a wavelet shrinkage approach to tomographic image reconstruction abstract a method is proposed for reconstructing images from tomographic data with respect to a two dimensional wavelet basis the wavelet vaguelette decomposition wvd is used as a framework within which expressions for the necessary wavelet coefficients may be derived these coefficients are calculated using a version of the filtered back projection algorithm as a computational tool in a multiresolution fashion the necessary filters are defined in terms of the underlying wavelets denoising is accomplished through an adaptation of the wavelet shrinkage ws approach of donoho et al and amounts to a form of regularization combining these two steps yields the proposed wvd ws reconstruction algorithm which is compared to the traditional filtered backprojection method in a small study

a geometric approach to the linear penrose transform on montre que sous une certaine condition cohomologique le theoreme de witten iseuberg yasskin et green sur la transformee de penrose inverse d une connexion est vraie dans une situation generale

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

physical space time and nonrealizable text cr structures space time views leading up to einstein s general relativity are described in relation to some of poincare s early ideas on the subject the basic geometry of twistor theory is introduced as it arises both from minkowski space time and the more general curved einstein models it is shown how this provides a cr structure this being in essence another of poincare s poioneering concepts in a natural way nonrealizable cr structure can arise and an example is presented due to c d hill g a j sparling and the author of a complex manifold with boundary which cannot be extended as a complex manifold beyond its c sup infinity boundary

radon penrose transform for d modules abstract let formula be a correspondence of complex analytic manifolds fbe a sheaf onx and m be a coherent d x module consider the associated sheaf theoretical and d module integral transforms given by sf rg f 1f d and s m g f 1 m whererg andf 1 resp g and f 1 denote the direct and inverse image functors for sheaves resp for d modules andd ds dyis the difference of dimension betweensandy in this paper assuming thatfis smooth gis proper and f g is a closed embedding we prove some general adjunction formulas for the functors sand s moreover under an additional geometrical hypothesis we show that the transformation sestablishes an equivalence of categories between coherent d x modules modulo flat connections and coherent d y modules with regular singularities along an involutive manifoldv modulo flat connections herevis determined by the geometry of the correspondence applications are given to the case of penrose s twistor correspondence

nonuniform sampling and antialiasing in image representation a unified approach to the representation and processing of a class of images which are not bandlimited but belong to the space of locally bandlimited signals is presented a nonuniform sampling theorem clark et al 1985 for functions belonging to this space is extended and a class of nonstationary stochastic processes is considered the space of locally bandlimited signals is shown to be a reproducing kernel space a generalized projection theorem can therefore be applied yielding either a continuous or a discrete projection filter the former can be used for image conditioning prior to nonuniform sampling while the latter provides a general tool for image representation by nonuniform sampling schemes the problem of finding the local bandwidth of a given signal in order to generate an optimal sampling scheme is addressed in the context of signal representation in the combined position frequency space the stochastic estimation of parameters which characterize the local bandwidth is discussed bounds on the error resulting from the utilization of nonexact position varying signal parameters are derived

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

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

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

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

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

spectral theory of the klein gordon equation in pontryagin spaces in this paper we investigate an abstract klein gordon equation by means of indefinite inner product methods we show that under certain assumptions on the potential which are more general than in previous works the corresponding linear operator a is self adjoint in the pontryagin space mathcal k induced by the so called energy inner product the operator a possesses a spectral function with critical points the essential spectrum of a is real with a gap around 0 and the non real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity the number of these pairs is related to the size of the potential moreover a generates a group of bounded unitary operators in the pontryagin space mathcal k finally the conditions on the potential required in the paper are illustrated for the klein gordon equation in mathbb r n they include potentials consisting of a coulomb part and an lp part with n p

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

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

time frequency distribution inversion of the radon transform image reconstruction in using filtered backprojection to compute the inverse radon transform the ramp filter amplifies noise spatially invariant noise filters reduce resolution it is desirable to filter noise where projections have no local high frequency components using the short time fourier transform the authors apply a time frequency mask filter that zeroes out projections where local signal energy is below a threshold results show improvement over reconstructions using spatially invariant smoothing filters

variational principles for eigenvalues of the klein gordon equation in this paper variational principles for eigenvalues of an abstract model of the klein gordon equation with electromagnetic potential are established they are used to characterize and estimate eigenvalues in cases where the essential spectrum has a gap around 0 even in the presence of complex eigenvalues as a consequence a comparison between eigenvalues of the klein gordon equation in rd and eigenvalues of certain schrodinger operators is obtained the results are illustrated on examples including the klein gordon equation with coulomb and square well potential

optimal filtering in fractional fourier domains for time invariant degradation models and stationary signals and noise the classical fourier domain wiener filter which can be implemented in o n log n time gives the minimum mean square error estimate of the original undistorted signal for time varying degradations and nonstationary processes however the optimal linear estimate requires o n sup 2 time for implementation we consider filtering in fractional fourier domains which enables significant reduction of the error compared with ordinary fourier domain filtering for certain types of degradation and noise especially of chirped nature while requiring only o n log n implementation time thus improved performance is achieved at no additional cost expressions for the optimal filter functions in fractional domains are derived and several illustrative examples are given in which significant reduction of the error by a factor of 50 is obtained

boundary values of resolvents of selfadjoint operators in krein spaces we prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a krein space if h is a selfadjoint operator on a krein space ch equipped with the krein scalar product langle cdot cdot rangle a is the generator of a c 0 group on ch and i subset rr is an interval such that begin itemize item 1 h admits a borel functional calculus on i item 2 the spectral projection one i h is positive in the krein sense item 3 the following em positive commutator estimate holds re langle u h i a u rangle geq c langle u u rangle u in rm ran one i h c 0 end itemize then assuming some smoothness of h with respect to the group e i t a the following resolvent estimates hold sup z in i pm i 0 nu langle a rangle s h z 1 langle a rangle s 12 as an application we consider abstract klein gordon equations p t 2 phi t 2 i k phi t h phi t 0 and obtain resolvent estimates for their generators in em charge spaces of cauchy data

weak solutions of the massless field equations we consider weak solutions of the zero rest mass z r m equations described in eastwood et al 1981 the space of hyperfunctions which contains the space of distributions is defined and we consider hyper function solutions of the equations on real minkowski space m i and its conformal compactification m we define a hyperfunction z r m field to be future or past analytic if it is the boundary value of a holomorphic z r m field on the future or past tube of complex minkowski space respectively and we demonstrate that any field on m i that is the sum of future and past analytic fields extends as a hyperfunction z r m field to all of m it is shown that any distribution solution on m i splits as required and hence extends as a hyperfunction solution to m twistor methods are then used to show that the same applies in the more general case of hyperfunction solutions on m i this leads to an alternative proof of the main result of wells 1981 a hyperfunction z r m field on compactified real minkowski space is a unique sum of future and past analytic solutions

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

spatially varying spectral filtering of signals on the unit sphere this paper presents a general framework for spatially varying spectral filtering of signals defined on the unit sphere as an analogy to joint time frequency filtering for this purpose we first map spherical signals from spatial domain into joint spatio spectral domain where a spatio spectral signal transformation or modification is introduced for mapping spatial signals into joint spatio spectral domain we use the spatially localized spherical harmonic transform slsht from the literature we then propose a suitable scheme to transform the modified signal from the spatio spectral domain back to an admissible signal in the spatial domain using the least squares approach we also show that the overall action of the slsht and spatio spectral signal modification can be described through a single transformation matrix which is useful in practice finally we discuss two specific and useful instances of spatially varying spectral filtering defined through multiplicative and convolutive modification of the slsht distribution and show through numerical examples their effectiveness in selective spectral filtering of different spatial regions of the signal

on eigenfunction expansion of solutions to the hamilton equations we establish a spectral representation for solutions to linear hamilton equations with positive definite energy in a hilbert space our approach is a special version of m krein s spectral theory of j selfadjoint operators in the hilbert spaces with indefinite metric our main result is an application to the eigenfunction expansion for the linearized relativistic ginzburg landau equation

hyperfunction solutions of the zero rest mass field equations in this paper it is shown how the penrose transform maps tangential hyperfunction dolbeault groups with coefficients in a power of the hyperplane section bundle on the hyperquadric of null twistors in projective twistor space isomorphically to all hyperfunction solutions of the massless field equations of nonnegative helicity on compactified minkowski space this is an extension of the penrose transform which generated real analytic solutions of the same field equations on the same space cf eastwood m penrose r wells r o 10 in additions one obtains the result that each hyperfunction solution of the massless field equations of nonnegative helicity is the sum of massless fields of positive and negative frequency a generalization of the usual fourier decomposition for solutions with appropriate growth conditions

scattering theory for the klein gordon equation with nondecreasing potentials the klein gordon equation is considered in the case of nondecreasing potentials the energy inner product is nonpositive on a subspace of infinite dimension not consisting entirely of eigenvectors of the associated operator a scattering theory for this case is developed and asymptotic completeness for generalized moller operators is proven

time variant filtering of signals in the mixed time frequency domain a new approach to the problem of time variant filtering is presented this approach is based upon the generation of the mixed time frequency representation mtfr of a signal multiplication of that representation by a time frequency function h t and obtaining a filtered output by an inverse operation the resultant filter is linear if the time frequency representation used is the complex spectrogram in contrast the filter is nonlinear if the wigner distribution function is used not every function of two variables is an allowed mtfr of a signal some conditions must be satisfied if the function produced by the product of the signal mtfr and the filter function is not an allowed mtfr an approximation based on projection onto the space of allowed mtfr functions is investigated this approximation yields a filtered function whose mtfr is as close as possible in the least squared sense to the desired

a conformal holomorphic field theory a formulation of a field theory on the complex minkowski space in terms of complex differential geometry is proposed it is also shown that our model of field theory differs from the standard model on the real minkowski space only in the limit of high energy

the hyperbolic class of quadratic time frequency representations ii subclasses intersection with the affine and power classes regularity and unitarity for pt i see ibid vol 41 p 3425 444 1993 part i introduced the hyperbolic class hc of quadratic bilinear time frequency representations qtfrs the present paper defines and studies four subclasses of the hc 1 the focalized kernel subclass of the hc is related to a time frequency concentration property of qtfrs it is analogous to the localized kernel subclass of the affine qtfr class 2 the affine subclass of the hc affine hc consists of all hc qtfrs that satisfy the conventional time shift covariance property it forms the intersection of the hc with the affine qtfr class 3 the power subclasses of the hc consist of all hc qtfrs that satisfy a power time shift covariance property they form the intersection of the hc with the recently introduced power classes 4 the power warp subclass of the hc consists of all hc qtfrs that satisfy a covariance to power law frequency warpings it is the hc counterpart of the shift scale covariant subclass of cohen s class all of these subclasses are characterized by 1d kernel functions the affine hc is contained in both the localized kernel hyperbolic subclass and the localized kernel affine subclass and that any affine hc qtfr can be derived from the bertrand unitary po distribution by a convolution we furthermore consider the properties of regularity and unitarity in the hc the calculus of inverse kernels is developed and important implications of regularity and unitarity are summarized the results comprise a general method for least squares signal synthesis and new relations for the altes marinovich q distribution

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

time varying filtering and signal estimation using wigner distribution synthesis techniques the short time fourier transform stft the ambiguity function af and the wigner distribution wd are mixed time frequency signal representations that use fourier transform techniques to map a one dimensional function of time into a two dimensional function of time and frequency these mixed time frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems although much work has also been done to develop stft and af synthesis algorithms that can be used to implement a variety of time varying signal processing operations no such synthesis techniques have thus far been developed for the wd in this paper a signal synthesis algorithm that works directly with the real valued high resolution wd will be derived examples of how this wd synthesis procedure can be used to perform time varying filtering operations or signal separation will be given

wavelet methods for inverting the radon transform with noisy data because the radon transform is a smoothing transform any noise in the radon data becomes magnified when the inverse radon transform is applied among the methods used to deal with this problem is the wavelet vaguelette decomposition wvd coupled with wavelet shrinkage as introduced by donoho 1995 we extend several results of donoho and others here first we introduce a new sufficient condition on wavelets to generate a wvd for a general homogeneous operator whose class includes the radon transform we show that a variant of donoho s method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a besov norm as the smoothing functional we give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise corrupted data we conduct tomographic reconstruction computations that support the hypothesis that near optimal shrinkage parameters can be derived if one can estimate only two besov space parameters about an image f both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than kolaczyk s 1996 variant of donoho s method and the classical filtered backprojection method

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

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

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

reconstruction from projections under time frequency constraints low pass filtering computed tomography ct images to reduce noise may smooth or modify image features which are very important to the physician image features are often more easily identified and processed in the time frequency plane the authors use time frequency distributions for spatially varying filtering of noisy ct images constraining time frequency representation coefficients of the projection data or of the reconstructed image to be zero in certain regions of the time frequency plane the authors consider two different applications 1 filtering the projection data and then performing image reconstruction and 2 filtering the reconstructed image directly criteria minimized subject to constraints may be either a deterministic minimum weighted perturbation of the given projection data or a stochastic minimum mean square error in colored gaussian noise results show improvement over processing the image with a linear spatially invariant filter

a computational study of reconstruction algorithms for diffraction tomography interpolation versus filtered backpropagation from the standpoint of reporting a new contribution this paper shows that by using bilinear interpolation followed by direct two dimensional fourier inversion one can obtain reconstructions of quality which is comparable to that produced by the filtered backpropagation algorithm proposed recently by devaney for an n n image reconstructed from n diffracted projections the former approach requires approximately 4n fft s whereas the backpropagation technique requires approximately n2fft s we have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography an area that is becoming increasingly important not only in medical imaging but also in underwater and seismic mapping with microwaves and sound the main feature of the tutorial part is the statement of the fourier diffraction projection theorem which is an extension of the traditional fourier slice theorem to the case of image formation with diffracting illumination

the wigner distribution of linear time variant systems the wigner distribution relation between the two input and output time varying spectra of the linear time variant system is derived this mathematical relationship can show how a time variant system modifies the time varying spectrum of the input to determine that of the output and this result will be very useful for time variant filtering in the mixed time frequency domain a linear time variant digital filtering example is used to show the effect of time variant filtering by using the wigner distribution

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

signal synthesis from modified discrete short time transform the discrete short time transform dstt is a generalization of the discrete short time fourier transform dstft the necessary and sufficient conditions on the analysis filter under which perfect reconstruction of the input signal is possible when the dstt is not modified are presented the class of linear modifications for which the original input can be reconstructed when the modification is applied is characterized the synthesis of an optimal in the minimum mean square error sense signal from a modified dstt mdstt of finite duration is presented it is shown that for an analysis filter length that does not exceed a given value the optimal synthesis scheme is independent of the duration of the given mdstt and is an extension of the weighted overlap add wola synthesis method for longer analysis filters the optimal synthesis scheme becomes quite cumbersome and therefore a steady state solution as the duration of the mdstt approaches infinity is presented for this case it is shown that this solution can be approximated with arbitrarily small reconstruction error

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

analysis and filtering using the optimally smoothed wigner distribution the authors consider the analysis and filtering of a deterministic signal with slowly time varying spectra using the optimally smoothed wigner distribution oswd they compare this mixed time frequency representation mtfr to other mtfrs such as the spectrogram the short time fourier transform stft and the wigner and pseudo wigner distributions the authors propose an approach to designing linear time varying filters for slowly time varying signals which is based on the concept of local nonstationarity cancellation and show that it is equivalent to masking the optimal stft the performance of the filter in suppressing white noise and in decomposing a slowly time varying signal into its components is studied and compared to the performance of the techniques based on the stft

the damped wave equation with unbounded damping abstract we analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity we prove the generation of a contraction semigroup study the relation between the spectra of the semigroup generator and the associated quadratic operator function the convergence of non real eigenvalues in the asymptotic regime of diverging damping on a subdomain and we investigate the appearance of essential spectrum on the negative real axis we further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential these analytic results are illustrated by examples

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

a variational approach to recovering depth from defocused images in this paper we propose a regularized solution to the depth from defocus dfd problem using the space frequency representation sfr framework a smoothness constraint is imposed on the estimates of the blur parameter and a variational approach to the dfd problem is developed among the numerous sfrs we study the applicability of the complex spectrogram and the wigner distribution in particular for depth recovery the performance of the proposed variational method is tested on both synthetic and real images the method yields good results and the quality of the estimates is significantly better than that obtained without the smoothness constraint on the blur parameter

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

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