A fast transform for spherical harmonics

Mohlenkamp, Martin J.

doi:10.1007/bf01261607

Cited by 185 publications

(144 citation statements)

References 11 publications

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“…The complexity of these algorithms scales linearly in the desired accuracy. We refer the interested reader to [63], where may be found a detailed and rigorous comparison and analysis of the performances of a variety of Legendre projection algorithms, including those based on the fast multipole method, as well as other methods [49].…”

Section: Historymentioning

confidence: 99%

FFTs for the 2-Sphere-Improvements and Variations

Healy

Rockmore

Kostelec

et al. 2003

J. Fourier Anal. Appl.

258

247

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Section: Historymentioning

confidence: 99%

FFTs for the 2-Sphere-Improvements and Variations

Healy

Rockmore

Kostelec

et al. 2003

J. Fourier Anal. Appl.

258

247

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show abstract

“…One can use fast transforms for both longitude (Fast Fourier Transform) and latitude (Fast Legendre Transform [FLT]) so that the spherical harmonic coefficients can be computed in O(p 2 (log p) 2 + p 2 log p) (see [44]) 3 . The pseudocode for the forward spherical harmonic transform is given in Appendix C. A similar algorithm can be used for the inverse transform.…”

Section: Spatial Schemementioning

confidence: 99%

A fast algorithm for simulating vesicle flows in three dimensions

Veerapaneni

Rahimian

Biros

et al. 2011

Journal of Computational Physics

137

143

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Vesicles are locally-inextensible fluid membranes that can sustain bending. In this paper, we extend "A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows", Veerapaneni et al. Journal of Computational Physics, 228(19), 2009 to general non-axisymmetric vesicle flows in three dimensions.Although the main components of the algorithm are similar in spirit to the axisymmetric case (spectral approximation in space, semi-implicit time-stepping scheme), important new elements need to be introduced for a full 3D method. In particular, spatial quantities are discretized using spherical harmonics, and quadrature rules for singular surface integrals need to be adapted to this case; an algorithm for surface reparameterization is neeed to ensure sufficient of the timestepping scheme, and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs. To characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi-implicit time-stepping scheme, we perform a spectral analysis of the evolution operator on the unit sphere.By introducing these algorithmic components, we obtain a time-stepping scheme that, in our numerical experiments, is unconditionally stable. We present results to analyze the cost and convergence rates of the overall scheme. To illustrate the applicability of the new method, we consider a few vesicle-flow interaction problems: a single vesicle in relaxation, sedimentation, shear flows, and many-vesicle flows.

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“…Here we employ an efficient sampling scheme proposed by Mohlenkamp (1999) that requires fewer samplings of the function than the sampling theorem described in Section 3.1 does. This sampling scheme is derived to sample a function that is ensured to be bandlimited with a particular bandwidth.…”

Section: Modeling Appearance By Using Elsmentioning

confidence: 99%

Appearance Sampling of Real Objects for Variable Illumination

2007

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Abstract. The appearance of an object greatly changes under different lighting conditions. Even so, previous studies have demonstrated that the appearance of an object under varying illumination conditions can be represented by a linear subspace. A set of basis images spanning such a linear subspace can be obtained by applying the principal component analysis (PCA) for a large number of images taken under different lighting conditions. Since little is known about how to sample the appearance of an object in order to correctly obtain its basis images, it was a common practice to use as many input images as possible. In this study, we present a novel method for analytically obtaining a set of basis images of an object for varying illumination from input images of the object taken properly under a set of light sources, such as point light sources or extended light sources. Our proposed method incorporates the sampling theorem of spherical harmonics for determining a set of lighting directions to efficiently sample the appearance of an object. We further consider the issue of aliasing caused by insufficient sampling of the object's appearance. In particular, we investigate the effectiveness of using extended light sources for modeling the appearance of an object under varying illumination without suffering the aliasing caused by insufficient sampling of its appearance.

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A fast transform for spherical harmonics

Cited by 185 publications

References 11 publications

FFTs for the 2-Sphere-Improvements and Variations

FFTs for the 2-Sphere-Improvements and Variations

A fast algorithm for simulating vesicle flows in three dimensions

Appearance Sampling of Real Objects for Variable Illumination

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