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Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fr\`{e}chet distance. In this…
The Voronoi diagrams technique was introduced by Cabello to compute the diameter of planar graphs in subquadratic time. We present novel applications of this technique in static, fault-tolerant, and partially-dynamic undirected unweighted…
In this paper, we present a new approach to obtain symmetric tiles with curved edges. Our approach is based on using higher-order Voronoi sites that are closed under wallpaper symmetries. The resulting Voronoi tessellations provide us with…
Recent work has demonstrated that the interior material layout of a 3D model can be designed to make a fabricated replica satisfy application-specific demands on its physical properties such as resistance to external loads. A widely used…
Formulating boundary value problems for multidimensional partial derivative equations in terms of invariant operators of vector (tensor) analysis is convenient. Computational algorithms for approximate solutions are based on constructing…
Although Poisson-Voronoi diagrams have interesting mathematical properties, there is still much to discover about the geometrical properties of its grains. Through simulations, many authors were able to obtain numerical approximations of…
Voronoi diagrams are highly compact representations that are used in various Graphics applications. In this work, we show how to embed a differentiable version of it -- via a novel deep architecture -- into a generative deep network. By…
In this paper we develop a numerical method for solving a class of optimization problems known as optimal location or quantization problems. The target energy can be written either in terms of atomic measures and the Wasserstein distance or…
Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a…
The rapid advances in 3D scanning and acquisition techniques have given rise to the explosive increase of volumetric digital models in recent years. This dissertation systematically trailblazes a novel volumetric modeling framework to…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
The use of numerical simulations in science is ever increasing and with it the computational size. In many cases single processors are no longer adequate and simulations are run on multiple core machines or supercomputers. One of the key…
Let $P$ be a planar set of $n$ sites in general position. For $k\in\{1,\dots,n-1\}$, the Voronoi diagram of order $k$ for $P$ is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest $k$…
This paper introduces a new approach toward characterizing local structural features of two-dimensional particle systems. The approach can accurately identify and characterize defects in high-temperature crystals, distinguish a wide range…
This paper presents the Voronoi diagram-based evolutionary algorithm (VorEAl). VorEAl partitions input space in abnormal/normal subsets using Voronoi diagrams. Diagrams are evolved using a multi-objective bio-inspired approach in order to…
The 3D network originated by the faces of irregular Poissonian Voronoi Polyhedrons may represent the backbone on which the galaxies are originated. As a consequence the spatial appearance of the catalogs of galaxies can be reproduced. The…
For the analysis of systems consisting of small, regular objects, the methods of mathematical morphology applied to images of these systems are well-suited. One of these methods is the use of Voronoi polygons. It was found that the Voronoi…
Unlike other schemes that locally violate the essential stability properties of the analytic parabolic and elliptic problems, Voronoi finite volume methods (FVM) and boundary conforming Delaunay meshes provide good approximation of the…
Numerical simulations of the air in the atmosphere and water in the oceans are essential for numerical weather prediction. The state-of-the-art for performing these fluid simulations relies on an Eulerian viewpoint, in which the fluid…
Embedding methods for product spaces are powerful techniques for low-distortion and low-dimensional representation of complex data structures. Here, we address the new problem of linear classification in product space forms -- products of…