Interpolation on spherical geodesic grids: A comparative study

Carfora, Maria Francesca

doi:10.1016/j.cam.2006.10.068

Cited by 34 publications

(28 citation statements)

References 27 publications

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“…This situation is different from the problem dealt in [6] where only an interpolation in the sphere is required. To take into account these differences, we propose to modify Equation 1 by replacing the single RBF with the product of an spatial ( φ ) and an angular ( ψ ) RBF:

y ((X, Θ)) = \sum_{i = 0}^{N - 1} w_{i} φ false(∣ X - X_{i} ∣ false) ψ false(∣ normalΘ - {normalΘ}_{i} ∣ false)

where X = ( x , y , z ) are the spatial coordinates, and Θ = ( φ, θ ) the spherical coordinates of the sampling vector U i .…”

Section: Methodsmentioning

confidence: 83%

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Reconstruction of scattered data in fetal diffusion MRI

Oubel

Koob

Studholme

et al. 2012

Medical Image Analysis

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In this paper we present a method for reconstructing diffusion-weighted MRI data on regular grids from scattered data. The proposed method has the advantage that no specific diffusion model needs to be assumed. Previous work assume the tensor model, but this is not suitable under certain conditions like intravoxel orientational heterogeneity (IVOH). Data reconstruction is particularly important when studying the fetal brain in utero, since registration methods applied for movement and distortion correction produce scattered data in spatial and diffusion domains. We propose the use of a groupwise registration method, and a dual spatio-angular interpolation by using radial basis functions (RBF). Leave-one-out experiments performed on adult data showed a high accuracy of the method. The application to fetal data showed an improvement in the quality of the sequences according to objective criteria based on fractional anisotropy (FA) maps, and differences in the tractography results.

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y ((X, Θ)) = \sum_{i = 0}^{N - 1} w_{i} φ false(∣ X - X_{i} ∣ false) ψ false(∣ normalΘ - {normalΘ}_{i} ∣ false)

where X = ( x , y , z ) are the spatial coordinates, and Θ = ( φ, θ ) the spherical coordinates of the sampling vector U i .…”

Section: Methodsmentioning

confidence: 83%

“…RBFs have already been applied for interpolation on spherical geodesic grids in the context of numerical weather prediction, outperforming linear interpolation strategies [6]. The idea behind RBF interpolation is that every point has an influence on a neighborhood according to some functional φ ( r ), where r is the distance from the point.…”

Section: Methodsmentioning

confidence: 99%

Reconstruction of scattered data in fetal diffusion MRI

Oubel

Koob

Studholme

et al. 2012

Medical Image Analysis

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“…Because the Earth's surface is approximately a spherical manifold, geodesics are relevant to the oceanographic reconstruction problem when a grid-based approach is used [18]. However, we note that geodesics are not directly applicable to the problem described in this paper.…”

Section: Related Workmentioning

confidence: 99%

“…As an example, consider the problem of reconstructing scalar fields representing oceanographic quantities such as 18 O/ 16 O and 13 C/ 12 C isotope ratios for the oceans of the distant past. Stable isotope data are obtained from measurements on benthic foraminifera obtained from deep sea core samples taken from the ocean floor [1].…”

Section: Introductionmentioning

confidence: 99%

Interpolating sparse scattered data using flow information

Streletz

Gebbie

Kreylos

et al. 2016

Journal of Computational Science

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a b s t r a c tScattered data interpolation and approximation techniques allow for the reconstruction of a scalar field based upon a finite number of scattered samples of the field. In general, the fidelity of the reconstruction with respect to the original scalar field tends to deteriorate as the number of samples decreases. For the situation of very sparse sampling, the results may not be acceptable at all. However, if it is known that the scalar field of interest is correlated with a known flow field -as is the case when the scalar field represents the value of an oceanographic tracer that propagates under the influence of the ocean's flow -then this knowledge can be exploited to enhance the scattered data reconstruction method. One way to exploit flow field information is to use it to construct a modified notion of distance between points. Replacing the standard Euclidean distance metric with a flow-field-aware notion of distance provides a method for extending standard scattered data interpolation methods into flow-based methods that produce superior results for very sparse data. The resulting reconstructions typically have lower root-mean-square errors than reconstructions that do not use the flow information, and qualitatively they often appear physically more realistic.

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“…Then, in a short study recently accepted for publication [15] we compared the extension of several different interpolation procedures (linear, distance based and cubic schemes) to the spherical surface. For completeness, we recall a short definition of the interpolation schemes we consider here.…”

Section: Interpolationmentioning

confidence: 99%

Semi‐Lagrangian advection on a spherical geodesic grid

Carfora

2007

Numerical Methods in Fluids

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SUMMARYA simple and efficient numerical method for solving the advection equation on the spherical surface is presented. To overcome the well-known 'pole problem' related to the polar singularity of spherical coordinates, the space discretization is performed on a geodesic grid derived by a uniform triangulation of the sphere; the time discretization uses a semi-Lagrangian approach. These two choices, efficiently combined in a substepping procedure, allow us to easily determine the departure points of the characteristic lines, avoiding any computationally expensive tree-search. Moreover, suitable interpolation procedures on such geodesic grid are presented and compared. The performance of the method in terms of accuracy and efficiency is assessed on two standard test cases: solid-body rotation and a deformation flow.

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Interpolation on spherical geodesic grids: A comparative study

Cited by 34 publications

References 27 publications

Reconstruction of scattered data in fetal diffusion MRI

Reconstruction of scattered data in fetal diffusion MRI

Interpolating sparse scattered data using flow information

Semi‐Lagrangian advection on a spherical geodesic grid

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