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16
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.
Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
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Cited by 121 (22 self)
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The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
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.
Nonuniform interpolation of noisy signals using support vector machines
 IEEE Transactions on Signal Processing
, 2007
"... Abstract—The problem of signal interpolation has been intensively studied in the Information Theory literature, in conditions such as unlimited band, nonuniform sampling, and presence of noise. During the last decade, support vector machines (SVM) have been widely used for approximation problems, i ..."
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Cited by 7 (2 self)
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Abstract—The problem of signal interpolation has been intensively studied in the Information Theory literature, in conditions such as unlimited band, nonuniform sampling, and presence of noise. During the last decade, support vector machines (SVM) have been widely used for approximation problems, including function and signal interpolation. However, the signal structure has not always been taken into account in SVM interpolation. We propose the statement of two novel SVM algorithms for signal interpolation, specifically, the primal and the dual signal model based algorithms. Shiftinvariant Mercer’s kernels are used as building blocks, according to the requirement of bandlimited signal. The sinc kernel, which has received little attention in the SVM literature, is used for bandlimited reconstruction. Wellknown properties of general SVM algorithms (sparseness of the solution, robustness, and regularization) are explored with simulation examples, yielding improved results with respect to standard algorithms, and revealing good characteristics in nonuniform interpolation of noisy signals. Index Terms—Dual signal model, interpolation, Mercer’s kernel, nonuniform sampling, primal signal model, signal, sinc
Implementations of Shannon’s sampling theorem, a timefrequency approach
 Sampl. Theory Signal Image Process
, 2005
"... Shannon's sampling theorem quanti¯es the Fourier domain periodization introduced by the equidistant sampling of a bandlimited signal when the sampling rate is at least as fast as the Nyquist rate dictated by the signal's bandwidth. If sampled faster than the Nyquist rate, i.e., oversampli ..."
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Cited by 6 (1 self)
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Shannon's sampling theorem quanti¯es the Fourier domain periodization introduced by the equidistant sampling of a bandlimited signal when the sampling rate is at least as fast as the Nyquist rate dictated by the signal's bandwidth. If sampled faster than the Nyquist rate, i.e., oversampling, a reconstruction composed of highly localized atoms is possible, allowing for practical applications where only a truncated set of samples is available. More speci¯cally, it is known that rootexponential accuracy can be achieved by constructing atoms whose Fourier transform (¯lter) is in¯nitely di®erentiable and compactly supported in the appropriate bandwidth. Unfortunately, there is no known compactly supported in¯nitely smooth ¯lter whose corresponding atom has a known explicit representation; and as such, an approximation of the atom is required for the implementation in the time domain. By considering ¯lters with Gevrey regularity, we obtain rootexponential localization for the atom, and an e±cient truncated Gabor approximation of the ¯lter and atom. Furthermore, we present an alternative error decomposition that allows for the complete rigorous analysis of the error in truncating the signal, and of the error introduced in approximating the ¯lter and atom. By scaling the approximation order appropriately, the rootexponential convergence rate is not adversely a®ected by the ¯lter's approximation.
Nonuniform Sampling: Exact Reconstruction From Nonuniformly Distributed . . .
, 2002
"... this article, we discuss the problem of reconstructing a function f in a latticeinvariant subspace of L (IR ) from a family of nonuniformly distributed, weightedaverages fhf; x j i : j 2 Jg using an approximationprojection iterative algorithm ..."
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Cited by 3 (0 self)
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this article, we discuss the problem of reconstructing a function f in a latticeinvariant subspace of L (IR ) from a family of nonuniformly distributed, weightedaverages fhf; x j i : j 2 Jg using an approximationprojection iterative algorithm
An realtime algorithm for time decoding machines
, 2005
"... Timeencoding is a realtime asynchronous mechanism of mapping the information contained in the amplitude of a bandlimited signal into a time sequence. Time decoding algorithms recover the signal from the time sequence. Under an appropriate Nyquisttype rate condition the signal can be perfectly rec ..."
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Cited by 2 (2 self)
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Timeencoding is a realtime asynchronous mechanism of mapping the information contained in the amplitude of a bandlimited signal into a time sequence. Time decoding algorithms recover the signal from the time sequence. Under an appropriate Nyquisttype rate condition the signal can be perfectly recovered. The algorithm for perfect recovery calls, however, for the computation of a pseudoinverse of an infinite dimensional matrix. We present a simple algorithm for local signal recovery and construct a stitching algorithm for realtime signal recovery. We also provide a recursive algorithm for computing the pseudoinverse of a family of finitedimensional matrices.
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
, 2014
"... In this paper, we consider the problem of recovering a compactlysupported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give ..."
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Cited by 2 (1 self)
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In this paper, we consider the problem of recovering a compactlysupported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Gröchenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing this dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm – based on the existing generalized sampling framework – that allows for stable and quasioptimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice – jittered, radial and spiral sampling schemes – and provide several examples illustrating the effectiveness of our approach when tested on these schemes. 1
Fast reconstruction from nonuniform samples in shiftinvariant spaces
 in Proc. of EUSIPCO
, 2006
"... We propose a new approach for signal reconstruction from nonuniform samples, without constraints on their locations. We look for a function that belongs to a linear shiftinvariant space, andminimizes a variational criterion that is a weighted sum of a leastsquares data term and a quadratic term p ..."
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Cited by 1 (1 self)
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We propose a new approach for signal reconstruction from nonuniform samples, without constraints on their locations. We look for a function that belongs to a linear shiftinvariant space, andminimizes a variational criterion that is a weighted sum of a leastsquares data term and a quadratic term penalizing the lack of smoothness. This leads to a resolutiondependent solution, that can be computed exactly by a fast noniterative algorithm. 1.