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54
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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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.
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 54 (20 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.
Linear interpolation revitalized
 IEEE Trans. Image Processing
, 2004
"... Abstract—We present a simple, original method to improve piecewiselinear interpolation with uniform knots: we shift the sampling knots by a fixed amount, while enforcing the interpolation property. We determine the theoretical optimal shift that maximizes the quality of our shifted linear interpol ..."
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Cited by 25 (2 self)
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Abstract—We present a simple, original method to improve piecewiselinear interpolation with uniform knots: we shift the sampling knots by a fixed amount, while enforcing the interpolation property. We determine the theoretical optimal shift that maximizes the quality of our shifted linear interpolation. Surprisingly enough, this optimal value is nonzero and close to 1 5. We confirm our theoretical findings by performing several experiments: a cumulative rotation experiment and a zoom experiment. Both show a significant increase of the quality of the shifted method with respect to the standard one. We also observe that, in these results, we get a quality that is similar to that of the computationally more costly “highquality ” cubic convolution. Index Terms—Approximation methods, error analysis, interpolation, piecewise linear approximation, recursive digital filters, spline functions. I.
Consistent Sampling and Signal Recovery
"... Abstract—An attractive formulation of the sampling problem is based on the principle of a consistent signal reconstruction. The requirement is that the reconstructed signal is indistinguishable from the input in the sense that it yields the exact same measurements. Such a system can be interpreted a ..."
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Abstract—An attractive formulation of the sampling problem is based on the principle of a consistent signal reconstruction. The requirement is that the reconstructed signal is indistinguishable from the input in the sense that it yields the exact same measurements. Such a system can be interpreted as an oblique projection onto a given reconstruction space. The standard formulation requires a onetoone relationship between the input measurements and the reconstructed model. Unfortunately, this condition fails when the crosscorrelation matrix between the analysis and reconstruction basis functions is not invertible; in particular, when there are less measurements than the number of reconstruction functions. In this paper, we propose an extension of consistent sampling that is applicable to those singular cases as well, and that yields a unique and welldefined solution. This solution also makes use of projection operators and has a geometric interpretation. The key idea is to exclude the null space of the sampling operator from the reconstruction space and to enforce consistency on its complement. We specify a class of consistent reconstruction algorithms corresponding to different choices of complementary reconstruction spaces. The formulation includes the MoorePenrose generalized inverse, as well as other potentially more interesting reconstructions that preserve certain preferential signals. In particular, we display solutions that preserve polynomials or sinusoids, and therefore perform well in practical applications. Index Terms—Consistency, preferential components, sampling method, signal reconstruction, underdetermined scenario. I.
Discretization of the Radon Transform and of its Inverse by Spline Convolutions
, 2002
"... We present an explicit formula for Bspline convolution kernels; these are defined as the convolution of several Bsplines of variable widths hi and degrees rzl. We apply our results to derive splineconvolutionbased algorithms for two closely related problems: the computation of the Radon transfor ..."
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We present an explicit formula for Bspline convolution kernels; these are defined as the convolution of several Bsplines of variable widths hi and degrees rzl. We apply our results to derive splineconvolutionbased algorithms for two closely related problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the leastsquares sense. We then consider the reverse problem and introduce a new splineconvolution version of the filtered backprojection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of highdegree splines; these offer better approximation performance than the conventional lowerdegree formulations (e.g., piecewise constant or piecewise linear models). We present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity. In particular, we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate.
Fourdimensional cardiac imaging in living embryos via postacquisition synchronization of nongated slice sequences
 JOURNAL OF BIOMEDICAL OPTICS 10(5)
, 2005
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Estimating intensity variance due to noise in registered images: applications to diffusion tensor MRI
 Neuroimage
, 2005
"... Image registration refers to the process of finding the spatial correspondence between two or more images. This is usually done by applying a spatial transformation, computed automatic or manually, to a given image using a continuous image model computed either with interpolation or approximation me ..."
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Cited by 13 (0 self)
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Image registration refers to the process of finding the spatial correspondence between two or more images. This is usually done by applying a spatial transformation, computed automatic or manually, to a given image using a continuous image model computed either with interpolation or approximation methods. We show that noise induced signal variance in interpolated images differs significantly from the signal variance of the original images in native space. We describe a simple approach to compute the signal variance in registered images based on the signal variance and covariance of the original images, the spatial transformations computed by the registration procedure, and the interpolation or approximation kernel chosen. Our approach is applied to diffusion tensor (DT) MRI data. We show that incorrect noise variance estimates in registered diffusion weighted images can affect the estimated DT parameters, their estimated uncertainty, as well as indices of goodness of fit such as chisquare maps. In addition to DTMRI, we believe that this methodology would be useful any time parameter extraction methods are applied to registered or interpolated data.
A note on cubic convolution interpolation
 IEEE Trans. Image Process
, 2003
"... Abstract—We establish a link between classical osculatory interpolation and modern convolutionbased interpolation and use it to show that two wellknown cubic convolution schemes are formally equivalent to two osculatory interpolation schemes proposed in the actuarial literature about a century ago. ..."
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Abstract—We establish a link between classical osculatory interpolation and modern convolutionbased interpolation and use it to show that two wellknown cubic convolution schemes are formally equivalent to two osculatory interpolation schemes proposed in the actuarial literature about a century ago. We also discuss computational differences and give examples of other cubic interpolation schemes not previously studied in signal and image processing. I
Elastic Image Registration using Parametric Deformation Models
, 2001
"... The main topic of this thesis is elastic image registration for biomedical applications. We start with an overview and classification of existing registration techniques. We revisit the landmark interpolation which appears in the landmarkbased registration techniques and add some generalizations. ..."
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Cited by 10 (1 self)
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The main topic of this thesis is elastic image registration for biomedical applications. We start with an overview and classification of existing registration techniques. We revisit the landmark interpolation which appears in the landmarkbased registration techniques and add some generalizations. We develop a general elastic image registration algorithm. It uses a grid of uniform Bsplines to describe the deformation. It also uses Bsplines for image interpolation. Multiresolution in both image and deformation model spaces yields robustness and speed. First we describe a version of this algorithm targeted at finding unidirectional deformation in EPI magnetic resonance images. Then we present the enhanced and generalized version of this algorithm which is significantly faster and capable of treating multidimensional deformations. We apply this algorithm to the registration of SPECT data and to the motion estimation in ultrasound image sequences. A semiautomatic version of the registration algorithm 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. In the second part of this thesis, we deal with the problem of generalized sampling and variational reconstruction. We explain how to reconstruct an object starting from several measurements using arbitrary linear operators. This comprises the case of traditional as well as generalized sampling. Among all possible reconstructions, we choose the one minimizing an a priori given quadratic variational criterion. We give an overview of the method and present several examples of applications. We also provide the mathematical details of the theory and discuss the choice of the variational criterio...
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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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.