Fast B-spline transforms for continuous image representation and interpolation

In this paper a correspondence is derived between regularization operators used in regularization networks and support vector kernels. We prove that the Green's Functions associated with regularization operators are suitable support vector kernels with equivalent regularization properties. Moreover, the paper provides an analysis of currently used support vector kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of radial basis functions, namely conditionally positive definite functions, may be used as support vector kernels. ᭧

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“…For the sake of simplicity, we consider the case n ¼ 1. Recalling the definition (up to scaling factors) by Unser et al (1991) …”

Section: Translation Invariant Kernelsmentioning

confidence: 99%

The connection between regularization operators and support vector kernels

Smola¹,

Schölkopf²,

Müller³

1998

Neural Networks

564

316

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“…This can be done either by matrix manipulations [20,27], or by means of recursive filtering techniques [55][56][57], the latter of which are easier to implement and computationally much more efficient. When using the latter approach, nth-degree spline interpolation is carried out by evaluating the following expression:…”

Section: Cardinal Spline Kernelsmentioning

confidence: 99%

“…where (b n ) −1 denotes the recursive prefilter, also known as the direct B-spline filter [55][56][57]. Note that the resulting interpolant is an element of C n−1 .…”

Section: Cardinal Spline Kernelsmentioning

confidence: 99%

Quantitative evaluation of convolution-based methods for medical image interpolation

Meijering

Niessen

Viergever

2001

Medical Image Analysis

254

165

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Abstract-Interpolation is required in a variety of medical image processing applications. Although many interpolation techniques are known from the literature, evaluations of these techniques for the specific task of applying geometrical transformations to medical images are still lacking. In this paper we present such an evaluation. We consider convolution-based interpolation methods and rigid transformations (rotations and translations). A large number of sincapproximating kernels are evaluated, including piecewise polynomial kernels and a large number of windowed sinc kernels, with spatial supports ranging from two to ten grid intervals. In the evaluation we use images from a wide variety of medical image modalities. The results show that spline interpolation is to be preferred over all other methods, both for its accuracy and its relatively low computational cost.

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“…In [24], Unser, Aldroubi, and Eden presented a method for the image interpolation using fast B-spline transform as follows. The key steps are to take the Z-transform to (3.2) as…”

Section: Applications In B-spline Image Interpolation and Computationmentioning

confidence: 99%

Eulerian polynomials and B-splines

2012

Journal of Computational and Applied Mathematics

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Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, Bsplines, respectively. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.

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Fast B-spline transforms for continuous image representation and interpolation

Cited by 438 publications

References 15 publications

The connection between regularization operators and support vector kernels

The connection between regularization operators and support vector kernels

Quantitative evaluation of convolution-based methods for medical image interpolation

Eulerian polynomials and B-splines

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