Manuel Gräf

Sampling Sets and Quadrature Formulae on the Rotation Group

In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature 𝕊2 × 𝕊1 of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on 𝕊2 and 𝕊1, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on 𝕊2 and 𝕊1 lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.

A Continuous Approach to Discrete Ordering on S^2

We consider the classical problem to find optimal distributions of interacting particles on a sphere by solving an evolution problem for a particle density. The higher order surface partial differential equation is an approximation of a surface dynamic density functional theory. We motivate the approach phenomenologically and sketch a derivation of the model starting from an interatomic potential. Different numerical approaches are discussed to solve the evolution problem: (a) an implicit approach to describe the surface using a phase-field description, (b) a parametric finite element approach, and (c) a spectral method based on nonequispaced fast Fourier transforms on the sphere. Results for computed minimal energy configurations are discussed for various particle numbers and are compared with known rigorous asymptotic results. Furthermore extensions to other more complex and evolving surfaces are mentioned.

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

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 fu...

A unified approach to scattered data approximation on

In this paper we use the connection between the rotation group SO(3) and the three-dimensional Euclidean sphere

\mathbb{S}^{\bf 3}

\mathbb{S}^{3}

and SO(3)

in order to carry over results on the sphere

On the computation of nonnegative quadrature weights on the sphere

\mathbb{S}^{3}

Efficient Algorithms for the Computation of Optimal Quadrature Points on Riemannian Manifolds

directly to the rotation group SO(3) and vice versa. More precisely, these results connect properties of sampling sets and quadrature formulae on SO(3) and

On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms

\mathbb{S}^{3}

Quadrature nodes meet stippling dots

respectively. Furthermore we relate Marcinkiewicz–Zygmund inequalities and conditions for the existence of positive quadrature formulae on the rotation group SO(3) to those on the sphere