Short communicationAnalytically exact spiral scheme for generating uniformly distributed points on the unit sphere
Research highlights
▶ Analytically exact spiral scheme can generate highly uniform distribution of points on the unit sphere. ▶ It does not require ad hoc experimentation or unnecessary approximation. ▶ It is applicable to 3D image reconstruction in MR and CT and many other areas of research.
Introduction
The problem of constructing a set of uniformly distributed points on the surface of a sphere has a long and interesting history, which dates back to J.J. Thomson in 1904 [13] and also [5]. A particular variant of the Thomson problem that is of great importance to biomedical imaging is that of generating a nearly uniform distribution of points on the sphere via a deterministic scheme. Although the point set generated through the minimization of electrostatic potential based on Coulomb's law is the gold standard, minimizing the electrostatic potential of one thousand points (or charges) or more remains a formidable task. Therefore, a deterministic scheme capable of generating efficiently and accurately a set of uniformly distributed points on the sphere has an important role to play in many scientific and engineering applications such as 3D projection reconstruction of Computed Tomography (CT) or Magnetic Resonance (MR) images [10], [1], analysis of the distribution of stars [3] and material science [14], not the least of which is to serve as an initial solution (with random perturbation) for the nonlinear minimization of its electrostatic potential energy or the iterative scheme of Centroidal Voronoi Tessellation on the sphere [4]. Many deterministic schemes have been proposed and most notable of which are the spiral, the equal solid angle and the quasi Monte-Carlo-based schemes, e.g., [1], [3], [11], [12], [16].
In this work, we will present an analytically exact spiral scheme for generating a highly uniform set of points on the unit sphere. This analytically exact spiral scheme is intuitive and geometrically motivated and its geometric flavor is similar to those of [12] and [3]. By analytically exact, we mean that our formulation and implementation of the spiral scheme do not depend on asymptotic approximations such as that of Bauer and on arbitrary parameters that need some experimentation, which was the case with the spiral scheme of [11], [12].
Section snippets
Methods
Between the spiral schemes of [11] and [3], Bauer's formulation of his spiral scheme is simpler and more direct, and therefore, we will adopt his formulation in presenting our analytically exact spiral scheme. Fig. 1 shows a surface element and a line element on the unit sphere in spherical coordinates. The line element, ds, can be expressed as
The first step in constructing a spiral scheme is to set the slope of the spiral curve, given by dϕ/dθ, to some
Discussion
The proposed scheme is analytically exact and does not require further experimentation. Although the proposed scheme contains two nonlinear equations that need to be solved iteratively, we have provided here two highly optimized and efficient iterative algorithms. For example, it took only 1.6 and 3.2 s to generate respectively 5000 and 10000 points in Mathematica 7 [15], running on a laptop with Intel® Core i7 CPU at 1.73 GHz. Most notably, the gain in performance is at least three order of
Acknowledgments
The author would like to express his sincere thanks to Dr. Peter J. Basser and Prof. Beth Meyerand for their continued support. Software related to this work is available freely at http://sites.google.com/site/hispeedpackets/. The author would like to acknowledge that the vertices of the spherical Voronoi tessellation were generated from software developed by Dr. John Burkardt, see. http://people.sc.fsu.edu/∼jburkardt/m_src/sphere_cvt/sphere_cvt.html.
Cheng Guan Koay is currently an assistant scientist in the Department of Medical Physics, University of Wisconsin-Madison. After completing his undergraduate studies in Mathematics at Berea College in 2002 and graduate studies in Physics at the University of Wisconsin-Madison in 2005, he held a research position as an IRTA fellow at the National Institutes of Health until August 2010. His current research focuses on diffusion Magnetic Resonance Imaging (diffusion MRI), analysis of MR signals
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Cheng Guan Koay is currently an assistant scientist in the Department of Medical Physics, University of Wisconsin-Madison. After completing his undergraduate studies in Mathematics at Berea College in 2002 and graduate studies in Physics at the University of Wisconsin-Madison in 2005, he held a research position as an IRTA fellow at the National Institutes of Health until August 2010. His current research focuses on diffusion Magnetic Resonance Imaging (diffusion MRI), analysis of MR signals and noise and applications of optimization techniques in biomedical problems.