Short kernel fifth-order interpolation

Abstract-We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly-supported basis function ( ). We first give the expression of the 's that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we call maximal-order-minimal-support functions (MOMS) is made of linear combinations of the B-spline of same order and of its derivatives.We provide the explicit form of the MOMS that maximize the approximation accuracy when the step-size is small enough. We compute the sampling gain obtained by using these optimal basis functions over the splines of same order. We show that it is already substantial for small orders and that it further increases with the approximation order . When is large, this sampling gain becomes linear; more specifically, its exact asymptotic expression is 2 ( ) . Since the optimal functions are continuous, but not differentiable, for even orders, and even only piecewise continuous for odd orders, our result implies that regularity has little to do with approximating performance.These theoretical findings are corroborated by experimental evidence that involves compounded rotations of images.

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“…This means that . Due to the link , we thus have By induction on , and by using , we now easily get [16].…”

Section: Appendix B Proof Of Theorem 2 ("Optimal-moms")mentioning

confidence: 99%

“…Moreover, the minimal constant is given by the explicit expression (16) (See the proof in Appendix B.) Some remarks concerning this result are as follows.…”

Section: A Solving the Minimization Problemmentioning

confidence: 99%

MOMS: maximal-order interpolation of minimal support

Blu

ThCvenaz²,

Unser

2001

IEEE Trans. on Image Process.

181

164

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“…Concerning the former, they discussed the approximation-theoretical aspects of such schemes and pointed at the importance of having a high approximation order, rather than a high regularity and a small value for the asymptotic constant, as discussed in the previous subsection. Indeed, the results of their experiments, which included all of the aforementioned piecewise polynomial kernels, as well as the quartic convolution kernel by German [259], several types of windowed-sinc kernels and the O-MOMS [278], confirmed the theoretical predictions and clearly showed the superiority of kernels with optimized properties in these terms.…”

Section: Evaluation Studies and Their Conclusionmentioning

confidence: 65%

“…Its order of approximation, however, is one less. German [259] used Keys' ideas to develop a continuously differentiable quartic interpolator with fifth order of approximation. Also, the present author [260] combined them with Park and Schowengerdt's frequency-domain error analysis to derive a class of odd-degree piecewise polynomial interpolation kernels with increasing regularity.…”

Section: J Development Of Alternative Piecewise Polynomial Kernelsmentioning

confidence: 99%

A chronology of interpolation: from ancient astronomy to modern signal and image processing

Meijering

Falk²

2002

Proc. IEEE

527

252

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“…Other noteworthy kernels include an additional cubic (W = 6, L = 4) by Keys [1981], the C 0 -continuous (interpolating) and the C 1 -continuous (not interpolating) quadratics (N = 2, W = 3, L = 2) of Dodgson [1997], a quartic (L = 5, W = 7, R = 1) by German [1997], and the quintic and septic kernels (W = 6 and 8, but L = 3) by Meijering et al [1999b]. The support of these kernels is larger than necessary for their approximation order.…”

Section: Introductionmentioning

confidence: 99%