Application of Voronoi Tessellation of Spherical Surface to Geometrical Models of Skeleton Forms of Spherical Radiolaria (original) (raw)
Related papers
Pores in Spherical Radiolarian Skeletons Directly Determined from Three-Dimensional Data
Forma, 2014
We propose a method that uses three-dimensional data to directly determine the pores (holes) in the skeleton of a spherical radiolarian. Our goal is to automatically determine both the number and the distribution of the pores. We used a set of grid points on a spherical surface to approximate the skeletal structure, which was obtained from a micro X-ray CT scan. Next, we counted the number of pores by using an algorithm for counting clusters on a grid. Finally, we used Voronoi tessellation to determine the distribution of the pores. For noisy data, a smoothing filter was applied before the procedure. We applied our method to three real data sets, and the results showed that our method worked well.
Computer Aided Design, 2019
This paper investigates the relevance of the representation of polycrystalline aggregates using Radical Voronoï (RV) tessellation, computed from Random Close Packs (RCP) of spheres with radius distribution following a lognormal distribution. A continuous relationship between the distribution of sphere radii with that of RV cell volumes is proposed. The stereology problem (deriving the 3D grain size distributions from 2D sections) is also investigated: two statistical methods are proposed, giving analytical continuous relationships between the apparent grain size distribution and the sphere radius distribution. In order to assess the proposed methods, a 3D aggregate has been generated based on a EBSD map of a real polycrystalline microstructure.
Geometric Characteristics of Random Spatial Voronoi Tessellations and Planar Sections
The Voronoi tessellation (Voronoi mosaic, Dirichlet diagram, Thiessen polygons. Wigner-Seitz cells) is a well—known model for the division of the plane or space into convex regions. In many fields of science and technology it is applied for modelling and analysing cell-like structures. Furthermore, in computational geometry it plays an important role as an extremely versatile tool for solving various proximity problems. The Voronoi tessellation can also be used as a stochastic model if the division of space is assumed to be random. This paper compares various distributional properties of geometric characteristics of different random spatial Voronoi tessellations. The results are based on a large-scale simulation study of Voronoi tessellations with respect to the Poisson, Matérn cluster, and Matérn hard-core point processes. Moments, correlation coefficients, and probability densities of geometric characteristics are estimated of both the spatial tessellation (e.g. surface area and vo...
Spherical Laguerre Voronoi diagram approximation to tessellations without generators
Graphical Models /graphical Models and Image Processing /computer Vision, Graphics, and Image Processing, 2018
This paper presents a method for approximating spherical polygonal tessellations with spherical Laguerre Voronoi diagrams when the generators of the tessellations are not available. The approximation method uses a polyhedron corresponding to the spherical Laguerre Voronoi diagram, and the problem is reduced to an optimization problem. The method is implemented on planar photographic images.
Constrained Centroidal Voronoi Tessellations for Surfaces
SIAM Journal on Scientific Computing, 2003
Centroidal Voronoi tessellations are useful for subdividing a region in Euclidean space into Voronoi regions whose generators are also the centers of mass, with respect to a prescribed density function, of the regions. Their extensions to general spaces and sets are also available; for example, tessellations of surfaces in a Euclidean space may be considered. In this paper, a precise definition of such constrained centroidal Voronoi tessellations (CCVTs) is given and a number of their properties are derived, including their characterization as minimizers of an "energy." Deterministic and probabilistic algorithms for the construction of CCVTs are presented and some analytical results for one of the algorithms are given. Computational examples are provided which serve to illustrate the high quality of CCVT point sets. Finally, CCVT point sets are applied to polynomial interpolation and numerical integration on the sphere.
Approximation of a Spherical Tessellation by the Laguerre Voronoi Diagram
2016
This paper presents a method for approximating spherical tessellations, the edges of which are geodesic arcs, using spherical Laguerre Voronoi diagrams. The approximation method involves fitting the polyhedron corresponding to the spherical Laguerre Voronoi diagram to the observed tessellation using optimization techniques.
Modeling of the material structure using Voronoi diagrams and tessellation methods
Tessellation methods are a relatively new approach for modeling the structure of a material. In this paper, such structures are interpreted as sphere packing models, where molecules and atoms represent spheres of equal or different size. Based on the review of the literature, it is shown that the tessellation approach is a powerful method for modeling and simulating such structures with desirable metric and topological properties. Two basic tessellation methods are considered more in detail: the Delaunay tessellation and the Voronoi diagram in Laguerre geometry, as well as some of their generalizations. The principal concepts of both tessellation methods are briefly explained for a better understanding of the application details. It is noted that packing models created by tessellation methods are not based on the use of the gravity camp effect, which is a difference to numerical and mathematic programming modeling approaches. Therefore, tessellation methods permit the development of structures without taking into account the gravitation, what is important for modeling the structure on the microscopic and nano levels, where the influence of the gravitation is studied insufficiently. A review of the related literature is given, focusing on the details of the tessellation method and the particle size distribution.
The Radiolaria Project Structural Tessellation of Double Curved Surfaces
2007
The Radiolaria Project aims to rethink architectural design and manufacturing techniques-it explores the filigree and beautiful skeletons of radiolarians, tiny marine organisms, with their striking hexagonal patterns, and transfers this concept to architectural scale and materializes it in a large scale structure.
Modeling of Spherical Particle Packing Structures Using Mathematical Tessellation
In recent years, the literature shows an increasing interest to tessellation methods based on Voronoi diagrams to model different structures as packing of spheres. Voronoi diagrams have found numerous practical and theoretical applications in a large number of fields in science and technology as well as in computer graphics. A useful property of Voronoi diagrams is that they represent cellular structures found in the nature and technology in a natural manner, easily to understand and to design. Although this approach is really not new, meanwhile its intensive use and, consecutively, a systematical study started around 2000 with advances in nanoscience and nanotechnology. In this chapter, two basic tessellation methods are considered in more detail: the Voronoi-Delaunay tessellation and the Voronoi diagram in Laguerre geometry, as well as some of their generalizations. The principal concepts of both tessellation methods are briefly explained for a better understanding of this approac...
A simple geometric method for navigating the energy landscape of centroidal Voronoi tessellations
SIAM J. Sci. Comput., 2021
Finding optimal centroidal Voronoi tessellations (CVTs) of a 2D domain presents a paradigm for navigating an energy landscape whose desirable critical points have sufficiently small basins of attractions that they are inaccessible with Monte-Carlo initialized gradient descent methods. We present a simple deterministic method for efficiently navigating the energy landscape in order to access these low energy CVTs. The method has two parameters and is based upon each generator moving away from the closest neighbour by a certain distance. We give a statistical analysis of the performance of this hybrid method comparing with the results of a large number of runs for both Lloyd's method and state of the art quasi-Newton methods.