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93
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
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Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 340 (27 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
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 102 (8 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.
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 57 (26 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.
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 48 (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.
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 43 (24 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.
Unwarping of Unidirectionally Distorted EPI Images
, 2000
"... Echoplanar imaging (EPI) is a fast nuclear magnetic resonance imaging method. Unfortunately, local magnetic field inhomogeneities induced mainly by the subject 's presence cause significant geometrical distortion, predominantly along the phaseencoding direction, which must be undone to allow ..."
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Cited by 40 (7 self)
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Echoplanar imaging (EPI) is a fast nuclear magnetic resonance imaging method. Unfortunately, local magnetic field inhomogeneities induced mainly by the subject 's presence cause significant geometrical distortion, predominantly along the phaseencoding direction, which must be undone to allow for meaningful further processing. So far, this aspect has been too often neglected.
Fresnelets: new multiresolution wavelet bases for digital holography
 IEEE Trans. Image Process
, 2003
"... Abstract—We propose a construction of new waveletlike bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCDarrays. The starting point is a wavelet basis of P to which we apply a unitary Fresnel transform. The transformed basis fun ..."
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Cited by 38 (7 self)
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Abstract—We propose a construction of new waveletlike bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCDarrays. The starting point is a wavelet basis of P to which we apply a unitary Fresnel transform. The transformed basis functions are shiftinvariant on a levelbylevel basis but their multiresolution properties are governed by the special form that the dilation operator takes in the Fresnel domain. We derive a Heisenberglike uncertainty relation that relates the localization of Fresnelets with that of their associated wavelet basis. According to this criterion, the optimal functions for digital hologram processing turn out to be Gabor functions, bringing together two separate aspects of the holography inventor’s work. We give the explicit expression of orthogonal and semiorthogonal Fresnelet bases corresponding to polynomial spline wavelets. This special choice of Fresnelets is motivated by their nearoptimal localization properties and their approximation characteristics. We then present an efficient multiresolution Fresnel transform algorithm, the Fresnelet transform. This algorithm allows for the reconstruction (backpropagation) of complex scalar waves at several userdefined, wavelengthindependent resolutions. Furthermore, when reconstructing numerical holograms, the subband decomposition of the Fresnelet transform naturally separates the image to reconstruct from the unwanted zeroorder and twin image terms. This greatly facilitates their suppression. We show results of experiments carried out on both synthetic (simulated) data sets as well as on digitally acquired holograms. Index Terms—Bsplines, digital holography, Fresnel transform, Fresnelet transform, Fresnelets, wavelets.
The Fractional Spline Wavelet Transform: Definition And Implementation
 in Proc. IEEE: International Conference on Acoustics, Speech, and Signal Processing
, 2000
"... We dene a new wavelet transform that is based on a recently dened family of scaling functions: the fractional Bsplines. The interest of this family is that they interpolate between the integer degrees of polynomial Bsplines and that they allow a fractional order of approximation. ..."
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Cited by 35 (13 self)
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We dene a new wavelet transform that is based on a recently dened family of scaling functions: the fractional Bsplines. The interest of this family is that they interpolate between the integer degrees of polynomial Bsplines and that they allow a fractional order of approximation.
Denoising functional MR images: a comparison of wavelet denoising and Gaussian smoothing
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2004
"... We present a general waveletbased denoising scheme for functional magnetic resonance imaging (fMRI) data and compare it to Gaussian smoothing, the traditional denoising method used in fMRI analysis. Onedimensional WaveLab thresholding routines were adapted to twodimensional images, and applied ..."
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Cited by 31 (2 self)
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We present a general waveletbased denoising scheme for functional magnetic resonance imaging (fMRI) data and compare it to Gaussian smoothing, the traditional denoising method used in fMRI analysis. Onedimensional WaveLab thresholding routines were adapted to twodimensional images, and applied to 2D wavelet coefficients. To test the effect of these methods on the signaltonoise ratio (SNR), we compared the SNR of 2D fMRI images before and after denoising, using both Gaussian smoothing and waveletbased methods. We simulated a fMRI series with a time signal in an active spot, and tested the methods on noisy copies of it. The denoising methods were evaluated in two ways: by the average temporal SNR inside the original activated spot, and by the shape of the spot detected by thresholding the temporal SNR maps. Denoising methods that introduce much smoothness are better suited for low SNRs, but for images of reasonable quality they are not preferable, because they introduce heavy deformations. Waveletbased denoising methods that introduce less smoothing preserve the sharpness of the images and retain the original shapes of active regions. We also performed statistical parametric mapping (SPM) on the denoised simulated time series, as well as on a real fMRI data set. False discovery rate control was used to correct for multiple comparisons. The results show that the methods that produce smooth images introduce more false positives. The less smoothing waveletbased methods, although generating more false negatives, produce a smaller total number of errors than Gaussian smoothing or waveletbased methods with a large smoothing effect.