Application of Gaussian Radial Basis Functions for Fast Spatial Imaging of Ground Penetration Radar Data Obtained on an Irregular Grid

Spatial imaging of ground penetrating radar (GPR) measurement data is a difficult computational problem that is time consuming and requires substantial memory resources. The complexity of the problem increases when the measurements are performed on an irregular grid. Such grid irregularities are typical for handheld or flying GPR systems. In this paper, a fast and efficient method of GPR data imaging based on radial basis functions is described. A compactly supported modified Gaussian radial basis function (RBF) and a hierarchical approximation method were used for computation. The approximation was performed in multiple layers with decreasing approximation radius, where in successive layers, increasingly finer details of the imaging were exposed. The proposed method provides high flexibility and accuracy of approximation with a computational cost of N·log (N) for model building and N·M for function evaluation, where N is the number of measurement points and M is the number of approximation centres. The method also allows for the control smoothing of measurement noise. The computation of one high-quality imaging using 5000 measurement points utilises about 5 s on an Intel Core i5-7200U CPU 2.5 GHz, 8 GB RAM computer. Such short time enables real-time image processing during field measurements.

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Use of Multiquadric Functions for Multivariable Representation of the Aerodynamic Coefficients of Airfoils

Mathematical Problems in Engineering ◽

10.1155/2021/6615601 ◽

2021 ◽

Vol 2021 ◽

pp. 1-21

Author(s):

Filipe Ribeiro ◽

Pedro Albuquerque ◽

Pedro Gamboa ◽

Kouamana Bousson

Keyword(s):

Radial Basis Functions ◽

Piecewise Linear ◽

Research Work ◽

Computational Cost ◽

Basis Functions ◽

Aerodynamic Coefficients ◽

Independent Variables ◽

Advantages And Disadvantages ◽

Radial Basis ◽

Multivariable Interpolation

Given an array (or matrix) of values for a function of one or more variables, it is often desired to find a value between two given points. Multivariable interpolation and approximation by radial basis functions are important subjects in approximation theory that have many applications in Science and Engineering fields. During the last decades, radial basis functions (RBFs) have found increasingly widespread use for functional approximation of scattered data. This research work aims at benchmarking two different approaches: an approximation by radial basis functions and a piecewise linear multivariable interpolation in terms of their effectiveness and efficiency in order to conclude about the advantages and disadvantages of each approach in approximating the aerodynamic coefficients of airfoils. The main focus of this article is to study the main factors that affect the accuracy of the multiquadric functions, including the location and quantity of centers and the choice of the form factor. It also benchmarks them against piecewise linear multivariable interpolation regarding their precision throughout the selected domain and the computational cost required to accomplish a given amount of solutions associated with the aerodynamic coefficients of lift, drag and pitching moment. The approximation functions are applied to two different multidimensional cases: two independent variables, where the aerodynamic coefficients depend on the Reynolds number (Re) and the angle-of-attack (α), and four independent variables, where the aerodynamic coefficients depend on Re, α, flap chord ratio (cflap), and flap deflection (δflap).

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Numerical Investigation of Electromagnetic Scattering Problems Based on the Compactly Supported Radial Basis Functions

Zeitschrift für Naturforschung A ◽

10.1515/zna-2016-0070 ◽

2016 ◽

Vol 71 (8) ◽

pp. 677-690 ◽

Cited By ~ 6

Author(s):

Hadi Roohani Ghehsareh ◽

Seyed Kamal Etesami ◽

Maryam Hajisadeghi Esfahani

Keyword(s):

Integral Equation ◽

Radial Basis Functions ◽

Cross Section ◽

Computational Cost ◽

Basis Functions ◽

Test Problems ◽

Em Scattering ◽

Compactly Supported ◽

Radial Basis ◽

Field Integral

AbstractIn the current work, the electromagnetic (EM) scattering from infinite perfectly conducting cylinders with arbitrary cross sections in both transverse magnetic (TM) and transverse electric (TE) modes is numerically investigated. The problems of TE and TM EM scattering can be mathematically modelled via the magnetic field integral equation (MFIE) and the electric field integral equation (EFIE), respectively. An efficient technique is performed to approximate the solution of these surface integral equations. In the proposed numerical method, compactly supported radial basis functions (RBFs) are employed as the basis functions. The radial and compactly supported properties of these basis functions substantially reduce the computational cost and improve the efficiency of the method. To show the accuracy of the proposed technique, it has been applied to solve three interesting test problems. Moreover, the method is well used to compute the electric current density and also the radar cross section (RCS) for some practical scatterers with different cross section geometries. The reported numerical results through the tables and figures demonstrate the efficiency and accuracy of the proposed technique.

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Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

Mathematics ◽

10.3390/math10020223 ◽

2022 ◽

Vol 10 (2) ◽

pp. 223

Author(s):

Pedro González-Rodelas ◽

Miguel Pasadas ◽

Abdelouahed Kouibia ◽

Basim Mustafa

Keyword(s):

Integral Equation ◽

Radial Basis Functions ◽

Volterra Integral Equation ◽

Computational Cost ◽

Basis Functions ◽

Numerical Examples ◽

Acceptable Accuracy ◽

Convergence Results ◽

Radial Basis ◽

Low Computational Cost

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

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GPU-Based Sphere Tracing for Radial Basis Function Implicits

International Journal of Image and Graphics ◽

10.1142/s0219467814500041 ◽

2014 ◽

Vol 14 (01n02) ◽

pp. 1450004

Author(s):

Harlen Costa Batagelo ◽

João Paulo Gois

Keyword(s):

Ray Tracing ◽

Radial Basis Functions ◽

Computational Cost ◽

Partition Of Unity ◽

Basis Functions ◽

Regular Partition ◽

Ambient Occlusion ◽

Radial Basis ◽

Data Points ◽

Normal Vectors

Ray tracing of implicit surfaces based on radial basis functions can demand high computational cost in the presence of a large number of radial centers. Recently, it was presented the least squares hermite radial basis functions (LS-HRBF) Implicits, a method for implicit surface reconstruction from Hermitian data (points equipped with their normal vectors) which makes use of iterative center selection in order to reduce the number of centers. In the present work, we propose an antialiazed sphere tracing algorithm fully implemented in OpenGL Shader Language for ray tracing LS-HRBF Implicits, which exploits a regular partition of unity for strong parallelization. We show that interactive frame rates can be achieved for surfaces composed of thousands of centers even when rendering effects such as cube mapping, soft shadows and ambient occlusion are used.

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A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

Mathematics ◽

10.3390/math7100964 ◽

2019 ◽

Vol 7 (10) ◽

pp. 964 ◽

Cited By ~ 1

Author(s):

Zhiyong Liu ◽

Qiuyan Xu

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Radial Basis Functions ◽

Computational Cost ◽

Basis Functions ◽

Partial Differential ◽

Compactly Supported ◽

Linear Partial Differential Equations ◽

Numerical Experimentation ◽

Radial Basis

In this paper, we derive and discuss the hierarchical radial basis functions method for the approximation to Sobolev functions and the collocation to well-posed linear partial differential equations. Similar to multilevel splitting of finite element spaces, the hierarchical radial basis functions are constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the compactly supported radial basis functions approximation and stationary multilevel approximation, the new method can not only solve the present problem on a single level with higher accuracy and lower computational cost, but also produce a highly sparse discrete algebraic system. These observations are obtained by taking the direct approach of numerical experimentation.

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