Convex Hull Research Articles

Convex hulls of surfaces in fourspace

AbstractThis is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces.

Quantized Iterative Learning Bipartite Containment Tracking Control for Unknown Nonlinear Multi-agent Systems

This paper proposes a quantized model-free adaptive iterative learning control (MFAILC) algorithm to solve the bipartite containment tracking problem of unknown nonlinear multi-agent systems, where the interactions between agents include cooperation and antagonistic interactions. To design the controller, the agent’s dynamics is transformed into the linear data model based on the dynamic linearization method, and then a quantized MFAILC algorithm is established based on the quantized values of the relative output measurements. The designed controller only depends on the input and output data of the agent. We prove that under the quantized MFAILC algorithm, the multi-agent systems can achieve the bipartite containment, that is, the output trajectories of followers converge to the convex hull formed by the leaders’ trajectories and the leaders’ symmetric trajectories. Finally, we provide simulations to illustrate the effectiveness of our theoretical results.

Group‐weighted oscillatory containment for multiple robots under heterogeneous cooperation and competition

AbstractThe group‐weighted containment (GWC) problem is studied for swarms of robots, where the information flow among agents is heterogeneous (weighted) cooperative‐competitive and the whole networked systems are composed of multiple different groups. Each group is formed by multiple harmonic oscillator leaders and multiple robot followers governed by Euler‐Lagrange (EL) equations. After introducing the definition of group‐weighted containment to the networked robot systems, the control algorithms for the agents containing followers and leaders are formulated over newly weighted cooperative‐competitive network. Some necessary conditions for solving the weighted containment control problem are established. It is shown that the followers in every group can achieve a kind of novel coordinated behaviors. Specifically, the followers can converge to the dynamic convex hull spanned by the corresponding leaders' weighted coordinates under some conditions. Simulation results are provided to illustrate the effectiveness of the proposed control schemes.

Deephullnet: a deep learning approach for solving the convex hull and concave hull problems with transformer

ABSTRACT Convex and concave hulls originating from computational geometry are widely applied in practice. For instance, to determine the boundaries of a geographical area within a group of cities, convex hulls can represent the approximate boundaries of the areas. Concave hulls can accurately describe the shape of the area. The traditional methods for solving two-dimensional convex hull problems include the Jarvis March algorithm and the Graham scanning algorithm. The K-nearest neighbours and alpha-shape methods are designed for solving the concave hull problem. Other than traditional algorithms, we consider using deep neural networks to handle the convex and concave hull problems. General neural networks cannot deal with the problem whose output is a variable-length sequence. Our study provides a machine-learning approach for solving the convex hull and concave hull problems. We combine the Pointer network (Ptr-Net) with the Transformer model and propose the DeepHullNet. The trained DeepHullNet can provide a feasible solution in the majority of cases. The experimental results show that DeepHullNet outperforms the original Ptr-Net regarding accuracy. Compared with traditional methods, DeepHullNet can generate a good solution quickly, and the running time is shortened by nearly 100 times based on GPU.

3D geometric kernel computation in polygon mesh structures

This paper introduces a novel approach to compute the geometric kernel of a polygon mesh embedded in 3D. The geometric kernel defines the set of points inside or on the shape’s boundary, ensuring visibility of the entire shape. The proposed method utilizes scattered rays to identify a sufficient number of sample points on the kernel surface and subsequently leverages these points to locate as many surface vertices as possible. By computing the convex hull of these identified points, we derive an approximation of the kernel. Notably, the output of our method consists exclusively of interior or boundary points of the actual kernel. Comparative evaluations against established CGAL and Polyhedron Kernel algorithms highlight our method’s superior computational speed and high approximation accuracy. The parametric structure of our solution allows for different levels of accuracy to be obtained, enabling the user to tailor the approximation to their specific needs. This property sets our algorithm apart from others and provides greater flexibility in its use. Additionally, adjusting the algorithmic settings also enables the computation of the kernel itself with a trade-off in computational speed. Furthermore, our algorithm swiftly and accurately identifies an empty kernel for non-star-shaped configurations.

A relaxation viewpoint to Unbalanced Optimal Transport: Duality, optimality and Monge formulation

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed.

Simulating Thin Shells by Bicubic Hermite Elements

In this study, we present the bicubic Hermite element method (BHEM), a new computational framework devised for the elastodynamic simulation of thin-shell structures. The BHEM is constructed based on quadrilateral Hermite patches, which serve as a unified representation for shell geometry, simulation, collision avoidance, as well as rendering. Compared with the commonly utilized linear FEM, the BHEM offers higher-order solution spaces, enabling the capture of more intricate and smoother geometries while employing significantly fewer finite elements. In comparison to other high-order methods, the BHEM achieves conforming C1 continuity for Kirchhoff–Love (KL) shells with minimal complexity. Furthermore, by leveraging the subdivision and convex hull properties of Hermite patches, we develop an efficient algorithm for ray-patch intersections, facilitating collision handling in simulations and ray tracing in rendering. This eliminates the need for laborious remodeling of the pre-existing surface as the conventional approaches do. We substantiate our claims with comprehensive experiments, which demonstrate the high accuracy and versatility of the proposed method.

Research on segmentation and reconstruction of overlapping ore contours based on EAM-SOLOv2 and convex hulls

Research on segmentation and reconstruction of overlapping ore contours based on EAM-SOLOv2 and convex hulls

The Robust Communication Method Against Dependent Jamming Based on Convex Hull in Communication Countermeasures

The Robust Communication Method Against Dependent Jamming Based on Convex Hull in Communication Countermeasures

Existence of a Cauchy surface and compactness of causally convex hulls in a spacetime

In this paper it is shown that, if a spacetime contains a Cauchy surface then the causally convex hulls of compact sets are compact. The converse is not true in general, however if the spacetime is causal then the compactness of causally convex hulls of compact sets of the spacetime ascertain the existence of a Cauchy surface. Also, the convex hull operator is denoted and some results regarding it has been explored.

An alternative for the robust assessment of the repeatability and reproducibility of analytical measurements using bivariate dispersion

IntroductionAssessing repeatability and reproducibility in analytical chemistry is commonly based on parametric dispersion indicators, such as relative standard deviation and standard deviation, calculated for each detected variable using repeated measurements of Quality Control (QC) samples collected throughout the data acquisition sequence. However, their reliability strongly relies on the assumption of normality distribution. Knowing that analytical variability is conditional to many sources, the use of such parametric estimators is not always suitable. There is therefore a need for robust indicators of data quality independent of central values and any parametric assumption. MethodsThree specific indicators were developed: (i) intra-group dispersion, based on the median area of the convex hull of QC samples within an analytical batch; (ii) inter-group dispersion, defined as the gradient of the deviation between analytical batches; and (iii) dispersion index. Mathematical properties of these indicators, including positivity, stability, and translation invariance, were then evaluated using synthetic data under normal and non-normal distributions. Finally, the relevance of these indicators and the associated visualization methods were highlighted based on a metabolomics case study involving liquid chromatography coupled to mass spectrometry measurements of the NIST SRM1950 reference material analyzed over more than one year within different projects. ResultsThe proposed indicators were shown to be translation invariant and always positive, while first investigations performed on synthetic data revealed a high stability for multiplication. Moreover, their application to experimental data revealed specific behaviors depending on the characteristics of the signal associated with the different detected analytes, showing their ability to capture the variability observed either in parametric or non-parametric conditions. Moreover, this investigation showed different structures of sensitivity to analytical variability all along the data processing steps. The proposed indicators also allowed a visualization of the analytical drift in two dimensions, to facilitate result interpretation. ConclusionThese indicators open the way to a better and more robust assessment of repeatability and reproducibility but also to improvements of long-term data comparability involving suitability testing.

Matching the building footprints of different vector spatial datasets at a similar scale based on one-class support vector machines

Automatic matching of multisource data is an important technique for achieving change detection, fusion and updating spatial data. However, most current learning methods for building footprint matching require a large number of samples, and labeling these samples is costly in terms of labor and time. Moreover, multisource building footprint data are complex and diverse leading to recognizing the different matching relationships is a hard task. Thus, this study proposes a learning-based method for recognizing multisource building footprints matching relationships by using a one-class support vector machine (OCSVM). The OCSVM was trained using only positive samples. First, a set of geometric indicators was designed to train a model and realize initial matching recognition. Then, a contextual metric was calculated based on the rough matching results, and geometric and contextual metrics were combined to train the model and realize relaxed matching recognition. Relaxed matching is an optimization process implemented after initial matching to recognize more relaxed matching relationships. In relaxed matching, a convex hull is used to recognize matching relationships besides 1:1, such as 1:n, m:1 and m:n. The experimental results showed that the proposed method outperformed indicator-weighted (weighted average) and learning-based matching methods, such as traditional SVMs and decision trees (DTs). The precision scores of the proposed model were 97.1%, 95% and 97.2% for the Wuhan (China), Beijing (China) and Richmond Hill (Canada) datasets, respectively. Furthermore, the proposed model identified the matching relationships of buildings with complex geometric features and high-density spatial distributions.

Composite anti-disturbance dynamic positioning for mass-switched unmanned marine vehicles with multisource disturbances and actuator saturation: A switched model method

This article focuses on the composite anti-disturbance dynamic positioning (CADDP) control problem for the unmanned marine vehicle (UMV) system subject to mass switching, multisource disturbances and actuator saturation. The multisource disturbances include the unmeasurable modeled disturbances and measurable unmodeled disturbances. First, a switched model is introduced to describe the mass-switched UMV (MSUMV) system. Then, a series of switching observers are built to capture the unmeasurable disturbances. Meanwhile, the convex hull technique is exploited to address the actuator saturation. Further, based on the disturbance observer, the CADDP control scheme is developed to alleviate the influences of the multisource disturbances and actuator saturation, and to achieve the dynamic positioning (DP) of the MSUMV model. In the end, the established CADDP control strategy is carried on a supply vessel to illustrate the feasibility of the developed theory.

Shortest closed curve to contain a sphere in its convex hull

AbstractWe show that in Euclidean 3‐space any closed curve which contains the unit sphere within its convex hull has length , and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in dimensions, we include the estimate by Nazarov, which is sharp up to the constant .

Precision Phenotyping of Wild Rocket (Diplotaxis tenuifolia) to Determine Morpho-Physiological Responses under Increasing Drought Stress Levels Using the PlantEye Multispectral 3D System

The PlantEye multispectral scanner is an optoelectrical sensor automatically applied to a mechatronic platform that allows the non-destructive, accurate, and high-throughput detection of morphological and physiological plant parameters. In this study, we describe how the advanced phenotyping platform precisely assesses changes in plant architecture and growth parameters of wild rocket salad (Diplotaxis tenuifolia L. [DC.]) under drought stress conditions. Four different irrigation supply levels from moderate to severe, required to keep 100, 70, 50, and 30% of the water-holding capacity, were adopted. Growth rate and plant architecture were recorded through the digital measure of biomass, leaf area, Canopy Light Penetration Depth, five convex hull traits, plant height, Surface Angle Average, and Voxel Volume Total. Vegetation color assessments included hue, lightness, and saturation. Vegetation and senescence indices were calculated from canopy reflectance in the red (620–645 nm), green (530–540 nm), blue (peak wavelength 460–485 nm), near-infrared (820–850 nm), and 3D laser (940 nm) ranges. The temperature, relative humidity, and solar radiation of the environment were also recorded. Overall, morphological parameters, color, multispectral data, and vegetation indices provided over 7200 data points through daily scans over three weeks of cultivation. Although a general decrease in growth parameters with increasing stress severity was observed, plants were able to maintain the same morpho-physiological performances as the control during the early growth stages, keeping both 70% and 50% of the total water-holding capacity. Among indices, the Normalized Differential Vegetation Index (NDVI) contributed the most to the differentiation between different stress levels during the cultivation cycle. Across the 3 weeks of growth, statistically significant differences were observed for all traits except for the Saturation Average. Comparisons with respect to the control highlighted the strong impact of drought stress on morphological plant traits. This study provided meaningful insights into the health status of wild rocket salad under increasing drought stress.

An efficient method for modeling and evaluating the bench terrain of open-pit mines

In order to quantitatively analyze the roughness of the bench floor during open-pit mine blasting, this study proposes a real-time measuring method for the three-dimensional terrain of the bench floor during the excavation process. Real-time monitoring is conducted at the boundary and discrete internal points of the workbench floor during electric shovel operation, utilizing real-time kinematic global navigation satellite system (RTK-GNSS) positioning technology. An improved convex hull algorithm is introduced to automatically extract the optimal boundary of discrete point clouds based on their spatial distribution characteristics. This study establishes a digital elevation model (DEM) using five interpolation algorithms for 3D terrain visualization simulation. Through cross-validation, a comparative analysis of the DEM accuracy, the simulation results of the ordinary kriging interpolation algorithm were found to be optimized. The optimized interpolation algorithm is applied to simulate the 3D terrain in the Dexing open-pit copper mine, and the relevant terrain parameters were calculated. This dataset can serve as a precise foundation for the real-time path planning of elevation blasting design and ground leveling operations.

Colour gamut analysis of low‐cost dye‐sensitised solar cells using natural dyes

AbstractDue to the increasing effects of greenhouse gases and environmental pollution, the need for new clean energies like solar energy attracts much attention. Although the characteristics and the efficiency of green photovoltaic devices have been researched, the colour gamut achievable from those devices has not been studied. This study investigates the colour gamut of low‐cost dye‐sensitised solar cells (DSSCs). To do this, 14 natural dye extracts were obtained from herbal resources including dragon fruit, blueberry, mango, radish, yellow rose, red cabbage, sour pomegranate, beetroot, olive, green cabbage pepper, eggplant, parsley, bramble and cherry, and employed as photosensitiser in solar cells. Then, the colorimetric attributes of the photosensitisers were studied in three‐dimensional (3D)‐colour space, that is, CIELab, CIELCH and CIE1931 chromaticity diagram. Additionally, the convex hull method was employed to determine the colour gamut boundary and the corresponding colour gamut volume. Results showed that the majority of samples benefited from approximately 3°–82° of hue angle in a*b* diagram of CIELab colour space and showed the yellowish to reddish tint effects. In CIELCH colour order system, cherry and parsley showed the lowest and highest lightness attributes while the chroma property of samples varied from minimum 1.2 for eggplant to maximum 60.2 for the dragon fruit. Moreover, the results of using the convex hull method showed the volume of 7.73 × 104 that is bounded by the colour gamut of 3D colour points over the CIELab colour space.

A hierarchical LFFC control scheme for underactuated unmanned surface vessels based on unreliable communication interaction

This article investigates the leader-follower formation-containment (LFFC) control issue of underactuated unmanned surface vessels (USVs) system subject to unreliable communication interaction and the external disturbances. Firstly, an adaptive control scheme is established for the leaders to track the desired trajectory associated with preset formation configuration. Subsequently, by means of convex hull theory, a formation-containment control algorithm is developed to guide the followers into the specific area generated by the leaders. Nevertheless, the communication interaction among USVs is frequently unreliable, which is constrained by communication distance, equipment quality, and indeterminate cyber-attack. Worse still, this unreliability will magnify the unpredictable impact on system stability. Moreover, the signals generally undergo multilevel transmission from virtual leader to followers under LFFC structure, which will result in the amplification of unreliable ratio during the information transmissions. To address the aforementioned issues, an online-update auxiliary signal is constructed through the quasi-sliding mode manifold, which can suppress the impact of unreliable communication interaction. Theoretical analysis and numerical simulations are demonstrated to verify the feasibility and validity of the proposed control strategies.

Rectilinear convex hull of points in 3D and applications

Let P be a set of n points in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} time and O(n) space algorithm to compute RCH(P). We also obtain an efficient O(nlog2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log ^2 n)$$\\end{document} time and O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}. Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Collaborative orchard pesticide spraying routing problem with multi-vehicles supported multi-UAVs

The application of UAVs in orchard management presents a substantial opportunity to enhance efficiency and promote environmentally friendly agricultural production. While individual UAVs encounter limitations, such as endurance constraints, these challenges can be mitigated through collaborative efforts involving multiple vehicles and UAVs. This study addresses the issue of multi-UAV cooperative orchard path planning with multiple vehicles by formulating a mathematical optimization model aimed at minimizing the maximum UAV operation time. In the proposed model, the vehicle solely provides batteries and supplies to the UAV. To efficiently solve this model, we propose an improved Simulated Annealing-Lin Kernighan Helsgaun (SA-LKH) algorithm. The proposed algorithm utilizes the K-means and convex hull algorithm to generate the initial solution. Subsequently, the paths of each group of UAVs and vehicles are optimized based on the LKH algorithm. Additionally, we construct a directed perturbation operator that perturbs the maximum and minimum time-consuming paths to achieve fast convergence of the algorithm. The results from four different cases demonstrate that, compared to the general variable neighborhood search algorithm, the improved Ant Colony Algorithm and Improved Genetic Algorithm, the proposed algorithm has demonstrated an average improvement in solution quality of 3.41%, 5.72% and 9.67%, respectively. The proposed method effectively enhances the solution quality, which can provide valuable insights into green cleaning management of orchards.