Minimum Volume Enclosing Ellipsoid Research Articles

Three-Dimensional Angularity Characterization of Coarse Aggregates by the Optimal Ellipsoid Approximation Algorithm

The three-dimensional (3D) angularity of coarse aggregates was investigated in this study by an optimal ellipsoid approximation algorithm. The minimum volume enclosing ellipsoid (MVEE) of the 3D coarse aggregate particle represented by the surface point cloud was computed by a Python program. An algorithm was proposed to compute the optimal ellipsoid that is inscribed in the aggregate particle as far as possible. In the algorithm, a fitting index was proposed and a cycle program was coded to search the optimal ellipsoid. Based on the optimal ellipsoid and the point cloud of the coarse aggregate particle, a 3D angularity index was proposed. The critical discriminant coefficient was introduced into the angularity index to filter out the tiny convex parts on the particle. Six 3D geometric models were generated and five 3D aggregate particles were randomly selected to verify the effectiveness of the angularity index. The results demonstrate that the angularity index proposed in this study can effectively distinguish the angularity of different geometric models. The effect of mesh precision on the angularity index was analyzed and the critical discriminant coefficient was studied by the parameter analysis method and Delphi expert investigation method.

Computing minimum-volume enclosing ellipsoids

For a given multidimensional data set (point cloud), we investigate methods for computing the minimum-volume enclosing ellipsoid (MVEE), which provides an efficient representation of the data that is useful in many applications, including data analysis, optimal design, and computational geometry. Contrary to conventional wisdom, we demonstrate that careful exploitation of problem structure can enable high-order (Newton and Newton-like) methods with superlinear convergence rates to scale to very large MVEE problems. We also introduce a hybrid method that combines the benefits of both high-order and low-order methods, along with new initialization schemes that further enhance performance. Observing that computational cost depends significantly on the particular distribution of the data, we demonstrate that kurtosis serves as an excellent indicator of problem difficulty and provides useful guidance in choosing an appropriate solution algorithm and initialization.

Three-dimensional particle tracking algorithm based on the special ellipsoids

A 3D PTV algorithm based on the minimum volume enclosing ellipsoid (MVEE) and the inertia-equivalent ellipsoid (IEE) is proposed. Firstly, different versions of this algorithm are proposed by incorporating the features of MVEE and Voronoi polyhedron through weighted summation. Secondly, according to the test by homemade particle flows, among these versions, the Voronoi polyhedron feature expressed by IEE has the best performance, thus leading to a completely ellipsoid-based version named E-PTV. Thirdly, a series of tests and analyses are conducted to compare E-PTV with some classical and newly developed 3D PTV algorithms. The results show that, E-PTV is parameter-independent and optimally efficient in addition to being close to these algorithms in accuracy. Finally, combining E-PTV with the volumetric 3-component velocimetry (V3V) system successfully reconstructs an actual 3D vortex ring flow, thereby confirming the practicality of E-PTV.

A robust reliability evaluation model with sequential acceleration method for power systems considering renewable energy temporal-spatial correlation

As the penetration of renewable energy increases, it becomes critical to evaluate the power system reliability, while considering the correlations among renewable energy sources (RESs) and their uncertainties. In this paper, the robust reliability evaluation model based on the minimum volume enclosing ellipsoid (MVEE) algorithm is established, and the sequential acceleration method is proposed to improve the convergence. First, the robust reliability evaluation model is built, where the robust state analysis is performed under each sampling. Second, the sequential Cross-Entropy-Latin Hypercube Sampling (CE-LHS) acceleration method is presented, which first obtains the optimal probability distribution of system parameters, and then forces the sampling values to be evenly distributed to overcome the truncated tail effect. Correspondingly, the reliability indices are improved. Third, for state analysis, the robust multi-period optimal load shedding (RMPOLS) model is established, where the dynamic performance of energy storage and system ramping rate constraints are described in detail. Therefore, the periods are strongly coupled, and the impact of RES uncertainties is carefully considered. Besides, to address the temporal-spatial correlation of RESs, the MVEE algorithm is introduced to generate the convex hull of RESs scenarios. Therefore, the RMPOLS model can be transformed into a second-order cone form. Finally, to further reduce the computational complexity, the branch flow constraints scanning strategy is proposed, which can quickly remove inactive constraints in advance. Numerical results prove that the MVEE-based robust reliability evaluation model considers the temporal-spatial correlation of RES and thus improves the system reliability by almost 30 %. Besides, branch flow constraint scanning can eliminate more than 65 % of the SOC constraints. Moreover, CE-LHS can improve the convergence by about 8 times compared with Monte Carlo Sampling (MCS) while the error is less than 1 %.

Improved Assessment of Globularity of Protein Structures and the Ellipsoid Profile of the Biological Assemblies from the PDB.

In this paper, we present an update to the ellipsoid profile algorithm (EP), a simple technique for the measurement of the globularity of protein structures without the calculation of molecular surfaces. The globularity property is understood in this context as the ability of the molecule to fill a minimum volume enclosing ellipsoid (MVEE) that approximates its assumed globular shape. The more of the interior of this ellipsoid is occupied by the atoms of the protein, the better are its globularity metrics. These metrics are derived from the comparison of the volume of the voxelized representation of the atoms and the volume of all voxels that can fit inside that ellipsoid (a uniform unit Å cube lattice). The so-called ellipsoid profile shows how the globularity changes with the distance from the center. Two of its values, the so-called ellipsoid indexes, are used to classify the structure as globular, semi-globular or non-globular. Here, we enhance the workflow of the EP algorithm via an improved outlier detection subroutine based on principal component analysis. It is capable of robust distinguishing between the dense parts of the molecules and, for example, disordered chain fragments fully exposed to the solvent. The PCA-based method replaces the current approach based on kernel density estimation. The improved EP algorithm was tested on 2124 representatives of domain superfamilies from SCOP 2.08. The second part of this work is dedicated to the survey of globularity of 3594 representatives of biological assemblies from molecules currently deposited in the PDB and analyzed by the 3DComplex database (monomers and complexes up to 60 chains).

Spatial Characterization of Single-Cracked Space Based on Microcrack Distribution in Sandstone Failure

To further understand the rock damage zone, an approach based on microcrack distribution was proposed to characterize the crack space of rock specimens in this research. Acoustic emission (AE) technology was utilized on sandstone to obtain the spatial distribution of microcracks in which uniaxial compression forms the single-cracked fracture. The proposed theoretical distribution pattern space (TDPS), 3D convex hull, and the minimum volume enclosing ellipsoid (MVEE) algorithms were adopted to analyze the geometric features of the crack space. It was found that the 3D convex hull method returned the smallest results in both area and volume of the crack space, and the largest results were provided by the proposed TDPS method. The difference between the results of the proposed TDPS method and the MVEE method became smaller after 85%. The deviation angle of the principal axis of the cracked space gradually decreased as the spatial scale decreased, while the other two major axes exhibited a tendency to increase at the 65% scale. The results indicate that a spatial scale from 65% to 85% is a reliable range for the characterization of crack space.

Computing minimum-volume enclosing ellipsoids for large datasets

An algorithm is provided for calculating the minimum-volume enclosing ellipsoid (MVEE) for a large dataset stored in a separate database, for which the existing algorithms run out of memory or become prohibitively slow. The focus is on tall datasets, i.e., those consisting of huge numbers of data points of moderate dimensionality. The proposed Big Index Batching algorithm works in an optimization-deletion-adaptation cycle that consists of: using an existing algorithm by applying it on a smaller batch of data; pruning the vector of the indices of the data points by removing the points that are guaranteed to not lie on the boundary of the MVEE; and efficiently adapting the choice of the batch. The algorithm is provably convergent, and simple to describe and implement. The reading of tall data from the database is very time consuming, therefore the amount of reading during an MVEE computation should be as small as possible. It is shown on examples that Big Index Batching tends to find the MVEE after reading all data points just two or three times. As a consequence, the proposed algorithm usually converges to the MVEE reasonably fast. Its usefulness in robust statistics and anomaly detection is demonstrated by finding the potential outliers in a large dataset by using so-called ellipsoidal trimming.

A Hyper-Ellipsoid Approach for Inverse Lack-of-Knowledge Uncertainty Quantification

Abstract This paper presents a novel methodology to solve an inverse uncertainty quantification problem where only the variation of the system response is provided by a small set of experimental data. Furthermore, the method is extended for cases where the uncertainty of the response quantities is given by an incomplete set of statistical moments. For both cases, the uncertainty on the output space is represented by a minimum volume enclosing ellipsoid (MVEE). The actual inverse uncertainty quantification is conducted by identifying also a hyper-ellipsoid for the input parameters, which has an image on the output space that matches the MVEE as close as possible. Hence, the newly introduced approach is a contribution to the field of nonprobabilistic uncertainty quantification methods. Compared to literature, the new approach has often superior accuracy and especially an improved efficiency for high-dimensional problems. The method is validated first by an analytical test case and subsequently applied to a jet engine performance model, where this type of inverse uncertainty quantification has to be solved to allow for a consistent and integrated solution procedure. In both cases, the results are compared with an inverse method where the variability on the input side is quantified by a multidimensional interval. It can be shown that the hyper-ellipsoid approach is superior with respect to the computation time in high-dimensional problems encountered not only in jet engine design.

A coordinate gradient descent algorithm for computing the minimum volume enclosing ellipsoid

The minimum volume enclosing ellipsoid (MVEE) is a basic convex optimization problem. This paper describes some new properties such as “algorithmic coordinate-wise smoothness” of this model and proposes a steepest descent type algorithm, the coordinate gradient descent (CGD) algorithm, to address the MVEE problem. We prove that the CGD algorithm is sublinearly convergent. Moreover, it is shown to converge in a linear rate locally, and slightly faster than the FW type algorithms, especially in the cases of large dimensions. At last, we compare our algorithm with the random coordinate descent (RCD) method and show that the RCD algorithm is less efficient than the CGD algorithm in computing the MVEE. The numerical tests support our theoretical results.

Rank-two update algorithm versus Frank–Wolfe algorithm with away steps for the weighted Euclidean one-center problem

The weighted Euclidean one-center (WEOC) problem is one of the classic problems in facility location theory, which is also a generalization of the minimum enclosing ball (MEB) problem. Given m points in $${\mathbb {R}}^{n}$$, the WEOC problem computes a center point $$c\in {\mathbb {R}}^{n}$$ that minimizes the maximum weighted Euclidean distance to m given points. The rank-two update algorithm is an effective method for solving the minimum volume enclosing ellipsoid (MVEE) problem. It updates only two components of the solution at each iteration, which was previously proposed in Cong et al. (Comput Optim Appl 51(1):241–257, 2012). In this paper, we further develop and analyze the rank-two update algorithm for solving the WEOC problem. At each iteration, the calculation of the optimal step-size for the WEOC problem needs to distinguish four different cases, which is a challenge in comparison with the MVEE problem. We establish the theoretical results of the complexity and the core set size of the rank-two update algorithm for the WEOC problem, which are the generalizations of the currently best-known results for the MEB problem. In addition, by constructing an important inequality for the WEOC problem, we establish the linear convergence of this rank-two update algorithm. Numerical experiments show that the rank-two update algorithm is comparable to the Frank–Wolfe algorithm with away steps for the WEOC problem. In particular, the rank-two update algorithm is more efficient than the Frank–Wolfe algorithm with away steps for problem instances with $$m\gg n$$ under high precision.

Distribution and Detection of Radiopaque Beads after Hepatic Transarterial Embolization in Swine: Cone-Beam CT versus MicroCT

PurposeTo determine the true distribution of radiopaque beads (ROBs) after hepatic embolization in swine as imaged by micro-computed tomography (microCT) compared with in vivo cone-beam computerized tomography (CT) imaged at different kVp settings. Materials and MethodsSwine (n = 3) underwent hepatic transarterial embolization (n = 6) with the use of 70–150-μm ROBs under fluoroscopic guidance. After stasis, in vivo cone-beam CT was performed at 120, 100, and 80 kVp. The animal was euthanized, the liver resected, and microCT with 17 μm resolution performed on embolized tissue samples. The resulting cone-beam CT and microCT data were segmented and registered. Total vessel length, minimum volume–enclosing ellipsoid (MVEE), and number of independent volumes were measured. Maximum-intensity projections (MIPs) were generated for each cone-beam CT. ResultsMetrics for all cone-beam CT segmentations differed significantly from microCT segmentations. Segmentations at 80 kVp presented significantly greater vessel length, MVEE, and number of independent volumes compared with 100 kVp and 120 kVp. In addition, 100 kVp segmentations presented significantly greater vessel length than 120 kVp. MIPs presented greater visualization than cone-beam CT segmentations and improved as kVp decreased. ConclusionsThe full ROB distribution was more extensive than was apparent on cone-beam CT. Quantitative measures of embolic distribution demonstrated significantly better correlation with microCT with decreasing kVp. Similarly, qualitative analysis of MIPs showed improved visualization of beads with decreasing kVp. These findings demonstrate the clinical value of 80 kVp and 100 kVp protocols in the imaging of radiopaque embolizations compared with 120 kVp. However, considerations on X-ray penetration and dose may favor use of 100 kVp imaging over 80 kVp.

Ellipsoidal data description

Support vector data description (SVDD) is a leading classification method for novelty detection, which minimizes the volume of a spherically shaped decision boundary around the normal class. While SVDD has achieved promising performance, it will lead to a loose boundary for multivariate datasets of which the input dimensions are usually correlated. Inspired by the relationship between kernel principal component analysis (kernel PCA) and the best-fit ellipsoid for a dataset, this study proposes the ellipsoidal data description (ELPDD) which considers feature variance of each dimension adaptively. A minimum volume enclosing ellipsoid (MVEE) is constructed around the target data in the kernel PCA subspace which can be learned via a SVM-like objective function with log-determinant penalty. We also provide the Rademacher complexity bound for our model. Some relating problems are investigated in detail. Experiments on artificial and real-world datasets validate the effectiveness of our method.

Lift-and-project MVEE based convex hull for robust SCED with wind power integration using historical data-driven modeling approach

This paper presents an adjustable robust security constrained economic dispatch (SCED) model with wind power uncertainties. First, the scenario based adjustable robust SCED model is presented. It considers multiple scenarios from historical data as well as the spatial correlation among wind farms. Then, the proposed SCED model becomes an optimization problem with a large amount of constraints which is skillfully solved using a lift-and-project minimum volume enclosing ellipsoid (MVEE) based convex hull. Furthermore, the proposed model is transformed into a second order cone programming (SOCP) model by the use of participation factors to generate adjustable generation outputs and thus guarantee the energy balance. In order to further reduce the computational complexity, the inactive constraints reduction strategy is proposed to quickly eliminate inactive SOC security constraints before solving the model. Numerical results of IEEE 14-bus and 118-bus test systems as well as the practical Polish power systems with several wind farms show that the proposed model can achieve better economies. Moreover, more than 82% of security constraints are identified as inactive in various cases of the simulation, and the proposed inactive constraints reduction strategy is promising for improving the computational performance.

Robust methods of inclusive outlier analysis for structural health monitoring

The key novel element of this work is the introduction of robust multivariate statistical methods into the structural health monitoring (SHM) field through use of the minimum covariance determinant estimator (MCD) and the minimum volume enclosing ellipsoid (MVEE). In this paper, robust outlier statistics are investigated, focussed mainly on a high level estimation of the “masking effect” of inclusive outliers, not only for determining the presence or absence of novelty-something that is of fundamental interest but also to examine the normal condition set under the suspicion that it may already include multiple abnormalities. By identifying and detecting variability at an early stage, the prospects of achieving good generalisation and establishing a correct normal condition classifier may be increased. It is critical to highlight that there is no a priori division between the damaged and the undamaged condition data when the algorithms are implemented, offering a significant advantage over other methodologies. In summary, this paper introduces a new scheme for SHM by exploiting robust multivariate outlier statistics in order to investigate if the selected features are free from multiple outliers before such features can be selected for either supervised or unsupervised analysis.

Rank-two update algorithms for the minimum volume enclosing ellipsoid problem

We consider the problem of computing a (1+?)-approximation to the minimum volume enclosing ellipsoid (MVEE) of a given set of m points in R n . Based on the idea of sequential minimal optimization (SMO) method, we develop a rank-two update algorithm. This algorithm computes an approximate solution to the dual optimization formulation of the MVEE problem, which updates only two weights of the dual variable at each iteration. We establish that this algorithm computes a (1+?)-approximation to MVEE in O(mn 3/?) operations and returns a core set of size O(n 2/?) for ??(0,1). In addition, we give an extension of this rank-two update algorithm. Computational experiments show the proposed algorithms are very efficient for solving large-scale problem with a high accuracy.

Enclosing machine learning: concepts and algorithms

A novel machine learning paradigm, i.e., enclosing machine learning based on regular geometric shapes was proposed. First, it adopted regular minimum volume enclosing and bounding geometric shapes (sphere, ellipsoid, box) or their unions and so on to obtain one class description model. Second, Data description, two class classification, learning algorithms based on the one class description model were presented. The most obvious feature was that enclosing machine learning emphasized one class description and learning. To illustrate the concepts and algorithms, a minimum volume enclosing ellipsoid (MVEE) case for enclosing machine learning was then investigated in detail. Implementation algorithms for enclosing machine learning based on MVEE were presented. Subsequently, we validate the performances of MVEE learners using real world datasets. For novelty detection, a benchmark ball bearing dataset is adopted. For pattern classification, a benchmark iris dataset is investigated. The performance results show that our proposed method is comparable even better than Support Vector Machines (SVMs) in the datasets studied.