Combinatorial Aspects of Multivariate Splines

Adjoint image warping using multivariate splines with application to four‐dimensional computed tomography

Medical Physics ◽

10.1002/mp.14765 ◽

2021 ◽

Author(s):

Jens Renders ◽

Jan Sijbers ◽

Jan De Beenhouwer

Keyword(s):

Computed Tomography ◽

Image Warping ◽

Multivariate Splines

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Approximation of noisy data using multivariate splines and finite element methods

Journal of Algorithms & Computational Technology ◽

10.1177/17483026211008405 ◽

2021 ◽

Vol 15 ◽

pp. 174830262110084

Author(s):

Bishnu P Lamichhane ◽

Elizabeth Harris ◽

Quoc Thong Le Gia

Keyword(s):

Finite Element ◽

Differential Operators ◽

Impulsive Noise ◽

Noisy Data ◽

Multivariate Splines ◽

The Finite Element Method ◽

Data Smoothing ◽

Multivariate Spline ◽

Partial Differential Operators ◽

Mixture Of Gaussian

We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differential operators, and are integrated using the finite element method. We compare three methods to two problems: to remove the mixture of Gaussian and impulsive noise from an image, and to recover a continuous function from a set of noisy observations.

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Multivariate piecewise polynomials

Acta Numerica ◽

10.1017/s0962492900002348 ◽

1993 ◽

Vol 2 ◽

pp. 65-109 ◽

Cited By ~ 20

Author(s):

C. de Boor

Keyword(s):

Approximation Theory ◽

Variational Approach ◽

Polynomial Function ◽

Piecewise Polynomial ◽

Multivariate Splines ◽

Piecewise Polynomial Function ◽

Multivariate Spline ◽

Piecewise Polynomials ◽

The Subject ◽

Function Of Several Arguments

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

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