Radial Basis Functions

In the essay a short description of the biological systems that are the models of artificial neural networks is given before a general, but brief, overview of artificial neural networks. After these fundamental background concepts, radial basis functions are described. The main part of the essay treats radial basis function networks. It starts with the descriptions of the RBFNs (Radial Basis Function Networks) by presenting the theory and it ends with some brief accounts applications of RBFNs. The applications consist often of simulations of real applications. These simulations are then shown to what an extent the RBFNs may be useful and whether they present any advantages over artificial neural networks not of the RBFN type.

In this work, we present a comparative study of meshless method, modified Bernstein polynomials (BP) and B-Spline finite element method (BS-FEM) for the numerical solution of two different models of Korteweg–de Vries (KdV) equation. The multiquadric (MQ) and Gaussian (GA) Radial basis functions (RBFs) are used due to exponential convergence rate. Excellent agreement is found between exact and RBFs solutions and same accuracy is obtained as Bernstein polynomials and B-Spline finite element method.

Level set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two-dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.

The main aims and contributions of the present paper are to use new soft computing methods for the simulation of scour geometry (depth/height and locations) in a comparative framework. Five models were used for the prediction of the dimension and location of the scour pit. The five developed models in this study are multilayer perceptron (MLP) neural network, radial basis functions (RBF) neural network, adaptive neuro fuzzy inference systems (ANFIS), multiple linear regression (MLR), and multiple non-linear regression (MNLR) in comparison with empirical equations. Four non-dimensional geometry parameters of scour hole shape are predicted by these models including the maximum scour depth (S), the distance of S from the weir (XS), the maximum height of downstream deposited sediments (hd), and distance of hd from the weir (XD). The best results over train data derived for XS/Z and hd/Z by the MLP model with R 2 are 0.95 and 0.96 respectively; the best predictions for S/Z and XD/Z are from the ANFIS model with R 2 0.91 and 0.96 respectively. The results indicate that the application of MLP and ANFIS results in the accurate prediction of scour geometry for the designing of stable grade control structures in alluvial irrigation channels.

It is well known that nonlinear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron (in press) [2] for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore–Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information abou...

This paper introduces and motivates the use of Gaussian mixture models (GMM) for robust text-independent speaker identification. The individual Gaussian components of a GMM are shown to represent some general speaker-dependent spectral shapes that are effective for modeling speaker identity. The focus of this work is on applications which require high identification rates using short utterance from unconstrained conversational speech and robustness to degradations produced by transmission over a telephone channel. A complete experimental evaluation of the Gaussian mixture speaker model is conducted on a 49 speaker, conversational telephone speech database. The experiments examine algorithmic issues (initialization, variance limiting, model order selection), spectral variability robustness techniques, large population performance, and comparisons to other speaker modeling techniques (uni-modal Gaussian, VQ codebook, tied Gaussian mixture, and radial basis functions). The Gaussian mixture speaker model attains 96.8% identification accuracy using 5 second clean speech utterances and 80.8% accuracy using 15 second telephone speech utterances with a 49 speaker population and is shown to outperform the other speaker modeling techniques on an identical 16 speaker telephone speech task

We present techniques for rendering implicit surfaces in di fferent pen-and-ink styles. The implicit models are rendered using point-based primitives to depict shape and tone using silhouettes with hidden-line attenuation, drawing directions, and stippling. We present sample renderings obtained for a variety of models. Furthermore, we describe simple and novel methods to control point placement and rendering style. Implicit surfaces provide important, mathematically precise information about surface properties and allow global calculations, such as point pertinence and distance evaluation, while, at the same time, also allow obtaining local di fferential properties such as normals and principal curvatures. Our approach is implemented using HRBF Implicits, a simple and compact representation, which has three fundamental qualities: a small number of point-normal samples as input for surface reconstruction, good projection of points near the surface, and smoothness of the gradient fiel...

2007 Jaime A. Echeverri A. / Rodrigo Cañaveral Uribe / Alexánder Vélez Vásquez RECONSTRUCCIÓN TRIDIMENSIONAL DE ROSTROS A PARTIR DE IMÁGENES DE RANGO POR MEDIO DE FUNCIONES DE BASE RADIAL DE SOPORTE COMPACTO Revista de Ingenierías Universidad de ...

The use of Mercer kernel methods in statistical learning theory provides for strong learning capabilities, as seen in kernel principal component analysis and support vector machines. Unfortunately, after learning, the computational complexity of execution through a kernel is of the order of the size of the training set, which is quite large for many applications. This paper proposes a two-step procedure for arriving at a compact and computationally efficient execution procedure. After learning in the kernel space, the proposed extension exploits the universal approximation capabilities of generalized radial basis function neural networks to efficiently approximate and replace the projections onto the empirical kernel map used during execution. Sample applications demonstrate significant compression of the kernel representation with graceful performance loss.