Voronoi Regions Research Articles

A Nearest Neighbor Algorithm for Imbalanced Classification

Due to the inability of the accuracy-driven methods to address the challenging problem of learning from imbalanced data, several alternative measures have been proposed in the literature, like the Area Under the ROC Curve (AUC), the Average Precision (AP), the F-measure, the G-Mean, etc. However, these latter measures are neither smooth, convex nor separable, making their direct optimization hard in practice. In this paper, we tackle the challenging problem of imbalanced learning from a nearest-neighbor (NN) classification perspective, where the minority examples typically belong to the class of interest. Based on simple geometrical ideas, we introduce an algorithm that rescales the distance between a query sample and any positive training example. This leads to a modification of the Voronoi regions and thus of the decision boundaries of the NN classifier. We provide a theoretical justification about this scaling scheme which inherently aims at reducing the False Negative rate while controlling the number of False Positives. We further formally establish a link between the proposed method and cost-sensitive learning. An extensive experimental study is conducted on many public imbalanced datasets showing that our method is very effective with respect to popular Nearest-Neighbor algorithms, comparable to state-of-the-art sampling methods and even yields the best performance when combined with them.

On the Analysis of Spatially Constrained Power of Two Choice Policies

We consider a class of power of two choice based assignment policies for allocating users to servers, where both users and servers are located on a two-dimensional Euclidean plane. In this framework, we investigate the inherent tradeoff between the communication cost, and load balancing performance of different allocation policies. To this end, we first design and evaluate a Spatial Power of two (sPOT) policy in which each user is allocated to the least loaded server among its two geographically nearest servers sequentially. When servers are placed on a two-dimensional square grid, sPOT maps to the classical Power of two (POT) policy on the Delaunay graph associated with the Voronoi tessellation of the set of servers. We show that the associated Delaunay graph is 4-regular and provide expressions for asymptotic maximum load using results from the literature. For uniform placement of servers, we map sPOT to a classical balls and bins allocation policy with bins corresponding to the Voronoi regions associated with the second order Voronoi diagram of the set of servers. We provide expressions for the lower bound on the asymptotic expected maximum load on the servers and prove that sPOT does not achieve POT load balancing benefits. However, experimental results suggest the efficacy of sPOT with respect to expected communication cost. Finally, we propose two non-uniform server sampling based POT policies that achieve the best of both the performance metrics. Experimental results validate the effectiveness of our proposed policies.

Approximation of Gaussian Curvature by the Angular Defect: An Error Analysis

It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian curvature at a vertex on the surface is approximated by the quotient of the angular defect and the area of the Voronoi region. The Voronoi region is the subset of the mesh that contains all points that are closer to the vertex than to any other vertex. Numerical error analyses suggest that the Gauss–Bonnet scheme always converges with quadratic convergence speed. However, the general validity of this conclusion remains uncertain. We perform an analytical error analysis on the Gauss–Bonnet scheme. Under certain conditions on the mesh, we derive the convergence speed of the Gauss–Bonnet scheme as a function of the maximal distance between the vertices. We show that the conditions are sufficient and necessary for a linear convergence speed. For the special case of locally spherical surfaces, we find a better convergence speed under weaker conditions. Furthermore, our analysis shows that the Gauss–Bonnet scheme, while generally efficient and effective, can give erroneous results in some specific cases.

Identification, Computational Examination, Critical Assessment and Future Considerations of Spatial Tactical Variables to Assess the Use of Space in Team Sports by Positional Data: A Systematic Review.

The aim of the review was to identify the spatial tactical variables used to assess the use of space in team sports using positional data. In addition, we examined computational methods, performed a critical assessment and suggested future considerations. We considered four electronic databases. A total of 3973 documents were initially retrieved and only 15 articles suggested original spatial variables or different computation methods. Spatial team sport tactical variables can be classified into 3 principal types: occupied space, total field coverage by several players; exploration space, the mean location (±standard deviations in X- and Y-directions) of the player/team during the entire game; and dominant/influence space, the region the players can reach before any other players. Most of the studies, i.e., 55%, did not include goalkeepers (GKs) and total playing space to assess occupied space, however, several proposed new variables that considered that all playing space could be “played” (i.e. effective free-space, normalized surface area). Only a collective exploration space variable has been suggested: the major range of the geometrical centre (GC). This suggestion could be applied to assess collective exploration space variables at a sub-system level. The measurement of the dominant/influence space has been based on the Voronoi region (i.e. distance d criteria), but several studies also based their computation on the time (t). In addition, several weighted dominant areas have been suggested. In conclusion, the use of spatial collective tactical variables considering the principal structural traits of each team sport (e.g. players of both teams, the location of the space with respect to the goal, and the total playing space) is recommended.

Multiple Importance Sampling for Symbol Error Rate Estimation of Maximum-Likelihood Detectors in MIMO Channels

In this paper we propose a multiple importance sampling (MIS) method for the efficient symbol error rate (SER) estimation of maximum likelihood (ML) multiple-input multipleoutput (MIMO) detectors. Given a transmitted symbol from the input lattice, obtaining the SER requires the computation of an integral outside its Voronoi region in a high-dimensional space, for which a closed-form solution does not exist. Hence, the SER must be approximated through crude or naive Monte Carlo (MC) simulations. This practice is widely used in the literature despite its inefficiency, particularly severe at high signal-to-noise-ratio (SNR) or for systems with stringent SER requirements. It is well-known that more sophisticated MC-based techniques such as MIS, when carefully designed, can reduce the variance of the estimators in several orders of magnitude with respect to naive Monte Carlo in rare-event estimation, or equivalently, they need significantly less samples for attaining a desired performance. The proposed MIS method provides unbiased SER estimates by sampling from a mixture of components that are carefully chosen and parametrized. The number of components, the parameters of the components, and their weights in the mixture, are automatically chosen by the proposed method. As a result, the proposed method is flexible, easy-to-use, theoretically sound, and presents a high performance in a variety of scenarios. We show in our simulations that SERs lower than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-8</sup> can be accurately estimated with just 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> random samples.

Qualitative-geometric ‘surrounds’ relations between disjoint regions

ABSTRACT This paper explores a class of qualitative-geometric relations between disjoint regions, embedded in the surface of a sphere or in the plane. These relations combine topological information about the configuration of the regions themselves as well as of the geometric Voronoi regions surrounding them. The method uses maptrees to construct a rigorous and exhaustive framework for n-ary relations between ensembles of regions. The resulting uniquely defined relations are akin to the verbal spatial relation ‘surrounds,’ in one common interpretation. The paper provides an exhaustive formal classification of such extended spatial relations between up to three regions. A further exhaustive computational exploration of up to seven regions in the sphere also provides an algorithm to enumerate all possible configurations. The results demonstrate how the approach can be used to provide fine-grained and salient descriptions of qualitative-geometric relations for complex scenes involving ensembles of multiple regions.

Voronoi Partition to Support Data Search in Uncertain Database with k-Bound Filtering

With the development of mobile technology, one of its main functions is to have mobile navigation capabilities, this is one of the Location-Based Services (LBS). Mobile navigation is designed as a mobile device to be able to monitor access objects from users. Because of the mobile characteristic, the impact on the points of objects to be found always occur a not certain change. Efficient query processing is needed on a set of mobile data due to the movement of users. This movement has an impact on searching in an uncertain database. For this uncertain database, the important query method is Probabilistic k-Nearest Neighbor query (PkNN), which calculates the probability of the set of k objects to be closest to the given query point. Several studies have been conducted at this time in order to contribute to the search results in a mobile database and uncertain with the support of certain algorithms to produce better performance. In this study, we propose a method called voronoi partitioning to support searching in uncertain database (Partition threshold k Aggregate Nearest Neighbor query method-Partition_PANN). In designing the partition of the voronoi region, it starts by using voronoi local networks to calculate query answers for small areas around the request point. A query point that meets the threshold used as a value as a set of threshold queries. So, all of the query points that meet the thresholds are the basis for forming local network voronoi partitions. Thus, this method does not require preliminary calculations also evaluation of distances at each intersection. The purpose of this research is to improve the performance of queries in an uncertain database by making the aggregate process and trimming the probability value as one phase of the search algorithm. In the first stage, objects which not able to form the answers are filtered by calculating the minimum closed circle from the dataset of query which prepared for the trimming phase. The second step called probabilistic candidate selection, its cut significant set of candidates for inspected as the aggregate function of the nearest neighbor’s demand. The remaining set is sent for verification which obtains possible answers at the lower and upper limits so that candidates whose probabilities are not less than the user-specified threshold are saved in the result set and returned to the user. We also examine the efficient data structures spatially which support this method. Our solution can be applied to uncertain data with an ever-changing probability density function.

A Generalized Asymmetric Dual-Front Model for Active Contours and Image Segmentation.

The Voronoi diagram-based dual-front scheme is known as a powerful and efficient technique for addressing the image segmentation and domain partitioning problems. In the basic formulation of existing dual-front approaches, the evolving contour can be considered as the interfaces of adjacent Voronoi regions. Among these dual-front models, a crucial ingredient is regarded as the geodesic metrics by which the geodesic distances and the corresponding Voronoi diagram can be estimated. In this paper, we introduce a new dual-front model based on asymmetric quadratic metrics. These metrics considered are built by the integration of the image features and a vector field derived from the evolving contour. The use of the asymmetry enhancement can reduce the risk for the segmentation contours being stuck at false positions, especially when the initial curves are far away from the target boundaries or the images have complicated intensity distributions. Moreover, the proposed dual-front model can be applied for image segmentation in conjunction with various region-based homogeneity terms. The numerical experiments on both synthetic and real images show that the proposed dual-front model indeed achieves encouraging results.

Retinal image mosaicking using scale-invariant feature transformation feature descriptors and Voronoi diagram.

Purpose: Peripheral retinal lesions substantially increase the risk of diabetic retinopathy and retinopathy of prematurity. The peripheral changes can be visualized in wide field imaging, which is obtained by combining multiple images with an overlapping field of view using mosaicking methods. However, a robust and accurate registration of mosaicking techniques for normal angle fundus cameras is still a challenge due to the random selection of matching points and execution time. We propose a method of retinal image mosaicking based on scale-invariant feature transformation (SIFT) feature descriptor and Voronoi diagram. Approach: In our method, the SIFT algorithm is used to describe local features in the input images. Then the input images are subdivided into regions based on the Voronoi method. Each pair of Voronoi regions is matched by the method zero mean normalized cross correlation. After matching, the retinal images are mapped into the same coordinate system to form a mosaic image. The success rate and the mean registration error (RE) of our method were compared with those of other state-of-the-art methods for the P category of the fundus image registration database. Results: Experimental results show that the proposed method accurately registered 42% of retinal image pairs with a mean RE of 3.040pixels, while a lower success rate was observed in the other four state-of-the-art retinal image registration methods GDB-ICP (33%), Harris-PIIFD (0%), HM-2016 (0%), and HM-2017 (2%). Conclusions: The proposed method outperforms state-of-the-art methods in terms of quality and running time and reduces the computational complexity.

A Parallelizing Method for Generation of Voronoi Diagram Using Contact Zone

Due to the recent popularization of the Geographic Information System (GIS), spatial network environments that can display the changes of spatial axes on mobile devices are receiving great attention. In spatial network environments, since a query object that seeks location information selects several candidate target objects based on the search conditions, we often use a k-nearest neighbor (kNN) search, which seeks several target objects near the query object. However, since a kNN search needs to find the kNN by calculating the distance from the query to all the objects, the computational complexity might become too large based on the number of objects. To reduce this computation time in a kNN search, many researchers have proposed a search method that divides regions using a Voronoi diagram. However, since conventional methods generate Voronoi diagrams for objects in order, the processing time for generating Voronoi diagrams might become too large when the number of objects is increased. In this paper, we propose a generation method of the Voronoi diagram by parallelizing the generation of Voronoi regions using a contact zone. Our proposed method can reduce the processing time of generating the Voronoi diagram by generating Voronoi regions in parallel based on the number of targets. Our evaluation confirmed that the processing time under the proposed method was reduced about 15.9\% more than conventional methods that are not parallelized.

Enhancing Downlink QoS and Energy Efficiency Through a User-Centric Stienen Cell Architecture for mmWave Networks

This paper presents an analytical framework for performance characterization of a novel Stienen cell based user-centric architecture operating in millimeter wave spectrum. In the proposed architecture, at most one remote radio head (RRH) is activated within non overlapping user equipment (UE)-centric Stienen cells (S-cells) generated within the Voronoi region around each UE. Under the presented framework, we derive analytical models for the three key performance indicators (KPIs): i) SINR distribution (used as an indicator for quality of service (QoS)), ii) area spectral efficiency (ASE), and iii) energy efficiency (EE) as a function of the three major design parameters in the proposed architecture, namely UE service probability, S-cell radius coefficient and RRH deployment density. The analysis is validated through extensive Monte Carlo simulations. The simulation results provide practical design insights into the interplay among the three design parameters, tradeoffs among the three KPIs, sensitivity of each KPI to the design parameters as well as optimal range of the design parameters. Results show that compared to current non user-centric architectures, the proposed architecture not only offers significant SINR gains, but also the flexibility to meet diverse UE specific QoS requirements and trade between EE and ASE by dynamically orchestrating the design parameters.

Voronoi Decomposition of Cardiovascular Dependency Structures in Different Ambient Conditions: An Entropy Study

This paper proposes a method that maps the coupling strength of an arbitrary number of signals D, D ≥ 2, into a single time series. It is motivated by the inability of multiscale entropy to jointly analyze more than two signals. The coupling strength is determined using the copula density defined over a [0 1]D copula domain. The copula domain is decomposed into the Voronoi regions, with volumes inversely proportional to the dependency level (coupling strength) of the observed joint signals. A stream of dependency levels, ordered in time, creates a new time series that shows the fluctuation of the signals’ coupling strength along the time axis. The composite multiscale entropy (CMSE) is then applied to three signals, systolic blood pressure (SBP), pulse interval (PI), and body temperature (tB), simultaneously recorded from rats exposed to different ambient temperatures (tA). The obtained results are consistent with the results from the classical studies, and the method itself offers more levels of freedom than the classical analysis.

Robust Construction of Voronoi Diagram of Circles by Region-Expansion Algorithm

This paper presents a numerically robust algorithm to construct a Voronoi diagram of circles in the plane. The circles are allowed to have intersections among them, but one circle cannot fully contain another circle. The Voronoi diagram is a tessellation of the plane into Voronoi regions of given circles. Each circle has its Voronoi region which is defined by a set of points in the plane closer to the circle than any other circles. The distance from a point p to a circle ci of center pi and radius ri is ||p-pi||-ri, which is the closest Euclidean distance from p to the circle boundary. The proposed algorithm first constructs the point Voronoi diagram of centers of given circles, then it enlarges each point to the circle and expands its Voronoi region accordingly. This region-expansion process is done by local modifications and after completing this process for the whole circles the desired circle Voronoi diagram can be obtained. The proposed algorithm is numerically robust and we provide with a few examples to show its robustness. The algorithm runs in O(n2) time in the worst case and O(n) time on average where n is the number of the circles. The experiment shows that the region-expansion algorithm is robust and runs fast with strong linear time behavior.

An analysis of pitch-class segmentation in John Cage's Ryoanji for oboe using morphological image analysis and formal concept analysis

In 1983, John Cage used the traditional stone garden, or karesansui at the Zen temple, Ryōan-ji in Kyoto as a model to generate a series of visual and musical works that utilized tracings of a collection of his own rocks. In this article, I analyze the first of the musical works, Ryoanji for oboe, using mixed methods drawn from morphological image analysis and formal concept analysis (FCA). I introduce the aesthetics of the karesansui and then examine the previous work of van Tonder and Lyons regarding the medial axis transform (MAT) of the garden at Ryōan-ji. This leads to the use of the distance transform, local maxima, and Voronoi diagram in order to decompose the two-dimensional image plane of Cage's Ryoanji for oboe. Finally, using the technique of FCA for constructing a number of formal concept lattices, the pitch-class segmentation of Ryoanji for oboe is investigated in regard to the sound gardens and the classes of Voronoi regions found across sound gardens.

On the Density of Sets Avoiding Parallelohedron Distance 1

The maximal density of a measurable subset of $${{\mathbb {R}}}^n$$ avoiding Euclidean distance 1 is unknown except in the trivial case of dimension 1. In this paper, we consider the case of a distance associated to a polytope that tiles space, where it is likely that the sets avoiding distance 1 are of maximal density $$2^{-n}$$ , as conjectured by Bachoc and Robins. We prove that this is true for $$n=2$$ , and for the Voronoi regions of the lattices $$A_n$$ , $$n\ge 2$$ .

Properties of tight frames that are regular schemes

Finite frames are sequences of vectors in finite dimensional Hilbert spaces that play a key role in signal processing and coding theory. In this paper, we study the class of tight unit-norm frames for $\mathbb {C}^{d}$ that also form regular schemes, which we call tight regular schemes (TRS). Many common frames that arise in applications such as equiangular tight frames and mutually unbiased bases fall in this class. We investigate characteristic properties of TRSs and prove that for many constructions, they are intimately connected to weighted 1-designs—arising from cubature rules for integrals over spheres in $\mathbb {C}^{d}$ —with weights dependent on the Voronoi regions of each frame element. Aided by additional numerical evidence, we conjecture that all TRSs in fact satisfy this property.

Center of mass and the optimal quantizers for some continuous and discrete uniform distributions

In this paper, we first consider a flat plate (called a lamina) with uniform density ρ that occupies a region of the plane. We show that the location of the center of mass, also known as the centroid, of the region equals the expected vector of a bivariate continuous random variable with a uniform probability distribution taking values on the region . Using this property, we prove that the Voronoi regions of an optimal set of two-means with respect to the uniform distribution defined on a disc partition the disc into two regions bounded by the semicircles. Besides, we show that if an isosceles triangle is partitioned into an isosceles triangle and an isosceles trapezoid in the Golden ratio, then their centers of mass form a centroidal Voronoi tessellation of the triangle. In addition, using the properties of center of mass we determine the optimal sets of two-means and the corresponding quantization error for a uniform distribution defined on a region with uniform density bounded by a rhombus. Further, we determine the optimal sets of n-means, and the nth quantization errors for two different discrete uniform distributions for some positive integers n ≤ card(supp(P)).

Statistical geometry characterization of local structure of TMAO, TBA and urea aqueous solutions

Recently, we have revealed that trimethylamine-N-oxide (TMAO) molecules in aqueous solutions are distributed generally like random spheres in space (A.V. Anikeenko et al. J. Mol. Liq. 245 (2017) 35–41). We have shown that the variance of the distribution of Voronoi region volumes for the TMAO molecules and the hard random spheres are identical at the corresponding concentrations up to the mole fraction x ~ 0.1 (or packing fraction η ~ 0.2). In the current work, we study clusters in TMAO solutions and calculate the weighted mean cluster size and some topological cluster characteristics. For all these parameters, a great matching of TMAO and random spheres is also observed. The same analysis for tert-butyl alcohol (TBA) and urea aqueous solutions is carried out. These molecules have a tendency to cluster even at very low concentrations. However, basic geometrical features of the clusters are the same both for solutions and for the system of random spheres, indicating that geometrical laws governing the allocation of non-overlapping spheres in space should be taken into account when describing the structure of solutions even at rather low concentrations.

Computation of Compact Distributions of Discrete Elements

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.

Randomized incremental construction for the Hausdorff Voronoi diagram revisited and extended

This paper applies the randomized incremental construction (RIC) framework to computing the Hausdorff Voronoi diagram of a family of k clusters of points in the plane. The total number of points is n. The diagram is a generalization of Voronoi diagrams based on the Hausdorff distance function. The combinatorial complexity of the Hausdorff Voronoi diagram is O(n+m), where m is the total number of crossings between pairs of clusters. For non-crossing clusters (m=0), our algorithm works in expected O(n log n + k log n log k) time and deterministic O(n) space. For arbitrary clusters (m=O(n^2)), the algorithm runs in expected O((m+n log k) log n) time and O(m +n log k) space. When clusters cross, bisectors are disconnected curves resulting in disconnected Voronoi regions that challenge the incremental construction. This paper applies the RIC paradigm to a Voronoi diagram with disconnected regions and disconnected bisectors, for the first time.