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

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

on binary cyclic codes of odd lengths from 101 to 127 all binary cyclic codes of odd lengths are checked from 101 to 127 to find codes which are better than those in a table by t verhoeff 1989 there are five such cases namely 117 36 32 117 37 29 117 42 26 117 49 24 and 127 36 35 cyclic codes according to verhoeff s table the previously known ranges of the highest minimum distance were 28 40 28 40 25 37 22 32 and 32 46 respectively applying constructions x and y1 120 37 32 and 108 28 32 codes were found moreover the highest minimum distances that cyclic codes of length 127 can attain are determined

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

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

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

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

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

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

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

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

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

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

lengthening and the gilbert varshamov bound we use lengthening and an enhanced version of the gilbert varshamov lower bound for linear codes to construct a large number of record breaking codes our main theorem may be seen as a closure operation on databases

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

generalizations of the projection methods with application to sor theory for hermitian positive semidefinite linear systems ann n complex matrixb is calledparacontracting if b 2 1 and 0 x n i b bx 2 x 2 we show that a productb b k b k 1 b1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofb in terms of the subdominant singular values of theb i s and in terms of the angles between certain subspaces our results here extend earlier results due to halperin and due to smith solomon and wagner we also determine necessary and sufficient conditions forn numbers in the interval 0 1 to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product in the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the sor iteration matrix associated with ann n hermitian positive semidefinite matrixa none of whose diagonal entries vanish

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

further results on paracontracting matrices and correction to optimal semistable control in ad hoc network systems a sequential two stage approach this technical note has twofold aspects the first aspect is to present some new results on characterization of matrix paracontraction and the second aspect is to correct some errors in the author s previous publication by means of these newly developed results for paracontraction specifically new necessary and or sufficient conditions to characterize paracontracting matrices using discrete time semistability singular value decomposition normal matrices and range symmetric matrices are provided using these results one can rectify an error in the author s previous publication

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

on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

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

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

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

a finite gilbert varshamov bound for pure stabilizer quantum codes a finite gilbert varshamov gv bound for pure stabilizer binary and nonbinary quantum error correcting codes is presented in analogy to the gv bound for classical codes by using several enumerative results in finite unitary geometry from this quantum gv bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes

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