the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

dual gabor frames theory and computational aspects we consider a general method for constructing dual gabor elements different from the canonical dual our approach is based on combining two gabor frames such that the generated frame type operator s sub g spl gamma is nonsingular we provide necessary and sufficient conditions on the gabor window functions g and spl gamma such that s sub g spl gamma is nonsingular for rational oversampling considering both the continuous time and the discrete time settings in contrast to the frame operator the operator s sub g spl gamma is in general not positive therefore all results in gabor analysis that are based on the positivity of the frame operator cannot be applied directly the advantage of the proposed characterization is that the algebraic system for computing the gabor dual elements preserves the high structure of usual gabor frames leading to computationally efficient algorithms in particular we consider examples in which both the condition number and the computational complexity in computing the proposed dual gabor elements decrease in comparison to the canonical dual gabor elements

a modified split radix fft with fewer arithmetic operations recent results by van buskirk have broken the record set by yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power of two discrete fourier transform dft here we present a simple recursive modification of the split radix algorithm that computes the dft with asymptotically about 6 fewer operations than yavne matching the count achieved by van buskirk s program generation framework we also discuss the application of our algorithm to real data and real symmetric discrete cosine transforms where we are again able to achieve lower arithmetic counts than previously published algorithms

aspects of an adaptive finite element method for the fractional laplacian a priori and a posteriori error estimates efficient implementation and multigrid solver abstract we develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations on non trivial domains in d 1 dimensions our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and where necessary modifying them so that they can be applied to the fractional pde case improved a priori error estimates are derived for the case of quasi uniform meshes which are seen to deliver sub optimal rates of convergence owing to the presence of singularities attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure we assume that the resulting refined meshes are locally quasi uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods including the multigrid method the storage of the dense matrices along with efficient techniques for computing the dense matrix vector products needed for the iterative solution is also considered the performance and efficiency of the resulting algorithm is illustrated for a variety of examples

discrete gabor structures and optimal representations the idea of gabor s 1946 signal expansion is to represent a signal in terms of a discrete set of time shifted and frequency modulated signals that are localized in the time frequency or phase space we present detailed descriptions of the block and banded structures for the gabor matrices based on the explicit descriptions of the sparsity of such matrices we can establish the sparse form of the gabor matrix and obtain the dual gabor atom mother wavelet the inverse of the gabor frame operator and carry out the discrete finite gabor transform in a very efficient way some explicit sufficient and also necessary conditions are derived for a gabor atom g to generate a gabor frame with respect to a tf lattice a b

complete discrete 2 d gabor transforms by neural networks for image analysis and compression a three layered neural network is described for transforming two dimensional discrete signals into generalized nonorthogonal 2 d gabor representations for image analysis segmentation and compression these transforms are conjoint spatial spectral representations which provide a complete image description in terms of locally windowed 2 d spectral coordinates embedded within global 2 d spatial coordinates in the present neural network approach based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights the network finds coefficients for complete conjoint 2 d gabor transforms without restrictive conditions in wavelet expansions based on a biologically inspired log polar ensemble of dilations rotations and translations of a single underlying 2 d gabor wavelet template image compression is illustrated with ratios up to 20 1 also demonstrated is image segmentation based on the clustering of coefficients in the complete 2 d gabor transform

the undersampled discrete gabor transform conventional studies on discrete gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the gabor families span the whole signal space in this paper we investigate undersampled discrete gabor transforms for an undersampled gabor triple g a b i e a spl middot b n we show that the associated generalized dual gabor window gdgw function is the same as the one associated with the oversampled g n b n a except for the constant factor ab n computations of undersampled gabor transforms are made possible by applying the methods algorithms developed in oversampled settings the undersampled gdgw is determined then we are able to obtain the best approximation of a signal x by linear combinations of vectors in the gabor family

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

interpreting image curve from multiframes abstract a method of reconstructing the structure of a rigid space curve from multiframes is presented the problem is formulated as does there exist a unique reconstruction of a general smooth space curve from images if the curve is moving constantly in the space what is the minimum information needed to allow such a reconstruction the motion here is taken to be general but rotates about a fixed axis uniformly the only featured points are the two endpoints of the curve we first establish a necessary and sufficient condition on the number of frames for determining the motion with only two feature points observable also whether the underlying motion meets the formulation of the problem can be checked from the observables next we show that the ambiguities of matching a given nonfeatured point on the curve in one frame are limited to the intersections of a straight line and the image curve on the other frame the ambiguities of matching nonfeature points can then be resolved and the reconstruction of the space curve follows readily experiments and examples are provided to illustrate each step of the method furthermore we find that conclusions obtained by yuille and poggio 8 generalized ordering constraint in the case of parallel projection are either special cases of or can be easily derived from this method

a comment on analysis of video image sequences using point and line correspondences abstract in this paper we would like to contest the results of y f wang n karandikar and j k aggarwal pattern recognition 24 1065 1085 1991 raising two fundamental claims a line does not contribute anything to recognition of motion parameters from two images four traceable points are not sufficient to recover motion parameters from two perspective projections to be constructive however we show that four traceable points are sufficient to recover motion parameters from two frames under orthogonal projection and that five points are sufficient to simplify the solution of the two frame problem under orthogonal projection to solving a linear equation system

calculating the dual gabor window for general sampling sets methods for the calculation of the so called dual gabor window for general sampling sets are developed first we consider the standard situation where the time frequency plane tf plane is over sampled on a grid i e a rectangular lattice is considered with more points than the dimension of the signal space second we show that the proposed method can be easily extended to general subgroups of the tf plane one typical example of such a sampling set is the so called quincunx lattice

modeling and identification of nonlinear systems in the short time fourier transform domain in this paper we introduce a novel approach for improved nonlinear system identification in the short time fourier transform stft domain we first derive explicit representations of discrete time volterra filters in the stft domain based on these representations approximate nonlinear stft models which consist of parallel combinations of linear and nonlinear components are developed the linear components are represented by cross band filters between subbands while the nonlinear components are modeled by multiplicative cross terms we consider the identification of quadratically nonlinear systems and show that a significant reduction in computational cost as well as substantial improvement in estimation accuracy can be achieved over a time domain volterra model particularly when long memory systems are considered experimental results validate the theoretical derivations and demonstrate the effectiveness of the proposed approach

a computational method for solving variable order fractional nonlinear diffusion wave equation abstract in this paper we generalize a one dimensional fractional diffusion wave equation to a one dimensional variable order space time fractional nonlinear diffusion wave equation v os tfnd we where the variable order fractional derivatives are considered in the caputo type to solve this introduced equation an easy to follow method is proposed which is based on the chebyshev cardinal functions coupling with the tau and collocation methods to carry out the method an operational matrix of variable order fractional derivative omv ofd is derived for the chebyshev cardinal functions to be employed for expanding the unknown function the proposed method can provide highly accurate approximate solutions by reducing the problem under study to a system of nonlinear algebraic equations which is technically simpler for handling the experimental results confirm the applicability and effectiveness of the method

a wavelet approach for solving multi term variable order time fractional diffusion wave equation abstract we firstly generalize a multi term time fractional diffusion wave equation to the multi term variable order time fractional diffusion wave equation m v tfd e by the concept of variable order fractional derivatives then we implement the chebyshev wavelets cws through the operational matrix method to approximate its solution in the unit square in fact we apply the operational matrix of variable order fractional derivative omv fd of the cws to derive the unknown solution we proceed with coupling the collocation and tau methods to reduce m v tfd e to a system of algebraic equations the important privilege of method is handling different kinds of conditions i e initial boundary conditions and dirichlet boundary conditions by implementing the same techniques the convergence and error estimation of the cws expansion in two dimensions are theoretically investigated we also examine the applicability and computational efficiency of the new scheme through the numerical experiments

approximation of the erdelyi kober operator with application to the time fractional porous medium equation this paper describes a method of approximating equations with the erdelyi kober fractional operator which arise in mathematical descriptions of anomalous diffusion we prove a theorem on the exact form of the approximating series and provide an illustration by considering the fractional porous medium equation applied to model moisture diffusion in building materials we obtain some approximate analytical solutions of our problem which accurately fit the experimental data better than other models found in the literature this accuracy is also verified numerically since they are very quick and easy to implement our approximations can be valuable for practitioners and experimentalists

discrete fourier invariant signals design and application in this paper two methods for the design of discrete fourier invariant signals are proposed the direct design method provides splitting between independent and dependent signal parts and calculation of the dependent part for any given independent part the iterative design method generates a family of discrete fourier invariant signals by a successive approach further we show how the proposed direct design method can be combined with the gabor uncertainty principle to generate discrete fourier invariant signals with the minimum product of their bandwidth b and their time width t we show that these signals as well as signal families generated with the iterative design method achieve the theoretical lower bt bound also it is shown that the bt product of discrete hermite gauss signals converges to the theoretical lower bound finally possible applications are illustrated in the case of time frequency spectral analysis using the obtained discrete fourier invariant signals as the window that provides isoresolution

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

vector valued weak gabor dual frames on discrete periodic sets the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples

robust transient signal detection using the oversampled gabor representation the gabor representation has been investigated previously as an effective and efficient means of detecting transient signals the major problem associated with the most commonly used form of the gabor representation is that it assumes that signals have integer arrival times and modulation frequencies an assumption that if violated causes signal miss match and deterioration in detector performance the present correspondence presents the concept of oversampling the gabor representation which makes the representation more robust to certain types of signal mismatch to show this the performance of an oversampled gabor detector is compared to that of critically sampled gabor detectors gabor coefficients are calculated in the oversampled representation via the least squares method which has been shown to be superior to the more commonly used biorthogonal function method in terms of corresponding detector performance

structure from motion of rigid and jointed objects a method for recovering the three dimensional structure of moving rigid and jointed objects from several single camera views is presented the method is based on the fixed axis assumption all movement consists of translations and rotations about an axis that is fixed in direction for short periods of time this assumption makes it possible to recover the structure of any group of two or more rigidly connected points the structure of jointed objects is recovered by analyzing them as collections of rigid parts and then unifying the structures proposed for the parts the method presented here has been tested on several sets of data including movies used to demonstrate human perception of structure from motion

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

on characterization of the optimal biorthogonal window functions for gabor transform gabor transforms have been recognized as useful tools in signal analysis it is known that the solutions for the biorthogonal analysis window function spl gamma t given a synthesis window function h t in gabor transforms are not unique in general among these solutions the minimum norm solution has already been given by wexler and raz 1990 in the discrete time case and has been studied by janssen ron and shen and daubechies et al 1995 in the continuous time case the minimum norm solution in the discrete time case was also proved to be equal to the most orthogonal like solution by qian and chen 1993 in this note we consider a general optimal solution problem where the minimum norm and the most orthogonal like solutions are two special cases we prove that these optimal solutions in many cases are equal we also prove that it remains true in the continuous time case

a short fe implementation for a 2d homogeneous dirichlet problem of a fractional laplacian in acosta etal 2017 a complete n dimensional finite element analysis of the homogeneous dirichlet problem associated to a fractional laplacian was presented here we provide a comprehensive and simple 2d matlab finite element code for such a problem the code is accompanied with a basic discussion of the theory relevant in the context the main program is written in about 80 lines and can be easily modified to deal with other kernels as well as with time dependent problems the present work fills a gap by providing an input for a large number of mathematicians and scientists interested in numerical approximations of solutions of a large variety of problems involving nonlocal phenomena in two dimensional space

robustness of oversampled gabor transient detectors a comparison of energy and known location detectors this correspondence provides a context for previous studies on the use of linear time frequency transforms for transient signal detection in the case where signal component locations are assumed to be known the robustness of a known location detector kld based on the oversampled gabor transform is compared with that of the corresponding energy detector in the gabor domain as expected the kld was sensitive to location mismatch unexpectedly it was extremely sensitive to location mismatch

discrete zak transforms polyphase transforms and applications we consider three different versions of the zak 1967 transform zt for discrete time signals namely the discrete time zt the polyphase transform and a cyclic discrete zt in particular we show that the extension of the discrete time zt to the complex z plane results in the polyphase transform an important and well known concept in multirate signal processing and filter bank theory we discuss fundamental properties relations and transform pairs of the three discrete zt versions and we summarize applications of these transforms in particular the discrete time zt and the cyclic discrete zt are important for discrete time gabor 1946 expansion weyl heisenberg frame theory since they diagonalize the weyl heisenberg frame operator for critical sampling and integer oversampling the polyphase representation plays a fundamental role in the theory of filter banks especially dft filter banks simulation results are presented to demonstrate the application of the discrete zt to the efficient calculation of dual gabor windows tight gabor windows and frame bounds

efficient calculation of finite gabor transforms the gabor transform may be viewed as a collection of localized fourier transforms and as such is useful for analysis of nonstationary signals and images we present a new approach to analyzing the gabor transform and use it to study the various critically sampled discretizations that form the infinite discrete periodic finite discrete and nonperiodic finite discrete versions of the transform in particular we distinguish between the analysis and synthesis forms of the transform and introduce an intermediate operation that decomposes both forms into collections of independent toeplitz operators in the continuous the infinite discrete and the periodic finite discrete cases this decomposition allows us to show that for appropriate windows the analysis and synthesis transforms are inverses of each other in the nonperiodic finite discrete case this relation no longer holds but we are still able to use the decomposition and results on toeplitz matrices to show that both the transform and the inverse transform of p discrete samples are computable in o p log p operations after a setup cost of o plog sup 2 p furthermore we use the decomposition to study in detail the differences between the periodic and nonperiodic versions of the transform and to compare their conditioning

a fast learning algorithm for gabor transformation an adaptive learning approach for the computation of the coefficients of the generalized nonorthogonal 2 d gabor 1946 transform representation is introduced the algorithm uses a recursive least squares rls type algorithm the aim is to achieve minimum mean squared error for the reconstructed image from the set of the gabor coefficients the proposed rls learning offers better accuracy and faster convergence behavior when compared with the least mean squares lms based algorithms applications of this scheme in image data reduction are also demonstrated

an efficient algorithm to compute the complete set of discrete gabor coefficients the discrete gabor 1946 transform algorithm is introduced that provides an efficient method of calculating the complete set of discrete gabor coefficients of a finite duration discrete signal from finite summations and to reconstruct the original signal exactly from the computed expansion coefficients the similarity of the formulas between the discrete gabor transform and the discrete fourier transform enables one to employ the fft algorithms in the computation the discrete 1 d gabor transform algorithm can be extended to 2 d as well

adjoint state method for fractional diffusion parameter identification fractional partial differential equations provide models for sub diffusion among which the fractal mobile immobile model fmim is often used to represent solute transport in complex media the fmim involves four parameters among which we have the order of an integro differential operator that accounts for the possibility for solutes to be sequestered during very long times to guess fmim parameters from experiments an accurate method consists in optimizing an objective function that measures how much model solutions deviate from data we show that solving an adjoint problem helps accurate computing of the gradient of such an objective function with respect to the parameters we illustrate the method by applying it on experimental data issued from tracing tests in porous media

complete gabor transformation for signal representation properties of the gabor transformation used for image representation are discussed the properties can be expressed in matrix notation and the complete gabor coefficients can be found by multiplying the inverse of the gabor 1946 matrix and the signal vector the gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function a fast algorithm is suggested to compute the inverse of the window function matrix enabling discrete signals to be transformed into generalized nonorthogonal gabor representations efficiently a comparison is made between this method and the analytical method the relation between the window function matrix and the biorthogonal functions is demonstrated a numerical computation method for the biorthogonal functions is proposed

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

efficient algorithms for discrete gabor transforms on a nonseparable lattice the discrete gabor transform dgt is the most commonly used transform for signal analysis and synthesis using a linear frequency scale it turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup most of the literature focuses on separable subgroups in this paper we will survey existing methods for a generalization to arbitrary groups as well as present an improvement on existing methods comparisons are made with respect to the computational complexity and the running time of optimized implementations in the c programming language the new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download by summarizing general background information on the state of the art this article can also be seen as a research survey sharing with the readers experience in the numerical work in gabor analysis

fast discrete orthonormal stockwell transform we present an efficient method for computing the discrete orthonormal stockwell transform dost the stockwell transform st is a time frequency decomposition transform that is showing great promise in various applications but is limited because its computation is infeasible for most applications the dost is a nonredundant version of the st solving many of the memory and computational issues however computing the dost of a signal of length n using basis vectors is still mathcal o n 2 the computational complexity of our method is mathcal o n log n putting it in the same category as the fft the algorithm is based on a simple decomposition of the dost matrix we also explore the way to gain conjugate symmetry for the dost and propose a variation of the parameters that exhibits symmetry akin to the conjugate symmetry of the fft of a real valued signal our fast method works equally well on this symmetric dost in this paper we provide a mathematical proof of our results and derive that the computational complexity of our algorithm is mathcal o n log n timing tests also confirm that the new method is orders of magnitude faster than the brute force dost and they demonstrate that our fast dost is indeed mathcal o n log n in complexity

isogeometric collocation method for the fractional laplacian in the 2d bounded domain abstract we consider the isogeometric analysis for fractional pdes involving the fractional laplacian in two dimensions an isogeometric collocation method is developed to discretize the fractional laplacian and applied to the fractional poisson problem and the time dependent fractional porous media equation numerical studies exhibit monotonous convergence with a rate of o n 1 where n is the degrees of freedom a comparison with finite element analysis shows that the method enjoys higher accuracy per degree of freedom and has better convergence rate we demonstrate that isogeometric analysis offers a novel and promising computational tool for nonlocal problems

discrete gabor transform a feasible algorithm for implementing the gabor expansion the coefficients of which are computed by the discrete gabor transform dgt is presented for a given synthesis window and sampling pattern computing the auxiliary biorthogonal function of the dgt is nothing more than solving a linear system the dgt presented applies for both finite as well as infinite sequences by exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like dgt is obtained as the discrete fourier transform dft is a discrete realization of the continuous time fourier transform similarly the dgt introduced provides a feasible vehicle to implement the useful gabor expansion

discrete multiwindow gabor type transforms the discrete finite gabor scheme is generalized by incorporating multiwindows two approaches are presented for the analysis of the multiwindow scheme the signal domain approach and the zak transform domain approach issues related to undersampling critical sampling and oversampling are considered the analysis is based on the concept of frames and on generalized moore penrose inverses the approach based on representing the frame operator as a matrix valued function is far less demanding from a computational complexity viewpoint than a straightforward matrix algebra in various operations such as the computation of the dual frame dft based algorithms including complexity analysis are presented for the calculation of the expansion coefficients and for the reconstruction of the signal in both signal and transform domains the scheme is further generalized and incorporates kernels other than the complex exponential representations other than those based on the dual frame and nonrectangular sampling of the combined space are considered as well an example that illustrates the advantages of the multiwindow scheme over the single window scheme is presented

novel dct based real valued discrete gabor transform and its fast algorithms the oversampled gabor transform is more effective than the critically sampled one in many applications the biorthogonality relationship between the analysis window and the synthesis window of the gabor transform represents the completeness condition however the traditional discrete cosine transform dct based real valued discrete gabor transform rgdt is available only in the critically sampled case and its biorthogonality relationship for the transform has not been unveiled to bridge these important gaps this paper proposes a novel dct based rdgt which can be applied in both the critically sampled case and the oversampled case and their biorthogonality relationships can be derived the proposed dct based rdgt involves only real operations and can utilize fast dct algorithms for computation which facilitates computation and implementation by hardware or software as compared to that of the traditional complex valued discrete gabor transform this paper also develops block time recursive algorithms for the efficient and fast computation of the rdgt and its inverse transform unified parallel lattice structures for the implementation of these algorithms are presented computational complexity analysis and comparisons have shown that the proposed algorithms provide a more efficient and faster approach for discrete gabor transforms as compared to those of the existing discrete gabor transform algorithms in addition an application in the noise reduction of the nuclear magnetic resonance free induction decay signals is presented to show the efficiency of the proposed rdgt for time frequency analysis

some results on discrete gabor transforms for finite periodic sequences in this correspondence using linear system theory and frame theory we address computation methods for analysis and synthesis sequences of the discrete gabor transform dgt where all sequences are periodic with the same period l and show that the minimum energy solution to the wexler and raz 1990 biorthogonal condition and the solution to the frame operator are the same via linear system theory and frame theory

adaptive gabor transformation for image processing an algorithm that computes the gabor coefficients of an image is presented an adaptive filter that uses the complex least mean square algorithm for their computation is proposed and its numerical characteristics are discussed it is shown that the filter is stable under certain conditions because the gabor transformation seems to be an excellent tool for image compression the efficiency of information coding using the gabor coefficients is investigated and compared with coding that uses the coefficients of the discrete cosine transformation dct properties of both transformations are discussed

discrete multi gabor expansions a discrete multi gabor expansion dmge is developed to meet the requirements of localized and refined time frequency tf representation of signals the dmge uses multiple windows and their translations and complex modulations as synthesis or analysis waveforms it includes and generalizes the metaplectic translation modulation and dilation representations which are useful in signal analysis uniform nonuniform and proportional time sampling schemes are analyzed the fundamental features and the importance of the dmge are discussed we focus on the construction of dmge and deriving fast algorithms for the computation of related multi analysis sequences with matrix algebra the algorithms derived apply to both multi gabor expansions and uni window gabor expansions another useful feature of the dmge lies in the fact that the multi gabor transform can be realized in a parallel fft based implementation structure examples of dmge and their applications to tf analysis are also discussed

multirate based fast parallel algorithms for 2 d dht based real valued discrete gabor transform novel algorithms for the multirate and fast parallel implementation of the 2 d discrete hartley transform dht based real valued discrete gabor transform rdgt and its inverse transform are presented in this paper a 2 d multirate based analysis convolver bank is designed for the 2 d rdgt and a 2 d multirate based synthesis convolver bank is designed for the 2 d inverse rdgt the parallel channels in each of the two convolver banks have a unified structure and can apply the 2 d fast dht algorithm to speed up their computations the computational complexity of each parallel channel is low and is independent of the gabor oversampling rate all the 2 d rdgt coefficients of an image are computed in parallel during the analysis process and can be reconstructed in parallel during the synthesis process the computational complexity and time of the proposed parallel algorithms are analyzed and compared with those of the existing fastest algorithms for 2 d discrete gabor transforms the results indicate that the proposed algorithms are the fastest which make them attractive for real time image processing

the black box fast multipole method a new o n fast multipole formulation is proposed for non oscillatory kernels this algorithm is applicable to kernels k x y which are only known numerically that is their numerical value can be obtained for any x y this is quite different from many fast multipole methods which depend on analytical expansions of the far field behavior of k for x y large other black box or kernel independent fast multipole methods have been devised our approach has the advantage of requiring a small pre computation time even for very large systems and uses the minimal number of coefficients to represent the far field for a given l 2 tolerance error in the approximation this technique can be very useful for problems where the kernel is known analytically but is quite complicated or for kernels which are defined purely numerically

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

gabor transform of time varying systems exact representation and approximation approximate modeling of discrete time linear time varying systems is studied based on a representation of linear systems in the gabor time frequency space the time varying system is assumed to be given in input output or kernel representation this representation has previously received attention because of its applicability in frozen time analysis and design of optimal control for time varying systems but requires a large number of coefficients motivated by its application to signal analysis the gabor transform is considered as a tool for the representation and approximation of linear time variant systems in order to show the main results the class of systems considered is restricted to the one usually considered in the frozen time approach an example is included to illustrate the potential application of the technique

detection of transient signals by the gabor representation gabor representation is used for the detection of transient signals with unknown arrival times a one sided exponential window function is used which seems to be most appropriate for transient modelling explicit expressions for the gabor coefficients are given for this window function when the given signal is random so are the coefficients the second order moments of the gabor coefficients are computed for a white noise signal these are then used to introduce a detection statistic based on the gabor coefficients the proposed detector is capable of separating transients having different arrival times even in this case where their waveforms partially overlap

the generalized gabor transform the generalized gabor transform for image representation is discussed for a given function f t t spl isin r the generalized gabor transform finds a set of coefficients a sub mr such that f t spl sigma sub m spl infin sup spl infin spl sigma sub r spl infin sup spl infin spl alpha sub mr g t mt exp i2 spl pi rt t the original gabor transform proposed by d gabor 1946 is the special case of t t the computation of the generalized gabor transform with biorthogonal functions is discussed the optimal biorthogonal functions are discussed a relation between a window function and its optimal biorthogonal function is presented based on the zak 1967 transform when t t is rational the finite discrete generalized gabor transform is also derived methods of computation for the biorthogonal function are discussed the relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized gabor transform can be extended to the finite discrete case efficient algorithms for the optimal biorthogonal function and generalized gabor transform for the finite discrete case are proposed

a parametric class of discrete gabor expansions the gabor expansion and its discretization have been widely studied and many potential applications have been suggested in various signal processing problems a new approach to the study of the discrete gabor expansion dge is introduced and analyzed in detail using the theory of pseudoframe decompositions a parametric and analytical formula for a class of different gabor analysis sequences is derived it is a simple algebraic formula rather than another abstract system of equations for the first time the structure of analysis sequences is questioned we show that while there is a class of infinite analysis sequences that possess the gabor translation and complex modulation structure there are also infinite analysis sequences of arbitrary forms simulation results are provided to demonstrate the proposed algorithms the study of the dge by means of the theory of pseudoframe decompositions reveals a much broader mathematical perspective on the dge the general algorithm derived provides a feasible platform for optimizations in discrete gabor expansions arising from various applications this is an area that can surely be exploited as algorithms of dges become known and applications become more and more intensive

can one evaluate the gabor expansion using gabor s iterative algorithm d gabor s original 1946 approach for calculation of expansion coefficients was examined and it was found that convergence of the suggested algorithm is conditional on the window function chosen the method of analysis enables a process of synthesis whereby eligible windows can be defined ensuring convergence of the algorithm for every represented signal for windows having a finite support of width d the coefficients calculated according to the algorithm are obtained after a single iteration it is proved that in the special case of a gaussian window the algorithm converges only for specific signals on the basis of the conclusions drawn conditions are formulated which ensure convergence of the gabor algorithm permitting examination of alternative windows

optimal biorthogonal analysis window function for discrete gabor transform like the dft spl minus a discrete version of the fourier transform spl minus the recently developed discrete gabor transform dgt provides a feasible vehicle to implement the very useful gabor expansion in general the choice of the biorthogonal window function spl gamma i is not unique the authors discuss the solution of spl gamma i that is optimally close to an arbitrary desired function d i when the synthesis window h i and sampling pattern are given for d i h i the resulting spl gamma sub opt i directly leads to so called orthogonal like dgt combining the dgt they believe that the result presented in this paper is rather significant for digital signal processing

the inverse fast multipole method using a fast approximate direct solver as a preconditioner for dense linear systems although some preconditioners are available for solving dense linear systems there are still many matrices for which preconditioners are lacking particularly in cases where the size of the matrix n becomes very large examples of preconditioners include incomplete lu ilu preconditioners that sparsify the matrix based on some threshold algebraic multigrid preconditioners and specialized preconditioners e g calderon and other analytical approximation methods when available despite these methods there remains a great need to develop general purpose preconditioners whose cost scales well with the matrix size n in this paper we propose a preconditioner with broad applicability and with cost mathcal o n for dense matrices when the matrix is given by a smooth as opposed to a highly oscillatory kernel extending the method using the same framework as general mathcal h 2 matrices i e algebraic instead of defined in terms of an analytical kernel is relatively straightforward but

decomposition of the wigner ville distribution and time frequency distribution series using the orthogonal like gabor expansion the authors decompose the wigner ville distribution wvd to a linear combination of localized and symmetric functions wvd sub h h t w the wvd of the gabor elementary functions h sub m n t and h sub m n t since the influence of the wvd sub h h t w to the useful properties is inversely proportional to the distance between h sub m n t and h sub m n t the wvd sub h h t w are further grouped as a series of the function p sub d t w the authors name the resulting representation the time frequency distribution series tpds also known as the gabor spectrogram in industry the tfdsd consists of up to a dth order p sub d t w while tfds sub 0 t w p sub 0 t w is similar to the spectrogram tfds sub spl infin t w converges to the wvd sub s t w numerical simulations demonstrate that adjusting the order d of the tfds one could effectively balance the cross term interference and the useful properties

fast parallel approach for 2 d dht based real valued discrete gabor transform two dimensional fast gabor transform algorithms are useful for real time applications due to the high computational complexity of the traditional 2 d complex valued discrete gabor transform cdgt this paper presents two block time recursive algorithms for 2 d dht based real valued discrete gabor transform rdgt and its inverse transform and develops a fast parallel approach for the implementation of the two algorithms the computational complexity of the proposed parallel approach is analyzed and compared with that of the existing 2 d cdgt algorithms the results indicate that the proposed parallel approach is attractive for real time image processing

oversampled gabor representation for transient signals considers the gabor representation that uses a one sided exponential window for detection and analysis of transient signals earlier results on the critically sampled case are extended to the more practically useful oversampled case for oversampling by an integer factor the authors derive an explicit analytical expression for the dual window dual frame function required for computing the gabor representation based on this expression they develop an efficient procedure for computing the gabor coefficients finally they demonstrate the performance of the method by numerical examples

weyl heisenberg frames translation invariant systems and the walnut representation abstract we present a comprehensive analysis of the convergence properties of the frame operators of weyl heisenberg systems and shift invariant systems and relate these to the convergence of the walnut representation we give a deep analysis of necessary conditions and sufficient conditions for convergence of the frame operator we show that symmetric norm and unconditional convergence of the walnut series are all different but that weak and norm convergence are the same while there are wh systems for which the walnut representation has none of these convergence properties we make a detailed study of the cc condition a sufficient condition for wh systems to have finite upper frame bounds and show that for ab rational a uniform version of this passes to the wexler raz dual we also show that a condition of tolimieri and orr implies the uniform cc condition we obtain stronger results in the case when g a b is a wh system and ab is rational for example if ab is rational then the cc condition becomes equivalent to the unconditional convergence of the walnut representation even in a more general setting many of the results are generalized to shift invariant systems we give classifications for numerous important classes of wh systems including 1 the wh systems for which the frame operator extends to a bounded operator on l p r for all 1 p 2 the wh systems for which the frame operator extends to a bounded operator on the wiener amalgam space 3 the families of frames which have the same frame operator

contact treatment in isogeometric analysis with nurbs abstract we study nurbs based isogeometric analysis of contact problems and compare with standard c 0 continuous lagrange finite elements a knot to surface kts algorithm is developed to treat the contact constraints with nurbs contact surface discretizations qualitative studies deliver satisfactory results for various finite deformation frictionless thermoelastic contact problems quantitative studies based on the hertz problem suggest the need for a relaxation of the mechanical contact constraints that appear in the standard kts approach the improved mortar based kts algorithm delivers robust and accurate results for nurbs discretizations based on numerical examples we conclude that nurbs based isogeometric analysis is a viable technology for contact problems and offers potential accuracy as well as convergence improvements over c 0 continuous finite elements

a linear cost algorithm to compute the discrete gabor transform in this paper we propose an alternative efficient method to calculate the gabor coefficients of a signal given a synthesis window with a support of size much lesser than the length of the signal the algorithm uses the canonical dual of the window which does not need to be calculated beforehand and achieves a computational cost that is linear with the signal length in both analysis and synthesis this is done by exploiting the block structure of the matrices and using an ad hoc cholesky decomposition of the gabor frame matrix

optimal ofdm design for time frequency dispersive channels transmission over wireless channels is subject to time dispersion due to multipath propagation and to frequency dispersion due to the doppler effect standard orthogonal frequency division multiplexing ofdm systems using a guard time interval or cyclic prefix combat intersymbol interference isi but provide no protection against interchannel interference ici this drawback has led to the introduction of pulse shaping ofdm systems we first present a general framework for pulse shape design our analysis shows that certain pulse shapes proposed in the literature are in fact optimal in a well defined sense furthermore our approach provides a simple way to adapt the pulse shape to varying channel conditions we then show that pulse shaping ofdm systems based on rectangular time frequency lattices are not optimal for time and frequency dispersive wireless channels this motivates the introduction of lattice ofdm lofdm systems which are based on general time frequency lattices using results from sphere packing theory we show how to design lofdm systems lattice and pulse shape optimally for timeand frequency dispersive channels in order to minimize the joint isi ici our theoretical analysis is confirmed by numerical simulations showing that lofdm systems outperform traditional pulse shaping ofdm systems with respect to robustness against isi ici

self sorting in place fft algorithm with minimum working space presents a modification of temperton s 1991 self sorting in place radix p fft algorithm this modification reduces the required temporary working space from order of p sup 2 to p 1 providing a better match to the limited number of registers in a cpu

multiwindow real valued discrete gabor transform and its fast algorithms based on the biorthogonal analysis approach a multiwindow real valued discrete gabor transform m rdgt for periodic sequences is presented to efficiently analyze the dynamic time frequency content of a signal containing components with multiple and or time varying frequencies the m rdgt offers a computationally efficient implementation as well as a real valued formulation of the multiwindow complex valued discrete gabor transform m cdgt the completeness condition of the m rdgt is proved to be equivalent to its biorthogonality constraint between analysis windows and synthesis windows the m rdgt can utilize the fast discrete hartley transform algorithms for fast computation and has a simple relationship with the m cdgt such that its coefficients can be directly computed from the m rdgt coefficients therefore the m rdgt offers an efficient method to compute the m cdgt since the analyzed sequence analysis and synthesis windows in the existing m cdgt must have an equal period if the period of a sequence is very long solving its windows requires a huge amount of computation and memory and could lead to numerical instability to overcome this problem a modified m rdgt for long periodic or even infinite sequences is presented and its corresponding biorthogonality constraint between analysis windows and synthesis windows is modified in which the period of the analysis and synthesis windows is independent of the period of a analyzed sequence so that one can apply short windows to process any long periodic or even in finite sequence finally the multirate based parallel implementation of the m rdgt is presented which has shown to be effective and fast for time frequency analysis

interpreting line drawings as three dimensional surfaces understanding how line drawings convey tri dimensionality is of fundamental importance in explaining surface perception when photometry is either uninformative or too complex to model analytically we put forward here a computational model for interpreting line drawings as three dimensional surfaces based on constraints on local surface orientation along extremal and discontinuity boundaries specific techniques are described for two key processes recovering the three dimensional conformation of a space curve e g a surface boundary from its two dimensional projection in an image and interpolating smooth surfaces from orientation constraints along extremal boundaries the relevance of the model to a general theory of low level vision is discussed

double preconditioning for gabor frames the authors present an application of the general idea of preconditioning in the context of gabor frames while most iterative algorithms aim at a more or less costly exact numerical calculation of the inverse gabor frame matrix we propose here the use of cheap methods to find an approximation for it based on double preconditioning we thereby obtain good approximations of the true dual gabor atom at low computational costs since the gabor frame matrix commutes with certain time frequency shifts it is natural to make use of diagonal and circulant preconditioners sharing this property part of the efficiency of the proposed scheme results from the fact that all the matrices involved share a well known block matrix structure at least for the smooth gabor atoms typically used the combination of these two preconditioners leads consistently to good results these claims are supported by numerical experiments in this paper for numerical evaluations we introduce two new matrix norms which can be calculated efficiently by exploiting the structure of the frame matrix

convergence of an iterative time variant filtering based on discrete gabor transform an iterative time variant filtering based on the discrete gabor transform dgt has been previously proposed by the authors in this article we present a proof of the convergence of the iterative algorithm under a sufficient condition on the analysis and synthesis window functions of the dgt in the meantime we show that the iterative algorithm refines the least squares solution

the zak transform and decimated time frequency distributions in this paper the interrelation between the zak transform and the generalized discrete time frequency distribution gdtfd is examined starting with the discrete zak transform its definition is broadened to include arbitrary windows creating the windowed zak transform wzt the wzt is then combined with the spectrogram it is demonstrated that the spectrogram based upon the wzt called the zak spectrogram zs is a generalization of the standard spectrogram next building upon the idea of the weighted spectrogram the weighted zs is used to produce a new class of gdtfd called the decimated gdtfd d gdtfd the decimated gdtfd is similar to the gdtfd except it trades bandwidth for computational speed and requires significantly less storage in order to be implemented the reduction in discrete bandwidth is from 2 spl pi for the gdtfd to 2 spl pi m for the d gdtfd an important attribute of the dgdtfd is that it requires significantly less storage than the gdtfd the d gdtfd requires only 1 m sup 2 of the storage of the gdtfd an example using the binomial distribution is given to illustrate the connection between the d gdtfd and the gdtfd throughout the paper examples of multirate systems are given which could be used to implement the building blocks of the dgdtfd this makes the d gdtfd practical to implement using slower and less expensive elements

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

system identification using chirp signals and time variant filters in the joint time frequency domain we propose a novel method to identify an unknown linear time invariant lti system in low signal to noise ratio snr environment the method is based on transmitting chirp signals for the transmitter and using linear time variant filters in the joint time frequency tf domain for the receiver to reduce noise before identification due to the tf localization property of chirp signals a large amount of additive white noise can be reduced and therefore the snr before identification can be significantly increased this however cannot be achieved in the conventional methods where pseudo random signals are used and therefore noise reduction techniques do not apply our simulation results indicate that the method proposed outperforms the conventional methods significantly in a low snr environment this paper provides a good application of time frequency analysis and synthesis

continuity properties of the gabor frame operator abstract for a gabor frame with small lattice it is shown that the frame operator is continuous and invertible on many banach spaces defined by smoothness and decay properties as a consequence it is shown that there exist gabor type frame dual frame pairs and tight frames with good decay and smoothness properties also it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly e g in lp and sobolev norms also explicit estimates on lattice sizes for which such results hold can be obtained

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions

a gabor expansion based positive time dependent power spectrum this correspondence presents a novel approach to the synthesis of positive time frequency distributions with marginals corresponding to the instantaneous power and the power spectrum this class of distribution has mainly been constructed using an entropy approach whereas this correspondence presents a noniterative approach based on the gabor expansion the resulting distribution has been named the positive gabor spectrogram and is evaluated using different test signals interpretation and applicability of positive distributions in connection with signal analysis are discussed