Thin-plate Radial Basis Function Research Articles

Turbulence-preserving interpolation of aero-optical wavefront data using radial basis functions: application and fractal analysis

Spatial interpolation using radial basis functions (RBFs) is compared with standard natural neighbor interpolation for the purpose of populating dead zones in optical path difference (OPD) wavefront data collected through aero-optics turbulence. Both methods, and some other commonly used basis functions, are described in detail. These candidates are applied to experimental subscale turbulent wind tunnel data collected with a conformal window hemispherical turret over a range of turret look-back angles spanning 60 deg to 135 deg. Interpolation performance is evaluated using both OPD magnitude and fractal (Hurst exponent) analysis. Interpolating aero-optical OPDs using the thin-plate RBF reduces the symmetric mean absolute percent error of the OPD magnitude and the Hurst exponent by an average of 28% and 50%, respectively, compared with the natural neighbor interpolation method. The improved interpolation accuracy should likewise increase the quality of subsequent postprocessing analysis requiring uniformly populated rectangular regions as inputs, such as optical flow algorithms. A MATLAB implementation of the pseudocubic spline RBF is provided.

A not-a-knot meshless method with radial basis functions for numerical solutions of Gilson–Pickering equation

In this paper, a numerical meshless method for solving the Gilson–Pickering equation is considered. The method is based on thin-plate radial basis function using collocation points. The scheme proposed works in a similar fashion as finite-difference method and a predictor–corrector scheme is proposed to avoid solving the nonlinear system. In addition, we use the super not-a-knot method for improving the accuracy at the boundaries. The results of numerical experiments are compared with the existing results in illustrative examples to confirm the accuracy and efficiency of the presented scheme and the norm of the error functions is obtained to show the convergence of the method.

Heterogeneous object modeling using the radial basis functions

This paper presented an effective method for the three-dimensional (3D) heterogeneous objects modeling. First, an implicit solid representing the smooth 3D scalar field for the material property distribution was reconstructed by interpolating the geometric positions of control points and property values defined at those points using an implicit interpolation algorithm based on the thinplate radial basis function (RBF). After generating the implicit solid representing the smooth 3D scalar field for the material property distribution, a heterogeneous object model with accurately controlled material property distribution was designed using the generated implicit solid. A variety of experimental results showed that the proposed heterogeneous objects modeling method has the potential benefits for accurately controlling the spatial material property distribution within an arbitrarily-shaped geometrical model.

Heterogeneous minimal surface porous scaffold design using the distance field and radial basis functions

This paper presented an effective method for the 3D heterogeneous porous scaffold design of human tissue using triply periodic minimal surface (TPMS) internal pore architectures. First, an implicit solid representing the smooth 3D scalar field for the porosity distribution was reconstructed by interpolating the geometric positions of control points and porosity values defined at those points using an implicit interpolation algorithm based on the thin-plate radial basis function. After generating the implicit solid representing the smooth 3D scalar field for the porosity distribution, a functionally graded tissue scaffold with accurately controlled porosity distribution was designed using the TPMS-based unit cell libraries. Numerical results showed that the proposed scaffold design method has the potential benefits for accurately controlling the spatial porosity distribution within an arbitrarily shaped scaffold while keeping the advantage of the TPMS-based unit cell libraries.

Three-dimensional surface reconstruction of human bone using a [formula omitted]-spline based interpolation approach

An effective method for the 3D Bio-CAD model reconstruction of human bone from a scanned point cloud data or a sequence of CT image data is presented based on a B -spline interpolation scheme. In the method, a base surface is generated by creating a smooth implicit surface from the given point cloud data or a sequence of CT image data. The implicit surface is defined by a combination of the well-known thin plate radial basis functions (RBFs) using the domain decomposition method (DDM). After generating the base implicit surface, various types of CAD models such as surface and solid are constructed by using the base implicit surface. In order to reconstruct a complex model, voxel data which can be extracted easily from the base implicit surface are used to generate a rectangular curve net with good quality using the projection and smoothing scheme. After generating the interior points and tangential vectors in each rectangular region considering the required accuracy, a complex B -spline surface is reconstructed by interpolating the rectangular array of points. Experimental results show that the proposed method creates the three dimensional shapes of human bones suitable for bone scaffold design, finite element analysis, and medical diagnosis.

A collocation and Cartesian grid methods using new radial basis function to solve class of partial differential equations

In the present paper, a class of partial differential equation represented by Poisson's type problems are solved using a proposed Cartesian grid method and a collocation technique using a new radial basis function. The advantage of using this new radial basis function represented by overcoming singularity from the diagonal elements when thin plate radial basis function is used. The new function is a combination of both multiquadric and thin plate radial basis functions. The new radial basis function contains a control parameter ϵ, that takes one when evaluating the singular elements and equals zero elsewhere. Collocation of the approximate solution of the potential over the governing and boundary condition equations leads to a double linear system of equations. A proposed algebraic procedure is then developed to solve the double system. Examples of Poisson and Helmholtz equations are solved and the present results are compared with the their analytical solutions. A good agreement with analytical results is achieved.

Meshfree Approximation for Multi-Asset Options

We price multi-asset options by solving their price partial differential equations using a meshfree approach with radial basis functions under jump-diffusion and geometric Brownian motion frameworks. In the geo- metric Brownian motion framework, we propose an effective technique that breaks the multi-dimensional problem to multiple 3D problems. We solve the price PDEs or PIDEs with an implicit meshfree scheme using thin-plate radial basis functions. Meshfree approach is very accurate, has high order of convergence and is easily scalable and adaptable to higher dimensions and different payoff profiles. We also obtain closed form approximations for the option Greeks. We test the model on American crack spread options traded on NYMEX.

An analysis of the linear advection–diffusion equation using mesh-free and mesh-dependent methods

The numerical solution of advection–diffusion equations has been a long standing problem and many numerical methods that attempt to find stable and accurate solutions have to resort to artificial methods to stabilize the solution. In this paper, we present a meshless method based on thin plate radial basis functions (RBF). The efficiency of the method in terms of computational processing time, accuracy and stability is discussed. The results are compared with the findings from the dual reciprocity/boundary element and finite difference methods as well as the analytical solution. Our analysis shows that the RBFs method, with its simple implementation, generates excellent results and speeds up the computational processing time, independent of the shape of the domain and irrespective of the dimension of the problem.

Preconditioned Iterative Methods for Scattered Data Interpolation

The scattered data interpolation problem in two space dimensions is formulated as a partial differential equation with interpolating side conditions. The system is discretized by the Morley finite element space. The focus of this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis functions for the equations corresponding to the interpolation conditions. The effect of the preconditioners are documented by numerical experiments.