Computation of Compact Distributions of Discrete Elements

Abstract

In our daily lives, many plane patterns can actually be regarded as a compact distribution of a number of elements with certain shapes, like the classic pattern mosaic. In order to synthesize this kind of pattern, the basic problem is, with given graphics elements with certain shapes, to distribute a large number of these elements within a plane region in a possibly random and compact way. It is not easy to achieve this because it not only involves complicated adjacency calculations, but also is closely related to the shape of the elements. This paper attempts to propose an approach that can effectively and quickly synthesize compact distributions of elements of a variety of shapes. The primary idea is that with the seed points and distribution region given as premise, the generation of the Centroidal Voronoi Tesselation (CVT) of this region by iterative relaxation and the CVT will partition the distribution area into small regions of Voronoi, with each region representing the space of an element, to achieve a compact distribution of all the elements. In the generation process of Voronoi diagram, we adopt various distance metrics to control the shape of the generated Voronoi regions, and finally achieve the compact element distributions of different shapes. Additionally, approaches are introduced to control the sizes and directions of the Voronoi regions to generate element distributions with size and direction variations during the Voronoi diagram generation process to enrich the effect of compact element distributions. Moreover, to increase the synthesis efficiency, the time-consuming Voronoi diagram generation process was converted into a graphical rendering process, thus increasing the speed of the synthesis process. This paper is an exploration of elements compact distribution and also carries application value in the fields like mosaic pattern synthesis.

Modeling a domain wall network in BiFeO3 with stochastic geometry and entropy-based similarity measure

“…The algorithm stops when the maximum distance between the seed points and centroids is less than an arbitrary value. The Voronoi tessellation is a distancebased method, which means that, based on the distance metric that is used to compute the tessellation, several types of patterns can be realized (Chen et al, 2019). As an example, if a L 1 -metric (Manhattan distance) is used, the Voronoi domains will converge toward a pattern of squares rotated at 45° (Efros and Leung, 1999), while the generated Voronoi region will be close to a regular hexagon if a L 2 -metric distance is used.…”

“…1 ) and β is the scale factor. Rectangleshaped Voronoi regions are generated with this metric (Chen et al, 2019).…”