Fast evaluation of quadrature formulae on the sphere

Keiner, Jens; Potts, Daniel

doi:10.1090/s0025-5718-07-02029-7

Cited by 36 publications

(31 citation statements)

References 23 publications

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“…We have used the libraries FFTW 3.0.1 [9] and NFFT 3.0 [12], the latter with default settings (pre-computed Kaiser-Bessel window function, cut-off parameter m = 6, oversampling factor σ = 2, and stabilization threshold κ = 1000 for the NFFT on the sphere). Note that the NFFT 3.0 includes the nonequispaced fast spherical Fourier transforms from [15,13]. Example 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In a sense, it is the inverse problem to the matrix vector multiplication f f f = Y Y Yfff which corresponds to evaluate a spherical polynomial on the sampling set. An efficient realization of this matrix vector multiplication is known as fast spherical Fourier transform at arbitrary nodes, and has been proposed in [15,13].…”

Section: Prerequisitesmentioning

confidence: 99%

“…We solve problem (3.1) by a factorized variant of conjugate gradients (CGNR, N for "Normal equation" and R for "Residual minimization,") where we use the nonequispaced fast spherical Fourier transform for fast matrix vector multiplications with Y Y Y and its adjoint Y Y Y , see [15,13] for details. Note that for 154Nδ < 1 a constant number of iterations suffices to decrease the residual to a certain fraction, i.e., the total number of floating point operations is bounded by the complexity O(N 2 log 2 N + M) of the fast spherical Fourier transform.…”

Section: Eigenvalue Estimatementioning

confidence: 99%

“…The most prominent approaches rely on so-called zonal basis function methods [28] or on finite expansions into spherical harmonics [19,23]. We focus on the latter, i.e., the use of spherical polynomials since these allow for the application of the fast spherical Fourier transforms, see, for example, [15,13] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Reconstruction of Functions on the Sphere from Scattered Data

Keiner

Kunis

Potts

2007

J Fourier Anal Appl

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Recently, fast and reliable algorithms for the evaluation of spherical harmonic expansions have been developed. The corresponding sampling problem is the computation of Fourier coefficients of a function from sampled values at scattered nodes. We consider a least squares approximation and an interpolation of the given data. Our main result is that the rate of convergence of the two proposed iterative schemes depends only on the mesh norm and the separation distance of the nodes. In conjunction with the nonequispaced FFT on the sphere, the reconstruction of N 2 Fourier coefficients from M reasonably distributed samples is shown to take O(N 2 log 2 N + M) floating point operations. Numerical results support our theoretical findings.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Prerequisitesmentioning

confidence: 99%

Section: Eigenvalue Estimatementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient Reconstruction of Functions on the Sphere from Scattered Data

Keiner

Kunis

Potts

2007

J Fourier Anal Appl

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“…More precisely, the algorithms for the first two cases S 1 and T 2 , called nonequispaced fast Fourier transform, (NFFT) or unequally spaced fast Fourier transform can be found, e.g., in [8,18,34,46]. The algorithms on the sphere S 2 , called nonequispaced fast spherical Fourier transform (NFSFT), were developed in [35,38]; see also [16,30]. In our numerical examples we have applied the software package [33].…”

Section: A Least Squares Setting Let X ∈ {Smentioning

confidence: 99%

Quadrature Errors, Discrepancies, and Their Relations to Halftoning on the Torus and the Sphere

Gräf¹,

Potts²,

Steidl³

2012

SIAM J. Sci. Comput.

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Abstract. This paper deals with continuous-domain quantization, which aims to create the illusion of a gray-value image by appropriately distributing black dots. For lack of notation, we refer to the process as halftoning, which is usually associated with the quantization on a discrete grid. Recently a framework for this task was proposed by minimizing an attraction-repulsion functional consisting of the difference of two continuous, convex functions. The first one of these functions describes attracting forces caused by the image gray values, the second one enforces repulsion between the dots. In this paper, we generalize this approach by considering quadrature error functionals on reproducing kernel Hilbert spaces (RKHSs) with respect to the quadrature nodes, where we ask for optimal distributions of these nodes. For special reproducing kernels these quadrature error functionals coincide with discrepancy functionals, which leads to a geometric interpretation. It turns out that the original attraction-repulsion functional appears for a special RKHS of functions on R 2 . Moreover, our more general framework enables us to consider optimal point distributions not only in R 2 but also on the torus T 2 and the sphere S 2 . For a large number of points the computation of such point distributions is a serious challenge and requires fast algorithms. To this end, we work in RKHSs of bandlimited functions on T 2 and S 2 . Then the quadrature error functional can be rewritten as a least squares functional. We use a nonlinear conjugate gradient method to compute a minimizer of this functional and show that each iteration step can be computed in an efficient way by fast Fourier transforms at nonequispaced nodes on the torus and the sphere. Numerical examples demonstrate the good quantization results obtained by our method. 1. Introduction. Halftoning is a method for creating the illusion of a continuous tone image having only a small number of tones available. It is usually associated with the quantization on a discrete grid. In this paper, we use the name halftoning for continuous-domain quantization; cf. [48,28]. We focus just on two tones, black and white, and ask for appropriate distributions of the black "dots." Applications of halftoning include printing and geometry processing [54] as well as sampling problems occurring in rendering [60], relighting [36], and artistic nonphotorealistic image visualization [4,49]. Halftoning has been an active field of research for many years.Dithering methods which place the black dots at image grid points include, e.g., ordered dithering [7,47], error diffusion [22,32,52,53], global or direct binary search [1,6], and structure-aware halftoning [45]. Ostromoukhov [44] extended the error

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Efficient reconstruction of functions on the sphere from scattered data

Keiner

Kunis

Potts

2007

Proc Appl Math and Mech

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Motivated by the fact that most data collected over the surface of the earth is available at scattered nodes only, the least squares approximation and interpolation of such data has attracted much attention, see e.g. [1,2,5]. The most prominent approaches rely on so-called zonal basis function methods [16] or on finite expansions into spherical harmonics [12,14]. We focus on the latter, i.e., the use of spherical polynomials since these allow for the application of the fast spherical Fourier transform, see for example [8,9] and the references therein.

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Fast evaluation of quadrature formulae on the sphere

Cited by 36 publications

References 23 publications

Efficient Reconstruction of Functions on the Sphere from Scattered Data

Efficient Reconstruction of Functions on the Sphere from Scattered Data

Quadrature Errors, Discrepancies, and Their Relations to Halftoning on the Torus and the Sphere

Efficient reconstruction of functions on the sphere from scattered data

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