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27
Generalized smoothing splines and the optimal discretization of the Wiener filter
 IEEE Trans. Signal Process
, 2005
"... Abstract—We introduce an extended class of cardinal L Lsplines, where L is a pseudodifferential operator satisfying some admissibility conditions. We show that the L Lspline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional L ..."
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Cited by 27 (14 self)
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Abstract—We introduce an extended class of cardinal L Lsplines, where L is a pseudodifferential operator satisfying some admissibility conditions. We show that the L Lspline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional L P, 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 Lspline. We show that this smoothing spline estimator has a stable representation in a Bsplinelike 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 L smoothing splines and the minimum mean square error (MMSE) estimation of a stationary signal corrupted by white Gaussian noise. In this modelbased 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. Index Terms—Nonparametric estimation, recursive filtering, smoothing splines, splines (polynomial and exponential), stationary processes, variational principle, Wiener filter. I.
Nonideal sampling and interpolation from noisy observations in shiftinvariant spaces
 IEEE Trans. Signal Processing
, 2006
"... Abstract—Digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuoustime signal from a sequence of corrupted discretetime samples. In this paper, a general formulation of this problem is developed that treats the interpolation problem fro ..."
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Cited by 25 (15 self)
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Abstract—Digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuoustime signal from a sequence of corrupted discretetime 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 shiftinvariant 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 (leastsquares, 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. Index Terms—Deconvolution, interpolation, minimax reconstruction, sampling. I.
Multichannel sampling of pulse streams at the rate of innovation
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... We consider minimalrate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of ..."
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Cited by 18 (4 self)
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We consider minimalrate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of infinite pulse streams was treated in previous works, either the rate of innovation was not achieved, or the pulse shape was limited to Diracs. In this paper we propose a multichannel architecture for sampling pulse streams with arbitrary shape, operating at the rate of innovation. Our approach is based on modulating the input signal with a set of properly chosen waveforms, followed by a bank of integrators. This architecture is motivated by recent work on subNyquist sampling of multiband signals. We show that the pulse stream can be recovered from the proposed minimalrate samples using standard tools taken from spectral estimation in a stable way even at high rates of innovation. In addition, we address practical implementation issues, such as reduction of hardware complexity and immunity to failure in the sampling channels. The resulting scheme is flexible and exhibits better noise robustness than previous approaches.
Exact Sampling Results for Signals with Finite Rate of Innovation Using StrangFix Conditions and Local Kernels
 Proc. IEEE ICASSP
, 2005
"... Recently, it was shown that it is possible to sample classes of signals with finite rate of innovation [7]. These sampling schemes, however, use kernels with infinite support and this leads to complex and instable reconstruction algorithms. ..."
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Cited by 15 (11 self)
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Recently, it was shown that it is possible to sample classes of signals with finite rate of innovation [7]. These sampling schemes, however, use kernels with infinite support and this leads to complex and instable reconstruction algorithms.
Sampling Piecewise Sinusoidal Signals With Finite Rate of Innovation Methods
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2009
"... We consider the problem of sampling piecewise sinusoidal signals. Classical sampling theory does not enable perfect reconstruction of such signals since they are not bandlimited. However, they can be characterized by a finite number of parameters namely the frequency, amplitude and phase of the sinu ..."
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Cited by 11 (5 self)
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We consider the problem of sampling piecewise sinusoidal signals. Classical sampling theory does not enable perfect reconstruction of such signals since they are not bandlimited. However, they can be characterized by a finite number of parameters namely the frequency, amplitude and phase of the sinusoids and the location of the discontinuities. In this paper, we show that under certain hypotheses on the sampling kernel, it is possible to perfectly recover the parameters that define the piecewise sinusoidal signal from its sampled version. In particular, we show that, at least theoretically, it is possible to recover piecewise sine waves with arbitrarily high frequencies and arbitrarily close switching points. Extensions of the method are also presented such as the recovery of combinations of piecewise sine waves and polynomials. Finally, we study the effect of noise and present a robust reconstruction algorithm that is stable down to SNR levels of 7 [dB].
Dynamic PET Reconstruction Using Wavelet Regularization with Adapted Basis Functions
 IEEE Trans. on Medical Imaging
, 2008
"... Abstract—Tomographic reconstruction from positron emission tomography (PET) data is an illposed problem that requires regularization. An attractive approach is to impose an 1regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thank ..."
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Cited by 7 (4 self)
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Abstract—Tomographic reconstruction from positron emission tomography (PET) data is an illposed problem that requires regularization. An attractive approach is to impose an 1regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatiotemporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponentialspline wavelets (Espline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponentialspline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the “optimal” Espline parameters (poles and zeros) and we investigate their
A Generalized Sampling Method for FiniteRateofInnovationSignal Reconstruction
 IEEE Signal Processing Letters
, 2008
"... Abstract—The problem of sampling signals that are not admissible within the classical Shannon framework has received much attention in the recent past. Typically, these signals have a parametric representation with a finite number of degrees of freedom per time unit. It was shown that, by choosing s ..."
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Cited by 7 (0 self)
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Abstract—The problem of sampling signals that are not admissible within the classical Shannon framework has received much attention in the recent past. Typically, these signals have a parametric representation with a finite number of degrees of freedom per time unit. It was shown that, by choosing suitable sampling kernels, the parameters can be computed by employing highresolution spectral estimation techniques. In this letter, we propose a simple acquisition and reconstruction method within the framework of multichannel sampling. In the proposed approach, an infinite stream of nonuniformlyspaced Dirac impulses can be sampled and accurately reconstructed provided that there is at most one Dirac impulse per sampling period. The reconstruction algorithm has a low computational complexity, and the parameters are computed on the fly. The processing delay is minimal—just the sampling period. We propose sampling circuits using inexpensive passive devices such as resistors and capacitors. We also show how the approach can be extended to sample piecewiseconstant signals with a minimal change in the system configuration. We provide some simulation results to confirm the theoretical findings. Index Terms—Exponential splines, finite rate of innovation, generalized sampling, localizing filter, piecewiseconstant functions, streams of Dirac impulses. I.
Feature Extraction for Image Superresolution using Finite Rate of Innovation Principles
, 2008
"... I certify that this thesis, and the research to which it refers, are the product of my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline. I acknowledge the h ..."
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Cited by 5 (0 self)
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I certify that this thesis, and the research to which it refers, are the product of my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline. I acknowledge the helpful guidance and support of my supervisor, Dr. Pier Luigi Dragotti. The material of this thesis has not been submitted for any degree at any other academic or professional institution.
Estimating Signals With Finite Rate of Innovation From Noisy Samples: A Stochastic Algorithm
, 2008
"... As an example of the recently introduced concept of rate of innovation, signals that are linear combinations of a finite number of Diracs per unit time can be acquired by linear filtering followed by uniform sampling. However, in reality, samples are rarely noiseless. In this paper, we introduce a ..."
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Cited by 5 (2 self)
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As an example of the recently introduced concept of rate of innovation, signals that are linear combinations of a finite number of Diracs per unit time can be acquired by linear filtering followed by uniform sampling. However, in reality, samples are rarely noiseless. In this paper, we introduce a novel stochastic algorithm to reconstruct a signal with finite rate of innovation from its noisy samples. Even though variants of this problem have been approached previously, satisfactory solutions are only available for certain classes of sampling kernels, for example, kernels that satisfy the Strang–Fix condition. In this paper, we consider the infinitesupport Gaussian kernel, which does not satisfy the Strang–Fix condition. Other classes of kernels can be employed. Our algorithm is based on Gibbs sampling, a Markov chain Monte Carlo method. Extensive numerical simulations demonstrate the accuracy and robustness of our algorithm.
SAMPLING SIGNALS WITH FINITE RATE OF INNOVATION IN THE PRESENCE OF NOISE
"... Recently, it has been shown that it is possible to sample nonbandlimited signals that possess a limited number of degrees of freedom and uniquely reconstruct them from a finite number of uniform samples. These signals include, amongst others, streams of Diracs. In this paper, we investigate the pro ..."
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Cited by 4 (3 self)
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Recently, it has been shown that it is possible to sample nonbandlimited signals that possess a limited number of degrees of freedom and uniquely reconstruct them from a finite number of uniform samples. These signals include, amongst others, streams of Diracs. In this paper, we investigate the problem of estimating the innovation parameters of a stream of Diracs from its noisy samples taken with polynomial or exponential reproducing kernels. For the oneDirac case, we provide exact expressions for the Cramér Rao bounds of this estimation problem. Furthermore, we propose methods to reconstruct the location of a single Dirac that reach the optimal performance given by the unbiased CramérRao bounds down to noise levels of