Non-regular Grid Research Articles

A TWO-POINT COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF ONE-DIMENSIONAL HYPERSINGULAR INTEGRAL EQUATIONS ON NONUNIFORM PARTITIONS

A quadrature formula has been constructed for calculating the hypersingular integral over a segment, which uses the ends of the segment partition intervals as nodes of piecewise constant interpolation of the integral density, as well as specially selected collocation points. A distinctive feature of the proposed quadrature formula is the ability to calculate the integral of functions that suffer a finite number of discontinuities of the first kind on the integration interval. On the basis of quadrature formula constructed, a numerical scheme for solving the characteristic hypersingular integral equation on non-regular grid is developed. Estimate of the rate of convergence of approximate solutions to exact ones is proved in the class of piecewise Ho¨lder functions.

An automatic eddy detection algorithm based on quadrant angle of velocity vector and its application

ABSTRACT Eddies assume a pivotal role in both the oceanic heat cycle and marine dynamic processes. The development of real-time satellite observations, coupled with advancements in computer intelligence, has propelled automatic eddy detection algorithms to the forefront of ocean remote sensing research. Nevertheless, target omissions and accuracy degradations are inevitable for multiple detection algorithms due to strong morphological variations in ocean eddies. This paper proposes an automatic detection algorithm based on the quadrant angle of the velocity vector in the flow field. Firstly, a rectangular search box is established, and the corresponding quadrant angles at the four vertices are calculated. Secondly, the centres and types of eddies are determined according to the cumulative sum and variety rules of quadrant angles. Then the outermost boundary is identified by regularity of quadrant angles in eight directions expanding outward from the centre of the eddy using the stream function equation. The new algorithm is assessed and verified using geostrophic flow data derived from the CMEMS standard gridded sea level anomaly product. Furthermore, its detection capabilities are demonstrated through a comparative analysis with several other algorithms. All methods exhibit consistent efficacy in detecting the majority of eddies with similar spatial distributions. In cases where the new algorithm identifies a specific eddy not detected by the FF15 and ND10 algorithms, sea surface temperature data is employed for verification. The sea surface temperature distribution map, along with the obtained results, illustrates the superiority of the new algorithm and its adaptability to products with varying resolutions. Additionally, the results undergo verification through a manual detection method, revealing that the new algorithm achieves SDR of 91.73% and an EDR of 3.54%. This percentage significantly surpasses the lower acceptable limit of 80% for the SDR parameter. The new algorithm is applied to study eddies with lifetimes exceeding ten days, spanning from March 2022 to February 2023, within the East China Sea and the South China Sea. The radius of SCS eddies is generally larger than that of ECS eddies, both cyclonic and anticyclonic. This further confirms the significant influence of regional ocean dynamics on the structure of eddies in different regions. Finally, the algorithm is applied to SWOT data to test the adaptability of the algorithm to non-regular grid data. The results show that the algorithm can effectively analyse SWOT data within 60° latitudes.

On the approximation of derivative values using a WENO algorithm with progressive order of accuracy close to discontinuities

In this article, we introduce a new WENO algorithm that aims to calculate an approximation to derivative values of a function in a non-regular grid. We adapt the ideas presented in [Amat et al., SIAM J. Numer. Anal. (2020)] to design the nonlinear weights in a manner such that the order of accuracy is maximum in the intervals close to the discontinuities. Some proofs, remarks on the choice of the stencils and explicit formulas for the weights and smoothness indicators are given. We also present some numerical experiments to confirm the theoretical results.

Grid-free least-squares method for pressure evaluation from LPT data

Lagrangian particle tracking Shake-the-box (STB) method (Schanz et al., 2016) acquires the 3D positions of tracer particles from the temporal sequences of their 2D projection images even for rather high seeding densities. Approximation of tracks by analytical functions (Gesemann, 2015) provides an accurate evaluation of tracers’ local velocity and acceleration. This data, which is obtained on non-regular grid, can be used to estimate local pressure fluctuations based on the Navier–Stokes equation. The present paper describes a grid-free least-squares method for the gradients and pressure evaluation based on irregularly scattered LPT data with random noise minimization.

Automatic calibration and simplified decoding pipeline for plenoptic camera

Plenoptic cameras attract extensive attention these days for their unique information acquisition and postcapture processing capability. In prior work, the decoding pipeline of plenoptic cameras mainly consisted of calibration, aligning, slicing, and resampling. Besides, slicing and resampling are performed on a generated grid of projected centers. Such methods consist of a series of processing steps, and the errors from different steps are accumulated. We propose a simplified decoding pipeline for plenoptic cameras. We first propose a coarse-to-fine strategy to calibrate the microlens array accurately and automatically. Then, unlike prior work, we perform slicing and resampling using a nonregular grid of projected centers. Such a strategy avoids using aligning and excludes the dark regions among different microlenses from decoding. Experiments on published datasets and real-world scenes demonstrate the validation of the proposed method.

Irregular Grid Interpolation using Radial Basis Function for Large Cylindrical Volume

Irregular grid interpolation is one of the numerical functions that often used to approximate value on an arbitrary location in the area closed by non-regular grid pivot points. In this paper, we propose method for achieving efficient computation time of radial basis function-based non-regular grid interpolation on a cylindrical coordinate. Our method consist of two stages. The first stage is the computation of weights from solving linear RBF systems constructed by known pivot points. We divide the volume into many subvolumes. At second stages, interpolation on an arbitrary point could be done using weights calculated on the first stage. At first, we find the nearest point with the query point by structuring pivot points in a K-D tree structure. After that, using the closest pivot point, we could compute the interpolated value with RBF functions. We present the performance of our method based on computation time on two stages and its precision by calculating the mean square error between the interpolated values and analytic functions. Based on the performance evaluation, our method is acceptable.

High-Resolution Representation for Mobile Mapping Data in Curved Regular Grid Model.

As mobile mapping systems become a mature technology, there are many applications for the process of the measured data. One interesting application is the use of driving simulators that can be used to analyze the data of tire vibration or vehicle simulations. In previous research, we presented our proposed method that can create a precise three-dimensional point cloud model of road surface regions and trajectory points. Our data sets were obtained by a vehicle-mounted mobile mapping system (MMS). The collected data were converted into point cloud data and color images. In this paper, we utilize the previous results as input data and present a solution that can generate an elevation grid for building an OpenCRG model. The OpenCRG project was originally developed to describe road surface elevation data, and also defined an open file format. As it can be difficult to generate a regular grid from point cloud directly, the road surface is first divided into straight lines, circular arcs, and and clothoids. Secondly, a non-regular grid which contains the elevation of road surface points is created for each road surface segment. Then, a regular grid is generated by accurately interpolating the elevation values from the non-regular grid. Finally, the curved regular grid (CRG) model files are created based on the above procedures, and can be visualized by OpenCRG tools. The experimental results on real-world data show that the proposed approach provided a very-high-resolution road surface elevation model.

Direct meshless local Petrov–Galerkin (DMLPG) method for time-fractional fourth-order reaction–diffusion problem on complex domains

A new numerical scheme has been developed based on the fast and efficient meshless local weak form i.e direct meshless local Petrov–Galerkin (DMLPG) method for solving the fractional fourth-order partial differential equation on computational domains with complex shape. The fractional derivative is the Riemann–Liouville fractional derivative. At first, a finite difference scheme with the second-order accuracy has been employed to discrete the time variable. Then, the DMLPG technique is employed to achieve a full-discrete scheme. The time-discrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the L2 space. Also, some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and non-regular grid points.

Efficient Lazy Theta* Path Planning over a Sparse Grid to Explore Large 3D Volumes with a Multirotor UAV.

Exploring large, unknown, and unstructured environments is challenging for Unmanned Aerial Vehicles (UAVs), but they are valuable tools to inspect large structures safely and efficiently. The Lazy Theta* path-planning algorithm is revisited and adapted to generate paths fast enough to be used in real time and outdoors in large 3D scenarios. In real unknown scenarios, a given minimum safety distance to the nearest obstacle or unknown space should be observed, increasing the associated obstacle detection queries, and creating a bottleneck in the path-planning algorithm. We have reduced the dimension of the problem by considering geometrical properties to speed up these computations. On the other hand, we have also applied a non-regular grid representation of the world to increase the performance of the path-planning algorithm. In particular, a sparse resolution grid in the form of an octree is used, organizing the measurements spatially, merging voxels when they are of the same state. Additionally, the number of neighbors is trimmed to match the sparse tree to reduce the number of obstacle detection queries. The development methodology adopted was Test-Driven Development (TDD) and the outcome was evaluated in real outdoors flights with a multirotor UAV. In the results, the performance shows over 90 percent decrease in overall path generation computation time. Furthermore, our approach scales well with the safety distance increases.

Modeling nanoindentation using the Material Point Method

Abstract

Modeling of imaging fiber bundles and adapted signal processing for fringe projection

Abstract Fringe projection profilometry is an established technique for capturing three-dimensional (3-D)-geometry data with high-point densities in short time. By combining fringe projection with endoscopy techniques, it is possible to perform inline inspection of industrial manufacturing processes. A new fringe projection system is presented, which uses flexible image fiber bundles to achieve versatile positioning of a compact sensor head. When measuring specimens with highly varying reflectivity, such as technical surfaces on tool geometries, measurement errors increase especially due to the crosstalk between individual fibers in the bundle. A detailed analysis of the transmission properties of the utilized fiber bundles is presented. It is shown that aliasing is avoided due to the non-regular grid structure of a bundle. Different techniques are demonstrated to reduce the effect of crosstalk on the phase evaluation. Measurements of highly reflective technical surfaces with different geometrical properties are shown.

Approximating the trace of iterative solutions at the interfaces with Nonuniform Fourier transform and singular value decomposition for cost-effectively accelerating the convergence of Schwarz domain decomposition

This paper deals with the representation of the trace of iterative Schwarz solutions at the interfaces of domain decomposition to approximate adaptively the interface error operator. This allows to build a cost-effectively accelerating of the convergence of the iterative method by extending to the vectorial case the Aitken’s accelerating convergence technique. The first representation is based on the building of a nonuniform discrete Fourier transform defined on a non-regular grid. We show how to construct a Fourier basis of dimension N+1 on this grid by building numerically a sesquilinear form, its exact accuracy to represent trigonometric polynomials of degree N / 2, and its spectral approximation property that depends on the continuity of the function to approximate. The decay of Fourier-like modes of the approximation of the trace of the iterative solution at the interfaces provides an estimate to adaptively select the modes involved in the acceleration. The drawback of this approach is to be dependent on the continuity of the trace of the iterated solution at the interfaces. The second representation, purely algebraic, uses a singular value decomposition of the trace of the iterative solution at the interfaces to provide a set of orthogonal singular vectors of which the associated singular values provide an estimate to adapt the acceleration. The resulting Aitken-Schwarz methodology is then applied to large scale computing on 3D linear Darcy flow where the permeability follows a log normal random distribution.

Visual object tracking by an evolutionary self-organizing neural network

In this paper, a recently proposed evolutionary self-organizing map is extended and applied to visual tracking of objects in video sequences. The proposed approach uses a simple geometric template to track an object executing a smooth movement represented by affine transformations. The template is selected manually in the first frame and consists of a small number of keypoints and the neighborhood relations among them. The coordinates of the keypoints are used as the coordinates of the nodes of a non-regular grid defining a self-organizing map that represents the object. The weight vectors of each node in the output grid are updated by an evolutionary algorithm and used to locate the object frame by frame. Qualitative and quantitative evaluations indicate that the proposed approach present better results than those obtained by a direct method approach. Additionally, the proposed approach is evaluated under situations of partial occlusion and self-occlusion, and outliers, also presenting good results.

Algorithm of the analytic solution of magnetostatic inverse problem in magnetocardiography

Event Abstract Back to Event Algorithm of the analytic solution of magnetostatic inverse problem in magnetocardiography Michael Primin1*, Igor Nedayvoda1 and Yury Maslennikov2 1 Academy of Science of Ukraine, Institute of cybernetics, Ukraine 2 Cryoton Ltd, Russia Development of analytical method of magnetostatic inverse problem solution is proposed for investigation of magnetic field of human heart. Set of N different dipole sources is ascribed to spatial distribution of heart magnetic field parameters, measured in observation plane. Every dipole is supposed to have at least one point in the measurement plane, where the dipole’s magnetic field coincides with known (calculated on the basis of observed data) magnetic induction vector values and with all of it's spatial first-order derivates. Estimations of sources quantity for investigated timepoint of cardiocycle, their mutual locations and relative powers are obtained as a result of magnetic gradient intensity's spatial analysis in the signal's measuring plane and as a result of the Voronoy's diagram construction.Conditions and the correct magnetostatic inverse problem solution for several signal sources distributed independently in heart's three-dimesional volume are obtained through investigations. An algorithm of a system of cross-sections corresponding to a set of F-planes parallel to the measurement plane and forming a virtual three-dimensional space with the magnetic signal source was developed for the visualization of magnetostatic inverse problem solution results. In standard display mode 20 equidistant cross-sections with 1 centimeter step are used. In the “nonregular grid” display mode the distance between cross-sections is 0,5 centimeter whereas their quantity is determined in the process of the inverse problem solution's spatial analysis. Algorithm of inverse problem solution is realized in magnetocardiograph software and is used in real magnetocardiogramm investigation and analysis. Conference: Biomag 2010 - 17th International Conference on Biomagnetism , Dubrovnik, Croatia, 28 Mar - 1 Apr, 2010. Presentation Type: Poster Presentation Topic: MCG: Instrumentation, Modeling, Basic and Clinical Studies Citation: Primin M, Nedayvoda I and Maslennikov Y (2010). Algorithm of the analytic solution of magnetostatic inverse problem in magnetocardiography. Front. Neurosci. Conference Abstract: Biomag 2010 - 17th International Conference on Biomagnetism . doi: 10.3389/conf.fnins.2010.06.00318 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 06 Apr 2010; Published Online: 06 Apr 2010. * Correspondence: Michael Primin, Academy of Science of Ukraine, Institute of cybernetics, Kiev, Ukraine, prima_m@voliacable.com Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Michael Primin Igor Nedayvoda Yury Maslennikov Google Michael Primin Igor Nedayvoda Yury Maslennikov Google Scholar Michael Primin Igor Nedayvoda Yury Maslennikov PubMed Michael Primin Igor Nedayvoda Yury Maslennikov Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Neural network approach to collision free path-planning for robotic manipulators

The paper deals with collision free path-planning for industrial robotic manipulators A new efficient approach is proposed that is based on the topologically ordered neural network model. This model describes harmonic potential field of the robot configuration space, sampled by the non-regular grid. The developed path-planning algorithm takes into account highly-irregular shape of the obstacles of welding and assembling robotic cells, and provides reduced number of collision checking. The stability of the topologically ordered neural network is investigated. The algorithm has been successfully applied to the off-line programming of a robotic manufacturing cell for the automotive industry.

Combining harmonic functions and random sampling in robot motion planning

This paper presents a robot motion planning method that combines harmonic functions and random sampling. The approach uses harmonic functions efficiently computed over a non-regular grid decomposition of Configuration Space. Random sampling allows the use of the harmonic functions approach without the explicit knowledge of the robot's Configuration Space. The combination of both results in a resolution and probabilistic complete planner.

Semi-Lagrangian treatment of advection on the sphere with accurate spatial and temporal approximations

This paper is devoted to the numerical solution of the transport equation on the sphere, aimed at the implementation of accurate spatial interpolation procedures and of accurate reconstruction of the characteristic lines in a semi-Lagrangian framework. Since nonregular grid and pole singularity can both affect accuracy of the numerical approximation, proper account is taken for these problems. It is shown on a literature test case that accurate spatial approximation highly improves the accuracy of the method, having a greater impact than accurate temporal approximation. The algorithms can be introduced into general circulation models, where accurate temporal approximation is expected to play a major role.

A Grid‐based approach to non‐regular mesh generation for automatic remeshing with metal forming analysis

An extension of grid-based approach with non-regular grid, as a master grid, is proposed for automatic remeshing during the finite element analysis of metal forming processes. In the proposed technique, the mesh is constructed by superimposing master grid over the object to be meshed, then taking internal cells of the grid as the elements, and finally modifying the rugged boundary with layers of surface elements and mesh smoothing. The construction method of surface element layers and the subsequent mesh smoothing technique are proposed to improve the shapes of severely distorted boundary elements. In order to verify the validity and effectiveness of the present technique, as a typical mesh generation example requiring a master grid and the finite element analysis of inner race forging is presented. Copyright © 2000 John Wiley & Sons, Ltd.

A Grid-based approach to non-regular mesh generation for automatic remeshing with metal forming analysis

An extension of grid-based approach with non-regular grid, as a master grid, is proposed for automatic remeshing during the finite element analysis of metal forming processes. In the proposed technique, the mesh is constructed by superimposing master grid over the object to be meshed, then taking internal cells of the grid as the elements, and finally modifying the rugged boundary with layers of surface elements and mesh smoothing. The construction method of surface element layers and the subsequent mesh smoothing technique are proposed to improve the shapes of severely distorted boundary elements. In order to verify the validity and effectiveness of the present technique, as a typical mesh generation example requiring a master grid and the finite element analysis of inner race forging is presented. Copyright © 2000 John Wiley & Sons, Ltd.

Reconstruction of MR images from data acquired on a general nonregular grid by pseudoinverse calculation

A minimum-norm least-squares image-reconstruction method for the reconstruction of magnetic resonance images from non-Cartesian sampled data is proposed. The method is based on a general formalism for continuous-to-discrete mapping and pseudoinverse calculation. It does not involve any regridding or interpolation of the data and therefore the methodology differs fundamentally from existing regridding-based methods. Moreover, the method uses a continuous representation of objects in the image domain instead of a discretized representation. Simulations and experiments show the possibilities of the method in both radial and spiral imaging. Simulations revealed that minimum-norm least-squares image reconstruction can result in a drastic decrease of artifacts compared with regridding-based reconstruction. Besides, both in vivo and phantom experiments showed that minimum-norm least-squares image reconstruction leads to contrast improvement and increased signal-to-noise ratio compared with image reconstruction based on regridding. As an appendix, an analytical calculation of the raw data corresponding to the well-known Shepp and Logan software head phantom is presented.