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223
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
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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.
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 102 (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.
Machine contouring using minimum curvature
 Geophysics
, 1974
"... Machine contouring must not introduce information which is not present in the data The onedimensional spline fit has well defined smoothness properties. These are duplicated for twodimensional interpolation in this paper, by solving the corresponding differential equation. Finite difference equ ..."
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Cited by 91 (0 self)
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Machine contouring must not introduce information which is not present in the data The onedimensional spline fit has well defined smoothness properties. These are duplicated for twodimensional interpolation in this paper, by solving the corresponding differential equation. Finite difference equations are deduced from a
Bspline snakes: a flexible tool for parametric contour detection
 IEEE Transactions on Image Processing
"... Abstract—We present a novel formulation for Bspline snakes that can be used as a tool for fast and intuitive contour outlining. We start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal, curvatureconstrained snake is a cubic spline, irrespe ..."
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Cited by 88 (16 self)
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Abstract—We present a novel formulation for Bspline snakes that can be used as a tool for fast and intuitive contour outlining. We start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal, curvatureconstrained snake is a cubic spline, irrespective of the form of the external energy field. Unfortunately, such regularized snakes suffer from slow convergence speed because of a large number of control points, as well as from difficulties in determining the weight factors associated to the internal energies of the curve. We therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori; this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies (i.e., the regularization term). In other words, we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots. The theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments. Validation results are presented, comparing the traditional snake using internal energies and the proposed approach without internal energies, showing the similar performance of the latter. Several biomedical examples of applications are included to illustrate the versatility of the method. I.
Biharmonic Spline Interpolation of GEOS3 and SEASAT Altimeter Data
 Georgia World Congress Center, Atlanta Georgia USA
, 1976
"... Abstract. Green functions of the biharmonic operator, in one and two dimensions, are used for minimum curvature interpolation of irregularly spaced data points. The interpolating curve (or surface) is a linear combination of Green functions centered at each data point. The amplitudes of the Green fu ..."
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Cited by 70 (1 self)
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Abstract. Green functions of the biharmonic operator, in one and two dimensions, are used for minimum curvature interpolation of irregularly spaced data points. The interpolating curve (or surface) is a linear combination of Green functions centered at each data point. The amplitudes of the Green functions are found by solving a linear system of equations. In one (or two) dimensions this technique is equivalent o cubic spline (or bicubic spline) interpolation while in three dimension it corresponds to multiquadric interpolation. Although this new technique is relatively slow, it is more flexible than the spline method since both slopes and values can be used to find a surface. Moreover, noisy data can be fit in a least squares ense by reducing the number of model parameters. These properties are well suited for interpolating irregularly spaced satellite altimeter profiles. The long wavelength radial orbit error is suppressed by differentiating each profile. The shorter wavelength noise is reduced by the least squares fit to nearby profiles. Using this technique with 0.5 million GEOS3 and SEASAT data points, it was found that the marine geoid of the Caribbean area is highly correlated with the sea floor topography. This suggests that similar applications, in more remote, areas may reveal new features of the sea floor.
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.
A clearsky spectral solar radiation model for snowcovered mountainous terrain
, 1980
"... A dearsky spectral solar radiation model for direct and diffuse fluxes, combined with topographic calculations from digital terrain data, computes either incident, net, or reflected solar radiation at any point on a snow surface in mountainous terrain. The radiation may be integrated over any wave ..."
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Cited by 35 (3 self)
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A dearsky spectral solar radiation model for direct and diffuse fluxes, combined with topographic calculations from digital terrain data, computes either incident, net, or reflected solar radiation at any point on a snow surface in mountainous terrain. The radiation may be integrated over any wavelength range from 250 to 5000 nm, or over any time step. Atmospheric attenuation parameters are ozone, water vapor, the Angstrom turbidity coefficient and exponent, and the absorptance to reflectance ratio of the atmospheric aerosols. The model derives these, from measurements which may contain both systematic and random errors, by finding the least squares solution to an overdetermined set of nonlinear equations. For calculations over a specified area, it employs table lookup procedures, so that computation speed for the spectral model approaches that for a lumped model. Thus it may be useful as part of a snow surface energy budget calculation over a drainage basin.
Improved error bounds for scattered data interpolation by radial basis functions
 Math. Comp
, 1999
"... Abstract. If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled. 1. ..."
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Cited by 32 (9 self)
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Abstract. If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled. 1.