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182
Robust Recovery of Signals From a Structured Union of Subspaces
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structu ..."
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Cited by 112 (43 self)
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Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a blocksparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed ℓ2/ℓ1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
ForWaRD: FourierWavelet Regularized Deconvolution for IllConditioned Systems
 IEEE Trans. on Signal Processing
, 2002
"... We propose an efficient, hybrid FourierWavelet Regularized Deconvolution (ForWaRD) al gorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's sparse representation of the colored noise inhere ..."
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Cited by 90 (2 self)
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We propose an efficient, hybrid FourierWavelet Regularized Deconvolution (ForWaRD) al gorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's sparse representation of the colored noise inherent in deconvolution, while the wavelet shrinkage exploits the wavelet do main's sparse representation of piecewise smooth signals and images. We derive the optimal balance between the amount of Fourier and wavelet regularization by optimizing an approxi mate meansquarederror (MSE) metric and find that signals with sparser wavelet representa tions require less Fourier shrinkage. ForWaRD is applicable to all illconditioned deconvolution problems, unlike the purely waveletbased Wavelet Vaguelette Deconvolution (WVD), and its es timate features minimal ringing, unlike purely Fourierbased Wiener deconvolution. We analyze ForWaRD's MSE decay rate as the number of samples increases and demonstrate its improved performance compared to the optimal WVD over a wide range of practical samplelengths.
A chronology of interpolation: From ancient astronomy to modern signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into histo ..."
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Cited by 61 (0 self)
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This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolutionbased interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.
Compressed Sensing of Analog Signals in ShiftInvariant Spaces
, 2009
"... A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that on ..."
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Cited by 50 (33 self)
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A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that only part of the bandwidth is used. In this paper, we develop methods for lowrate sampling of continuoustime sparse signals in shiftinvariant (SI) spaces, generated by m kernels with period T. We model sparsity by treating the case in which only k out of the m generators are active, however, we do not know which k are chosen. We show how to sample such signals at a rate much lower than m/T, which is the minimal sampling rate without exploiting sparsity. Our approach combines ideas from analog sampling in a subspace with a recently developed block diagram that converts an infinite set of sparse equations to a finite counterpart. Using these two components we formulate our problem within the framework of finite compressed sensing (CS) and then rely on algorithms developed in that context. The distinguishing feature of our results is that in contrast to standard CS, which treats finitelength vectors, we consider sampling of analog signals for which no underlying finitedimensional model exists. The proposed framework allows to extend much of the recent literature on CS to the analog domain.
Sampling and reconstruction of signals with finite rate of innovation in the presence of noise
 IEEE Transactions on Signal Processing
, 2005
"... Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely, certain signals of finite rate of innovation [24]. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time an ..."
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Cited by 45 (1 self)
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Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely, certain signals of finite rate of innovation [24]. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples. In order to prove sampling theorems, Vetterli et al. considered the case of deterministic, noiseless signals, and developed algebraic methods that lead to perfect reconstruction. However, when noise is present, many of those schemes can become illconditioned. In this paper, we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved, more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction. We analyze in detail a signal made up of a stream of Diracs and develop algorithmic tools that will be used as a basis in all constructions. While some of the techniques have been already encountered in the spectral estimation framework, we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components. For classes of periodic signals, such as piecewise polynomials and nonuniform splines, we propose novel algebraic approaches that solve the sampling problem in the Laplace domain, after appropriate windowing. Building on the results for periodic signals, we extend our analysis to finitelength signals and develop schemes based on a Gaussian kernel, which avoid the problem of illconditioning by proper weighting of the data matrix. Our methods use structured linear systems and robust algorithmic solutions, which we show through simulation results.
Wavelet theory demystified
 IEEE Trans. Signal Process
, 2003
"... Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to red ..."
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Cited by 45 (22 self)
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Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a selfcontained, accessible fashion. In particular, we prove that the Bspline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in thesense and a sharper theorem stating that smoothness implies order. Index Terms—Approximation order, Besov spaces, Hölder smoothness, multiscale differentiation, splines, vanishing moments, wavelets. I.
Robust recovery of signals from a union of subspaces
 IEEE TRANS. INFORM. THEORY
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees a unique signal consistent with the given measurements is that x lies in a known subspace. Recently, there has been growing interest in non ..."
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Cited by 39 (13 self)
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Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees a unique signal consistent with the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x is assumed to lie in a union of subspaces. An example is the case in which x is a finite length vector that is sparse in a given basis. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a finite union of finite dimensional spaces and the samples are modelled as inner products with an arbitrary set of sampling functions. We first develop conditions under which unique and stable recovery of x is possible, albeit with algorithms that have combinatorial complexity. To derive an efficient and robust recovery algorithm, we then show that our problem can be formulated as that of recovering a block sparse vector, namely a vector whose nonzero elements appear in fixed blocks. To solve this problem, we suggest minimizing a mixed ℓ2/ℓ1 norm subject to the measurement equations. We then develop equivalence conditions under which the proposed convex algorithm is guaranteed to recover the original signal. These results rely on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. A special case of the proposed framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Specializing our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
Generalizations of the sampling theorem: Seven decades after Nyquist
 IEEE Trans. Circuits and Systems
, 2001
"... Abstract. 1 The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well known form is Shannon’s uniform sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also wellkno ..."
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Cited by 37 (3 self)
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Abstract. 1 The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well known form is Shannon’s uniform sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also wellknown. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed.
Cardinal exponential splines: Part I—Theory and filtering algorithms
 IEEE Trans. Signal Process
, 2005
"... Abstract—Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and selfcontained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding Bspline basis functi ..."
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Cited by 36 (13 self)
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Abstract—Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and selfcontained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding Bspline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz bounds). We show that the exponential Bspline framework allows an exact implementation of continuoustime signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete Bspline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the thorder decay of the Papproximation error as a function of the knot spacing. Index Terms—Continuoustime signal processing, convolution, differential operators, Green functions, interpolation, modulation, multiresolution approximation, splines. I.
D.X.: Shannon sampling and function reconstruction from point values
 Bull. Am. Math. Soc
, 2004
"... then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my firs ..."
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Cited by 32 (8 self)
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then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my first years after my Ph.D. I studied his work in cobordism, singularities of maps, and transversality, gaining many insights. I also enjoyed listening to his provocations, for example his disparaging remarks on complex analysis, 19th century mathematics, and Bourbaki. There was also a stormy side in our relationship. Neither of us could hide the pain that our public conflicts over “catastrophe theory ” caused. René Thom was a great mathematician, leaving his impact on a wide part of mathematics. I will always treasure my memories of him.