Barycentric Coordinate System Research Articles

Application of Barycentric Coordinates and the Jacobian Matrix to the Analysis of a Closed Structure Robot

A new approach is presented to study the kinematic properties of stationary robots with a closed structure. It combines the application of conventional methods from kinematics with geometric parameters represented in a barycentric coordinate system. This allows examining the influence of the proportions of the robot’s links on its basic mechanical characteristics. Each point from the newly introduced barycentric space corresponds to a set of robots with the same link proportions. The proposed approach is used to study three aspects: the link proportions for which the robot can exist; the shape of the robot’s workspace; and the possible singular configurations. This is valuable when evaluating the qualities of existing robots and could be applied to the design of new mechanical systems. An example of a 5-link robot with a closed structure is considered. The conditions for the existence of the mechanism and the conditions under which certain types of singular configurations can occur are defined. The example reveals the great potential of combining barycentric coordinates and Jacobian properties. The barycentric coordinates of 10 robots with a 5-link closed structure known from the literature are determined, and their properties are analyzed. The results are presented graphically. An extension of the application area of the approach is discussed.

Nested barycentric coordinate system as an explicit feature map for polyhedra approximation and learning tasks

We introduce a new embedding technique based on a nested barycentric coordinate system. We show that our embedding can be used to transform the problems of polyhedron approximation, piecewise linear classification and convex regression into one of finding a linear classifier or regressor in a higher dimensional (but nevertheless quite sparse) representation. Our embedding maps a piecewise linear function into an everywhere-linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We explain the applications of our embedding to the problems of approximating separating polyhedra—in fact, it can approximate any convex body and unions of convex bodies—as well as to classification by separating polyhedra, and to piecewise linear regression.

Optimization of spherically arranged lens arrays based on class II and III geodesic polyhedra.

Spherical arrangement is a common way to pack lens arrays. Due to the restriction of original structures, the previous optimization methods based on icosahedral subdivision are only applicable to a few types of lens arrays adopting different numbers of sub-lenses. In order to design more types of lens arrays, an optimization packing method is proposed with the vertices of class II and class III geodesic polyhedra as the arrangement positions of sub-lenses. The vertices of the geodesic polyhedron are generated by projecting grid nodes on each face of the ortho-icosahedron to a sphere. The vertex translation function is established in a barycentric coordinate system, and then the chord ratio and packing density are used as evaluation functions to optimize the vertices' positions. The universality of the optimization method is tested, and its robustness is certified by analyzing the optimization results of different original structures with the same number of vertices. Finally, the effectiveness and feasibility of the optimization method in engineering applications are demonstrated in a monocentric multi-scale imaging system. The optimization method proposed in this paper can significantly increase the types of lens arrays from 9 to 37 types, adopting different numbers of sub-lenses within 1000 and effectively improved the distribution uniformity of lens arrays.

Method involving shape-morphing of multiple hull forms aimed at organizing and visualizing the propulsive performance of optimal ship designs

Most cases of successfully designed hull forms developed using conventional nonlinear optimization methods propose the optimal candidates in a particular design constraint. Furthermore, these optimal designs contain uncertainties in terms of the evaluation parameters and design conditions, thereby raising the issue of robustness of the design idea that the designer needs to investigate to arrive at a decision for building the expensive structure of a ship. This paper proposes a method to organize and visualize the propulsive performance of any number of hull form combinations. The shape morphing technique, which has been limited to the blending of two or three hull forms, is extended to the shape morphing of N hull forms by introducing the concept of the barycentric coordinate system. Thus, a hull-form coordinate system with each hull form as a basis is proposed. The effectiveness of this method is demonstrated by analyzing several hull-form design spaces. This method can contribute to improving the robustness of the optimal solution obtained by nonlinear optimization and machine learning, by analyzing and visualizing the design space around the optimal point.

On the Geometric Description of Nonlinear Elasticity via an Energy Approach Using Barycentric Coordinates

The deformation of a solid due to changing boundary conditions is described by a deformation gradient in Euclidean space. If the deformation process is reversible (conservative), the work done by the changing boundary conditions is stored as potential (elastic) energy, a function of the deformation gradient invariants. Based on this, in the present work we built a “discrete energy model” that uses maps between nodal positions of a discrete mesh linked with the invariants of the deformation gradient via standard barycentric coordinates. A special derivation is provided for domains tessellated by tetrahedrons, where the energy functionals are constrained by prescribed boundary conditions via Lagrange multipliers. The analysis of these domains is performed via energy minimisation, where the constraints are eliminated via pre-multiplication of the discrete equations by a discrete null-space matrix of the constraint gradients. Numerical examples are provided to verify the accuracy of the proposed technique. The standard barycentric coordinate system in this work is restricted to three-dimensional (3-D) convex polytopes. We show that for an explicit energy expression, applicable also to non-convex polytopes, the general barycentric coordinates constitute fundamental tools. We define, in addition, the discrete energy via a gradient for general polytopes, which is a natural extension of the definition for discrete domains tessellated by tetrahedra. We, finally, prove that the resulting expressions can consistently describe the deformation of solids.

A Model for Developing an Academic Activity Index for Higher Education Instructors Based on Composite Indicators

The assessment of the teacher performance is a subject of criticism due to the lack of a well-establish methodology. This study develops an overall score to measure the dimensions that encompass the academic activities. To that end, a Benefit-of-the-doubt model is proposed. The advantage of this technique is the flexibility in the weights, so that the model selects for each teacher the most favourable set of weights. Furthermore, the paper proposes the barycentric coordinate system as a method to classify the teachers in clusters depending on their contribution to the dimensions. A specific pie chart has been proposed as an efficient way to report the contribution of the teachers to the dimensions and the overall teacher performance.

Enhanced Active Power Balancing Capability of Grid-Connected Solar PV Fed Cascaded H-Bridge Converter

Cascaded H-bridge converter is a promising solution for the next generation large-scale photovoltaic (PV) system having both active and reactive power exchange capacity. However, one challenge with this converter is unbalanced power generation among the clusters or cells due to variation in radiation level, temperature, dusting on PV cells etc. In order to inject balanced power to the grid, in this paper, a controlled exchange of reactive power is proposed so that the converter can work with unbalanced power generation within the clusters. The magnitude of reactive power exchange is derived and necessary conditions to prevent overmodulation are discussed. In addition, the magnitude of zero voltage is derived taking into account both active and reactive power exchange with the grid. By exchanging reactive power, the converter can also be used during low-voltage ride-through condition. This paper introduces a novel representation of unbalanced power generation using the barycentric coordinate system which helps to identify regions in the power plane which require the exchange of reactive power. The proposed method is verified through computer simulation and in a laboratory prototype.

Optimal SUAS Path Planning in Three-Dimensional Constrained Environments

Small Unmanned Aircraft Systems have grown in autonomy and capability and continue to complement Department of Defense mission objectives. Teaming unmanned aircraft with manned vehicles can expand mission profiles and reduce risk to human life. To fully leverage unmanned systems, vehicles must be efficient and autonomous in path planning development. The work herein explores direct orthogonal collocation optimal control techniques combined with fast geometric path planning algorithms to reduce computation time and increase solution accuracy for small unmanned aircraft systems path planning missions. Previous work in the two-dimensional plane demonstrated a methodology to provide optimal flight paths through defined simplex corridors and simplified the optimal control parameter bounds by formulating the problem in the barycentric coordinate system. These methodologies are extended in this paper for three-dimensional flight and are solved with two different formulations for flight in an urban environment. The first formulation solves the constrained optimal control problem using a single phase while modeling the building constraints with superquadric functions. The second formulation implements the simplex methodology, eliminating polygonal constraints from the search domain, and solving the optimal path in a multiple phase approach. Results illustrate the benefits gained in computation time and accuracy when implementing simplex methods into the optimal control design and provide a foundation for closing the gap to real-time, onboard operations for unmanned vehicle path planning.

Fusion of thermal imagery with point clouds for building façade thermal attribute mapping

Thermal image data are widely used to assess the insulation quality of buildings and to detect thermal leakages. In our approach, we merge terrestrial thermal image data and 3D point clouds to perform thermal texture mapping for building facades. Since geo-referencing data of a hand-held thermal camera is usually not available in such applications, registration between thermal images and a 3D point cloud (for instance generated from RGB image data by structure-from-motion techniques) is essential. In our approach, thermal image data registration is conducted in four steps: First, another point cloud is generated from the thermal image data. Next, a coarse registration between thermal point cloud and RGB point cloud is performed using the fast global registration (FGR) algorithm. The best corresponding thermal-RGB image pairs are acquired by picking up the lowest Euclidean distance between the exterior orientation parameters of thermal images and transformed exterior orientation parameters of RGB images. Subsequently, radiation-invariant feature transform (RIFT), normalized barycentric coordinate system (NBCS) and random sample consensus (RANSAC) are employed to extract reliable matching features on thermal-RGB image pairs. Afterwards, a fine registration is performed by mono-plotting of the RGB image, followed by image resection of the thermal image. Finally, in terms of texture mapping algorithms, in order to remove the blur effects caused by small misalignments for different candidate images, a global image pose refinement approach, which aims to minimize the temperature disagreements provided by different images for the same object points, is proposed. In addition, in order to ensure high geometric and radiant accuracy, camera calibrations are performed. Experiments showed that the proposed method could not only achieve high geometric registration accuracy, but also provide a good radiometric accuracy with RMSE lower than 1.5 °C.

Mean value coordinates for quad cages in 3D

Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a cage , within its volume. These generalizations of the barycentric coordinate system have progressively expanded the range of eligible cages to triangle and planar polygon surface meshes with arbitrary topology, concave regions and a spatially-varying sampling ratio, while preserving a smooth diffusion of the prescribed on-surface functions. In spite of their potential for major computer graphics applications such as freeform deformation or volume texturing, current space coordinate systems have only found a moderate impact in applications. This follows from the constraint of having only triangles in the cage most of the time, while many application scenarios favor arbitrary (non-planar) quad meshes for their ability to align the surface structure with features and to naturally cope with anisotropic sampling. In order to use space coordinates with arbitrary quad cages currently, one must triangulate them, which results in large propagation distortion. Instead, we propose a generalization of a popular coordinate system - Mean Value Coordinates - to quad and tri-quad cages, bridging the gap between high-quality coarse meshing and volume diffusion through space coordinates. Our method can process non-planar quads, comes with a closed-form solution free from global optimization and reproduces the expected behavior of Mean Value Coordinates, namely smoothness within the cage volume and continuity everywhere. As a result, we show how these coordinates compare favorably to classical space coordinates on triangulated quad cages, in particular for freeform deformation.

Effects of hawkmoth-like flexibility on the aerodynamic performance of flapping wings with different shapes and aspect ratios

The effect of hawkmoth-like flexibility on the aerodynamic hovering performance of wings at a Reynolds number of 400 has been assessed by conducting fluid structure interaction simulations incorporating a finite difference based immersed boundary method coupled with a finite-element based structure solver. The stiffness distribution of a hawkmoth forewing was mapped onto three wing shapes (r¯1 = 0.43, 0.53, and 0.63) defined by the radius of the first moment of wing area each with aspect ratios, AR = 1.5, 2.96, 4.5, and 6.0 using elliptic mesh generation, the Jacobi method for iterations, and the concept of the barycentric coordinate system. The results show that there is a dominant chordwise deformation at AR = 1.5, and the wings also deform in the spanwise direction and their tips deviate from the horizontal stroke plane as AR increases. At AR = 1.5, 2.96, and 4.5, flexibility increases the mean lift (up to 39%, 18%, and 17.6%, respectively) for all wing shapes. At AR = 6.0, the r1¯ = 0.53 and 0.63 flexible wings give lesser lift than the rigid equivalents because of negative lift or small positive lift during the early stroke as the vortical structures remain on the bottom surface. This is attributed to the rapid pitch-down rotation, lesser stroke angular velocity than the rigid wing, and upward motion of the wingtip, away from the horizontal stroke plane. From the design perspective, the anisotropic flexible wings (except r1¯ = 0.53 and 0.63 with AR = 6.0) can be used in micro aerial vehicles for high lift requirements, such as for a high payload. Results here show that in nature, the hawkmoth wings with r1¯ and AR of 0.43-0.44 and 2.73-2.92, respectively, appear to have a combination of the shape, AR, and flexibility that optimizes power economy.

Economic Uncertainty: A Geometric Indicator of Discrepancy Among Experts’ Expectations

In this study we present a geometric approach to proxy economic uncertainty. We design a positional indicator of disagreement among survey-based agents’ expectations about the state of the economy. Previous dispersion-based uncertainty indicators derived from business and consumer surveys exclusively make use of the two extreme pieces of information: the percentage of respondents expecting a variable to rise and to fall. With the aim of also incorporating the information coming from the share of respondents expecting a variable to remain constant, we propose a geometrical framework and use a barycentric coordinate system to generate a measure of disagreement, referred to as a discrepancy indicator. We assess its performance both empirically and experimentally by comparing it to the standard deviation of the share of positive and negative responses. When applied in sixteen European countries, we find that both time-varying metrics co-evolve in most countries for expectations about the country’s overall economic situation in the present, but not in the future. Additionally, we obtain their simulated sampling distributions and we find that the proposed indicator gravitates uniformly towards the three vertices of the simplex representing the three answering categories, as opposed to the standard deviation, which tends to overestimate the level of uncertainty as a result of ignoring the no-change responses. Consequently, we find evidence that the information coming from agents expecting a variable to remain constant has an effect on the measurement of disagreement.

Visual-based analysis of classification measures and their properties for class imbalanced problems

With a plethora of available classification performance measures, choosing the right metric for the right task requires careful thought. To make this decision in an informed manner, one should study and compare general properties of candidate measures. However, analysing measures with respect to complete ranges of their domain values is a difficult and challenging task. In this study, we attempt to support such analyses with a specialized visualisation technique, which operates in a barycentric coordinate system using a 3D tetrahedron. Additionally, we adapt this technique to the context of imbalanced data and put forward a set of measure properties, which should be taken into account when examining a classification performance measure. As a result, we compare 22 popular measures and show important differences in their behaviour. Moreover, for parametric measures such as the Fβ and IBAα(G-mean), we analytically derive parameter thresholds that pinpoint the changes in measure properties. Finally, we provide an online visualisation tool that can aid the analysis of measure variability throughout their entire domains.

Proxying Economic Uncertainty: A Geometric Approach to Measure Disagreement Among Qualitative Expectations

In this study we present a geometric approach to proxy economic uncertainty. We design a positional indicator of disagreement among survey-based agents' expectations about the state of the economy. Previous dispersion-based uncertainty indicators derived from business and consumer surveys exclusively make of the information coming from the respondents expecting a variable to rise or to fall. With the aim of also incorporating the information coming from the share of respondents expecting a variable to remain constant, we propose a geometrical framework and use a barycentric coordinate system to generate a measure of disagreement, referred to as a discrepancy indicator. We assess its performance, both empirically and experimentally, by comparing it to the standard deviation of the share of positive and negative responses, which has been used by Bachman et al. (2013) as a proxy for economic uncertainty. When applied in sixteen European countries, we find that both time-varying metrics co-evolve in most countries for expectations about the country's overall economic situation in the present, but not in the future. Additionally, we obtain their simulated sampling distributions and we find that the proposed indicator gravitates uniformly towards the three vertices of the simplex representing the three answering categories, as opposed to the standard deviation, which tends to overestimate the level of uncertainty as a result of ignoring the no-change responses. Consequently, we find evidence that the information coming from agents expecting a variable to remain constant has an effect on the measurement of disagreement.

Locally optimal control of space tug motion between the libration points of the Earth-Moon system

The article deals with the locally optimal control of space tug transfers between the libration points of the Earth-Moon system using low-thrust engines. The control program was obtained using the Fedorenko method to find derivatives and a variable-step gradient method to optimize control laws. All flights were considered within the restricted three-body problem. The mathematical model of flight was described in a barycentric coordinate system. The total flight time was assumed as the optimization criterion. The disturbances from the Earth, the Moon and the Sun were taken into account. The influence of shadow zones created by the Earth and the Moon was examined. Laws of control, trajectories and values of the total flight time were obtained as a result of optimization.

Robust feature matching for geospatial images via an affine‐invariant coordinate system

AbstractFeature matching is a crucial stage for many photogrammetric and remote sensing applications. This paper proposes a novel and robust feature‐matching method based on a normalised barycentric coordinate system (NBCS), which is superior to a Cartesian system for this task. A scale‐invariant feature transform (SIFT) is performed to provide initial matches containing both correct matches (inliers) and false matches (outliers), with a focus on model estimation from contaminated observations (matches with outliers). An affine‐invariant coordinate system called NBCS is defined based on ratios of areas. The two feature points of a correct match have the same coordinates under NBCS while false correspondences do not. This principle is adapted into a hypothesise‐and‐verify framework. The proposed method is robust, efficient and effective. Extensive experiments on real geospatial image pairs show that it significantly outperforms six other state‐of‐the‐art approaches. The source code and datasets used in this paper have been made public.

Vector Barycentric Method in Computational Electrodynamics

The article presents a vector barycentric method for solving the internal problem of electrodynamics, i.e. solving Maxwell's equations or their respective wave equations in a bounded computational domain with prescribed boundary conditions. The developed method refers to the method of direct solution of the boundary value problems of mathematical physics, the basis for the formation of which are the results obtained by V. Ritz, I.G. Bubnov and B.G. Galerkin. The basic idea of the method lies in the synthesis of a procedure of the vector potential approximation, done by the polynomials of the Lagrange type. The approximating polynomial is formed in the barycentric coordinate system for the entire region of analysis as a whole without partitioning into elementary sub-areas. It is assumed that the scope of analysis is a region with a piecewise linear boundary, and the dimension of the barycentric coordinate system is determined by the number of vertices of the analyzed region. The vector barycentric method is implemented both in the frequency and time domains. The solution to the problem of controlling the electromagnetic field in the approximation of the vector barycentric method is considered.

The triangulated affine transformation parameters and barycentric coordinates of Turkish Permanent GPS Network

There are several computational approaches and methods to produce the complete solutions of geometry problems and they give sufficient computational strength for geodetic networks. Every approach and method has their advantages and drawbacks. Barycentric coordinates, also called areal coordinates, provide a new knock approach for geometry problems of such networks. This coordinate system offers triangular finite elements which are quite different from the usual Cartesian coordinate system. Actually, barycentric coordinates are a natural system of coordinates for the affine geometry. The modified approach for the affine transformation and barycentric coordinate system has been introduced and discussed, respectively in the study whether the affine transformation improves the results deteriorated by geometry of a GPS network. For this aim, general affine transformation of six-parameters between Eurasia Fixed Reference Frame (EUREF) and International Terrestrial Reference Frame (ITRF) for Turkish Permanent Global Positioning System Network (TPGN) has been computed. The affine transformation results show that the effects of the first and second translation parameters are in negative and positive directions, respectively. The third and sixth parameters are almost constant, on the other hand the fourth and fifth have a very small positive effect. Additionally, three triangular networks of GPS stations, at different locations in Turkey as a numerical example of barycentric coordinates, have also been presented. In the numerical example, the coordinate values of EUREF and ITRF have some differences after the decimal point, but the barycentric coordinates have very small amount of differences (sixth digit after decimal) which proves that the modified approach is meaningful.

Can interestingness measures be usefully visualized?

Abstract The paper presents visualization techniques for interestingness measures. The process of measure visualization provides useful insights into different domain areas of the visualized measures and thus effectively assists their comprehension and selection for different knowledge discovery tasks. Assuming a common domain form of the visualized measures, a set of contingency tables, which consists of all possible tables having the same total number of observations, is constructed. These originally four-dimensional data may be effectively represented in three dimensions using a tetrahedron-based barycentric coordinate system. At the same time, an additional, scalar function of the data (referred to as the operational function, e.g., any interestingness measure) may be rendered using colour. Throughout the paper a particular group of interestingness measures, known as confirmation measures, is used to demonstrate the capabilities of the visualization techniques. They cover a wide spectrum of possibilities, ranging from the determination of specific values (extremes, zeros, etc.) of a single measure, to the localization of pre-defined regions of interest, e.g., such domain areas for which two or more measures do not differ at all or differ the most.

TU‐F‐CAMPUS‐J‐02: Mooney‐Rivlin Biomechanical Modeling of Lung with Inhomogeneous Material Property

Purpose:The Mooney‐Rivlin material with hyperelastic strain energy has been proposed for realistic biomechanical modeling of lung. In this study, the lung is modeled as an inhomogeneous Mooney‐Rivlin material with the incompressibility factors being optimized to improve the tumor center of mass (TCM) motion simulation accuracy during respiration.Method:ITK‐SNAP was used to segment lungs of eight lung cancer patients from the 4D‐CT images and tetrahedral volume meshes of the lungs in phase 50% were created by using adaptive mesh generation toolkit. The interphase deformation vector fields (DVFs) are calculated by demons deformable registration algorithm and the barycentric coordinate system of tetrahedral elements is obtained from the resulted DVFs. Mooney‐Rivlin hyperelastic material is used to model the lung volume. Each element is considered unique where the incompressibility factor (k‐factor) for each element is assumed to be proportional to the magnitude of normalized DVF. The incompressibility factor for each element was optimized by minimizing the tumor center of mass motion simulation error.Results:If lung is considered as a homogenous material in Mooney‐Rivlin modeling, the average TCM motion simulation error is 2.26 mm. By considering inhomogeneous properties of lung in the proposed strategy, the average TCM motion simulation error is reduced to 2.04 mm.Conclusions:We proposed a method for assigning the inhomogeneous biomechanical material in the Mooney‐Rivlin model of lung based on the lung regional deformation vector fields. Inhomogeneous material property of lung improves the simulation accuracy.