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72
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 341 (27 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Nonuniform sampling and reconstruction in shiftinvariant spaces
 In SIAM Review
, 2001
"... Abstract. This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by br ..."
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Cited by 221 (13 self)
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Abstract. This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces 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 welldefined within the setting of shiftinvariant 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 Lpspaces. 1.
Efficient numerical methods in nonuniform sampling theory
, 1995
"... We present a new “second generation” reconstruction algorithm for irregular sampling, i.e. for the problem of recovering a bandlimited function from its nonuniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named ..."
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Cited by 93 (10 self)
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We present a new “second generation” reconstruction algorithm for irregular sampling, i.e. for the problem of recovering a bandlimited function from its nonuniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems. The choice of ”adaptive weights” can be seen as a simple but very efficient method of preconditioning. Further substantial acceleration is achieved by utilizing the Toeplitztype structure of the system matrix. This new algorithm can handle problems of much larger dimension and condition number than have been accessible so far. Furthermore, if some gaps between samples are large, then the algorithm can still be used as a very efficient extrapolation method across the gaps.
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator
"... We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically t ..."
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Cited by 79 (9 self)
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We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about nonuniform sampling in shiftinvariant spaces. 1.
Image reconstruction and enhanced resolution imaging from irregular samples
 IEEE Transactions on Geoscience and Remote Sensing
, 2001
"... Abstract—While high resolution, regularly gridded observations are generally preferred in remote sensing, actual observations are often not evenly sampled and have lowerthandesired resolution. Hence, there is an interest in resolution enhancement and image reconstruction. This paper discusses a ge ..."
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Cited by 79 (47 self)
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Abstract—While high resolution, regularly gridded observations are generally preferred in remote sensing, actual observations are often not evenly sampled and have lowerthandesired resolution. Hence, there is an interest in resolution enhancement and image reconstruction. This paper discusses a general theory and techniques for image reconstruction and creating enhanced resolution images from irregularly sampled data. Using irregular sampling theory, we consider how the frequency content in aperture functionattenuated sidelobes can be recovered from oversampled data using reconstruction techniques, thus taking advantage of the high frequency content of measurements made with nonideal aperture filters. We show that with minor modification, the algebraic reconstruction technique (ART) is functionally equivalent to Grochenig’s irregular sampling reconstruction algorithm. Using simple Monte Carlo simulations, we compare and contrast the performance of additive ART, multiplicative ART, and the scatterometer image reconstruction (SIR) (a derivative of multiplicative ART) algorithms with and without noise. The reconstruction theory and techniques have applications with a variety of sensors and can enable enhanced resolution image production from many nonimaging sensors. The technique is illustrated with ERS2 and SeaWinds scatterometer data. Index Terms—Irregular samples, reconstruction, resolution enhancement, sampling. I.
BeurlingLandauType Theorems For NonUniform Sampling In Shift Invariant Spline Spaces
, 1999
"... . Under the appropriate definition of sampling density D # , a function f that belongs to a shift invariant space can be reconstructed in a stable way from its nonuniform samples only if D # # 1. This result is similar to Landau's result for the PaleyWiener space B 1/2 . If the shift invari ..."
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Cited by 40 (9 self)
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. Under the appropriate definition of sampling density D # , a function f that belongs to a shift invariant space can be reconstructed in a stable way from its nonuniform samples only if D # # 1. This result is similar to Landau's result for the PaleyWiener space B 1/2 . If the shift invariant space consists of polynomial splines, then we show that D # < 1 is su#cient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B 1/2 . In the sampling problem one seeks to recover a function f from its samples {f(x j ) : j # ZZ}. Clearly, this problem is illposed, and becomes meaningful only after some a priori conditions on f . Until recently, the standard assumption is that f is bandlimited, i.e., supp f # [#, #] [7, 19, 29]. Although this is a reasonable assumption, in many applications di#erent hypothesis on the functions are desirable, e.g., for taking into account real acquisition and reconstruction devices, for obtaini...
Random sampling of multivariate trigonometric polynomials
 SIAM J. Math. Anal
, 2004
"... We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for th ..."
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Cited by 32 (3 self)
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We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for the associated Vandermondetype and Toeplitzlike matrices. The results provide a solid theoretical foundation for some efficient numerical algorithms that are already in use.
Stability Results for Scattered Data Interpolation by Trigonometric Polynomials
 SIAM J. Sci. Comput
, 2007
"... A fast and reliable algorithm for the optimal interpolation of scattered data on the torus Td by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main ..."
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Cited by 31 (14 self)
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A fast and reliable algorithm for the optimal interpolation of scattered data on the torus Td by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main result is that under mild assumptions the total complexity for solving the interpolation problem at M arbitrary nodes is of order O(M log M). This result is obtained by the use of localised trigonometric kernels where the localisation is chosen in accordance to the spatial dimension d. Numerical examples show the efficiency of the new algorithm.
Optimal SubNyquist Nonuniform Sampling and Reconstruction for Multiband Signals
, 2001
"... We study the problem of optimal subNyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau ..."
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Cited by 29 (3 self)
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We study the problem of optimal subNyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's lower bound equal to the measure of . For signals with sparse , this rate can be much smaller than the Nyquist rate. Unfortunately, the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a recent study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with that tiles under translation. For signals with nontiling , which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with subNyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate.
Nonuniform average sampling and reconstruction in multiply generated shiftinvariant spaces
 Constr. Approx
"... Abstract. From an average (ideal) sampling/reconstruction process, the question arises whether and how the original signal can be recovered from its average (ideal) samples. We consider the above question under the assumption that the original signal comes from a prototypical space modelling signals ..."
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Cited by 23 (12 self)
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Abstract. From an average (ideal) sampling/reconstruction process, the question arises whether and how the original signal can be recovered from its average (ideal) samples. We consider the above question under the assumption that the original signal comes from a prototypical space modelling signals with finite rate of innovation, which includes finitelygenerated shiftinvariant spaces, twisted shiftinvariant spaces associated with Gabor frames and Wilson bases, and spaces of polynomial splines with nonuniform knots as its special cases. We show that the displayer associated with an average (ideal) sampling/reconstruction process, that has welllocalized average sampler, can be found to be welllocalized. We prove that the reconstruction process associated with an average (ideal) sampling process is robust, locally behaved, and finitely implementable, and thus we conclude that the original signal can be approximately recovered from its incomplete average (ideal) samples with noise in real time. Most of our results in this paper are new even for the special case that the original signal comes from a finitelygenerated shiftinvariant space. average sampling, ideal sampling, signals with finite rate of innovation, shiftKey words. invariant spaces