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36
Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets StrangFix
 IEEE Trans. on Signal Processing
, 2007
"... Abstract—Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Cl ..."
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Cited by 92 (27 self)
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Abstract—Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. Recently, it was shown that, by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation, it is possible to reconstruct such signals uniquely [34]. These sampling schemes, however, use kernels with infinite support, and this leads to complex and potentially unstable reconstruction algorithms. In this paper, we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang–Fix conditions, exponential splines and functions with rational Fourier transform. This last class of kernels is quite general and includes, for instance, any linear electric circuit. We, thus, show with an example how to estimate a signal of finite rate of innovation at the output of an circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling. Index Terms—Analogtodigital conversion, annihilating filter method, multiresolution approximations, sampling methods, splines, wavelets. I.
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.
Complex Wavelet Bases, Steerability, and the MarrLike Pyramid
, 2008
"... Our aim in this paper is to tighten the link between wavelets, some classical imageprocessing operators, and David Marr’s theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the GradientLaplace operator. Starting from firs ..."
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Cited by 12 (6 self)
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Our aim in this paper is to tighten the link between wavelets, some classical imageprocessing operators, and David Marr’s theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the GradientLaplace operator. Starting from first principles, we show that a singlegenerator wavelet can be defined analytically and that it yields a semiorthogonal complex basis of, irrespective of the dilation matrix used. We also provide an efficient FFTbased filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translationinvariant and that is optimized for better steerability (Gaussianlike smoothing kernel). We call it the Marrlike wavelet pyramid because it essentially replicates the processing steps in Marr’s theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative
Information extraction from sound for medical telemonitoring
 IEEE Transactions on TITB
, 2006
"... Abstract—Today, the growth of the aging population in Europe needs an increasing number of health care professionals and facilities for aged persons. Medical telemonitoring at home (and, more generally, telemedicine) improves the patient’s comfort and reduces hospitalization costs. Using sound surve ..."
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Cited by 12 (6 self)
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Abstract—Today, the growth of the aging population in Europe needs an increasing number of health care professionals and facilities for aged persons. Medical telemonitoring at home (and, more generally, telemedicine) improves the patient’s comfort and reduces hospitalization costs. Using sound surveillance as an alternative solution to video telemonitoring, this paper deals with the detection and classification of alarming sounds in a noisy environment. The proposed sound analysis system can detect distress or everyday sounds everywhere in the monitored apartment, and is connected to classical medical telemonitoring sensors through a data fusion process. The sound analysis system is divided in two stages: sound detection and classification. The first analysis stage (sound detection) must extract significant sounds from a continuous signal flow. A new detection algorithm based on discrete wavelet transform is proposed in this paper, which leads to accurate results when applied to nonstationary signals (such as impulsive sounds). The algorithm presented in this paper was evaluated in a noisy environment and is favorably compared to the state of the art algorithms in the field. The second stage of the system is sound classification, which uses a statistical approach to identify unknown sounds. A statistical study was done to find out the most discriminant acoustical parameters in the input of the classification module. New wavelet based parameters, better adapted to noise, are proposed in this paper. The telemonitoring system validation is presented through various real and simulated test sets. The global sound based system leads to a 3 % missed alarm rate and could be fused with other medical sensors to improve performance. Index Terms—Gaussian mixture model (GMM), medical telemonitoring, sound classification, sound detection, wavelet transform. I.
Selfsimilarity: Part II  Optimal estimation of fractallike processes
 IEEE SIGNAL PROCESSING MAGAZINE, THE IEEE TRANSACTIONS ON IMAGE PROCESSING (1992 TO 1995), AND THE IEEE SIGNAL PROCESSING LETTERS
, 2007
"... In a companion paper (see SelfSimilarity: Part I—Splines and Operators), we characterized the class of scaleinvariant convolution operators: the generalized fractional derivatives of order. We used these operators to specify regularization functionals for a series of Tikhonovlike leastsquares da ..."
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Cited by 10 (9 self)
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In a companion paper (see SelfSimilarity: Part I—Splines and Operators), we characterized the class of scaleinvariant convolution operators: the generalized fractional derivatives of order. We used these operators to specify regularization functionals for a series of Tikhonovlike leastsquares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of these smoothing splines and proposed a fast Fourier transform (FFT)based implementation. Here, we present an alternative stochastic formulation to further justify these fractional spline estimators. As suggested by the title, the relevant processes are those that are statistically selfsimilar; that is, fractional Brownian motion (fBm) and its higher order extensions. To overcome the technical difficulties due to the nonstationary character of fBm, we adopt a distributional formulation due to Gel’fand. This allows us to rigorously specify an innovation model for these fractal processes, which rests on the property that they can be whitened by suitable fractional differentiation. Using the characteristic form of the fBm, we then derive the conditional probability density function (PDF) @ @ A A, where a @ AC ‘ “ are the noisy samples of the fBm @ A with Hurst exponent. We find that the conditional mean is a fractional spline of degree P, which proves that this class of functions is indeed optimal for the estimation of fractallike processes. The result also yields the optimal [minimum meansquare error (MMSE)] parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering.
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
Construction of Hilbert transform pairs of wavelet bases and Gaborlike transforms
 IEEE TRANS. SIGNAL PROCESS
, 2009
"... We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximationtheoretic characterization of scaling functions—the Bspline factorization theorem. In particular, starting from welllocalized scaling functions, we construct HT pairs of b ..."
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Cited by 7 (4 self)
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We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximationtheoretic characterization of scaling functions—the Bspline factorization theorem. In particular, starting from welllocalized scaling functions, we construct HT pairs of biorthogonal wavelet bases of by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimallylocalized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a onesided spectrum. Based on the tensorproduct of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of, we then discuss a methodology for constructing 2D directionalselective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT—the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)based filterbank algorithm for implementing the associated complex wavelet transform.
Analysis and VLSI architecture for 1D and 2D discrete wavelet transform
, 2005
"... Abstract—In this paper, a detailed analysis of very large scale integration (VLSI) architectures for the onedimensional (1D) and twodimensional (2D) discrete wavelet transform (DWT) is presented in many aspects, and three related architectures are proposed as well. The 1D DWT and inverse DWT (I ..."
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Cited by 7 (3 self)
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Abstract—In this paper, a detailed analysis of very large scale integration (VLSI) architectures for the onedimensional (1D) and twodimensional (2D) discrete wavelet transform (DWT) is presented in many aspects, and three related architectures are proposed as well. The 1D DWT and inverse DWT (IDWT) architectures are classified into three categories: convolutionbased, liftingbased, and Bsplinebased. They are discussed in terms of hardware complexity, critical path, and registers. As for the 2D DWT, the large amount of the frame memory access and the die area occupied by the embedded internal buffer become the most critical issues. The 2D DWT architectures are categorized and analyzed by different external memory scan methods. The implementation issues of the internal buffer are also discussed, and some reallife experiments are given to show that the area and power for the internal buffer are highly related to memory technology and working frequency, instead of the required memory size only. Besides the analysis, the Bsplinebased IDWT architecture and the overlapped stripebased scan method are also proposed. Last, we propose a flexible and efficient architecture for a onelevel 2D DWT that exploits many advantages of the presented analysis. Index Terms—Bspline factorization, discrete wavelet transform, lifting scheme, linebased method, VLSI architecture. I.
A scalable wavelet transform vlsi architecture for realtime signal processing in highdensity intracortical implants
 IEEE Trans. Circuits and Systems I
, 2007
"... Abstract—This paper describes an area and powerefficient VLSI approach for implementing the discrete wavelet transform on streaming multielectrode neurophysiological data in real time. The VLSI implementation is based on the lifting scheme for wavelet computation using the symmlet4 basis with quant ..."
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Cited by 6 (0 self)
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Abstract—This paper describes an area and powerefficient VLSI approach for implementing the discrete wavelet transform on streaming multielectrode neurophysiological data in real time. The VLSI implementation is based on the lifting scheme for wavelet computation using the symmlet4 basis with quantized coefficients and integer fixedpoint data precision to minimize hardware demands. The proposed design is driven by the need to compress neural signals recorded with highdensity microelectrode arrays implanted in the cortex prior to data telemetry. Our results indicate that signal integrity is not compromised by quantization down to 5bit filter coefficient and 10bit data precision at intermediate stages. Furthermore, results from analog simulation and modeling show that a hardwareminimized computational core executing filter steps sequentially is advantageous over the pipeline approach commonly used in DWT implementations. The design is compared to that of a Bspline approach that minimizes the number of multipliers at the expense of increasing the number of adders. The performance demonstrates that in vivo realtime DWT computation is feasible prior to data telemetry, permitting large savings in bandwidth requirements and communication costs given the severe limitations on size, energy consumption and power dissipation of an implantable device. Index Terms—Bspline, brain machine interface, lifting, microelectrode arrays, neural signal processing, neuroprosthetic devices, wavelet transform. I.
Invariances, Laplacianlike wavelet bases, and the whitening of fractal processes
, 2009
"... In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fraction ..."
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Cited by 6 (4 self)
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In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional Brownian motion (fBm) fields. In particular, using a rigorous and powerful distribution theoretic formulation, we extend previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator. We then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional Laplacian, which also turns out to be the whitening operator for fBm fields. This fact allows us to study the probabilistic properties of the wavelet transform coefficients of fBmlike processes, leading for instance to ways of estimating the Hurst exponent of a multiparameter process from its wavelet transform coefficients. We provide theoretical and experimental verification of these results. To complement the toolbox available for multiresolution processing of stochastic fractals, we also introduce an extended family of multidimensional multiresolution spaces for a large class of (separable and nonseparable) lattices of arbitrary dimensionality.