Convex Hull Research Articles

Customized m-RCNN and hybrid deep classifier for liver cancer segmentation and classification

Diagnosing liver disease presents a significant medical challenge in impoverished countries, with over 30 billion individuals succumbing to it each year. Existing models for detecting liver abnormalities suffer from lower accuracy and higher constraint metrics. As a result, there is a pressing need for improved, efficient, and effective liver disease detection methods. To address the limitations of current models, this method introduces a deep liver segmentation and classification system based on a Customized Mask-Region Convolutional Neural Network (cm-RCNN). The process begins with preprocessing the input liver image, which includes Adaptive Histogram Equalization (AHE). AHE helps dehaze the input image, remove color distortion, and apply linear transformations to obtain the preprocessed image. Next, a precise region of interest is segmented from the preprocessed image using a novel deep strategy called cm-RCNN. To enhance segmentation accuracy, the architecture incorporates the ReLU activation function and the modified sigmoid activation function. Subsequently, a variety of features are extracted from the segmented image, including ResNet features, shape features (area, perimeter, approximation, and convex hull), and enhanced median binary pattern. These extracted features are then used to train a hybrid classification model, which incorporates classifiers like SqueezeNet and DeepMaxout models. The final classification outcome is determined by averaging the scores obtained from both classifiers.

Consensus for nonlinear multi-agent systems with actuator saturation by adaptive event-triggered scheme

This paper studies the leader–follower consensus problem for nonlinear multi-agent systems with actuator saturation by adaptive event-triggered scheme. This adaptive mechanism allows parameters in triggered mechanism to change automatically compared to fixed ones in traditional strategies. The paper introduces two distinct approaches to address the issue of saturation. The first approach is sector bounded condition, while the second relies on convex hull representation. Compared with the former one, the latter one can reduce the conservatism of controller design effectively. Furthermore, it is assumed that the nonlinear function adheres to incremental quadratic limitations, thereby offering a more comprehensive depiction of its nonlinear properties. Finally, the validity of the proposed approaches is demonstrated by one numerical example in the background of mechanical systems.

Reducing the Chvátal rank through binarization

In a classical paper, Chvátal introduced a rounding procedure for strengthening the polyhedral relaxation P of an integer program; applied recursively, the number of iterations needed to obtain the convex hull of the integer solutions in P is known as the Chvátal rank. Chvátal showed that this rank can be exponential in the input size L needed to describe P. We give a compact extended formulation of P, described by introducing binary variables, whose rank is polynomial in L.

Iterative RNDOP-Optimal Anchor Placement for Beyond Convex Hull ToA-Based Localization: Performance Bounds and Heuristic Algorithms

Iterative RNDOP-Optimal Anchor Placement for Beyond Convex Hull ToA-Based Localization: Performance Bounds and Heuristic Algorithms

Calculating functional diversity metrics using neighbor‐joining trees

The study of functional diversity (FD) provides ways to understand phenomena as complex as community assembly or the dynamics of biodiversity change under multiple pressures. Different frameworks are used to quantify FD, either based on dissimilarity matrices (e.g. Rao entropy, functional dendrograms) or multidimensional spaces (e.g. convex hulls, kernel‐density hypervolumes), each with their own strengths and limits. Frameworks based on dissimilarity matrices either do not enable the measurement of all components of FD (i.e. richness, divergence, and regularity), or result in the distortion of the functional space. Frameworks based on multidimensional spaces do not allow for comparisons with phylogenetic diversity (PD) measures and can be sensitive to outliers.We propose the use of neighbor‐joining trees (NJ) to represent and quantify FD in a way that combines the strengths of current frameworks without many of their weaknesses. Importantly, our approach is uniquely suited for studies that compare FD with PD, as both share the use of trees (NJ or others) and the same mathematical principles.We test the ability of this novel framework to represent the initial functional distances between species with minimal functional space distortion and sensitivity to outliers. The results using NJ are compared with conventional functional dendrograms, convex hulls, and kernel‐density hypervolumes using both simulated and empirical datasets.Using NJ, we demonstrate that it is possible to combine much of the flexibility provided by multidimensional spaces with the simplicity of tree‐based representations. Moreover, the method is directly comparable with taxonomic diversity (TD) and PD measures, and enables quantification of the richness, divergence and regularity of the functional space.

Using fossil records to predict short‐term changes in niche and spatial dynamics in a broadly distributed coral reef: Niche conservatism and adaptation

AbstractAimGiven the rapid pace of climate change and its significant impact on species distribution and survival, understanding the dynamics of ecological niches over time becomes imperative. By employing ecological niche models and species distribution models, alongside analyses of historical occurrence records and palaeoclimatic data, we aimed to assess the extent of changes in the niche of Porites astreoides from the Holocene to the present. Specifically, we quantified the amount of: (1) niche stability, (2) niche expansion, (3) niche decline and (4) niche overlap in both the environmental and geographical space.LocationAtlantic Ocean.TaxonCnidarian, Scleractinia.Materials and MethodsThrough fossil and current records and environmental variables from the same periods, we use two ecological niche modelling approaches (overlap and maximum entropy) to analyse niche dynamics in environmental and geographical space, specifically the degree of expansion, stability and loss of niches over the last ~8000 years. We used a principal component analysis to build the models within two different calibration areas (minimum convex polygon and buffer).ResultsOur results were similar, except for the loss of habitat, across the calibration areas used to build the models. The PCA used to build the models accounts for more than 65% of the variation in the data. Most of the environmental niche remained stable with an increase of almost a third and a negligible loss. Models in geographical space showed that most of the suitable area remained stable, but the species expanded its range from the Caribbean to the coast of Brazil. The analysis also showed a high value of niche overlap in environmental and geographical spaces.Main ConclusionsThe species studied is one of the corals with the greatest capacity to adapt to new environments and locations, but it also has a high tendency to preserve niche traits. These characteristics helped the species to expand its environmental and geographical niche in a period of relative climatic stability. On the other hand, these same characteristics of niche conservatism can lead the species to population decline and habitat loss, if the speed of adaptation is lower than the rapid climate changes predicted for the middle and end of the 21st century.

Convex hull property for elliptic and parabolic systems of PDE

We study the convex hull property for systems of partial differential equations. This is a generalization of the maximum principle for a single equation. We show that the convex hull property holds for a class of elliptic and parabolic systems of non-linear partial differential equations. In particular, this includes the case of the parabolic p-Laplace system. The coupling conditions for coefficients are demonstrated to be optimal by means of respective counterexamples.

When D-companion matrix meets incomplete polynomials

In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we obtain a majorization relationship between the zeros of convex combinations of incomplete polynomials and the original polynomial. Moreover, we prove that the set of zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. Location of zeros of convex combinations of incomplete polynomials is determined.

A Geometric Approach to Polynomial and Rational Approximation

Abstract We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.

An online hyper‐volume action bounding approach for accelerating the process of deep reinforcement learning from multiple controllers

AbstractThis paper fuses ideas from reinforcement learning (RL), Learning from Demonstration (LfD), and Ensemble Learning into a single paradigm. Knowledge from a mixture of control algorithms (experts) are used to constrain the action space of the agent, enabling faster RL refining of a control policy, by avoiding unnecessary explorative actions. Domain‐specific knowledge of each expert is exploited. However, the resulting policy is robust against errors of individual experts, since it is refined by a RL reward function without copying any particular demonstration. Our method has the potential to supplement existing RLfD methods when multiple algorithmic approaches are available to function as experts, specifically in tasks involving continuous action spaces. We illustrate our method in the context of a visual servoing (VS) task, in which a 7‐DoF robot arm is controlled to maintain a desired pose relative to a target object. We explore four methods for bounding the actions of the RL agent during training. These methods include using a hypercube and convex hull with modified loss functions, ignoring actions outside the convex hull, and projecting actions onto the convex hull. We compare the training progress of each method using expert demonstrators, employing one expert demonstrator with the DAgger algorithm, and without using any demonstrators. Our experiments show that using the convex hull with a modified loss function not only accelerates learning but also provides the most optimal solution compared with other approaches. Furthermore, we demonstrate faster VS error convergence while maintaining higher manipulability of the arm, compared with classical image‐based VS, position‐based VS, and hybrid‐decoupled VS.

Correcting the Location Error of Persistent Scatterers in an Urban Area Based on Adaptive Building Contours Matching: A Case Study of Changsha

Persistent Scatterer InSAR (PS-InSAR) technology enables the monitoring of displacement in millimeters. However, without the use of external parameter correction, radar scatterers exhibit poor geopositioning precision in meters, limiting the correlation between observed deformation and the actual structure. The integration of PS-InSAR datasets and building databases is often overlooked in deformation research. This paper presents a novel strategy for matching between PS points and building contours based on spatial distribution characteristics. A convex hull is employed to simplify the building outline. Considering the influence of building height and incident angle on geometric distortion, an adaptive buffer zone is established. The PS points on a building are further identified through the nearest neighbor method. In this study, both ascending and descending TerraSAR-X orbit datasets acquired between 2016 and 2019 were utilized for PS-InSAR monitoring. The efficacy of the proposed method was evaluated by comparing the PS-InSAR results obtained from different orbits. Through a process of comparison and verification, it was demonstrated that the matching effect between PS points and building contours was significantly enhanced, resulting in an increase of 29.2% in the number of matching PS points. The results indicate that this novel strategy can be employed to associate PS points with building outlines without the need for complex calculations, thereby providing a robust foundation for subsequent building risk assessment.

Short fiber bundle filtering and test-retest reproducibility of the Superficial White Matter.

In recent years, there has been a growing interest in studying the Superficial White Matter (SWM). The SWM consists of short association fibers connecting near giry of the cortex, with a complex organization due to their close relationship with the cortical folding patterns. Therefore, their segmentation from dMRI tractography datasets requires dedicated methodologies to identify the main fiber bundle shape and deal with spurious fibers. This paper presents an enhanced short fiber bundle segmentation based on a SWM bundle atlas and the filtering of noisy fibers. The method was tuned and evaluated over HCP test-retest probabilistic tractography datasets (44 subjects). We propose four fiber bundle filters to remove spurious fibers. Furthermore, we include the identification of the main fiber fascicle to obtain well-defined fiber bundles. First, we identified four main bundle shapes in the SWM atlas, and performed a filter tuning in a subset of 28 subjects. The filter based on the Convex Hull provided the highest similarity between corresponding test-retest fiber bundles. Subsequently, we applied the best filter in the 16 remaining subjects for all atlas bundles, showing that filtered fiber bundles significantly improve test-retest reproducibility indices when removing between ten and twenty percent of the fibers. Additionally, we applied the bundle segmentation with and without filtering to the ABIDE-II database. The fiber bundle filtering allowed us to obtain a higher number of bundles with significant differences in fractional anisotropy, mean diffusivity, and radial diffusivity of Autism Spectrum Disorder patients relative to controls.

Efficient Target Classification Based on Vehicle Volume Estimation in High-Resolution Radar Systems

In this paper, we propose a method for efficient target classification based on the spatial features of the point cloud generated by using a high-resolution radar sensor. The frequency-modulated continuous wave radar sensor can estimate the distance and velocity of a target. In addition, the azimuth and elevation angle of the target can be estimated by using a multiple-input and multiple-output antenna system. Using the estimated distance, velocity, and angle, the 3D point cloud of target can be generated. From the generated point cloud, we extract the point cloud for each individual target using the density-based spatial clustering of application with noise method and a camera mounted on the radar sensor. Then, we define the convex hull boundaries that enclose these point clouds in both 3D and 2D spaces obtained by orthogonally projecting onto the xy, yz, and zx planes. Using the vertices of convex hull, we calculate the volume of the targets and the areas in 2D spaces. Several feature points, including the calculated spatial information, are numerized and configured into feature vectors. We design an uncomplicated deep neural network classifier based on minimal input information to achieve fast and efficient classification performance. As a result, the proposed method achieved an average accuracy of 97.1%, and the time required for training was reduced compared to the method using only point cloud data and the convolutional neural network-based method.

Distributed adaptive prescribed-time orbit containment control for satellite cluster flight

The distributed prescribed-time orbit containment control for the satellite cluster flight with multiple dynamic leaders is investigated. The directed information communication topology between followers is taken into account in the overall paper. When the satellite mass is assumed to be constant, a distributed prescribed-time orbit containment controller is, firstly, presented to drive the followers into the dynamic convex hull produced by multiple leaders. Then, the parameter uncertainty is considered, and a prescribed-time sliding mode estimator is introduced to estimate the desired velocity of each follower. Based on the estimated state, a novel distributed adaptive prescribed-time orbit containment control scheme is proposed. The Lyapunov stability theory is utilized to prove the prescribed-time stability of the closed-loop system. Finally, several numerical simulations and comparison of different control methods are provided to verify the effectiveness and superiority of the proposed control method.

MEDAVET: Traffic Vehicle Anomaly Detection Mechanism based on spatial and temporal structures in vehicle traffic

Road traffic anomaly detection is vital for reducing the number of accidents and ensuring a more efficient and safer transportation system. In highways, where traffic volume and speed limits are high, anomaly detection is not only essential but also considerably more challenging, given the multitude of fast-moving vehicles, often observed from extended distances and diverse angles, occluded by other objects, and subjected to variations in illumination and adverse weather conditions. This complexity has meant that human error often limits anomaly detection, making the role of computer vision systems integral to its success. In light of these challenges, this paper introduces MEDAVET - a sophisticated computer vision system engineered with an innovative mechanism that leverages spatial and temporal structures for high-precision traffic anomaly detection on highways. MEDAVET is assessed in its object tracking and anomaly detection efficacy using the UA-DETRAC and Track 4 benchmarks and has its performance compared with that of an array of state-of-the-art systems. The results have shown that, when MEDAVET’s ability to delimit relevant areas of the highway, through a bipartite graph and the Convex Hull algorithm, is paired with its QuadTree-based spatial and temporal approaches for detecting occluded and stationary vehicles, it emerges as superior in precision, compared to its counterparts, and with a competitive computational efficiency.

Convexification of Bilinear Terms over Network Polytopes

It is well-known that the McCormick relaxation for the bilinear constraint z = xy gives the convex hull over the box domains for x and y. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints [Formula: see text] where xi represents the arc-flow variable in a network polytope, and yj is in a simplex. For the case where the simplex contains a single y variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple y variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments conducted on different applications show the effectiveness of the proposed methods in improving the dual bounds obtained from alternative techniques. Funding: This work was supported by Air Force Office of Scientific Research [Grant FA9550-23-1-0183]; National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2338641]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0001 .

Remark on Algorithm 1012: Computing projections with large data sets

In ACM TOMS Algorithm 1012, the DELAUNAYSPARSE software is given for performing Delaunay interpolation in medium to high dimensions. When extrapolating outside the convex hull of the training set, DELAUNAYSPARSE calls the nonnegative least squares solver DWNNLS to compute projections onto the convex hull. However, DWNNLS and many other available sum of squares optimization solvers were not intended for usage with many variable problems, which result from the large training sets that are typical in machine learning applications. Thus, a new PROJECT subroutine is given, based on the highly customizable quadratic program solver BQPD . This solution is shown to be as robust as DELAUNAYSPARSE for projection onto both synthetic and real-world data sets, where other available solvers frequently fail. Although it is intended as an update for DELAUNAYSPARSE , due to the difficulty and prevalence of the problem, this solution is likely to be of external interest as well.

Finite Sequences Representing Expected Order Statistics

Characterizations of finite sequences β1<⋯<βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{1}<\\cdots <\\beta _{n}$$\\end{document} representing expected values of order statistics from a random sample of size n are given. As a by-product, a characterization of binomial mixtures, when the mixing random variable is supported in the open interval (0, 1), is presented; this enables the exact description of the convex hull of the open binomial curve, as well as the open moment curve.

Mixed-integer Nonlinear Optimization: A Hatchery for Modern Mathematics

Mixed-integer nonlinear programming (MINLP) is concerned with finding optimal solutions to mathematical formulations of optimization problems combining discrete and nonlinear phenomena. The scientific program was organized around three areas: convex envelopes and relaxation hierarchies, mixed-integer optimal control, and current trends. These topics were addressed with a variety of tutorials, talks, and short research announcements.

On some advantages of convex hull pricing for the European electricity auction

Since the liberalization of the power sector and the creation of wholesale electricity markets, the question of how to price the non-convexities that are present in the market has attracted the interest of both academics and practitioners. Over the years, US markets have studied and adopted different and evolving pricing rules. Since the “Trilateral Market Coupling” (2006), the European day-ahead market has opted for a notably different pricing rule. Recently, EU stakeholders have undertaken research to reform it, and have indicated an interest for some approaches that are discussed in the other side of the Atlantic. Our paper aims at contributing to the debate. We analyse six different pricing methods. We establish several mathematical properties for enabling their accurate comparison. Our findings are illustrated on stylized examples and numerical simulations that are performed on realistic datasets. Both theoretical and numerical evidences that are gathered in our paper point towards the advantages of convex hull pricing.