A survey on spherical designs and algebraic combinatorics on spheres

Spherical t-designs provide quadrature rules for the sphere which are exact for polynomials up to degree t. In this paper, we propose a computational algorithm based on interval arithmetic which, for given t, upon successful completion will have proved the existence of a t-design with (t + 1) 2 nodes on the unit sphere S 2 ⊂ R 3 and will have computed narrow interval enclosures which are known to contain these nodes with mathematical certainty. Since there is no theoretical result which proves the existence of a t-design with (t + 1) 2 nodes for arbitrary t, our method contributes to the theory because it was tested successfully for t = 1, 2, . . . , 100. The t-design is usually not unique; our method aims at finding a well-conditioned one. The method relies on computing an interval enclosure for the zero of a highly nonlinear system of dimension (t + 1) 2 . We therefore develop several special approaches which allow us to use interval arithmetic efficiently in this particular situation. The computations were all done using the MATLAB toolbox INTLAB. Mathematics Subject Classification (2000) 65H10 · 65G20Dedicated to Götz Alefeld on the occasion of his 70th birthday.

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“…A lower bound on the number of points N t needed to construct a spherical t-design for any t ≥ 1 and d = 2 was given in [5]: 2 if t is even.…”

Section: Introductionmentioning

confidence: 99%

Computational existence proofs for spherical t-designs

2010

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“…, 100, spherical N -designs with (N + 1) 2 nodes exist using the characterization of fundamental spherical N -designs and interval arithmetic. For further recent developments on spherical N -designs and related topics we refer to the survey article [5]. Finally, we remark that the equivalence between spherical N -designs and the relation…”

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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“…It is well known (see, for example, [4,9]) that there are many equivalent conditions for a set X N ⊂ S 2 to be a spherical t-design. Among these equivalent statements, one that plays an important role in the subsequent discussion is the following proposition.…”

mentioning

confidence: 99%

“…Since then, spherical t-designs have been studied extensively [2,3,6,7,13,17,29,30,32]. In 2009, Bannai and Bannai gave a comprehensive survey of research on spherical t-designs in the last three decades [4].…”

mentioning

confidence: 99%

Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

An¹,

Chen

Sloan³

et al. 2010

SIAM J. Numer. Anal.

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Abstract.A set X N of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over X N is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical t-designs on the unit sphere S 2 ⊂ R 3 when N ≥ (t + 1) 2 , the dimension of the space Pt of spherical polynomials of degree at most t. We show how to construct well conditioned spherical designs with N ≥ (t + 1) 2 points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints. Interval methods are then used to prove the existence of a true spherical t-design very close to the calculated points and to provide a guaranteed interval containing the determinant. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example.

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A survey on spherical designs and algebraic combinatorics on spheres

Cited by 147 publications

References 122 publications

Computational existence proofs for spherical t-designs

Computational existence proofs for spherical t-designs

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

Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

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