double series representation of bounded signals series representations of the form f t approximately sigma sub n infinity sup infinity sigma sub k infinity sup infinity a sub n k nu t n sub e sup 2 pi kts for bounded signals f t are studied as are conditions on the unit function nu t such that coefficients a sub n k reveal the energy content of f t in the time interval n 1 2

functions analytic on the half plane as quantum mechanical states a transform between the state space of one dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half plane is constructed it gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple generalizations to higher dimensions are also discussed

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

polygonal figures and their form spaces ii angle permutations in equilateral polygons abstract in a closed nonregular equilateral polygonal curve angles cannot be permuted arbitrarily without opening the curve here we investigate the possible angle permutations that leave a polygonal curve closed to this end we introduce angle sequences i e symbol sequences describing the number and arrangement of like angle values in classes of forms we find that there exist universal angle sequences giving closed polygons for arbitrary angle values provided that the angle sum is kept constant at one of certain permissible values the universal angle sequences are periodic they consist of at least two identical series of symbols within a series the symbols can be permuted freely on condition that in all other series the very same permutation is applied all these correlated permutations when applied to the angles of a corresponding polygonal curve leave this curve closed polygons belonging to a given angle sequence form class constitute a submanifold of the manifold m i of equilateral polygons within the form space these submanifolds structure m i in a hierarchical manner so that the form class concept gives a valuable complementary classification scheme for the forms of m i furthermore we find that polygons with constant angle composition the term composition is used here in its chemical sense show a wide variety of possible shapes from quasi circular to wormlike and even self intersecting ones corresponding to different degrees of segregation of like angles within the polygonal curve

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

family of scaling chirp functions diffraction and holography it is observed that diffraction is a convolution operation with a chirp kernel whose argument is scaled family of functions obtained from a prototype by shifting and argument scaling form the essential ground for wavelet framework therefore a connection between diffraction and wavelet transform is developed however wavelet transform is essentially prescribed for time frequency and or multiresolution analysis which is irrelevant in our case instead the proposed framework is useful in various location depth type of analysis in imaging the linear transform when the analyzing functions are the chirps is called the scaling chirp transform the scaled chirp functions do not satisfy the commonly used admissibility condition for wavelets however it is formally shown that these neither band nor time limited signals can be used as wavelet functions and the inversion is still possible diffraction and in line holography are revisited within the scaling chirp transform context it is formally proven that a volume in line hologram gives perfect reconstruction the developed framework for wave propagation based phenomena has the potential of advancing both signal processing and optical applications

crisp and fuzzy motif and arrangement symmetries in composite geometric figures abstract the notions of motif and arrangement symmetries within composite geometric figures are defined the relationships between these types of symmetry and the symmetry of the whole figure are clarified by making use of the crystallographic concepts of site symmetry and direction symmetry from this it has been deduced that a figure with arbitrary symmetry can be assembled from motifs of likewise arbitrary symmetries if a motif with symmetry g m is placed on a site having the site symmetry g s g m its contribution to the figure symmetry g is only a subgroup g mo of its direction symmetry g mo where g s g mo g mo g m supernumerary symmetry elements of the motif give rise to intermediate or latent symmetries of the figure a consequent decomposition of a geometric figure into its constituent points reveals that a large part of the o n symmetry of every single point is lost when assembling these points to build up the figure all lost symmetries can however be detected as intermediate symmetries of this figure they can be displayed as fuzzy symmetry landscapes and symmetry profiles for a given figure showing all crisp and intermediate symmetries of interest

linear systems in gabor time frequency space a new approach for the description and operation of linear time varying systems is presented the proposed formalism implements the superposition integral describing a general linear operation in gabor time frequency space enabling the use of its joint properties the method is based on transforming both the input and the time varying impulse response function into the gabor space the 2 and 4 d arrays corresponding to the gabor spectra of the input signal and the impulse response respectively are multiplied yielding a 2 d array corresponding to the output gabor spectrum from which the output time signal is subsequently reconstructed unlike other time varying operation methods carried out in different mixed time frequency spaces e g wigner space the suggested approach when employing the critically sampled gabor expansion generates a legal output spectrum and eliminates the need to resort to cumbersome and numerically taxing approximate synthesis techniques the proposed formalism can be implemented in the areas of time variant filtering speech scrambling and any other linear operations that vary their characterizations in time

two dimensional filter bank design for optimal reconstruction using limited subband information in this correspondence we propose design techniques for analysis and synthesis filters of 2 d perfect reconstruction filter banks prfb s that perform optimal reconstruction when a reduced number of subband signals is used based on the minimization of the squared error between the original signal and some low resolution representation of it the 2 d filters are optimally adjusted to the statistics of the input images so that most of the signal s energy is concentrated in the first few subband components this property makes the optimal prfb s efficient for image compression and pattern representations at lower resolutions for classification purposes by extending recently introduced ideas from frequency domain principal component analysis to two dimensions we present results for general 2 d discrete nonstationary and stationary second order processes showing that the optimal filters are nonseparable particular attention is paid to separable random fields proving that only the first and last filters of the optimal prfb are separable in this case simulation results that illustrate the theoretical achievements are presented

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

fixed points and boundary layers in asymmetric simple exclusion processes in this paper we show how a fixed point based boundary layer analysis can be used to understand phases and phase transitions in asymmetric simple exclusion processes aseps with open boundaries in order to illustrate this method we choose a two species asep which has interesting phase transitions not seen in the one species case we also apply this method to the single species problem where the analysis is simple but nevertheless quite insightful

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly examining the frequency content of signals is critical in many applications from neuroscience to astronomy many techniques have been proposed to accomplish this one of these the s transform provides simultaneous time and frequency information similar to the wavelet transform but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements it has shown promise in many medical imaging applications but has high computational requirements this paper presents a general transform that describes fourier family transforms including the fourier short time fourier and s transforms a discrete nonredundant formulation of this transform as well as algorithms for calculating the forward and inverse transforms are also developed these utilize efficient sampling of the time frequency plane and have the same computational complexity as the fast fourier transform when configured appropriately this new algorithm samples the continuous s transform spectrum efficiently and nonredundantly allowing signals to be transformed in milliseconds rather than days as compared to the original s transform algorithm the new and efficient algorithms make practical many existing signal and image processing techniques both in biomedical and other applications

improved impedance matrix localization method em problems moment method calculations have the well known limitations of requiring excessive storage and execution times for even modestly large electromagnetics problems the impedance matrix localization iml method was introduced as a modification to standard moment method calculations to ease these limitations it utilizes a matrix transformation which effectively changes the basis testing functions into ones resembling traveling waves an improved method that uses an orthogonal transformation to generate standing wave like basis functions is presented here remarkable improvements are achieved in the numerical stability of the method and in its compatibility with iterative solvers furthermore the correspondence of the large elements in this matrix to geometrical theory of diffraction gtd terms is strengthened as is the possibility of further increasing the speed of iterative solutions by constructing preconditioners based on the pattern of nonzero matrix elements

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

l stability for linear multiresolution algorithms a new explicit approach part ii the cases of symlets coiflets biorthogonal wavelets and supercompact multiwavelets abstract in amat and moncayo s amat m moncayo l stability for linear multiresolution algorithms a new explicit approach part i the basic rules and the daubechies case appl math comput 206 1 2008 74 91 was introduced a direct procedure to obtain an explicit computation of the error bounds for the mallat s multiresolution transform associated to orthogonal wavelet filters the general stability framework was applied to the specific case of daubechies filters in this paper we extend our approach giving the bounds related to others well known and used wavelet families as symlets coiflets biorthogonal wavelets and supercompact multiwavelets

image reconstruction from projections under wavelet constraints first the authors discuss how the wavelet transform can be used to perform spatially varying filtering of an image suppressing noise locally in smooth regions of the image and discuss detection of such regions in a noise corrupted image second they show how to compute the minimum mean square estimate of an image given 1 noisy projections of the image 2 statistics of additive noise in the projections and 3 constraints on wavelet coefficients of the image examples illustrate the resulting procedure

wavelet frames and admissibility in higher dimensions this paper is concerned with the relations between discrete and continuous wavelet transforms on k dimensional euclidean space we start with the construction of continuous wavelet transforms with the help of square integrable representations of certain semidirect products thereby generalizing results of bernier and taylor we then turn to frames of l2 rk and to the question when the functions occurring in a given frame are admissible for a given continuous wavelet transform for certain frames we give a characterization which generalizes a result of daubechies to higher dimensions

deformations isosymmetric manifolds and higher dimensional form space symmetries for point ensembles polygonal forms under o 2 symmetry i two and three points abstract the analysis of polygonal forms and their form transitions using normal deformations 1 has been extended to a global analysis of form spaces for polygonal forms specified by n points in a plane coincidences of the points are explicitly allowed and the origin of the form space is taken to correspond to a coincidence of all n points this choice is natural since for every form there exists a totally symmetric deformation leading to exactly this configuration the form space f for n points in a plane is a 2n dimensional euclidean space whose points represent all possible n vertex polygons simple and self intersecting ones by fixing the centres of gravity of the forms in the origin of the x y plane the dimension of the form space reduces to 2n 2 to give a reduced form space f within the form spaces isosymmetric manifolds geometric loci of all forms having the same symmetry g are determined these manifolds inscribed in a form space f define the corresponding symmetry space s the symmetry g s of the reduced symmetry space for three points has been determined in part the four dimensional symmetry group 21 04 has been found to be a subgroup of the full noncrystallographic symmetry of s for some cases the relation g f g s has been exemplified

multishocks in driven diffusive processes insights from fixed point analysis of the boundary layers boundary induced phase transitions in a driven diffusive process can be studied through a phase plane analysis of the boundary layer equations in this paper we generalize this approach further to show how various shapes including multishocks and downward shocks in the bulk particle density profile can be understood by studying the dependence of the fixed points of the boundary layer equation on an appropriate parameter this is done for a particular driven interacting particle system as a prototypical example the present analysis shows the special role of a specific bifurcation of the fixed points in giving rise to different kinds of shocks

wavelet transforms associated with finite cyclic groups multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers an important class of multiresolution decompositions are the laplacian pyramid schemes in which the resolution is successively halved by recursively low pass filtering the signal under analysis and decimating it by a factor of two in general the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete time fourier analysis of sequences of complex numbers an analogous framework is developed for the multiresolution analysis of finite length sequences of elements from arbitrary fields attention is restricted to sequences of length 2 sup n for n a positive integer so that the resolution may be recursively halved to completion as in finite length fourier analysis a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields

the growth of bargmann functions and the completeness of sequences of coherent states the growth of bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of bargmann functions of an order not exceeding and of a type not exceeding if of order these results generalize known results on the completeness of von neumann lattices the practical significance of this formalism in the context of quantum optics is also discussed

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

enhanced frequency analysis using wavelets an enhanced frequency analysis method is introduced using a phase shift wavelet representation to analyse individual frequency components and to reveal features of a given signal the frequency representation provides details at each time location without the loss caused by local orthogonality between the wavelet basis functions and the given signal which occurs when a normal wavelet decomposition is employed movement of the local orthogonality locations corresponding to the phase shift of the wavelet basis functions are studied and it is shown that these movements are important and can be used to reveal features of the signal in the frequency domain such as continuities smoothness and trends several examples are presented to illustrate the method and an application to a permanent magnet synchronous drive system is provided to demonstrate the approach

triangular wavelet based finite elements via multivalued scaling equations this paper develops a class of triangular finite elements based on l2 rn orthonormal compactly supported wavelets as in the case of recently derived tensor product wavelet elements the triangular elements are defined by deriving an algebraic eigenvalue problem characterizing the entries of the elemental matrices these entries are uniquely defined by normalization conditions derived from the polynomial reproducing properties of the wavelet scaling functions the work herein generalizes the development of the tensor product wavelet elements in that the geometry of the triangular elemental domain is characterized using a multivalued scaling equation the multivalued scaling function is used to represent the self similar tiling of rn onto itself by triangular domains

tensor least angle regression for sparse representations of multidimensional signals sparse signal representations have gained much interest recently in both signal processing and statistical communities compared to orthogonal matching pursuit omp and basis pursuit which solve

universal interpretation of efficacy parameter in perturbed nonequilibrium systems abstract the fluctuation theorems have remained one of the cornerstones in the study of systems that are driven far out of equilibrium and they provide strong constraints on the fraction of trajectories that behave atypically in light of the second law they have mainly been derived for a predetermined external drive applied to the system however to improve the efficiency of a process one needs to incorporate protocols that are modified by receiving feedbacks about the recent state of the system during its evolution in such a case the forms of the conventional fluctuation theorems get modified the correction term involving terms that depend on the way the reverse conjugate process is defined namely the rules of using feedback in order to generate the exact time reversed conjugate protocols we show in this paper that this can be done in a large number of ways and in each case we would get a different expression for the correction terms this would in turn lead to several lower bounds on the mean work that must be performed on the system or on the entropy changes here we analyze a form of the extended fluctuation theorems that involves the efficacy parameter and find that this form retains a consistent physical meaning regardless of the design of feedback along the conjugate process as opposed to the case of the previously mentioned form of the modified fluctuation theorems

a maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform in this paper a maximum likelihood ml method is presented for joint estimation of amplitude phase time delay and data symbols in a single user direct sequence spread spectrum communication system since maximization of the likelihood function is analytically intractable a novel coordinate ascent algorithm is used to obtain sequential updates of the data symbols and all unknown nuisance parameters the novelty of the algorithm is due to the use of a multiresolution expansion of the received signal and the use of polynomial rooting in the complex plane in place of a line search over the signal delay parameter the multiresolution structure of the algorithm is exploited to reduce sensitivity to impulsive noise via wavelet thresholding computer simulations of the single user system show that the algorithm has fast convergence and comparison with theoretical lower bounds establishes that the algorithm achieves nearly optimal error performance

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

on some identities and inequalities for frames in hilbert spaces we give new parseval type identities and inequalities for frames in hilbert spaces our results generalize the remarkable results obtained recently by r balan p g casazza d edidin and g kutyniok

balian low type theorems in finite dimensions we formulate and prove finite dimensional analogs for the classical balian low theorem and for a quantitative balian low type theorem that in the case of the real line we obtained in a previous work moreover we show that these results imply their counter parts on the real line

isosymmetric manifolds in form spaces and the normal deformations of polygonal forms abstract form variations are described in an appropriately constructed form space f typically an r n where every point of f represents a different form regarding the symmetries of the forms f can be divided into disjunct isosymmetric manifolds i e points lines surfaces and volumes whose points correspond to forms with equal symmetries these manifolds are derived from a symmetry analysis of possible deformations of the forms this analysis is comparable to the construction of symmetry coordinates in a normal coordinate analysis of molecules and results in normal modes of deformation normal deformations of these forms from the symmetry species of a normal deformation the symmetry of the resulting form can be inferred transformation of the form space coordinates into normal coordinates the differentials of which are the normal deformations facilitates the description of the high dimensional form spaces and can be made the basis of an easy symmetry diagnosis of forms furthermore the problem of an ascent in symmetry by deformation is discussed

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

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

polygonal figures and their form spaces i the structure of isodiastemic manifolds manifolds of equilateral polygons abstract the geometric structure of isodiastemic manifolds i e manifolds on which the equilaterality of polygons is preserved within a form space f spanned by n 2 consecutive vertex angles i internal coordinates of plane n gons is reported for n 5 and n 6 the curved isodiastemic manifolds are almost everywhere locally n 3 dimensional exceptions are singular points linear forms for even n where the local dimension is n 2 the isodiastemic manifolds for n 5 and n 6 are subdivided into five n 3 dimensional submanifolds of nondegenerate forms forms without coincident points each comprising forms with the same angle sum called a form family the submanifolds are bounded by n 4 dimensional dividing manifolds of degenerate forms which are structured hierarchically according to the types of degeneracy of their forms for n 6 the boundary polyhedra for two submanifolds are described in detail the manifold of spearhead shaped forms is equivalent to an octahedron whereas the hexagon manifold is equivalent to a special icosihexahedron 1

a fuzzy symmetry concept for forms with imperfect symmetries we develop a new symmetry concept for forms whose symmetries are only approximate the concept is based on fuzzy set theory the crisp set s g of objects having a certain symmetry g is replaced by a fuzzy set s g with a membership function 0 1 corresponding to the degree of fulfilment of the symmetry requirements of g degree of symmetry the degree of symmetry s g of a given form s relative to a symmetry group g is evaluated on the basis of a harmonic analysis of the corresponding form function r depending on the symmetry of the form certain coefficients of the fourier series vanish the extinction rules for the fourier coefficients are deduced from the symmetries of the individual cosine terms cos we define the degree of symmetry as the relative weight of all terms which are compatible with g thus s is a continous measure of symmetry it may quantitatively characterise continuous transitions of forms between different symmetries and also the degree of membership of s in s g depending on the values of s we propose a gradation of fuzzy symmetries actual apparent approximate distant hidden missing symmetry

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

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

acceleration of the frame algorithm shows how polynomial acceleration techniques which have been developed for the solution of large linear systems can be employed to improve and accelerate the frame algorithm these methods permit a reduction in the number of necessary iterations by an order of magnitude when the frame algorithm is slow the author gives several examples from the theory of irregular sampling from wavelet theory and from gabor theory where these methods are probably mandatory for efficient reconstruction

frame vector multipliers for finite group representations abstract a frame vector or generator for a group representation of a countable or finite group g on a hilbert space h is a vector h such that g g g is a parseval frame for h frame vector multipliers are the unitary operators on h that map frame vectors to frame vectors based on a characterization of frame vectors with respect to the standard decomposition of a group representation as the direct sums of irreducible subrepresentations with multiplicity we obtain explicit characterizations of frame generator multipliers for two basic cases for finite group representations with the help of these characterizations we obtain some necessary conditions of frame vector multipliers for general frame representations and present several examples to demonstrate how these results can be used to get explicit characterizations for frame vector multipliers

frames in the bargmann space of entire functions we look at the decomposition of arbitrary f in l2 r in terms of the family of functions cp x w exp iimnab iman i x nb 2 with a b 0 we derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab 2w n n an integer 2 2 transported to the hilbert space f of entire functions introduced by v bargmann these results are expressed as inequalities of the form

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

duality and phase diagram of one dimensional transport the idea of duality in one dimensional nonequilibrium transport is introduced by generalizing the observations by mukherji and mishra a general approach is developed for the classification and characterization of the steady state phase diagrams which are shown to be determined by the nature of the zeros of a set of coarse grained functions that encode the microscopic dynamics a new class of nonequilibrium multicritical points has been identified

pattern analysis with two dimensional spectral localisation applications of two dimensional s transforms abstract an image is a function f x y of the independent space variables x and y the global fourier spectrum of the image is a complex function f k x k y of the wave numbers k x and k y the global spectrum may be viewed as a construct of the spectra of an arbitrary number of segments of f x y leading to the concept of a local spectrum at every point of f x y the two dimensional s transform is introduced here as a method of computation of the local spectrum at every point of an image in addition to the variables x and y the 2 d s transform retains the variables k x and k y being a complex function of four variables visualisation of a function of four variables is difficult we skirt around this by removing one degree of freedom through examination of slices each slice of the 2 d s transform would then be a complex function of three variables with separate amplitude and phase components by ranging through judiciously chosen slice locations the entire s transform can be examined images with strictly periodic patterns are best analysed with a global fourier spectrum on the other hand the 2 d s transform would be more useful in spectral characterisation of aperiodic or random patterns

hierarchical active shape models using the wavelet transform active shape models asms are often limited by the inability of relatively few eigenvectors to capture the full range of biological shape variability this paper presents a method that overcomes this limitation by using a hierarchical formulation of active shape models using the wavelet transform the statistical properties of the wavelet transform of a deformable contour are analyzed via principal component analysis and used as priors in the contour s deformation some of these priors reflect relatively global shape characteristics of the object boundaries whereas some of them capture local and high frequency shape characteristics and thus serve as local smoothness constraints this formulation achieves two objectives first it is robust when only a limited number of training samples is available second by using local statistics as smoothness constraints it eliminates the need for adopting ad hoc physical models such as elasticity or other smoothness models which do not necessarily reflect true biological variability examples on magnetic resonance images of the corpus callosum and hand contours demonstrate that good and fully automated segmentations can be achieved even with as few as five training samples

generalized gabor expansions of discrete time signals in l sup 2 z via biorthogonal like sequences in this paper a biorthogonal like sequences bls theory and its application to the generalized gabor expansions equivalently the generalized short time fourier transform filterbank summation are presented a pair of bls s are defined to be two sequences satisfying a biorthogonal like condition blc which is a moment equation and equivalent to a linear difference equation we show that two collections in a hilbert space generated by a pair of bls s in the joint time frequency domain are complete either can be used as an analysis filter and the other can be used as a synthesis filter for a generalized gabor expansion of discrete time signals a sufficient and necessary condition on the existence of bls s based on the moment equation is presented which is simpler to use than frame theory given a filter generating a frame its bls s also generate frames the dual frame is one of them given a fir analysis synthesis filter there is a fir synthesis analysis filter if bls s exist the algorithm to compute fir analysis and synthesis filters based on the linear difference equation is presented in this paper which is simpler than frame operator

rational von neumann lattices a rational von neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers zeros of harmonic oscillator functions in the kq representation on such lattices are found it is shown that the number of zeros of the kq function determines the number of states by which a set on a von neumann lattice is overcomplete interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation

adaptive learning for event modeling and characterization adaptive learning of specific patterns or events of interest has been an area of significant research for various applications in the last two decades in developing diagnostic evaluation and safety monitoring applications of a propulsion system it is critical to detect characterize and model events of interest it is a challenging task since the detection system should allow adaptive characterization of potential events of interest and correlate them to learn new models for future detection for online health monitoring and diagnostic evaluation in this paper a novel framework is established using a hierarchical adaptive clustering approach with fuzzy membership functions to characterize specific events of interest from the measured and processed features raw engine measurement data is first analyzed using the wavelet transform to provide features for localization of frequency information for use in the classification system a method combining hierarchical and fuzzy k means clustering is then applied to a set of selected measurements and computed features to determine the events of interest during engine operations experimental results have shown that the proposed approach is effective and computationally efficient to detect characterize and model new events of interest from data collected through continuous operations

weyl heisenberg frames and balian low theorem in l2 z this paper focuses on the reconstruction of digital signals in multirate digital signal processing we study the discrete time weyl heisenberg dtwh system for oversampling schemes and describe the frame operator of dtwh frames as the composition of the sampling operator and interpolation operator using discrete time zak transform we characterize the dual discrete time weyl heisenberg tight frames dtwh frames and tight dtwh frames based on oversampling schemes in addition we discuss the applications of dtwh frames to the reconstruction of weyl heisenberg systems in periodic space l2 0 1 finally we give balian low theorem for an orthonormal basis formed by dtwh systems and weak balian low theorem for exact dtwh frames in sequence space l2 z

the analysis and design of windowed fourier frame based multiple description source coding schemes in this paper the windowed fourier encoding decoding scheme applied to the multiple description compression problem is analyzed in the general case four window functions are needed to define the encoder and decoder although this number can be reduced to three or two by using time shift or frequency shift division schemes the encoding coefficients are next divided into two groups according to the eveness of either the modulation or translation index the distortion on each channel is analyzed using the zak transform for the optimal windows explicit representation formulas are obtained and nonlocalization results are proved asymptotic formulas of the total distortion and transmission rate are established and the redundancy is shown to trade off between these two

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling

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