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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.
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 17 (11 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
Learning a potential function from a trajectory
 Signal Processing Letters
, 2006
"... Abstract — This letter concerns the use of stochastic gradient systems in the modeling of the paths of moving particles and the consequent estimation of a potential function. The work proceeds by setting down a model for the potential function which leads to a stochastic differential equation. The m ..."
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Cited by 13 (5 self)
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Abstract — This letter concerns the use of stochastic gradient systems in the modeling of the paths of moving particles and the consequent estimation of a potential function. The work proceeds by setting down a model for the potential function which leads to a stochastic differential equation. The method is simple, direct and flexible being based on a linear model and least squares. The estimated potential function may be used for: simple description, summary, comparison, seeking patterns, simulation, prediction, and model appraisal. Explanatories, attractors and repellors, may be included in the potential function directly. The large sample distribution of the estimated potential function is provided. There is an example analyzing the path of an elk. There are direct extensions to: updating, sliding window, adaptive, robust and real time variants. Index Terms—Mobility model, monitoring, potential function, stochastic differential equation, stochastic gradient system, surveillance, tracking, waypoint data. I.
INVARIANCE OF A SHIFTINVARIANT SPACE
"... Abstract. A shiftinvariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shiftinvariant subspaces S that are also invaria ..."
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Cited by 11 (5 self)
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Abstract. A shiftinvariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shiftinvariant subspaces S that are also invariant under additional (noninteger) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shiftinvariant spaces with a compactly supported generator cannot be invariant under any noninteger translations. 1.
Splines: A perfect fit for medical imaging
 In Progress in Biomedical Optics and Imaging
, 2002
"... Splines, which were invented by Schoenberg more than fifty years ago, constitute an elegant framework for dealing with interpolation and discretization problems. They are widely used in computeraided design and computer graphics, but have been neglected in medical imaging applications, mostly as a ..."
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Cited by 7 (0 self)
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Splines, which were invented by Schoenberg more than fifty years ago, constitute an elegant framework for dealing with interpolation and discretization problems. They are widely used in computeraided design and computer graphics, but have been neglected in medical imaging applications, mostly as a consequence of what one may call the &quot;bad press &quot; phenomenon. Thanks to some recent research efforts in signal processing and waveletrelated techniques, the virtues of splines have been revived in our community. There is now compelling evidence (several independent studies) that splines offer the best costperformance tradeoff among available interpolation methods. In this presentation, we will argue that the spline representation is ideally suited for all processing tasks that require a continuous model of signals or images. We will show that most forms of spline fitting (interpolation, least squares approximation, smoothing splines) can be performed most efficiently using recursive digital filters. We will also have a look at their multiresolution properties which make them prime candidates for constructing wavelet bases and computing image pyramids. Typical application areas where these techniques can be useful are: image reconstruction from projection data, sampling grid conversion, geometric correction, visualization, rigid or elastic image registration, and feature extraction including edge detection and active contour models.
The Haar–Wavelet Transform in Digital Image Processing: Its Status and Achievements
"... Abstract. Image processing and analysis based on the continuous or discrete image transforms are classic techniques. The image transforms are widely used in image filtering, data description, etc. Nowadays the wavelet theorems make up very popular methods of image processing, denoising and compressi ..."
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Cited by 5 (0 self)
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Abstract. Image processing and analysis based on the continuous or discrete image transforms are classic techniques. The image transforms are widely used in image filtering, data description, etc. Nowadays the wavelet theorems make up very popular methods of image processing, denoising and compression. Considering that the Haar functions are the simplest wavelets, these forms are used in many methods of discrete image transforms and processing. The image transform theory is a well known area characterized by a precise mathematical background, but in many cases some transforms have particular properties which are not still investigated. This paper for the first time presents graphic dependences between parts of Haar and wavelets spectra. It also presents a method of image analysis by means of the wavelets–Haar spectrum. Some properties of the Haar and wavelets spectrum were investigated. The extraction of image features immediately from spectral coefficients distribution were shown. In this paper it is presented that two–dimensional both, the Haar and wavelets functions products man be treated as extractors of particular image features. Furthermore, it is also shown that some coefficients from both spectra are proportional, which simplify slightly computations and analyses.
Distance function wavelets – Part II: Extended results and conjectures, CoRR preprint, Research report of Simula Research Laboratory
 CoRR preprint, http://xxx.lanl.gov/abs/cs.CE/0205063, Research report of Simula Research Laboratory
, 2002
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Distance function wavelets – Part I: Helmholtz and convectiondiffusion transforms and series
 CoRR preprint, http://xxx.lanl.gov/abs/cs.CE/0205019, Research report of Simula Research Laboratory
, 2002
"... This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and general solutions of the Helmholtz, modified Helmholtz, and convectiondiffusion equations, which include the isotropic HelmholtzFourier (HF) transform and series, the Helmholt ..."
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Cited by 2 (2 self)
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This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and general solutions of the Helmholtz, modified Helmholtz, and convectiondiffusion equations, which include the isotropic HelmholtzFourier (HF) transform and series, the HelmholtzLaplace (HL) transform, and the anisotropic convectiondiffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the HF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the HF and HL transforms, a new dimensiondependent function notation is introduced. The HF series is also used to evaluate the analytical solutions of linear diffusion problems of arbitrary dimensionality and geometry. The weakness of this report is lacking of rigorous mathematical analysis due to the author’s limited mathematical knowledge. Keywords: HelmholtzFourier transform and series, HelmholtzLaplace transform, distance function, radial basis function, distance function wavelet, ridgelets, Helmholtz equation, modified Helmholtz equation, convectiondiffusion equation, fundamental solution, general solution, edge effect. 1 1.
SelfSimilarity: Part I  Splines and Operators
 IEEE TRANS. SIGNAL PROCESS
, 2007
"... The central theme of this pair of papers (Parts I and II in this issue) is selfsimilarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of Lsplines; these are defined in the following terms: @ A is a c ..."
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Cited by 1 (0 self)
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The central theme of this pair of papers (Parts I and II in this issue) is selfsimilarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of Lsplines; these are defined in the following terms: @ A is a cardinal Lspline iff v @ A a ‘ “ @ A, where L is a suitable pseudodifferential operator. Our starting point for the construction of “selfsimilar” splines is the identification of the class of differential operators L that are both translation and scale invariant. This results into a twoparameter family of generalized fractional derivatives,, where is the order of the derivative and is an additional phase factor. We specify the corresponding Lsplines, which yield an extended class of fractional splines. The operator is used to define a scaleinvariant energy measure—the squared Pnorm of the th derivative of the signal—which provides a regularization functional for interpolating or fitting the noisy samples of a signal. We prove that the corresponding variational (or smoothing) spline estimator is a cardinal fractional spline of order P, which admits a stable representation in a Bspline basis. We characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical Butterworth filter of order P. We also establish a formal link between the regularization parameter and the cutoff frequency of the smoothing spline filter: H P. Finally, we present an efficient computational solution to the fractional smoothing spline problem: It uses the fast Fourier transform and takes advantage of the multiresolution properties of the underlying basis functions.
LOCAL BASES FOR REFINABLE SPACES
, 2005
"... Abstract. We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular these functions include all the homogeneous polynomials that are reproducible by the generator, what links ..."
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Abstract. We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular these functions include all the homogeneous polynomials that are reproducible by the generator, what links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension than the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling. 1.