Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications as, for instance, medical imaging. In this paper, we study an RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of kernel functions, information on the edges of the interpolated function is necessary. We characterize the native space spanned by these kernel functions and study error bounds in terms of the fill distance of the node set. To extract the location of the discontinuities, we use a segmentation method based on a classification algorithm from machine learning. The conducted numerical experiments confirm the theoretically derived convergence rates in case that the discontinuities are a priori known. Further, an application to interpolation in magnetic particle imaging shows that the presented method is very promising.

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“…with n = (n, n + 1), n ∈ N, are the generating curves of the Padua points [2,13]. If = 2, then the curve is non-degenerate.…”

Section: Lissajous Interpolation Nodesmentioning

confidence: 99%

Shape-Driven Interpolation With Discontinuous Kernels: Error Analysis, Edge Extraction, and Applications in Magnetic Particle Imaging

Marchi¹,

Erb²,

Marchetti³

et al. 2020

SIAM J. Sci. Comput.

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“…with a frequency vector m = (m 1 , m 2 ) ∈ N 2 and a rotation parameter α ∈ R. The curve (m) α lies in the unit sphere S 2 = {x ∈ R 3 : x 2 1 + x 2 2 + x 2 3 = 1} of the three-dimensional space R 3 . Similar as for bivariate Lissajous curves [7,8,11,12], the curve (m) α describes a superposition of a latitudinal and a longitudinal harmonic motion determined by the frequencies m 1 and m 2 .…”

Section: Introductionmentioning

confidence: 89%

“…Now applying Proposition 3 we can find i ∈ I (m) and v ∈ {−1, 1} such that the relations(11) and(12)are satisfied. In particular, this implies (m) 2ρ/m 2 (t (m) l ) = sin vi 1 π m 1…”

mentioning

confidence: 99%

A spectral interpolation scheme on the unit sphere based on the nodes of spherical Lissajous curves

Erb

2018

IMA Journal of Numerical Analysis

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For sampling values along spherical Lissajous curves we establish a spectral interpolation and quadrature scheme on the sphere. We provide a mathematical analysis of spherical Lissajous curves and study the characteristic properties of their intersection points. Based on a discrete orthogonality structure we are able to prove the unisolvence of the interpolation problem. As basis functions for the interpolation space we use a parity-modified double Fourier basis on the sphere which allows us to implement the interpolation scheme in an efficient way. We further show that the numerical condition number of the interpolation scheme displays a logarithmic growth. As an application, we use the developed interpolation algorithm to estimate the rotation of an object based on measurements at the spherical Lissajous nodes.

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“…In this case, the black and white dots indicate the two interlacing grids determining RD The frequency parameters m 1 and m 2 in the curve (m) α determine a superposition of a radial and an angular harmonic motion. For this reason, rose curves can also be regarded as polar variants of bivariate Lissajous curves [7,8,11,13]. If the numbers m 1 and m 2 are relatively prime, the minimal period P of (m) α is given by P = 2π if m 1 + m 2 is odd, and P = π if m 1 + m 2 is even (see Proposition 1).…”

Section: Comparison To Existing Workmentioning

confidence: 99%

“…However, if the rectangular spectral index set Γ (m) is used for the interpolation space, we can guarantee that P (m) f is continuous also at the center. (0, θ) (see also (54) in the proof of Theorem 10) is contained in a space of dimension larger than m 2 and the m 2 given boundary conditions can not guarantee that P (a) Interpolant P (10,11) R,f using Γ (10,11) .…”

Section: The Interpolating Functions Pmentioning

confidence: 99%

Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk

Erb

2020

Constr Approx

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Rhodonea curves are classical planar curves in the unit disk with the characteristic shape of a rose. In this work, we use point samples along such rose curves as node sets for a novel spectral interpolation scheme on the disk. By deriving a discrete orthogonality structure on these rhodonea nodes, we will show that the spectral interpolation problem is unisolvent. The underlying interpolation space is generated by a parity-modified Chebyshev-Fourier basis on the disk. This allows us to compute the spectral interpolant in an efficient way. Properties as continuity, convergence and numerical condition of the scheme depend on the spectral structure of the interpolation space. For rectangular spectral index sets, we show that the interpolant is continuous at the center, the Lebesgue constant grows logarithmically and that the scheme converges fast if the function under consideration is smooth. Finally, we derive a Clenshaw-Curtis quadrature rule using function evaluations at the rhodonea nodes and conduct some numerical experiments to compare different parameters of the scheme.Keywords: Spectral interpolation on the disk, rhodonea curves, intersection and boundary nodes of rhodonea curves, parity-modified Chebyshev-Fourier series, Clenshaw-Curtis quadrature on the disk, numerical condition and convergence of interpolation schemes 2010 MSC: 41A05,42A16,65D05,65T50

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Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

Cited by 20 publications

References 31 publications

Shape-Driven Interpolation With Discontinuous Kernels: Error Analysis, Edge Extraction, and Applications in Magnetic Particle Imaging

Shape-Driven Interpolation With Discontinuous Kernels: Error Analysis, Edge Extraction, and Applications in Magnetic Particle Imaging

A spectral interpolation scheme on the unit sphere based on the nodes of spherical Lissajous curves

Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk

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