Quickhull Algorithm Research Articles

Octagonal and hexadecagonal cut algorithms for finding the convex hull of finite sets with linear time complexity

This paper is devoted to an octagonal cut algorithm and a hexadecagonal cut algorithm for finding the convex hull of n points in R2, where some octagon and hexadecagon are used for discarding most of the given points interior to these polygons. In this way, the scope of the given problem can be reduced significantly. In particular, the convex hull of n points distributed bleast-bmost-boundedly in some rectangle can be determined with the complexity O(n). Computational experiments demonstrate that our algorithms outperform the Quickhull algorithm by a significant factor of up to 47 times when applied to the tested data sets. The speedup compared to the CGAL library is even more pronounced.

Long-term trends in functional diversity of exploited marine fish in the Azores’ archipelago: past and present

IntroductionEffective fisheries management requires monitoring and quantifying changes in exploited fish communities. Concerns about global fisheries sustainability have led to innovative approaches. Functional diversity, rooted in ecological theory, offers valuable insights into fishery activities and ecosystem processes. A trait-based approach was used to investigate the functional diversity of landed fish species in the Azores archipelago from 1980 to 2021.MethodsLandings data of exploited Actinopterygii and Elasmobranchii were provided by the Azores Fisheries Auction Services (LOTAÇOR/OKEANOS-UAc Fisheries Database). A trait matrix was built, incorporating 12 functional traits assigned to each species, capturing their importance in marine ecological processes. The Quickhull algorithm for convex hull was employed to calculate the volume occupied by the species in the four-dimensional functional space. Functional diversity (FD) was measured using three indices: functional richness (FRic), functional evenness (FEve), and functional divergence (FDiv). Trends in FD indices over the past 42 years were visualized using Generalized Additive Models (GAM) with interaction terms.Results and discussionGAM analysis revealed significant variations in the functional space and FD metrics over time. FRic exhibited peaks in the 1980s and 2010s, declining in the 1990s and from the 2010s onwards, indicating diversification in target species. The recent decrease in FRic can be attributed to the absence of catches of species with unique traits. The distribution of landings and trait combinations showed higher regularity in the functional space during the 1980s and 1990s (high FEve). Actinopterygii species targeted in the 1980s and 1990s had lower trait divergence (low FDiv) compared to those targeted from the 2000s onwards (high FDiv). Variability in FD can be linked to changes in fishing practices, species availability, market demand, environmental factors, and local regulations. This study underscores the importance of considering FD metrics alongside species richness and abundance when assessing the potential effects of fisheries on marine ecosystems and sustainable use of fishery resources.

Online temperature-monitoring technology for grain storage: a three-dimensional visualization method based on an adaptive neighborhood clustering algorithm.

Post-harvest quality assurance is a crucial link between grain production and end users. It is essential to ensure that grain does not deteriorate due to heating during storage. To visualize the temperature distribution of a grain pile, the present study proposed a three-dimensional (3D) temperature field visualization method based on an adaptive neighborhood clustering algorithm (ANCA). The ANCA-based visualization method contains four calculation modules. First, discrete grain temperature data, obtained by sensors, are collected and interpolated using back propagation (BP) neural networks to model the temperature field. Then a new adaptive neighborhood clustering algorithm is applied to divide interpolation data into different categories by combining spatial characteristics and spatiotemporal information. Next, the Quickhull algorithm is used to compute the boundary points of each cluster. Finally, the polyhedrons determined by boundary points are rendered into different colors and are constructed in a 3D temperature model of the grain pile. The experimental results show that ANCA is much better than the DBSCAN and MeanShift algorithms on compactness (around 95.7% of tested cases) and separation (approximately 91.3% of tested cases). Moreover, the ANCA-based visualization method for grain pile temperatures has a shorter rendering time and better visual effects. This research provides an efficient 3D visualization method that allows managers of grain depots to obtain temperature field information for bulk grain visually in real time to help them protect grain quality during storage. © 2023 Society of Chemical Industry.

Segmentation of Multiple Nuclei from Non-overlapping Immuno-histochemically Stained Histological Hepatic Images.

In this paper, we describe an algorithm for accurately segmenting multiple nudfclei from clumps of non-overlapping immuno-histochemically stained histological hepatic (liver) images. This problem is notoriously difficult because of the degree of presence of stains among the multi-nucleated cells, the poor contrast of cell cytoplasm, and the presence of mucus, blood, and inflammatory cells in the images. Hepatocellular carcinoma, characterized by cellular and nuclear enlargement, nuclear pleomorphism, and multi-nucleation, poses a prominent threat. Our proposed method addresses the aforementioned issues for an automated diagnosis system by judging the presence of multiple nuclei in a two-step process: the Quickhull algorithm defines the convex hull of each cell in the image and candidate nuclei regions are located with morphological operations. A combination of features containing local minima and shape-dependent features is extracted for the detection of single or multiple nuclei in each cell with a significant reduction in the number of false positives and false negatives providing an accuracy of 89.76%.

A fast and efficient algorithm for determining the connected orthogonal convex hulls

The Quickhull algorithm for determining the convex hull of a finite set of points was independently conducted by Eddy in 1977 and Bykat in 1978. Inspired by the idea of this algorithm, we present a new efficient algorithm, for determining the connected orthogonal convex hull of a finite set of points through extreme points of the hull, that still keeps advantages of the Quickhull algorithm. Consequently, our algorithm runs faster than the others (the algorithms introduced by Montuno and Fournier in 1982 and by An, Huyen and Le in 2020). We also show that the expected complexity of the algorithm is O(nlogn), where n is the number of points.

Determination of confidence bounds and artificial neural networks in non-linear optimization problems

Feedforward is a type of ANN (artificial neural network) widely used in modeling linear and non-linear systems. This type of ANN is found in system identification, system control, and optimization. In most works on optimization with ANN embedded, the purpose of the ANN is to provide a surrogate model that approximates the objective function. Since the ANN is an algorithm that is trained with an experimental/actual data set, it is imperative to know the limits of the data set and therefore avoid the extrapolation of the ANN. This work proposes a procedure where the network model domain is limited using the Quickhull algorithm, avoiding extrapolation. Also, this procedure allows for the quantification of the accuracy in the network's predictions, while taking into account the density of the training data within the network model domain. The procedure considers four essential steps: the modeling of the objective function, the model domain determination, the application of the operational constraints, and the optimization. Two examples of optimization are solved using this procedure, where the results are assessed according to the width of the confidence bounds, measuring how far or near the actual value is.

Variations on the theme of slacks-based measure of efficiency: Convex hull-based algorithms

Non-radial data envelopment analysis (DEA) models, such as the slacks-based measure (SBM) model, exert an important role in theoretical research and real applications on efficiency evaluation and improvement. However, those models maximize input and output slacks and may therefore find a rather far projection for each inefficient decision-making unit (DMU). This results in varying performance measures and inconveniences in improving the performance of inefficient DMUs. Tone (2016) proposed a new non-oriented SBM-Max model and several algorithms to overcome such limitations. However, those algorithms were computationally expensive and complicated, limiting their applications to different problems.In the present study, algorithms are proposed based on two convex hull algorithms to make a novel variation of (Tone, 2016) that is easily applicable to different problems. The proposal is based on the constant returns-to-scale (CRS) assumption. The algorithms can be further extended based on other assumptions of returns to scale. The used two convex hull algorithms were the Quickhull (Qhull) algorithm and the C++ (ANSI C) implementation of the double description (CDD) algorithm. The proposed algorithms were tested on a dataset from prior literature and a real dataset of Hong Kong hospitals. The results demonstrate that the proposed algorithms are effective for finding a close projection on efficiency evaluation, resulting in improvements in DMUs.

A geometric-based data reduction approach for large low dimensional datasets: Delaunay triangulation in SVM algorithms

Training a support vector machine (SVM) on large datasets is a slow daunting process. Further, SVM becomes slow in the testing phase, due to its large number of support vectors (SVs). This paper proposes an effective geometric algorithm based on construction of Delaunay triangulation (DT) algorithm using Quickhull algorithm with a novel strategy to exactly identify and extract the boundary data points laid between the two classes of a dataset, and later uses these most informative data points as a reduced dataset to solve various SVM algorithms and proposes new DT-SVM algorithms Two synthetic datasets with the size of 1K incrementally up to 500K datasets are generated to extensively verify the effectiveness of the proposed DT-SVM algorithms over various data sizes and for further assessment, the most efficient version of proposed DT-SVM is applied on well-known benchmark datasets from UCI Machine Learning Repository. Two variant of sequential minimization optimization (SMO) decomposition methods, in addition to Least Square form of SVM are implemented to present the scalability of new DT-SVM algorithms in linear/nonlinear separable/non-separable large low dimensional datasets. Moreover, the most efficient version of the proposed algorithm is compared to RCH-SK as a known geometric approach in the SVM literature. The results demonstrate that while the proposed approach improves the scalability of DT-SVM in large low dimensional datasets, it leads SVM algorithms to maintain the accuracy in an acceptable range with considerably lower time in both training and testing phases with using a noticeably fewer number of SVs.

Improved Convex Hull Algorithm Applied to Body Size Measurements

The Quickhull algorithm is a very efficient convex hull algorithm for many engineering application. Although it can be applied in the clothing industry, its conventional formulation requires massive computational resources when processing point clouds from concave parts of the human body. Thus, we improve the Quickhull algorithm and apply it to accurately estimate human body dimensions. Assuming body symmetry with respect to the sagittal plane, some concave points can be quickly deleted to reduce the number of recursions, enhancing the overall calculation efficiency. Compared with the traditional Quickhull algorithm, the efficiency of the improved algorithm increases by 15.1–24.4%. Additionally, the computation time of the improved Quickhull algorithm is about three-quarters that of Graham’s scan. We expect that this study provides insights on the application of body size measurement and estimation based on 3D scan data.

Efficiently Testing Digital Convexity and Recognizing Digital Convex Polygons

A set $$S \subset \mathbb {Z}^2$$ of integer points is digital convex if $${{\,\mathrm{conv}\,}}(S) \cap \mathbb {Z}^2 = S$$ , where $${{\,\mathrm{conv}\,}}(S)$$ denotes the convex hull of S. In this paper, we consider the following two problems. The first one is to test whether a given set S of n lattice points is digital convex. If the answer to the first problem is positive, then the second problem is to find a polygon $$P\subset \mathbb {Z}^2$$ with minimum number of edges and whose intersection with the lattice $$P\cap \mathbb {Z}^2$$ is exactly S. We provide linear-time algorithms for these two problems. The algorithm is based on the well-known quickhull algorithm. The time to solve both problems is $$O(n + h \log r) = O(n + n^{1/3} \log r)$$ , where $$h = \min (|{{\,\mathrm{conv}\,}}(S)|, n^{1/3})$$ and r is the diameter of S.

Hybrid Data Envelopment Analysis for Large-Scale Smartphone Data Modeling

This paper deals with the problem of improving the existing optimization techniques for Data Envelopment Analysis (DEA). The algorithm proposed herein is a combination of the “quickhull algorithm” and a DEA algorithm written in Python programming language. To the best of the authors’ knowledge no prior effort has been made to date to propose a methodology for reducing the running time of a DEA problem that incorporates multiple inputs and outputs. The algorithmic implementation is applied on the existing problem of driving efficiency evaluation by exploiting a driving data sample of 10,088 trips collected from smartphone devices. Results indicate that the proposed algorithm is performing relatively well for Big Data compared to other existing DEA algorithmic methodologies that yield the same optimal solution such as Standard DEA and RBE DEA methodologies. The results obtained are calculated for the test sets of 100, 500, 1000, 5,000 and 10,088 Decision-Making-Units (DMUs) and compared in terms of running time of each of the algorithms applied. The results of per trip analysis can be exploited in order to classify trips into different efficiency categories (such as efficient, less efficient, non-efficient) and present their main characteristics.

QuickhullDisk: A faster convex hull algorithm for disks

Convex hull is one of the most fundamental constructs in geometry and its construction has been extensively studied. There are many prior works on the convex hull of points. However, its counterpart for weighted points has not been sufficiently addressed despite important applications. Here, we present a simple and fast algorithm, QuickhullDisk, for the convex hull of a set of disks in R2 by generalizing the quickhull algorithm for points. QuickhullDisk takes O(nlog n) time on average and O(mn) time in the worst case where m represents the number of extreme disks which contribute to the boundary of the convex hull of n disks. These time complexities are identical to those of the quickhull algorithm for points in R2. Experimental result shows that the proposed QuickhullDisk algorithm runs significantly faster than the O(nlog n) time incremental algorithm, proposed by Devillers and Golin in 1995, particularly for big data. QuickhullDisk is approximately 2.6 times faster than the incremental algorithm for random disks and is 1.2 times faster even for the disk sets where all disks are extreme. This speed-up is because the basic geometric operation of the QuickhullDisk algorithm is a predicate for the location of a point w.r.t. a line and is much faster than that of the incremental algorithm. The source code of QuickhullDisk is freely available from Mendeley Data and a GUI-version from Voronoi Diagram Research Center, Hanyang University (http://voronoi.hanyang.ac.kr/).

Online Anomaly Detection of Transformer Vibration Based on SVDD Incremental Learning

The problems in theoretical model analysis and the lack of prior knowledge bring difficulties and challenges to the analysis of transformer by the vibration method. In this paper, an incremental learning algorithm based on support vector data description was proposed for on-line anomaly detection of transformer vibration. The anomaly detection model was established by using one-class support vector data description, the incremental data were reduced based on the Quickhull algorithm to achieve quick incremental learning. The simulation results of measured transformer vibration showed that the proposed method could quickly learn new samples and accurately capture transformer vibration changes.

Approximation of Voronoï cell attributes using parallel solution

This paper concerns an algorithm for fast parallel approximation of selected attributes of hyper-dimensional Voronoï cells in a unit hypercube. The presented algorithm does not require the construction of a corresponding Voronoï diagram (usually by employing the Quick Hull algorithm) which typically is a highly computationally demanding task, especially when performed in higher dimensions. The algorithm is suitable for both the clipped and periodic variants of Voronoï tessellation and provides a significant speedup in a convenient range of practical usage. For the purposes of approximation of selected scalar properties of Voronoï cells, only the distances of points in sample are evaluated using an adequately fine underlying orthogonal mesh. The algorithm estimates e.g. the volumes and centroids of Voronoï cells, radii and coordinates of centers of the corresponding Delaunay hyper-circles. The paper also provides quite accurate explicit error estimators for the extracted attributes due to rasterization and thus the user is given a control over the accuracy by selecting an appropriate discretization density. Among numerous fields in which Voronoï diagrams are being utilized, the authors are concerned with optimization of point samples for the Design of Experiments and also with weighting of integration points in Monte Carlo type integration. In these applications, selected scalar topological descriptors of Voronoï diagrams are being repeatedly computed. As the full Voronoï diagram is not of interest but the resulting cell volumes or shape descriptions have to be repeatedly computed, the presented parallel solution seems highly suitable for these applications.

A fast convex hull algorithm inspired by human visual perception

This paper proposes a convex hull algorithm for high dimensional point set, which is faster than the well-known Quickhull algorithm in many cases. The main idea of the proposed algorithm is to exclude inner points by early detection of global topological properties. The algorithm firstly computes an initial convex hull of $2*d + 2^{d}$ extreme points. Then, it discards all the inner points which are inside the inscribed ball of the initial convex hull. The other inner points are processed recursively according to the relationships of points and facets. Maximum inscribed circle affine transformations are also designed to accelerate the computation of the convex hull. Experimental results show that the proposed algorithm achieves a significant saving of computation time in comparison with the Quickhull algorithm in 3, 4 and 5 dimensional space. The space efficiency of the proposed algorithm is also demonstrated by experimental results.

Компьютерное визуальное представление цветового охвата систем воспроизведения многокрасочного изображения

Methods of surfaces’ geometric simulation are widely used for construction of different solid figures. Colours at the XYZ color system are located at the same distance stated geometrically. It doesn't generate a complete visual presentation for the color gamut of reproduction system for multicolor image. The computer visualization of correspondent pictures at multicolor image’s reproduction accuracy estimation is one of priority tasks. In this paper have been presented results related to practical implementation of geometrical model and software for evaluation of printing system’s color reproduction, construction of color gamut bodies, and a projection representation of reproducible colors’ geometric models for different playback systems: additive and autotype synthesis of color. The color gamut construction is performed over CIE L*a*b* color space. The color is presented as the one lightness coordinate and two color-difference coordinates of chromaticity. A color gamut body volume calculation has been performed in automated mode using software tools. Calculation of the playback system’s color gamut is carried out with Qhull (Quickhull) algorithm realised in the Matlab software. The practical algorithm realisation has been considered on example of ink jet printing based on Epson 3880. The color gamut bodies’ surfaces projections construction was performing in light tones (lightness L = 80), middle tones (L = 50), and shadow tones (L = 20). Has been performed a section of the color gamut bodies’ constructed surfaces. Have been demonstrated the results of computer visualization for geometrical models interpreting the multicolor image’s reproduction accuracy relative to the original.

Void handling using Geo-Opportunistic Routing in underwater wireless sensor networks

Underwater Wireless Sensor Networks (UWSNs) consist of a group of acoustic sensor nodes that report the sensed information to the sink nodes at the surface level. The factors like multipath fading, path loss, Doppler spread, limited battery power, and high mobility void region cause UWSNs to become spatially variable. In this paper, the idea of Geography-based Opportunistic Routing (GOR) is applied to advance the packet towards the destination and to handle the void problem. The proposed design is termed as Void Handling using Geo-Opportunistic Routing (VHGOR). In executing so, a heuristic algorithm comprising two metrics, namely OREPP and NCD for optimal forwarder selection is proposed. Additionally, a void handling mechanism is also proposed that handles void in two ways. Quick hull algorithm that makes a convex hull assists to identify whether the void is convex or concave. When the node approaches a convex void, reconstruction of convex hull helps to determine an alternative way to resume the greedy forwarding if the node has neighbors within its proximity. Failure of convex void handling during communication void makes VHGOR switch to concave void handling (or) recovery mode to recover the packet from local maximum node and to route the packet to the destination. At the end of simulation, the proposed VHGOR results considerably improve the network performance when compared with the existing solutions, both in sparse and very dense networks.

Partial convex hull algorithms for efficient grasp quality evaluation

We present two algorithms to efficiently determine the value of two grasp quality metrics formerly proposed in literature. The first one is heavily used in practice but has some drawbacks, e.g., it is not scale invariant and it does not focus on the disturbance forces that will occur in practice when the robot grasps an object. The second one overcomes these limitations, but is rarely used because it is computationally too demanding. The two algorithms we propose are based on the common intuition that both metrics can be efficiently computed through a modified version of the QuickHull algorithm that is commonly used to compute convex hulls. In both cases it is possible to establish when enough information has been generated to determine the desired value, and then stop the construction of a suitably defined convex hull. Extensive numerical evaluations demonstrate that our algorithms provide substantial computational gains when compared with the state of the art. The speedup provides an immediate benefit to planners using grasp quality metrics to guide the search through the space of possible grasps.

Visual nesting system for irregular cutting-stock problem based on rubber band packing algorithm

This article deals with the packing problem of irregular items allocated into a rectangular sheet to minimize the waste. Conventional solution is not visual during the packing process. It obtains a reasonable and relatively satisfactory solution between the nesting time and nesting solution. This article adopts a physical method that uses rubber band packing algorithm to simulate a rubber band wrapping those packing irregular items. The simulation shows a visual and fast packing process. The resultant rubber band force is applied in the packing items to translate, rotate, and slide them to make the area decrease and obtain a high packing density. An improved analogy QuickHull algorithm is presented to obtain extreme points of rubber band convex hull. An adaptive module could set a variable rubber band force and a variable time step to make a proper convergence and no intersection. A quick convex decomposition method is used to solve the problem of concave polygon. A plural vector expression approach is adopted to calculate the resultant vector of the rubber band force. Several cases are compared with the benchmark problems to prove rubber band packing algorithm performance.

Local convex hulls for a special class of integer multicommodity flow problems.

Based on previous work in rolling stock scheduling problems (Alfieri et al. in Transp Sci 40:378–391, 2006; Cacchiani et al. in Math Progr B 124:207–231, 2010; Lin and Kwan in Electron Notes Discret Math 41:165–172, 2013; Schrijver in CWI Q 6:205–217, 1993; Ziarati et al. in Manag Sci 45:1156–1168, 1999), we generalize a local convex hull method for a class of integer multicommodity flow problems, and discuss its feasibility range in high dimensional cases. Suppose a local convex hull can be divided into an up hull, a main hull and a down hull if certain conditions are met, it is shown theoretically that the main hull can only have at most two nonzero facets. The numbers of points in the up and down hull are explored mainly on an empirical basis. The above properties of local convex hulls have led to a slightly modified QuickHull algorithm (the “2-facet QuickHull”) based on the original version proposed by Barber et al. (ACM Trans Math Softw 22:469–483, 1996). As for the feasibility in applying this method to rolling stock scheduling, our empirical experiments show that for the problem instances of ScotRail and Southern Railway, two major train operating companies in the UK, even in the most difficult real-world or artificial conditions (e.g. supposing a train can be served by any of 11 compatible types of self-powered unit), the standard QuickHull (Barber et al. in ACM Trans Math Softw 22:469–483, 1996) can easily compute the relevant convex hulls. For some even more difficult artificial instances that may fall outside the scope of rolling stock scheduling (e.g. a node in a graph can be covered by more than 11 kinds of compatible commodities), there is evidence showing that the “2-facet QuickHull” can be more advantageous over the standard QuickHull for our tested instances. When the number of commodity types is even higher (e.g. >19), or the number of points in a high dimensional space (e.g. 15 dimensions) is not small (e.g. >2000), the local convex hulls cannot be computed either by the standard or the 2-facet QuickHull methods within practical time.