66 research outputs found
Probabilistic and parallel algorithms for centroidal Voronoi tessellations with application to meshless computing and numerical analysis on surfaces
Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, optimal allocation of resources, and grid generation. A detailed review is given in chapter 1. In chapter 2, some probabilistic methods for determining centroidal Voronoi tessellations and their parallel implementation on distributed memory systems are presented. The results of computational experiments performed on a CRAY T3E-600 system are given for each algorithm. These demonstrate the superior sequential and parallel performance of a new algorithm we introduce. Then, new algorithms are presented in chapter 3 for the determination of point sets and associated support regions that can then be used in meshless computing methods. The algorithms are probabilistic in nature so that they are totally meshfree, i.e., they do not require, at any stage, the use of any coarse or fine boundary conforming or superimposed meshes. Computational examples are provided that show, for both uniform and non-uniform point distributions that the algorithms result in high-quality point sets and high-quality support regions. The extensions of centroidal Voronoi tessellations to general spaces and sets are also available. For example, tessellations of surfaces in a Euclidean space may be considered. In chapter 4, a precise definition of such constrained centroidal Voronoi tessellations (CCVT\u27s) is given and a number of their properties are derived, including their characterization as minimizers of a kind of energy. Deterministic and probabilistic algorithms for the construction of CCVT\u27s are presented and some analytical results for one of the algorithms are given. Some computational examples are provided which serve to illustrate the high quality of CCVT point sets. CCVT point sets are also applied to polynomial interpolation and numerical integration on the sphere. Finally, some conclusions are given in chapter 5
Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional Archive of Phenotypic Elites Algorithm
The recently introduced Multi-dimensional Archive of Phenotypic Elites
(MAP-Elites) is an evolutionary algorithm capable of producing a large archive
of diverse, high-performing solutions in a single run. It works by discretizing
a continuous feature space into unique regions according to the desired
discretization per dimension. While simple, this algorithm has a main drawback:
it cannot scale to high-dimensional feature spaces since the number of regions
increase exponentially with the number of dimensions. In this paper, we address
this limitation by introducing a simple extension of MAP-Elites that has a
constant, pre-defined number of regions irrespective of the dimensionality of
the feature space. Our main insight is that methods from computational geometry
could partition a high-dimensional space into well-spread geometric regions. In
particular, our algorithm uses a centroidal Voronoi tessellation (CVT) to
divide the feature space into a desired number of regions; it then places every
generated individual in its closest region, replacing a less fit one if the
region is already occupied. We demonstrate the effectiveness of the new
"CVT-MAP-Elites" algorithm in high-dimensional feature spaces through
comparisons against MAP-Elites in maze navigation and hexapod locomotion tasks
Layered Fields for Natural Tessellations on Surfaces
Mimicking natural tessellation patterns is a fascinating multi-disciplinary
problem. Geometric methods aiming at reproducing such partitions on surface
meshes are commonly based on the Voronoi model and its variants, and are often
faced with challenging issues such as metric estimation, geometric, topological
complications, and most critically parallelization. In this paper, we introduce
an alternate model which may be of value for resolving these issues. We drop
the assumption that regions need to be separated by lines. Instead, we regard
region boundaries as narrow bands and we model the partition as a set of smooth
functions layered over the surface. Given an initial set of seeds or regions,
the partition emerges as the solution of a time dependent set of partial
differential equations describing concurrently evolving fronts on the surface.
Our solution does not require geodesic estimation, elaborate numerical solvers,
or complicated bookkeeping data structures. The cost per time-iteration is
dominated by the multiplication and addition of two sparse matrices. Extension
of our approach in a Lloyd's algorithm fashion can be easily achieved and the
extraction of the dual mesh can be conveniently preformed in parallel through
matrix algebra. As our approach relies mainly on basic linear algebra kernels,
it lends itself to efficient implementation on modern graphics hardware.Comment: Natural tessellations, surface fields, Voronoi diagrams, Lloyd's
algorith
AN ADAPTIVE POLYGONAL CENTROIDAL VORONOI TESSELLATION ALGORITHM FOR SEGMENTATION OF NOISY SAR IMAGES
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