolation procedure the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal to noise ratio the method can be generalized to include other classes of piecewise polynomial functions expressed as linear combinations of b splines and their derivatives

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

estimation of spectral gap for elliptic operators a variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time the main known results are either recovered or improved a large number of new examples with sharp estimate are illustrated moreover as an application of the march coupling the poincare inequality with respect to the absolute distribution of the process is also studied

logarithmic sobolev inequalities and exponential integrability we prove the exponential integrability of lipschitz functions in abstract wiener spaces by using logarithmic sobolev inequalities we introduce two notions of lipschitz continuity and apply the method of davis and simon j funct anal 59 1984 335 395 to prove our main theorem moreover we give several conditions under which the function is exponentially integrable

uniqueness of generalized schrodinger operators and applications abstract we prove necessary and sufficient conditions on r d r 0 a e so that the operator s 2 1 w1 n has exactly one self adjoint extension on w1 p which generates a sub markovian semigroup on w1 n these conditions are shown to be always fulfilled if d 1 and are verified for a large class of functions if d 1 we also prove an infinite dimensional analogue of this result where r d is replaced by some topological vector space e and the lebesgue measure dx by a gaussian measure the role of is taken by some ornstein uhlenbeck operator with general linear drift and dom s bounded smooth cylinder functions both results are derived as special cases from a general perturbation result we also give the corresponding probabilistic version i e we prove uniqueness for the associated martingale problem we present in some detail applications to the case where is a ground state of a hamiltonian in quantum field theory settling a problem which has been open for some time furthermore we apply our results to prove uniqueness in distribution of the diffusion first constructed by g jona lasinio and p k mitter for the stochastic quantization of f 2 field theory

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator we establish by simple semigroup arguments a levy gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the gaussian measure this produces in particular a new proof of the gaussian isoperimetric inequality this isoperimetric inequality strengthens the classical logarithmic sobolev inequality in this context a local version for the heat kernel measures is also proved which may then be extended into an isoperimetric inequality for the wiener measure on the paths of a riemannian manifold with bounded ricci curvature

approximation by translates of refinable functions the functions f 1 x dots f r x are refinable if they are combinations of the rescaled and translated functions f i 2x k this is very common in scientific computing on a regular mesh the space v 0 of approximating functions with meshwidth h 1 is a subspace of v 1 with meshwidth h 1 2 these refinable spaces have refinable basis functions the accuracy of the computations depends on p the order of approximation which is determined by the degree of polynomials 1 x dots x p 1 that lie in v 0 most refinable functions such as scaling functions in the theory of wavelets have no simple formulas the functions f i x are known only through the coefficients c k in the refinement equation scalars in the traditional case r times r matrices for multiwavelets the scalar sum rules that determine p are well known we find the conditions on the matrices c k that yield approximation of order p from v 0 these are equivalent to the strang fix conditions on the fourier transforms hat f i omega but for refinable functions they can be explicitly verified from the c k

l_2 rates of convergence for attractive reversible nearest particle systems the critical case reversible nearest particle systems are certain one dimensional interacting particle systems whose transition rates are determined by a probability density beta n with finite mean on the positive integers the reversible measure for such a system is the distribution nu of the stationary renewal process corresponding to this density in an earlier paper we proved under certain regularity conditions that the system converges exponentially rapidly in l 2 nu if and only if the system is supercritical this in turn is equivalent to beta n having exponential tails in this paper we consider the critical case and give moment conditions on beta n which are separately necessary and sufficient for the convergence of the process in l 2 nu at a specified algebraic rate in order to do so we develop conditions on the generator of a general markov process which correspond to algebraic l 2 convergence of the process the use of these conditions is also illustrated in the context of birth and death chains on the positive integers

bipanconnectivity of faulty hypercubes with minimum degree abstract in this paper we consider the conditionally faulty hypercube q n with n 2 where each vertex of q n is incident with at least m fault free edges 2 m n 1 we shall generalize the limitation m 2 in all previous results of edge bipancyclicity we also propose a new edge fault tolerant bipanconnectivity called k edge fault tolerant bipanconnectivity a bipartite graph is k edge fault tolerant bipanconnected if g f remains bipanconnected for any f e g with f k for every integer m under the same hypothesis we show that q n is n 2 edge fault tolerant edge bipancyclic and bipanconnected and the results are optimal with respect to the number of edge faults tolerated this not only improves some known results on edge bipancyclicity and bipanconnectivity of hypercubes but also simplifies the proof

approximation error of shifted signals in spline spaces spline signal spaces offer several advantages for the representation of signals compared with the more traditional signal spaces of bandlimited signals among them are the finite support of b splines simple manipulations like differentiation and integration etc a major disadvantage however is that spline signal spaces are not closed under signal shifts in order to assess the approximation error introduced by shifting a spline signal the approximation error norm and its average are evaluated furthermore an upper bound on the expected normalized approximation error is derived using reid s inequality

multiple regular graph embeddings into a hypercube with unbounded expansion as topological structures interconnection networks play an important role in parallel and distributed computer systems particularly in multicomputer systems which provide an effective mechanism for exchanging data between processors in this paper we study the node fault tolerant capability of an n dimensional hypercube with respect to multiple regular graph embeddings into a hypercube with unbounded expansion we present a fault tolerant method for multiple regular graph embeddings into a hypercube with dilation 3 congestion 1 and load 1 these results show that o n email protected log email protected 2 faults can be tolerated where n is the number of dimensions in a hypercube and l is the number of the nodes of the regular graph the presented embedding methods are mainly optimized for balancing the processor loads while minimizing load and congestion as far as possible furthermore our method expands on some previously known findings also this technology can be applied in grid computing and cloud computing

embedding meshes into twisted cubes the n dimensional twisted cube tn n is a variation of the hypercube in this paper we study embedding of meshes into tn n we prove three major results in this paper 1 for any integer n 1 a 2x2 n 1 mesh can be embedded into tn n with dilation 1 and expansion 1 2 for any integer n 4 an mxk m 3 k 3 mesh cannot be embedded into tn n with dilation 1 3 for any integer n 4 two node disjoint 4x2 n 3 meshes can be embedded into tn n with dilation 2 and expansion 1

embedding a hamiltonian cycle in the crossed cube with two required vertices in the fixed positions abstract a hamiltonian graph g is said to be panpositionably hamiltonian if for any two distinct vertices x and y of g there is a hamiltonian cycle c of g having dc x y l for any integer l satisfying d g x y l v g 2 where dg x y respectively dc x y denotes the distance between vertices x and y in g respectively c and v g denotes the total number of vertices of g as the importance of hamiltonian properties for data communication among units in an interconnected system the panpositionable hamiltonicity involves more flexible message transmission in this paper we study this property with respect to the class of crossed cubes which is a popular variant of the hypercube network

efficient energies and algorithms for parametric snakes parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity however they have a few drawbacks which limit their performance in this paper we identify some of these problems and propose efficient solutions to get around them the widely used gradient magnitude based energy is parameter dependent its use will negatively affect the parametrization of the curve and consequently its stiffness hence we introduce a new edge based energy that is independent of the parameterization it is also more robust since it takes into account the gradient direction as well we express this energy term as a surface integral thus unifying it naturally with the region based schemes the unified framework enables the user to tune the image energy to the application at hand we show that parametric snakes can guarantee low curvature curves but only if they are described in the curvilinear abscissa since normal curve evolution do not ensure constant arc length we propose a new internal energy term that will force this configuration the curve evolution can sometimes give rise to closed loops in the contour which will adversely interfere with the optimization algorithm we propose a curve evolution scheme that prevents this condition

on the isoperimetric deficit in gauss space we prove a sharp quantitative version of the isoperimetric inequality in the space bbb r n endowed with the gaussian measure

quantitative isoperimetric inequalities for log convex probability measures on the line abstract the purpose of this paper is to analyze the isoperimetric inequality for symmetric log convex probability measures on the line using geometric arguments we first re prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals a result due to bobkov and houdre then we give a quantitative form of the isoperimetric inequality leading to a somehow anomalous behavior indeed it could be that a set is very close to be optimal in the sense that the isoperimetric inequality is almost an equality but at the same time is very far in the sense of the symmetric difference between sets from any extremal sets from the results on sets we derive quantitative functional inequalities of weak cheeger type

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

sampling curves with finite rate of innovation in this paper we extend the theory of sampling signals with finite rate of innovation fri to a specific class of two dimensional curves which are defined implicitly as the zeros of a mask function here the mask function has a parametric representation as a weighted summation of a finite number of complex exponentials and therefore has finite rate of innovation an associated edge image which is discontinuous on the predefined parametric curve is proved to satisfy a set of linear annihilation equations we show that it is possible to reconstruct the parameters of the curve i e to detect the exact edge positions in the continuous domain based on the annihilation equations robust reconstruction algorithms are also developed to cope with scenarios with model mismatch moreover the annihilation equations that characterize the curve are linear constraints that can be easily exploited in optimization problems for further image processing e g image up sampling we demonstrate one potential application of the annihilation algorithm with examples in edge preserving interpolation experimental results with both synthetic curves as well as edges of natural images clearly show the effectiveness of the annihilation constraint in preserving sharp edges and improving snrs

iterated filter banks with rational rate changes connection with discrete wavelet transforms some properties of two band filter banks with rational rate changes rational filter banks are first reviewed focusing then on iterated rational filter banks compactly supported limit functions are obtained in the same manner as previously done for dyadic schemes allowing a characterization of such filter banks these functions are carefully studied and the properties they share with the dyadic case are highlighted they are experimentally observed to verify a shift property strictly verified in the dyadic ease up to an error which can be made arbitrarily small when their regularity increases in this case the high pass outputs of an iterated filter bank can be very close to samples of a discrete wavelet transform with the same rational dilation factor straightforward extension of the formalism of multiresolution analysis is also made finally it is shown that if one is ready to put up with the loss of the shift property rational iterated filter banks can be used in the same manner as if they were dyadic filter banks with the advantage that rational dilation factors can be chosen closer to 1

regular connected bipancyclic spanning subgraphs of hypercubes an n dimensional hypercube q n is a hamiltonian graph in other words q n n 2 contains a spanning subgraph which is 2 regular and 2 connected in this paper we explore yet another strong property of hypercubes we prove that for any integer k with email protected email protected n q n n 3 contains a spanning subgraph which is k regular k connected and bipancyclic we also obtain the result that every mesh p mxp n m n 2 is bipancyclic which is used to prove the property above

weak poincare inequalities and l2 convergence rates of markov semigroups abstract in order to describe l 2 convergence rates slower than exponential the weak poincare inequality is introduced it is shown that the convergence rate of a markov semigroup and the corresponding weak poincare inequality can be determined by each other conditions for the weak poincare inequality to hold are presented which are easy to check and which hold in many applications the weak poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes some typical examples are given to illustrate the general results in particular our results are applied to the stochastic quantization of field theory in finite volume moreover a sharp criterion of weak poincare inequalities is presented for poisson measures on configuration spaces

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

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

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

an exact method for computing the area moments of wavelet and spline curves we present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis using green s theorem we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product the multidimensional filter coefficients are precomputed exactly as the solution of a two scale relation to demonstrate the performance improvement of the new method we compare it with existing methods such as pixel based approaches and approximation of the region by a polygon we also propose an alternate scheme when the scaling function is sinc x

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

sampling of periodic signals a quantitative error analysis we present an exact expression for the l sub 2 error that occurs when one approximates a periodic signal in a basis of shifted and scaled versions of a generating function this formulation is applicable to a wide variety of linear approximation schemes including wavelets splines and bandlimited signal expansions the formula takes the simple form of a parseval s like relation where the fourier coefficients of the signal are weighted against a frequency kernel that characterizes the approximation operator we use this expression to analyze the behavior of the error as the sampling step approaches zero we also experimentally verify the expression of the error in the context of the interpolation of closed curves

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

functional inequalities for empty essential spectrum abstract in terms of the equivalence of poincare inequality and the existence of spectral gap the super poincare inequality is suggested in the paper for the study of essential spectrum it is proved for symmetric diffusions that such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator this inequality recovers known sobolev and nash type ones it is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below some results are also proved for a more general setting including symmetric jump processes moreover estimates of inequality constants are also presented which lead to a proof of a result on ultracontractivity suggested recently by d stroock finally concentration of reference measures for super poincare inequalities is studied the resulting estimates extend previous ones for poincare and log sobolev inequalities

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

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