Open AccessJournal Article
A survey on bivariate Lagrange interpolation on Lissajous nodes
TL;DR: In this paper, a survey on bivariate polynomial interpolation on the node points of Lissajous curves is presented, which is a generalization of the generating curve approach developed for Lagrange interpolation of the Padua points.
read more
Abstract: This article is a survey on recent research on bivariate polynomial interpolation on the node points of Lissajous curves. The resulting theory is a generalization of the generating curve approach developed for Lagrange interpolation on the Padua points. After classifying the different types of Lissajous curves, we give a short overview on interpolation and quadrature rules defined on the node points of the Lissajous curves. Further, we summarize some convergence results and show how the interpolating polynomials can be computed efficiently. Finally, the developed theory is applied to a practical problem from a medical imaging modality called Magnetic Particle Imaging.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
•Posted Content
Multivariate polynomial interpolation on Lissajous-Chebyshev nodes
Peter Dencker,Wolfgang Erb +1 more
TL;DR: In this paper, a discrete orthogonality structure on non-tensor product node sets related to Lissajous curves and Chebyshev varieties is derived, and polynomial interpolation and quadrature rules for these node sets are obtained.
16
Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous-Chebyshev nodes
Peter Dencker,Wolfgang Erb,Yurii Kolomoitsev,Yurii Kolomoitsev,Tetiana Lomako,Tetiana Lomako +5 more
TL;DR: To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous-Chebyshev node points, upper and lower bounds for the respective Lebesgue constant are derived.
15
•Posted Content
Trivariate polynomial approximation on Lissajous curves
TL;DR: Lissajous curves in the 3-cube are studied, that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices that are used to construct trivariate hyperinterpolation polynomials and to compute discrete extremal sets of Fekete and Leja type for trivARIate polynomial interpolation.
15
•Posted Content
Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in MPI
TL;DR: An RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function and an application to interpolation in magnetic particle imaging shows that the presented method is very promising.
11
On (β,γ)-Chebyshev functions and points of the interval
TL;DR: This paper introduces the class of ( β , γ ) -Chebyshev functions and corresponding points, which can be seen as a family of generalized ChebysheV polynomials and points, and proves that they are orthogonal in certain subintervals of [ − 1 , 1 ] with respect to a weighted arcsine measure.
6
References
Tomographic imaging using the nonlinear response of magnetic particles
TL;DR: A method for obtaining a high-resolution image of magnetic tracers that takes advantage of the nonlinear magnetization curve of small magnetic particles and has the potential to be developed into an imaging method characterized by both high spatial resolution as well as high sensitivity.
2.1K
•Book
Orthogonal Polynomials of Several Variables
Charles F. Dunkl,Yuan Xu +1 more
- 19 Mar 2001
TL;DR: In this article, the authors considered the properties of orthogonal polynomials on the unit sphere, root systems and Coxeter groups, and the Summability of Orthogonal expansions.
Polynomial interpolation in several variables
M. Gasca,Thomas Sauer +1 more
TL;DR: This is a survey of the main results on multivariate polynomial interpolation in the last twenty-five years, a period of time when the subject experienced its most rapid development.
Three-dimensional real-time in vivo magnetic particle imaging
TL;DR: First in vivo 3D real-time MPI scans are presented revealing details of a beating mouse heart using a clinically approved concentration of a commercially available MRI contrast agent.
686
Magnetic particle imaging: current developments and future directions
N Panagiotopoulos,RL Duschka,Mandy Ahlborg,Gael Bringout,Christina Debbeler,Matthias Graeser,Christian Kaethner,Kerstin Lüdtke-Buzug,Hanne Medimagh,Jan Stelzner,Thorsten M. Buzug,Jörg Barkhausen,Florian M. Vogt,Julian Haegele +13 more
TL;DR: This review aims at giving an overview of current developments and future directions in MPI about a decade after its first appearance.