aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

sampling from a system theoretic viewpoint part ii noncausal solutions this paper puts to use concepts and tools introduced in part i to address a wide spectrum of noncausal sampling and reconstruction problems particularly we follow the system theoretic paradigm by using systems as signal generators to account for available information and system norms l2 and l as performance measures the proposed optimization based approach recovers many known solutions derived hitherto by different methods as special cases under different assumptions about acquisition or reconstructing devices e g polynomial and exponential cardinal splines for fixed samplers and the sampling theorem and its modifications in the case when both sampler and interpolator are design parameters we also derive new results such as versions of the sampling theorem for downsampling and reconstruction from noisy measurements the continuous time invariance of a wide class of optimal sampling and reconstruction circuits etcetera

radial basis function approximation from gridded centres to scattered centres abstract the paper studies l infinity ir d norm approximations from a space spanned by a discrete set of translates of a basis function theta attention here is restricted to functions theta whose fourier transform is smooth on ird 0 and has a singularity at the origin examples of such basis functions are the thin plate splines and the multiquadrics as well as other types of radial basis functions that are employed in approximation theory the above approximation problem is well understood in case the set of points used for translating theta forms a lattice in ir d and many optimal and quasi optimal approximation schemes can already be found in the literature in contrast only few mostly specific results are known for a set of scattered points the main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non uniform case we introduce a single relatively simple conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature more precisely to almost all stationary schemes in anticipation of future introduction of new schemes for uniform grids an effort is made to impose only a few mild conditions on the function theta which still allow for a unified error analysis to hold in the course of the discussion here the recent results of budl on scattered center approximation are reproduced and improved upon

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

minimum probability of error for asynchronous gaussian multiple access channels consider a gaussian multiple access channel shared by k users who transmit asynchronously independent data streams by modulating a set of assigned signal waveforms the uncoded probability of error achievable by optimum multiuser detectors is investigated it is shown that the k user maximum likelihood sequence detector consists of a bank of single user matched filters followed by a viterbi algorithm whose complexity per binary decision is o 2 k the upper bound analysis of this detector follows an approach based on the decomposition of error sequences the issues of convergence and tightness of the bounds are examined and it is shown that the minimum multiuser error probability is equivalent in the iow noise region to that of a single user system with reduced power these results show that the proposed multiuser detectors afford important performance gains over conventional single user systems in which the signal constellation carries the entire burden of complexity required to achieve a given performance level

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

time delay estimation from low rate samples a union of subspaces approach time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel previous methods for time delay recovery either operate on the analog received signal or require sampling at the nyquist rate of the transmitted pulse in this paper we develop a unified approach to time delay estimation from low rate samples this problem can be formulated in the broader context of sampling over an infinite union of subspaces although sampling over unions of subspaces has been receiving growing interest previous results either focus on unions of finite dimensional subspaces or finite unions the framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate even in the presence of overlapping transmitted pulses and allows for a variety of different sampling methods the sampling rate depends only on the number of multipath components and the transmission rate but not on the bandwidth of the probing signal this result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces by properly manipulating the low rate samples we show that the time delays can be recovered using the well known esprit algorithm combining results from sampling theory with those obtained in the context of direction of arrival estimation we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

generalized sampling expansion on lattices the papoulis generalized sampling expansion gse can be combined with petersen middleton lattice sampling to obtain a gse for n dimensional bandlimited functions such expansions will be discussed for nonperiodic periodic hybrids the periodic and nonperiodic cases are incorporated as special cases for a restrictive special case a parseval like formula is proved for n dimensional bandlimited functions sampled on periodic nonuniform sample sets

quantitative error analysis for the reconstruction of derivatives we present a general fourier based method which provides an accurate prediction of the approximation error when the derivative of a signal s t is continuously reconstructed from uniform point samples or generalized measurements on s this formalism applies to a wide class of convolution based techniques it provides a key tool the frequency error kernel for designing computationally efficient reconstruction schemes which are near optimal in the least squares sense

near tip field for diffraction on square lattice by crack the displacement field near a tip of a finite crack due to the diffraction of a wave on a square lattice is studied the finite section method in the theory of toeplitz operators on ell 2 is invoked as the semi infinite crack diffraction problem is shown equivalent to the inversion of a toeplitz operator a truncation of which appears in the finite crack diffraction problem for the same incident wave the existence and uniqueness of the solution in ell 2 for the semi infinite crack problem is established by an application of the well known krein conditions continuum limit of the semi infinite crack diffraction problem is established in a discrete sobolev space a graphical illustration of convergence in the relevant sobolev norm is also included a low frequency asymptotic approximation of the normalized shear force in vertical bonds ahead of the crack tip recovers the classical crack tip singularity displacement of particles in the vicinity of the crack tip a closed form expression of wh

finite difference schemes on regular triangular grids the phase error and isotropy properties of various finite difference schemes on grids consisting of regular triangles are compared with similar schemes on square grids the comparisons are based on a fourier analysis of semidiscrete solutions to the two dimensional linear convection equation the finite difference schemes presented on the triangular grid include a second order method a compact fourth order method and a modified compact method designed to extend the accurate wave number range of the numerical approximation all of the schemes considered are centered and hence nondissipative in each case the finite difference scheme on the triangular grid reduces the anisotropy of the phase error in comparison with a similar scheme on the square grid

noninvertible gabor transforms time frequency analysis such as the gabor transform plays an important role in many signal processing applications the redundancy of such representations is often directly related to the computational load of any algorithm operating in the transform domain to reduce complexity it may be desirable to increase the time and frequency sampling intervals beyond the point where the transform is invertible at the cost of an inevitable recovery error in this paper we initiate the study of recovery procedures for noninvertible gabor representations we propose using fixed analysis and synthesis windows chosen e g according to implementation constraints and to process the gabor coefficients prior to synthesis in order to shape the reconstructed signal we develop three methods for signal recovery the first follows from the consistency requirement namely that the recovered signal has the same gabor representation as the input signal the second is based on minimization of a worst case error last we develop a recovery technique based on the assumption that the input signal lies in some subspace of l2 we show that for each of the criteria the manipulation of the transform coefficients amounts to a 2d twisted convolution which we show how to perform using a filter bank when the undersampling factor is an integer the processing reduces to standard 2d convolution we provide simulation results demonstrating the advantages and weaknesses of each of the algorithms

performance bounds and design criteria for estimating finite rate of innovation signals in this paper we consider the problem of estimating finite rate of innovation fri signals from noisy measurements and specifically analyze the interaction between fri techniques and the underlying sampling methods we first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method next we provide a bound on the performance achievable using any specific sampling approach essential differences between the noisy and noise free cases arise from this analysis in particular we identify settings in which noise free recovery techniques deteriorate substantially under slight noise levels thus quantifying the numerical instability inherent in such methods this instability which is only present in some families of fri signals is shown to be related to a specific type of structure which can be characterized by viewing the signal model as a union of subspaces finally we develop a methodology for choosing the optimal sampling kernels for linear reconstruction based on a generalization of the karhunen loeve transform the results are illustrated for several types of time delay estimation problems

isotropic polyharmonic b splines scaling functions and wavelets in this paper we use polyharmonic b splines to build multidimensional wavelet bases these functions are nonseparable multidimensional basis functions that are localized versions of radial basis functions we show that rabut s elementary polyharmonic b splines do not converge to a gaussian as the order parameter increases as opposed to their separable b spline counterparts therefore we introduce a more isotropic localization operator that guarantees this convergence resulting into the isotropic polyharmonic b splines next we focus on the two dimensional quincunx subsampling scheme this configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach however up until now the design of appropriate filters for the quincunx scheme has mainly been done using the mcclellan transform in our approach we start from the scaling functions which are the polyharmonic b splines and as such explicitly known and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi orthogonal wavelet transform e g orthonormal b spline and dual the filters are automatically specified by the scaling relations satisfied by these functions we prove that the isotropic polyharmonic b spline wavelet converges to a combination of four gabor atoms which are well separated in the frequency domain we also show that these wavelets are nearly isotropic and that they behave as an iterated laplacian operator at low frequencies we describe an efficient fast fourier transform based implementation of the discrete wavelet transform based on polyharmonic b splines

the structure of shift invariant subspaces of l2 rn using the range function approach to shift invariant spaces in l2 rn we give a simple characterization of frames and riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2 zn this in turn gives a simplified approach to the analysis of frames and riesz families done by gramians and dual gramians we prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator as an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

reconstruction of multidimensional bandlimited signals from nonuniform and generalized samples this paper addresses the problem of multidimensional signal reconstruction from nonuniform or generalized samples typical solutions in the literature for this problem utilize continuous filtering the key result of the current paper is a multidimensional interpolation identity which establishes the equivalence of two multidimensional processing operations one of these uses continuous domain filters whereas the other uses discrete processing this result has obvious benefits in the context of the afore mentioned problem the results here expand and generalize earlier work by other authors on the one dimensional 1 d case potential applications include two dimensional 2 d images and video signals

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

minimizers of cost functions involving nonsmooth data fidelity terms application to the processing of outliers we present a theoretical study of the recovery of an unknown vector x in mathbb r p such as a signal or an image from noisy data y in mathbb r q by minimizing with respect to x a regularized cost function cal f x y psi x y alpha phi x where psi is a data fidelity term phi is a smooth regularization term and alpha 0 is a parameter typically psi x y ax y 2 where a is a linear operator the data fidelity terms psi involved in regularized cost functions are generally smooth functions only a few papers make an exception to this and they consider restricted situations nonsmooth data fidelity terms are avoided in image processing in spite of this we consider both smooth and nonsmooth data fidelity terms our goal is to capture essential features exhibited by the local minimizers of regularized cost functions in relation to the smoothness of the data fidelity term in order to fix the context of our study we consider psi x y sum i psi a i tx y i where a i t are the rows of a and psi is cal c m on mathbb r setminus 0 we show that if psi 1 0 psi 1 0 then typical data y give rise to local minimizers hat x of cal f y which fit exactly a certain number of the data entries there is a possibly large set hat h of indexes such that a i t hat x y i for every i in hat h in contrast if psi is smooth on mathbb r for almost every y the local minimizers of cal f y do not fit any entry of y thus the possibility that a local minimizer fits some data entries is due to the nonsmoothness of the data fidelity term this is a strong mathematical property which is useful in practice by way of application we construct a cost function allowing aberrant data outliers to be detected and to be selectively smoothed our numerical experiments advocate the use of nonsmooth data fidelity terms in regularized cost functions for special purposes in image and signal processing

regularization theory in image restoration the stabilizing functional approach several aspects of the application of regularization theory in image restoration are presented this is accomplished by extending the applicability of the stabilizing functional approach to 2 d ill posed inverse problems inverse restoration is formulated as the constrained minimization of a stabilizing functional the choice of a particular quadratic functional to be minimized is related to the a priori knowledge regarding the original object through a formulation of image restoration as a maximum a posteriori estimation problem this formulation is based on image representation by certain stochastic partial differential equation image models the analytical study and computational treatment of the resulting optimization problem are subsequently presented as a result a variety of regularizing filters and iterative regularizing algorithms are proposed a relationship between the regularized solutions proposed and optimal wiener estimation is identified the filters and algorithms proposed are evaluated through several experimental results

polyharmonic smoothing splines and the multidimensional wiener filtering of fractal like signals motivated by the fractal like behavior of natural images we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the laplacian this type of functional was initially introduced by duchon for the approximation of nonuniformily sampled multidimensional data he proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions rbfs unfortunately this is tedious to implement for images because of the poor conditioning of rbfs and their lack of decay here we present a much more efficient method for the special case of a uniform grid the key idea is to express duchon s solution in a fractional polyharmonic b spline basis that spans the same space as the rbfs this allows us to derive an algorithm where the smoothing is performed by filtering in the fourier domain next we prove that the above smoothing spline can be optimally tuned to provide the mmse estimation of a fractional brownian field corrupted by white noise this is a strong result that not only yields the best linear filter wiener solution but also the optimal interpolation space which is not bandlimited it also suggests a way of using the noisy data to identify the optimal parameters order of the spline and smoothing strength which yields a fully automatic smoothing procedure we evaluate the performance of our algorithm by comparing it against an oracle wiener filter which requires the knowledge of the true noiseless power spectrum of the signal we find that our approach performs almost as well as the oracle solution over a wide range of conditions

interpolating multiwavelet bases and the sampling theorem this paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants as discussed by xia and zhang 1993 for an orthogonal scaling function to support such a sampling theorem the scaling function must be cardinal interpolating they also showed that the only orthogonal scaling function that is both cardinal and of compact support is the haar function which is not continuous this paper addresses the same question but in the multiwavelet context where the situation is different this paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal the scaling functions thereby support a shannon like sampling theorem such wavelet bases are appealing because the initialization of the discrete wavelet transform prefiltering is the identity operator

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

exploiting sparse user activity in multiuser detection the number of active users in code division multiple access cdma systems is often much lower than the spreading gain the present paper exploits fruitfully this a priori information to improve performance of multiuser detectors a low activity factor manifests itself in a sparse symbol vector with entries drawn from a finite alphabet that is augmented by the zero symbol to capture user inactivity the non equiprobable symbols of the augmented alphabet motivate a sparsity exploiting maximum a posteriori probability s map criterion which is shown to yield a cost comprising the l2 least squares error penalized by the p th norm of the wanted symbol vector p 0 1 2 related optimization problems appear in variable selection shrinkage schemes developed for linear regression as well as in the emerging field of compressive sampling cs the contribution of this work to such sparse cdma systems is a gamut of sparsity exploiting multiuser detectors trading off performance for complexity requirements from the vantage point of cs and the least absolute shrinkage selection operator lasso spectrum of applications the contribution amounts to sparsity exploiting algorithms when the entries of the wanted signal vector adhere to finite alphabet constraints

nonideal sampling and interpolation from noisy observations in shift invariant spaces digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous time signal from a sequence of corrupted discrete time samples in this paper a general formulation of this problem is developed that treats the interpolation problem from ideal noisy samples and the deconvolution problem in which the signal is filtered prior to sampling in a unified way the signal reconstruction is performed in a shift invariant subspace spanned by the integer shifts of a generating function where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter several alternative approaches to designing the correction filter are suggested which differ in their assumptions on the signal and noise the classical deconvolution solutions least squares tikhonov and wiener are adapted to our particular situation and new methods that are optimal in a minimax sense are also proposed the solutions often have a similar structure and can be computed simply and efficiently by digital filtering some concrete examples of reconstruction filters are presented as well as simple guidelines for selecting the free parameters e g regularization of the various algorithms

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

nonideal sampling and regularization theory shannon s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some shift invariant space which may or may not be bandlimited in this paper we present some further justification for this type of representation while addressing the issue of the specification of the best reconstruction space we consider a realistic setting where a multidimensional signal is prefiltered prior to sampling and the samples are corrupted by additive noise we adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a tikhonov like continuous domain l2 regularization to obtain the continuous space solution we present theoretical justification for the minimization of this cost functional and show that the globally minimal continuous space solution belongs to a shift invariant space generated by a function generalized b spline that is generally not bandlimited when the sampling is ideal we recover some of the classical smoothing spline estimators the optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a b spline like basis to make the scheme practical we specify the generating functions corresponding to the most popular families of regularization operators derivatives iterated laplacian as well as a new generalized one that leads to a new brand of matern splines we conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error mmse wiener solutions of the generalized sampling problem

linear multiuser detectors for synchronous code division multiple access channels under the assumptions of symbol synchronous transmissions and white gaussian noise the authors analyze the detection mechanism at the receiver comparing different detectors by their bit error rates in the low background noise region and by their worst case behavior in a near far environment where the received energies of the users are not necessarily similar optimum multiuser detection achieves important performance gains over conventional single user detection at the expense of computational complexity that grows exponentially with the number of users it is shown that in the synchronous case the performance achieved by linear multiuser detectors is similar to that of optimum multiuser detection attention is focused on detectors whose linear memoryless transformation is a generalized inverse of the matrix of signature waveform crosscorrelations and on the optimum linear detector it is shown that the generalized inverse detectors exhibit the same degree of near far resistance as the optimum multiuser detectors the optimum linear detector is obtained

reduced dimension multiuser detection we present a reduced dimension multiuser detector rd mud structure for synchronous systems that significantly decreases the number of required correlation branches at the receiver front end while still achieving performance similar to that of the conventional matched filter mf bank rd mud exploits the fact that in some wireless systems the number of active users may be small relative to the total number of users in the system hence the ideas of analog compressed sensing may be used to reduce the number of correlators the correlating signals used by each correlator are chosen as an appropriate linear combination of the users spreading waveforms we derive the probability of symbol error when using two methods for recovery of active users and their transmitted symbols the reduced dimension decorrelating rdd detector which combines subspace projection and thresholding to determine active users and sign detection for data recovery and the reduced dimension decision feedback rddf detector which combines decision feedback matching pursuit for active user detection and sign detection for data recovery we derive probability of error bounds for both detectors and show that the number of correlators needed to achieve a small probability of symbol error is on the order of the logarithm of the number of users in the system the theoretical performance results are validated via numerical simulations

multiscale semilocal interpolation with antialiasing aliasing is a common artifact in low resolution lr images generated by a downsampling process recovering the original high resolution image from its lr counterpart while at the same time removing the aliasing artifacts is a challenging image interpolation problem since a natural image normally contains redundant similar patches the values of missing pixels can be available at texture relevant lr pixels based on this we propose an iterative multiscale semilocal interpolation method that can effectively address the aliasing problem the proposed method estimates each missing pixel from a set of texture relevant semilocal lr pixels with the texture similarity iteratively measured from a sequence of patches of varying sizes specifically in each iteration top texture relevant lr pixels are used to construct a data fidelity term in a maximum a posteriori estimation and a bilateral total variation is used as the regularization term experimental results compared with existing interpolation methods demonstrate that our method can not only substantially alleviate the aliasing problem but also produce better results across a wide range of scenes both in terms of quantitative evaluation and subjective visual quality

decision feedback multiuser detection a systematic approach a systematic approach to decision feedback multiuser detection is introduced for the joint detection of symbols of k simultaneously transmitting users of a synchronous correlated waveform multiple access cwma channel with gaussian noise a new performance criterion called symmetric energy is defined which is a low noise indicator of the joint error rate that at least one user is detected erroneously even the best linear detectors can perform poorly in terms of symmetric energy compared to the maximum likelihood detector a general class of decision feedback detectors dfds is defined with o k implementational complexity per user the symmetric energy of arbitrary dfd and bounds on their asymptotic effective energy aee performance are obtained along with an exact bit error rate and aee analysis for the decorrelating dfd the optimum dfd that maximizes symmetric energy is obtained each one of the k optimum decorrelating and conventional dfds that correspond to the k orders in which the users can be detected are shown to outperform the linear optimum decorrelating and conventional detectors respectively in terms of symmetric energy moreover algorithms are obtained for determining the choice of order of detection for the three dfds which guarantee that they uniformly user wise outperform their linear counterparts in addition to optimality in symmetric energy it is also shown that under certain conditions the optimum dfd achieves the aee performance of the exponentially complex maximum likelihood detector for all users simultaneously none of the results of this paper make the perfect feedback assumption the implications of our work on power control for multiuser detection are also discussed

generalized smoothing splines and the optimal discretization of the wiener filter we introduce an extended class of cardinal l sup l splines where l is a pseudo differential operator satisfying some admissibility conditions we show that the l sup l spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional spl par ls spl par sub l2 sup 2 subject to the interpolation constraint next we consider the corresponding regularized least squares estimation problem which is more appropriate for dealing with noisy data the criterion to be minimized is the sum of a quadratic data term which forces the solution to be close to the input samples and a smoothness term that privileges solutions with small spline energies here too we find that the optimal solution among all possible functions is a cardinal l sup l spline we show that this smoothing spline estimator has a stable representation in a b spline like basis and that its coefficients can be computed by digital filtering of the input signal we describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of l is rational which corresponds to the case of exponential splines we justify these algorithms statistically by establishing an equivalence between l sup l smoothing splines and the minimum mean square error mmse estimation of a stationary signal corrupted by white gaussian noise in this model based formulation the optimum operator l is the whitening filter of the process and the regularization parameter is proportional to the noise variance thus the proposed formalism yields the optimal discretization of the classical wiener filter together with a fast recursive algorithm it extends the standard wiener solution by providing the optimal interpolation space we also present a bayesian interpretation of the algorithm

multiuser detection in a dynamic environment part i user identification and data detection in random access communication systems the number of active users varies with time and has considerable bearing on receiver s performance thus techniques aimed at identifying not only the information transmitted but also that number play a central role in those systems an example of application of these techniques can be found in multiuser detection mud in typical mud analyses receivers are based on the assumption that the number of active users is constant and known at the receiver and coincides with the maximum number of users entitled to access the system this assumption is often overly pessimistic since many users might be inactive at any given time and detection under the assumption of a number of users larger than the real one may impair performance the main goal of this paper is to introduce a general approach to the problem of identifying active users and estimating their parameters and data in a random access system where users are continuously entering and leaving the system the tool whose use we advocate is random set theory rst applying this we derive optimum receivers in an environment where the set of transmitters comprises an unknown number of elements in addition we can derive bayesian filter equations which describe the evolution with time of the a posteriori probability app density of the unknown user parameters and use this density to derive optimum detectors in this paper we restrict ourselves to interferer identification and data detection while in a companion paper we shall examine the more complex problem of estimating users parameters

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

spectral efficiency of cdma with random spreading the cdma channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods furthermore its analysis provides a comparison baseline for cdma channels with deterministic signature waveforms spanning one symbol period we analyze the spectral efficiency total capacity per chip as a function of the number of users spreading gain and signal to noise ratio and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading white gaussian background noise and equal power synchronous users are assumed the following receivers are analyzed a optimal joint processing b single user matched filtering c decorrelation and d mmse linear processing

diffraction of waves on square lattice by semi infinite crack in this paper a discrete analogue of sommerfeld half plane diffraction is investigated the two dimensional problem of diffraction on a square lattice of a plane transverse wave by a semi infinite crack is solved the discrete wiener hopf method has been used to obtain the exact solution of the discrete helmholtz equation with input data prescribed on the crack boundary sites due to a time harmonic incident wave it is established that there exists a unique saddle point for the diffraction integral and its properties are characterized an asymptotic approximation of the solution in the far field is provided and for some values of the frequency it is compared with the numerical solution of the diffraction problem using a finite grid a low frequency approximation of the solution in integral form recovers the classical continuum solution at sufficiently large frequency in the pass band the effects due to discreteness and anisotropy appear the analysis is relevant to a 5 point discretization bas

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

optimal lattices for sampling the generalization of the sampling theorem to multidimensional signals is considered with or without bandwidth constraints the signal is modeled as a stationary random process and sampled on a lattice exact expressions for the mean square error of the best linear interpolator are given in the frequency domain moreover asymptotic expansions are derived for the average mean square error when the sampling rate tends to zero and infinity respectively this makes it possible to determine the optimal lattices for sampling in the low rate sampling case or equivalently for rough processes the optimal lattice is the one which solves the packing problem whereas in the high rate sampling case or equivalently for smooth processes the optimal lattice is the one which solves the dual packing problem in addition the best linear interpolation is compared with ideal low pass filtering cardinal interpolation

finitely supported rm l _2 optimal kernels for digital signal interpolation interpolation is responsible for digital signal resampling and can significantly degrade the original signal quality if not done properly for many years optimal interpolation algorithms were sought within constrained classes of interpolation kernel functions we derive a new family of unconstrained finitely supported l 2 optimal interpolation kernels h l x and compare their properties to the previously known results our research demonstrates that l 2 optimal kernels provide superior interpolation quality and can be efficiently applied to any digital signal of arbitrary nature bandwidth and dimensionality

a probabilistic approach to pattern matching in the continuous domain the goal of this paper is to solve the following basic problem given discrete noisy samples from a continuous signal compute the probability distribution of its distance from a fixed template as opposed to the typical restoration problem which considers a single optimal signal the computation of the entire probability distribution necessitates integrating over the entire signal space to achieve this we apply path integration techniques the problem is studied in one and two dimensions and an accurate solution as well as an efficient approximation scheme are provided

l 2 sampled signal reconstruction with causality constraints part i setup and solutions this paper studies the problem of reconstructing an analog signal from its sampled measurements in which the sampler acquisition device is given and the reconstructor interpolator hold is the design parameter we formulate this problem as an l2 wiener kalman filtering like optimization problem and place the main emphasis on a systematic incorporation of causality constraints into the design procedure specifically the optimization problem is solved under the constraint that the interpolation kernel is l causal for a given i n i e that its impulse response is zero in the time interval ih where h is the sampling period we present a closed form state space solution of the problem whose computational complexity does not depend on i and which can be efficiently calculated and implemented

local approximation by certain spaces of exponential polynomials approximation order of exponential box splines and related interpolation problems local approximation order to smooth complex valued functions by a finite dimensional space 7b h spanned by certain products of exponentials by polynomials is investigated the results obtained together with a suitable quasi interpolation scheme are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline the analysis of a typical 7b h is based here on the identification of its dual with a certain space 7b p of multivariate polynomials this point of view allows us to solve a class of multivariate interpolation problems by the polynomials from 7b p

a perfectly matched layer for the helmholtz equation in a semi infinite strip the perfectly matched layer pml has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains we develop a discretization to solve the helmholtz equation in an infinite two dimensional strip we solve the interior equation using high order finite differences schemes the combined helmholtz pml problem is then analyzed for the parameters that give the best performance we show that the use of local high order methods in the physical domain coupled with a specific second order approximation in the pml yields global high order accuracy in the physical domain we discuss the impact of the parameters on the effectiveness of the pml numerical results are presented to support the analysis

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

efficient minimization methods of mixed l 2 l 1 and l 1 l 1 norms for image restoration image restoration problems are often solved by finding the minimizer of a suitable objective function usually this function consists of a data fitting term and a regularization term for the least squares solution both the data fitting and the regularization terms are in the ell 2 norm in this paper we consider the least absolute deviation lad solution and the least mixed norm lmn solution for the lad solution both the data fitting and the regularization terms are in the ell 1 norm for the lmn solution the regularization term is in the ell 1 norm but the data fitting term is in the ell 2 norm since images often have nonnegative intensity values the proposed algorithms provide the option of taking into account the nonnegativity constraint the lmn and lad solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods at each iteration of the interior point method a structured linear system must be solved the preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems experimental results are used to demonstrate the effectiveness of our approach we also show the quality of the restored images using the minimization of mixed ell 2 ell 1 and ell1 ell 1 norms is better than that using only the ell 2 norm

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

random sampling in shift invariant spaces abstract the set of sampling in a shift invariant space plays an important role in signal processing and has many applications this paper addresses the problem when some randomly chosen samples x x j j j form a set of sampling in a shift invariant space that is when the inequality of the form c p f l p r d p x j x f x j p c p f l p r d p holds uniformly for all functions f in a shift invariant space where c p and c p are positive constants 1 p we prove that with overwhelming probability the above sampling inequality holds for certain compact subsets of the shift invariant space when the sampling size is sufficiently large

regularized interpolation for noisy images interpolation is the means by which a continuously defined model is fit to discrete data samples when the data samples are exempt of noise it seems desirable to build the model by fitting them exactly in medical imaging where quality is of paramount importance this ideal situation unfortunately does not occur in this paper we propose a scheme that improves on the quality by specifying a tradeoff between fidelity to the data and robustness to the noise we resort to variational principles which allow us to impose smoothness constraints on the model for tackling noisy data based on shift rotation and scale invariant requirements on the model we show that the lp norm of an appropriate vector derivative is the most suitable choice of regularization for this purpose in addition to tikhonov like quadratic regularization this includes edge preserving total variation like tv regularization we give algorithms to recover the continuously defined model from noisy samples and also provide a data driven scheme to determine the optimal amount of regularization we validate our method with numerical examples where we demonstrate its superiority over an exact fit as well as the benefit of tv like nonquadratic regularization over tikhonov like quadratic regularization

a minimum squared error framework for generalized sampling we treat the problem of reconstructing a signal from its nonideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a hilbert space our formulation is general and includes as special cases reconstruction from finitely many samples as well as uniform sampling of continuous time signals which are not necessarily bandlimited to obtain a good approximation of the signal in the reconstruction space from its samples we suggest two design strategies that attempt to minimize the squared norm error between the signal and its reconstruction the approaches we propose differ in their assumptions on the input signal if the signal is known to lie in an appropriately chosen subspace then we propose a method that achieves the minimal squared error on the other hand when the signal is not restricted we show that the minimal norm reconstruction cannot generally be obtained instead we suggest minimizing the worst case squared error between the reconstructed signal and the best possible but usually unattainable approximation of the signal within the reconstruction space we demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem

many access for small packets based on precoding and sparsity aware recovery modern mobile terminals produce massive small data packets for these short length packets it is inefficient to follow the current multiple access schemes to allocate transmission resources due to heavy signaling overhead we propose a non orthogonal many access scheme that is well suited for the future communication systems equipped with many receive antennas the system is modeled as having a block sparsity pattern with unknown sparsity level i e unknown number of transmitted messages block precoding is employed at each single antenna transmitter to enable the simultaneous transmissions of many users the number of simultaneously served active users is allowed to be even more than the number of receive antennas sparsity aware recovery is designed at the receiver for joint user detection and symbol demodulation to reduce the effects of channel fading on signal recovery normalized block orthogonal matching pursuit bomp algorithm is introduced and based on its approximate performance analysis we develop interference cancellation based bomp icbomp algorithm the icbomp performs error correction and detection in each iteration of the normalized bomp simulation results demonstrate the effectiveness of the proposed scheme in small packet services as well as the advantages of icbomp in improving signal recovery accuracy and reducing computational cost

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

on the fundamental solutions for the difference helmholtz operator in this paper we construct the difference approximation of the helmholtz operator on a square grid and we analyze asymptotic properties of the difference operator fundamental solutions we show that for a sommerfeld type asymptotic condition there exists a fundamental solution that is determined in a unique way

sampling and reconstruction of spatial fields using mobile sensors spatial sampling is traditionally studied in a static setting where static sensors scattered around space take measurements of the spatial field at their locations in this paper we study the emerging paradigm of sampling and reconstructing spatial fields using sensors that move through space we show that mobile sensing offers some unique advantages over static sensing in sensing bandlimited spatial fields since a moving sensor encounters such a spatial field along its path as a time domain signal a time domain anti aliasing filter can be employed prior to sampling the signal received at the sensor such a filtering procedure when used by a configuration of sensors moving at constant speeds along equispaced parallel lines leads to a complete suppression of spatial aliasing in the direction of motion of the sensors we analytically quantify the advantage of using such a sampling scheme over a static sampling scheme by computing the reduction in sampling noise due to the filter we also analyze the effects of nonuniform sensor speeds on the reconstruction accuracy using simulation examples we demonstrate the advantages of mobile sampling over static sampling in practical problems we extend our analysis to sampling and reconstruction schemes for monitoring time varying bandlimited fields using mobile sensors we demonstrate that in some situations we require a lower density of sensors when using a mobile sensing scheme instead of the conventional static sensing scheme the exact advantage is quantified for a problem of sampling and reconstructing an audio field

on sampling in shift invariant spaces a necessary and sufficient condition for sampling in the general framework of shift invariant spaces is derived then this result is applied respectively to the regular sampling and the perturbation of regular sampling in shift invariant spaces a simple necessary and sufficient condition for regular sampling in shift invariant spaces is attained furthermore an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift invariant spaces the derived estimate is easy to calculate and shown to be optimal in some shift invariant spaces the algorithm to calculate the reconstruction frame is also presented

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

estimation and model selection for an ide based spatio temporal model a state space model of the stochastic spatio temporal integro difference equation ide is derived based on multidimensional sampling theory the dimensions of the state space and parameter space of the model are identified from the spatial bandwidth of the system and the support of the redistribution kernel of the ide when both the bandwidth and the kernel support are unknown a method to propose a number of state space and parameter space dimensions is presented these chosen dimensions result in a number of candidate model structures bayesian model selection making use of bayes factor the data augmentation algorithm and importance sampling is then used to identify the model best suited to represent the data in a maximum a posteriori sense

diffraction of waves on triangular lattice by a semi infinite rigid constraint and crack abstract an exact solution is provided for a discrete analog of each of the two sommerfeld diffraction problems using a triangular lattice model with nearest neighbor interactions deforming in the anti plane direction the discrete helmholtz equation with time harmonic data prescribed on semi infinite row s of lattice sites is solved using the discrete wiener hopf method an asymptotic expression of the exact solution given in the form of a contour integral has been obtained in far field using the standard approximation of the diffraction integrals based on the method of stationary phase in case of the rigid constraint diffraction the displacement of particles on the semi infinite row complementing the constrained lattice sites as well as on the adjacent row is presented in closed form as a discrete convolution for the crack diffraction problem the length of both types of slant bonds on the semi infinite row complementing the crack as well as the crack opening displacement is given in a similar form but in terms of the fourier coefficients of the associated wiener hopf kernel which are not available in closed form the displacement field associated with the scattered waves at sites far from the defect tip as well as few sites near the tip is compared graphically with that of a numerical solution on a finite grid both discrete sommerfeld problems are naturally relevant to their continuous counterparts involving the traditional helmholtz equation that model the diffraction of electromagnetic and acoustic waves by a semi infinite screen based on 7 point discretization on a triangular grid