Geometric Mapping Research Articles

Mapping techniques for collocation method of time-fractional convection–diffusion equations in domains with cracks

This paper proposes numerical methods that effectively deal with time-fractional convection–diffusion equations containing crack singularities. To deal with singularities, we design the geometrical mapping whose push-forward from the parameter space into the physical space generates point singularity functions based on the parametrization of the circular arc and NURBS (non-uniform rational B-spline). We adopt the collocation method with B-spline basis functions to approximate the solution in the spatial direction and enrich the approximation space by k-refinements in IGA (Isogeometric Analysis). For the discretization along the temporal direction, we employ the explicit Predictor-Corrector (PC) scheme that has the order 2−ν and 3−ν of the truncation error for the linear and quadratic interpolation, respectively. Taking advantage of the NURBS geometrical mapping, we demonstrate the performance of the proposed methods applying to time-fractional convection–diffusion equations with nonlinear terms on curved domains containing crack singularities.

BF3D: Bi-directional fusion 3D detector with semantic sampling and geometric mapping

3D object detection is a key task in environmental awareness, which plays a vital role in autonomous driving safety. Lidars and cameras are widely used sensors that provide complementary information for accurate 3D detection. However, due to the domain difference between the two modalities, it is challenging to leverage their respective strengths and fuse them perfectly. In this paper, an end-to-end detector termed BF3D is proposed, which integrates with the semantic sampling module, geometric point-image mapping module, and bi-directional attention fusion module. Specifically, the semantic sampling module incorporates a novel downsampling strategy to preserve more foreground points and pixels. Additionally, the geometric point-image mapping module is developed to find geometric correlation pixels of the point and take advantage of the high density of image features. We also introduce a bi-directional attention fusion module to combine useful information from the two modalities by attention mechanism. Extensive experiments demonstrate that BF3D outperforms both single- and multi-modal 3D detectors. Codes are available at: https://github.com/hustzyj/BF3D.

Interday Reliability of Upper-limb Geometric MyoPassivity Map for Physical Human-Robot Interaction.

The value of intrinsic energetic behavior of human biomechanics has recently been recognized and exploited in physical human-robot interaction (pHRI). The authors have recently proposed the concept of "Biomechanical Excess of Passivity," based on nonlinear control theory, to construct a user-specific energetic map. The map would assess the behavior of the upper-limb in absorbing the kinesthetic energy when interacting with robots. Integrating such knowledge into the design of pHRI stabilizers can reduce the conservatism of the control by unleashing hidden energy reservoirs indicating a less conservative margin of stability. The outcome would enhance the system's performance, such as rendering kinesthetic transparency of (tele)haptics systems. However, current methods require an offline data-driven identification procedure prior to each operation to estimate the energetic map of human biomechanics. This can be time-consuming and challenge users susceptible to fatigue. In this study, for the first time, we investigate the interday reliability of upper-limb passivity maps in a sample of five healthy subjects. Our statistical analyses indicate that the identified passivity map is highly reliable in estimating the expected energetic behavior based on Intraclass correlation coefficient analysis (conducted on different days and with various interactions). The results illustrate that a one-shot estimate is a reliable measure to be used repeatedly in biomechanics-aware pHRI stabilization, enhancing practicality in real-life scenarios.

Enhancing Occlusion Handling in Real-Time Tracking Systems through Geometric Mapping and 3D Reconstruction Validation

Object detection is a classic research problem in the area of Computer Vision. Many smart world applications, like, video surveillance or autonomous navigation systems require a high accuracy in pose detection of objects. One of the main challenges in Object detection is the problem of detecting occluded objects and its respective 3D reconstruction. The focus of this paper is inter-object occlusion where two or more objects being tracked occlude each other. A novel algorithm has been proposed for handling object occlusion by using the technique of geometric matching and its 3D projection obtained. The developed algorithm has been tested using sample data and the results are presented.

Geometrically transformed spectral methods to solve partial differential equations in circular geometries, application for multi-phase flow

Circular geometries are ubiquitously encountered in science and technology, and the polar coordinate provides the natural way to analyze them; However, its application is limited to symmetric cases, and it cannot be applied to segments that are formed in multiphase flow problems in pipes. To address that, spectral discretization of circular geometries via orthogonal collocation technique is developed using geometrical mapping. Two analytical mappings between the circle and square geometries, namely, elliptical and horizontally squelched mappings, are employed. Accordingly, numerical algorithms are developed for solving PDEs in circular geometries with different boundary conditions for both steady state and transient problems. Various implementation issues are thoroughly discussed, including vectorization and strategies to avoid solving the differential–algebraic system of equations. Moreover, several case studies for symmetric and asymmetric Poisson equations with different boundary conditions are performed to evaluate several aspects of these techniques, such as error properties, condition number, and computational time. For both steady state and transient solvers, it was revealed that the computation time scales quadratically with respect to the grid size for both mapping and polar discretization techniques. However, due to the presence of the second mixed derivative, mapping techniques are more computationally costly. Finally, the squelched mapping was successfully employed to discretize the two, and three-phase gravity flows in sloped pipes.

Portable confocal-controlled LIBS microscopy for topographic and LIBS mapping

The on-site detection of the geometrical topography and composition distribution of unknown minerals is a key bottleneck in the field of deep space exploration and geological material analysis. A novel portable confocal-controlled laser-induced breakdown spectroscopy (PCCLIBS) scanning three-dimensional (3D) imaging method with high spatial resolution and highly integrated structure was proposed to solve this problem. The method utilizes a confocal-controlled LIBS technique to achieve precise focusing of the sample, significantly improving the spatial resolution of the LIBS mapping system. In addition, we employ a dual two-dimensional (2D) MEMS mirror scanning technique to achieve a miniaturized design of LIBS mapping system. These two technologies are combined to achieve miniaturization, high spatial resolution, and 3D LIBS microelement imaging. The method had an axial focusing ability of ∼ 60 nm and a lateral resolution of geometrical and elemental maps of around 700 nm and 10 μm, respectively, using an objective lens with a numerical aperture (NA) of 0.4. Finally, we performed the LIBS and topographic imaging of carbonaceous chondrite (Northwest Africa 13323) using the PCCLIBS system to realize multi-information of the micro-regional chemical, element, and topographic information imaging in the sample. These results demonstrate that this method is an effective approach for the elemental characterization of LIBS in samples with complex morphologies, such as in planet landing exploration and geological exploration.

Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics

We present a mass lumping approach based on an isogeometric Petrov–Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is “close” to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We also address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

A new method for researching and constructing spherical bicentric polygons based on geometric mapping

A bicentric polygon is a special planar polygon that has both a circumcircle and an incircle. Many interesting properties of a planar bicentric polygon have been discovered. However, searching the geometric properties of a spherical bicentric polygon with elementary geometric method is very difficult. In this paper, we propose a special mapping for a pair of circles between a plane and a sphere. This mapping provides us a new method to research the spherical bicentric polygons. With this mapping, we can translate the Fuss relation, which describes the relation of the circumcircle and incircle of a bicentric polygon, from a planar bicentric polygon to a spherical polygon. Based on the mapping, we can derive the geometric properties of spherical bicentric polygons from the planar bicentric polygons, and construct a series of spherical bicentric n-polygons from a planar bicentric n-polygon.

Navigating to objects in the real world.

Semantic navigation is necessary to deploy mobile robots in uncontrolled environments such as homes or hospitals. Many learning-based approaches have been proposed in response to the lack of semantic understanding of the classical pipeline for spatial navigation, which builds a geometric map using depth sensors and plans to reach point goals. Broadly, end-to-end learning approaches reactively map sensor inputs to actions with deep neural networks, whereas modular learning approaches enrich the classical pipeline with learning-based semantic sensing and exploration. However, learned visual navigation policies have predominantly been evaluated in sim, with little known about what works on a robot. We present a large-scale empirical study of semantic visual navigation methods comparing representative methods with classical, modular, and end-to-end learning approaches across six homes with no prior experience, maps, or instrumentation. We found that modular learning works well in the real world, attaining a 90% success rate. In contrast, end-to-end learning does not, dropping from 77% sim to a 23% real-world success rate because of a large image domain gap between sim and reality. For practitioners, we show that modular learning is a reliable approach to navigate to objects: Modularity and abstraction in policy design enable sim-to-real transfer. For researchers, we identify two key issues that prevent today's simulators from being reliable evaluation benchmarks-a large sim-to-real gap in images and a disconnect between sim and real-world error modes-and propose concrete steps forward.

Comparison of Land Cover Categorical Data Stored in OSM and Authoritative Topographic Data

This study aims at a comparative analysis of quantitative data, namely, OSM and BDOT10k. Analyses were conducted in a 1 km2 hexagonal grid, in seven test counties located in different regions of Poland, differing in the degree of urbanization, land cover and natural environment. It is assumed that the authors’ consolidated regional classification of the Compound Correspondence Index CCIRn is attributed to the geometric mapping unit based on TOPSIS values, and their statistical measure of dispersion enables the comparison of datasets for individual geographically disjointed areas according to uniform criteria, e.g., the number of topographic features stored in analyzed datasets, both polygonal (buildings, forests, surface water) and linear (roads, watercourses, railroads). The final results of the regional assessment outperform the local classification giving a higher level of data compliance. Overestimation of regional concordance ranges from 9 to 20% of the county area, with an average of 3% reduction in the area where the two datasets (BDOT10k and OSM) have comparable information ranges. Areas of medium and high nonconformity are decreased by an average of 2.4%.

Electromagnetic Modeling Using Adaptive Grids – Error Estimation and Geometry Representation

AbstractThis review paper addresses the development of numerical modeling of electromagnetic fields in geophysics with a focus on recent finite element simulation. It discusses ways of estimating errors of our solutions for a perfectly matched modeling domain and the problems that arise from its insufficient representation. After a brief outline of early methods and modeling approaches, the paper mainly discusses the capabilities of the finite element method formulated on unstructured grids and the advantages of local h-refinement allowing for both a flexible and largely accurate representation of the geometries of the multi-scale geomaterial and an accurate evaluation of the underlying functions representing the physical fields. In summary, the accuracy of the solution depends on the geometric mapping, the choice of the mathematical model, and the spatial discretization. Although the available error estimators do not necessarily provide reliable error bounds for our complex geomodels, they are still useful to guide grid refinement. Therefore, an overview of the most common a posteriori error estimators is given. It will be shown that the sensitivity is the most important function in both guiding the geometric mapping and the local refinement.

Geometrically compatible integrated design method for conformal rotor and nacelle of distributed propulsion tilt-wing UAV

The inner rotors of distributed propulsion tilt-wing Unmanned Aerial Vehicles (UAVs) are often folded in the cruising state and deployed in vertical take-off and landing to cope with the huge difference in thrust requirements. However, the blades of the conventional rotor have poor conformality with the nacelle profile, which will greatly increase the drag of the UAV after folding. This paper proposes an integrated method for the design of rotor and nacelle considering geometric compatibility to reduce the drag of the folded rotor and nacelle, so as to further improve the aerodynamic efficiency in cruise while ensuring the rotor efficiency in the vertical flight mode. A geometric mapping model based on nacelle design parameters and rotor design parameters is established, and a parametric model and aerodynamic optimization model of the outer arc airfoil family are developed. In addition, a rotor performance analysis model and a neural network response surface model for nacelle drag prediction that meet the requirements of confidence level are established. Based on the oblique inflow blade element momentum theory method, numerical simulation method, and genetic algorithm, an integrated optimization framework of the design of the conformal rotor and nacelle is built. Then, a geometrically compatible integrated optimization for the rotor and nacelle is carried out with the objective of maximizing energy efficiency in the full mission profile. Finally, a conformal rotor and nacelle design solution is obtained, which satisfies geometric compatibility and thrust constraints while providing high thrust efficiency and low cruising drag. A comparison of the results of the integrated design and the conventional rotor optimization design shows that the drag of the conventional rotor is 3.45 times that of the conformal integrated design in the cruising state, which proves the effectiveness and necessity of the proposed method.

Reducing Redundancy in Maps without Lowering Accuracy: A Geometric Feature Fusion Approach for Simultaneous Localization and Mapping

Geometric map features, such as line segments and planes, are receiving increasing attention due to their advantages in simultaneous localization and mapping applications. However, large structures in different environments are very likely to appear repeatedly in several consecutive time steps, resulting in redundant features in the final map. These redundant features should be properly fused, in order to avoid ambiguity and reduce the computation load. In this paper, three criteria are proposed to evaluate the closeness between any two features extracted at two different times, in terms of their included angle, feature circle overlapping and relative distance. These criteria determine whether any two features should be fused in the mapping process. Using the three criteria, all features in the global map are categorized into different clusters with distinct labels, and a fused feature is then generated for each cluster by means of least squares fitting. Two competing methods are employed for comparative verification. The comparison results indicate that using the commonly used KITTI dataset and the commercial software PreScan, the proposed feature fusion method outperforms the competing methods in terms of conciseness and accuracy.

Co-Modulation of Spin Angular Momentum and High-Order Orbital Angular Momentum Based on Anisotropic Holographic Metasurfaces

In this communication, a new method for modulating spin angular momentum (SAM) and orbital angular momentum (OAM) based on anisotropic holographic metasurfaces is proposed, which can realize the co-modulation of SAM, high order OAM, and beam’s direction simultaneously. The tensor unit adopts an ellipse patch with three adjustable parameters without slotting, and a design and analysis method suitable for this kind of tensor unit is proposed based on the two-dimensional effective impedance database for geometrical size mapping. Based on the tensor unit and proposed methods, the vortex beams with a single SAM state, a single high order OAM mode, and different directions are generated at 22 GHz. Then, the deflected vortex beams with the same SAM state and mixed OAM modes, or the deflected vortex beams with different SAM states and OAM modes are realized using a single feed and a single metasurface integrated for the first time. Simulation and experimental measurement results show that the OAM mode purity can reach more than 80% of the theoretical best value in the main SAM state. The proposed design methods and the device can achieve arbitrary modulation of SAM and OAM, providing a solution for combining and applying the two to multiplexing communications.

Geometric Correction of BaySpec OCI-F Pushbroom Hyperspectral Images via the Sequential Geometric Mappings among Simultaneous Video Frames Estimated by Least Square Matching

ضرورت دسترسی به کاربردهای وسیع تصاویر ابرطیفی سبب توسعة سیستم‌های تصویربرداری نوآورانه و اقتصادی در ثبت این تصاویر شده است. به‌منظور استفاده از این تصاویر، لازم است ارتباط هندسی دقیقی میان آنها و فضای زمین برقرار شود و این فرایند نیازمند نقاط کنترلی بسیاری است. این نکته ضرورت توسعة راهکارهای اصلاح هندسی منطبق با ساختار هریک از این دوربین‌ها را بارز می‌کند. سنجندة (nm 400-1000) BaySpec OCI-F یکی از سیستم‌های نوآورانه‌ای است که تصاویر ابرطیفی را با هندسة تصویربرداری پوش‌بروم دریافت می‌کند. این سنجنده، علاوه‌بر یک سنسور پوش‌بروم، از یک سنسور فریم نیز بهره می‌برد که هم‌زمان با سنسور پوش‌بروم و با رزولوشن مکانی زمانی مشابه، تصویر را دریافت می‌کند. در این مقاله، روشی برای اصلاح هندسی تصاویر پوش‌برومِ این سنجنده بیان شده است. در بخش اول این روش، با توجه به ساختار تصویربرداری دوربین، ارتباط هندسی میان آرایة خطی و سنسور فریم در قالب پارامترهای کالیبراسیونی مشخص می‌شود. در ادامه، به‌کمک برآورد ارتباط هندسی میان تصاویر فریم متوالی، پیکسل‌های تصویر پوش‌بروم در کنار یکدیگر چیده و تصویر اصلاح‌شده تولید می‌شود. در این روش، ارتباط هندسی میان هر جفت فریم متوالی به‌طور مستقیم، ازطریق تناظریابی کمترین مربعات، محاسبه می‌شود. نتایج به‌دست‌آمده نشان می‌دهد که این روش، به‌طور متوسط، 2/62% از اعوجاجات هندسی تصویر خام را کاهش داده است. این کاهش سبب شده است متوسط دقت مدل‌های درون‌یاب عمومی سادة دوبعدی و سه‌بعدی بین فضای تصویر و زمین، به‌ترتیب، 9/39% و 1/34% افزایش یابد.

Geometric parameterization of unit cell configurations for reduced basis approximation

To facilitate rapid yet accurate multiscale homogenization, we formulate geometric mapping schemes that effectively handles the fiber radius changes of a unit cell to dispense with mesh regeneration. To further expedite unit cell analysis involving fiber radius variation, we integrate the formulated geometric transformations into reduced basis analysis using an empirical interpolation method. For demonstration, we examine the proposed geometric parameterization strategies with the linear elastic analyses of square and hexagonal unit cells under tension and shear loads, respectively. Overall, reduced basis analyses combined with the devised geometric mappings are found more efficient than the corresponding finite element analyses while producing insignificant errors.

Stability and bifurcations of complex vibrations in a nonlinear brush-seal rotor system.

A brush seal has the advantages of adapting to different vibration conditions and increasing the stability of the nonlinear rotor system. In this research, the stability and bifurcations of complex vibrations in a brush-seal rotor system are studied. An analytical seal force model is obtained through the beam theory and mutual coupling dynamics of the bristles and the rotor. The interaction between the bristles and the rotor is clearly depicted by a geometric map. Periodic and chaotic vibrations as well as the corresponding amplitude-frequency characteristics are first predicted by a numerical bifurcation diagram and 3D waterfalls. Discrete dynamic eigenvalue analysis is adopted for a detailed investigation of the stability and bifurcations of nonlinear vibrations. Jumping, quasi-periodic, and half-frequency vibrations are warned during the speeding up and down process. Four separate nonlinear vibration evolving routes are discovered. Two period-doubling bifurcation trees evolving to chaos are illustrated for the observation of global and independent periodic vibrations. Nonlinear vibration illustrations are presented through displacement orbits as well as harmonic amplitudes and phases. Chaotic vibration and unstable semi-analytical vibration solutions are compared. The obtained results and analysis methods provide new perspectives on nonlinear vibrations in the brush-seal rotor system.

Signal Recovery Performance Analysis in Wireless Sensing With Rectangular-Type Analog Joint Source-Channel Coding

The signal recovery performance of the rectangular-type Analog Joint Source-Channel Coding (AJSCC) is analyzed in this work for high and medium/low Signal-to-Noise Ratio (SNR) scenarios in a wireless sensing system with the Doppler channel. The analytical formulations of the Mean Square Error (MSE) performance are derived based on the geometry of the rectangular-type AJSCC on analog sensing and system with Doppler effects. The comprehensive listing of all cases in the three-dimensional geometric signal mapping curve is provided to obtain the theoretical formulations for the medium/low SNR scenario. Evaluation results indicate that there are optimal parameters in the rectangular-type AJSCC to minimize the signal recovery MSE for analog sensing and system with Doppler effects in both high and medium/low SNR scenarios. The analysis of the work provides insights on the optimization of the rectangular-type AJSCC in practical wireless sensing systems.

A benchmark analysis of data‐driven and geometric approaches for robot ego‐motion estimation

AbstractIn the last decades, ego‐motion estimation or visual odometry (VO) has received a considerable amount of attention from the robotic research community, mainly due to its central importance in achieving robust localization and, as a consequence, autonomy. Different solutions have been explored, leading to a wide variety of approaches, mostly grounded on geometric methodologies and, more recently, on data‐driven paradigms. To guide researchers and practitioners in choosing the best VO method, different benchmark studies have been published. However, the majority of them compare only a small subset of the most popular approaches and, usually, on specific data sets or configurations. In contrast, in this work, we aim to provide a complete and thorough study of the most popular and best‐performing geometric and data‐driven solutions for VO. In our investigation, we considered several scenarios and environments, comparing the estimation accuracies and the role of the hyper‐parameters of the approaches selected, and analyzing the computational resources they require. Experiments and tests are performed on different data sets (both publicly available and self‐collected) and two different computational boards. The experimental results show pros and cons of the tested approaches under different perspectives. The geometric simultaneous localization and mapping methods are confirmed to be the best performing, while data‐driven approaches show robustness with respect to nonideal conditions present in more challenging scenarios.

Second-Order Nédélec Curl-Conforming Hexahedral Element for Computational Electromagnetics

We follow a systematic approach to obtain mixed-order curl-conforming basis functions for the hexahedron that are compatible with basis functions for tetrahedra and triangular prisms previously published. The approach is mathematically sound since we obtain the functions as the dual basis with respect to properly discretized Nédélec degrees of freedom. Well-conditioned bases without the need for added orthogonalization procedures are obtained. We provide simple closed-form expressions for second-order basis functions in a reference hexahedron in terms of integer coefficients and monomials. The expressions are ready to use as long as the appropriate geometric mappings are made. We apply the Method of Manufactured Solutions (MMS) to a finite-element double curl vector wave formulation for verification purposes; specifically, we conduct a study of the non-symmetrical structure of the corresponding tensor product finite-element space. We also solve generalized eigenvalue problems for well-known cavities. We provide the open-source code for generating the coefficients, evaluating the basis functions, and computing the finite-element matrices involved in some of the numerical solutions shown in the article.