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198
Sampling signals with finite rate of innovation
 IEEE Transactions on Signal Processing
, 2002
"... Abstract—Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials ..."
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Cited by 218 (51 self)
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Abstract—Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic “bandlimited and sinc kernel ” case. In particular, we show how to sample and reconstruct periodic and finitelength streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinitelength signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and errorcorrection coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems. Index Terms—Analogtodigital conversion, annihilating filters, generalized sampling, nonbandlimited signals, nonuniform splines, piecewise polynomials, poisson processes, sampling. I.
Kernel regression for image processing and reconstruction
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2007
"... In this paper, we make contact with the field of nonparametric statistics and present a development and generalization of tools and results for use in image processing and reconstruction. In particular, we adapt and expand kernel regression ideas for use in image denoising, upscaling, interpolation, ..."
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Cited by 110 (50 self)
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In this paper, we make contact with the field of nonparametric statistics and present a development and generalization of tools and results for use in image processing and reconstruction. In particular, we adapt and expand kernel regression ideas for use in image denoising, upscaling, interpolation, fusion, and more. Furthermore, we establish key relationships with some popular existing methods and show how several of these algorithms, including the recently popularized bilateral filter, are special cases of the proposed framework. The resulting algorithms and analyses are amply illustrated with practical examples.
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.
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.
Fast parametric elastic image registration
 IEEE Transactions on Image Processing
, 2003
"... Abstract—We present an algorithm for fast elastic multidimensional intensitybased image registration with a parametric model of the deformation. It is fully automatic in its default mode of operation. In the case of hard realworld problems, it is capable of accepting expert hints in the form of so ..."
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Cited by 60 (4 self)
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Abstract—We present an algorithm for fast elastic multidimensional intensitybased image registration with a parametric model of the deformation. It is fully automatic in its default mode of operation. In the case of hard realworld problems, it is capable of accepting expert hints in the form of soft landmark constraints. Much fewer landmarks are needed and the results are far superior compared to pure landmark registration. Particular attention has been paid to the factors influencing the speed of this algorithm. The Bspline deformation model is shown to be computationally more efficient than other alternatives. The algorithm has been successfully used for several twodimensional (2D) and threedimensional (3D) registration tasks in the medical domain, involving MRI, SPECT, CT, and ultrasound image modalities. We also present experiments in a controlled environment, permitting an exact evaluation of the registration accuracy. Test deformations are generated automatically using a random hierarchical fractional waveletbased generator. Index Terms—Elastic registration, image registration, landmarks, splines. I.
MOMS: MaximalOrder Interpolation of Minimal Support
 IEEE Trans. Image Process
, 2001
"... We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactlysupported basis function ( ). We first give the expression of the 's that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we ..."
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Cited by 48 (17 self)
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We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactlysupported basis function ( ). We first give the expression of the 's that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we call maximal orderminimalsupport functions (MOMS) is made of linear combinations of the Bspline of same order and of its derivatives.
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.
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.
General framework for consistent sampling in Hilbert spaces
 Int. J. Wavelets, Multiresolution, Inf. Process
, 2005
"... We introduce a general framework for consistent linear reconstruction in infinitedimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinitedimensional frames. As we show, the linear reconstruction sche ..."
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Cited by 32 (16 self)
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We introduce a general framework for consistent linear reconstruction in infinitedimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinitedimensional frames. As we show, the linear reconstruction scheme coincides with the socalled oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.