Non-linear systems identification using radial basis functions

This paper investigates the identification of discrete-time non-linear systems using radial basis functions. A forward regression algorithm based on an orthogonal decomposition of the regression matrix is employed to select a suitable set of radial basis function centers from a large number of possible candidates and this provides, for the first time, fully automatic selection procedure for identifying parsimonious radial basis function models of structure-unknown non-linear systems. The relationship between neural networks and radial basis functions is discussed and the application of the algorithms to real data is included to demonstrate the effectiveness of this approach.     

Automated freehand sketch segmentation using radial basis functions

Freehand sketching is widely regarded as an efficient and natural way for interaction between computers and humans. We present a robust computerized scheme to automatically segment freehand sketches into a series of components with specific geometric meaning regardless of whether these are generated online or offline. This task is a necessary first step toward sketch understanding. By exploiting the interpolation/extrapolation characteristic of radial basis functions (RBFs), a greedy algorithm consisting of forward and backward operations is proposed for finding the minimum set of segmentation points that can be used to reconstruct with high fitting accuracy freehand sketches in the form of implicit functions. To obtain segmentation points, a simple angle-based rule is used to remove “bridging” points that provide a smooth transition between consecutive sketch components. Feasibility of the proposed algorithm is demonstrated by a preliminary performance assessment study using ten computer generated drawings. These experiments show that in this dataset sensitivity of the segmentation was higher than 97.5% with a false positive (FP) rate of approximately 25%. The majority of false positive identifications are located on arc regions where a larger number of segmentation points are needed for reconstruction purposes. The primary contribution of this algorithm is that it transforms an ambiguous problem, namely, freehand sketch segmentation, into an implicit function fitting operation. Therefore, this proposed approach has several advantages, including independence of the actual sketching activity, and the ability for a satisfactory detection of the transition point between a line and an arc or between two arcs.     

Shape-adaptive radial basis functions

Radial basis functions for discrimination and regression have been used with some success in a wide variety of applications. Here, we investigate the optimal choice for the form of the basis functions and present an iterative strategy for obtaining the function in a regression context using a conjugate gradient-based algorithm together with a nonparametric smoother. This is developed in a discrimination framework using the concept of optimal scaling. Results are presented for a range of simulated and real data sets.     

一种改进的RBF全局优化方法

径向基函数（Radial Basis Functions）由于具有良好的近似效果和运算简单的特点,被应用于全局优化中,成为解决黑箱函数全局优化问题的有效方法。然而现有的基于RBF的全局优化算法存在迭代过程中RBF模型重构效率低下,以及采样方法不合理导致函数估值次数过多等问题。在此提出几个改进思路：采用基于矩阵分块的增量RBF方法以减少模型重构时间提高效率;采用增量LHD采样方法以确保具有更好的空间填充性;采用算法重启策略以降低估值次数。通过实验验证改进方法的优势。     

Reconstructing surfaces by volumetric regularization using radial basis functions

We present a new method of surface reconstruction that generates smooth and seamless models from sparse, noisy, nonuniform, and low resolution range data. Data acquisition techniques from computer vision, such as stereo range images and space carving, produce 3D point sets that are imprecise and nonuniform when compared to laser or optical range scanners. Traditional reconstruction algorithms designed for dense and precise data do not produce smooth reconstructions when applied to vision-based data sets. Our method constructs a 3D implicit surface, formulated as a sum of weighted radial basis functions. We achieve three primary advantages over existing algorithms: (1) the implicit functions we construct estimate the surface well in regions where there is little data, (2) the reconstructed surface is insensitive to noise in data acquisition because we can allow the surface to approximate, rather than exactly interpolate, the data, and (3) the reconstructed surface is locally detailed, yet globally smooth, because we use radial basis functions that achieve multiple orders of smoothness.     

Approximation of insurance liability contracts using radial basis functions

ABSTRACT

We introduce the Option Interpolation Model (OIM) for accurate approximation of embedded option values in insurance liabilities. Accurate approximation is required for ex-ante risk management applications. The OIM is based on interpolation with radial basis functions, which can interpolate scattered data, and does not suffer from the curse of dimensionality. To reduce computation time we present an inversion method to determine the interpolation function weights. The robustness, accuracy and efficiency of the OIM are investigated in several numerical experiments. We show that the OIM results in highly accurate approximations.     

Robustness of radial basis functions

Neural networks are intended to be used in future nanoelectronic technology since these architectures seem to be robust to malfunctioning elements and noise in its inputs and parameters. In this work, the robustness of radial basis function networks is analyzed in order to operate in noisy and unreliable environment. Furthermore, upper bounds on the mean square error under noise contaminated parameters and inputs are determined if the network parameters are constrained. To achieve robuster neural network architectures fundamental methods are introduced to identify sensitive parameters and neurons.     

Metamodeling using extended radial basis functions: a comparative approach

The process of constructing computationally benign approximations of expensive computer simulation codes, or metamodeling, is a critical component of several large-scale multidisciplinary design optimization (MDO) approaches. Such applications typically involve complex models, such as finite elements, computational fluid dynamics, or chemical processes. The decision regarding the most appropriate metamodeling approach usually depends on the type of application. However, several newly proposed kernel-based metamodeling approaches can provide consistently accurate performance for a wide variety of applications. The authors recently proposed one such novel and effective metamodeling approach—the extended radial basis function (E-RBF) approach—and reported highly promising results. To further understand the advantages and limitations of this new approach, we compare its performance to that of the typical RBF approach, and another closely related method—kriging. Several test functions with varying problem dimensions and degrees of nonlinearity are used to compare the accuracies of the metamodels using these metamodeling approaches. We consider several performance criteria such as metamodel accuracy, effect of sampling technique, effect of sample size, effect of problem dimension, and computational complexity. The results suggest that the E-RBF approach is a potentially powerful metamodeling approach for MDO-based applications, as well as other classes of computationally intensive applications.     

Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

This paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.     

Parametric shape-from-shading by radial basis functions

We present a new method of shape from shading by using radial basis functions to parameterize the object depth. The radial basis functions are deformed by adjusting their centers, widths, and weights such that the intensity errors are minimized. The initial centers and widths are arranged hierarchically to speed up convergence and to stabilize the solution. Although the smoothness constraint is used, it can be eventually dropped out without causing instabilities in the solution. An important feature of our parametric shape-from-shading method is that it offers a unified framework for integration of multiple sensory information. We show that knowledge about surface depth and/or surface normals anywhere in the image can be easily incorporated into the shape from shading process. It is further demonstrated that even qualitative knowledge can be used in shape from shading to improve 3D reconstruction. Experimental comparisons of our method with several existing ones are made by using both synthetic and real images. Results show that our solution is more accurate than the others     

Implicit fitting of point cloud data using radial hermite basis functions

We describe a new technique for fitting noisy scattered point cloud data. The fitting surface is determined as zero level isosurface of a trivariate model which is an implicit least squares fit of the data based upon Radial Hermite Operators (RHO). We illustrate the value of these new techniques with several diverse applications.     

Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization

Gradient-based aerodynamic shape optimization using computational fluid dynamics (CFD), and time dependent problems in aeroelasticity, that is, coupled calculations between computational structural mechanics (CSM) and CFD, require repeated deformations of the CFD mesh.An interpolation scheme, based on radial basis functions (RBF), is devised in order to propagate the deformations from the boundaries to the interior of the CFD mesh. This method can lower the computational costs due to the deformation of the mesh, in comparison with the usual Laplace smoothing. Moreover, the algorithm is independent of the mesh connectivities. Therefore, structured and unstructured meshes are equally treated as well as hybrid meshes.The application of this interpolation scheme in problems of aerodynamic shape optimization is also carefully investigated. When the optimization is executed by a gradient-based algorithm the cost function is differentiated with respect to the design parameters in order to obtain the gradient. The gradient is most efficiently and accurately calculated by solving a certain adjoint equation derived from the discretized flow equations. The calculation of the gradient, which is detailed in this presentation, involves the Jacobian matrix of the mesh deformation.Finally, we present the results of an optimization of the ONERA M6 wing at transonic speed using the interpolation algorithm. The results are used for comparison with another technique of mesh deformation. The quality of the mesh obtained by the new algorithm, and the interpolation error, are analyzed with respect to the parameters of the interpolation scheme: the type of RBF, the RBF’s shape parameter, and the sets of control points.     

Single-shot 3D hand pose estimation using radial basis function networks trained on synthetic data

Pattern Analysis and Applications - In this work, we present a novel framework to perform single-shot hand pose estimation using depth data as input. The method follows a coarse to fine strategy...     

Approximate implicitization of parametric surfaces by using compactly supported radial basis functions

In this paper, a new approach is proposed to solve the approximate implicitization of parametric surfaces. It is primarily based on multivariate interpolation of scattered data by using compactly supported radial basis functions. Experimental results are provided to illustrate the proposed method is flexible and effective.     

An interactive approach for CBIR using a network of radial basis functions

An important requirement for constructing effective content-based image retrieval (CBIR) systems is accurate characterization of visual information. Conventional nonadaptive models, which are usually adopted for this task in simple CBIR systems, do not adequately capture all aspects of the characteristics of the human visual system. An effective way of addressing this problem is to adopt a "human-computer" interactive approach, where the users directly teach the system about what they regard as being significant image features and their own notions of image similarity. We propose a machine learning approach for this task, which allows users to directly modify query characteristics by specifying their attributes in the form of training examples. Specifically, we apply a radial-basis function (RBF) network for implementing an adaptive metric which progressively models the notion of image similarity through continual relevance feedback from users. Experimental results show that the proposed methods not only outperform conventional CBIR systems in terms of both accuracy and robustness, but also previously proposed interactive systems.     

Importance sampling of products from illumination and BRDF using spherical radial basis functions

In this paper, a new approach for the importance sampling of products from a complex high dynamic range (HDR) environment map and measured bidirectional reflectance distribution function (BRDF) data using spherical radial basis functions (SRBFs) is presented. In the pre-process, a complex HDR environment map and measured BRDF data are transformed into a scattered SRBF representation by using a non-uniform and non-negative SRBF fitting algorithm. An initial guess is determined for the fitting operation. In the run-time rendering process, after the product of the two SRBFs is evaluated, this is used to guide the number of samples. The sampling is done by mixing samples from the various “product” SRBFs using multiple importance sampling. Hence, the proposed approach efficiently renders images with multiple HDR environment maps and measured BRDFs.     

A new model for time-series forecasting using radial basis functions and exogenous data

In this paper, we present a new model for time-series forecasting using radial basis functions (RBFs) as a unit of artificial neural networks (ANNs), which allows the inclusion of exogenous information (EI) without additional pre-processing. We begin by summarizing the most well-known EI techniques used ad hoc, i.e., principal component analysis (PCA) and independent component analysis (ICA). We analyze the advantages and disadvantages of these techniques in time-series forecasting using Spanish bank and company stocks. Then, we describe a new hybrid model for time-series forecasting which combines ANNs with genetic algorithms (GAs). We also describe the possibilities when implementing the model on parallel processing systems.

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Sequential improvement for robust optimization using an uncertainty measure for radial basis functions

The performance of the sequential metamodel based optimization procedure depends strongly on the chosen building blocks for the algorithm, such as the used metamodeling method and sequential improvement criterion. In this study, the effect of these choices on the efficiency of the robust optimization procedure is investigated. A novel sequential improvement criterion for robust optimization is proposed, as well as an improved implementation of radial basis function interpolation suitable for sequential optimization. The leave-one-out cross-validation measure is used to estimate the uncertainty of the radial basis function metamodel. The metamodeling methods and sequential improvement criteria are compared, based on a test with Gaussian random fields as well as on the optimization of a strip bending process with five design variables and two noise variables. For this process, better results are obtained in the runs with the novel sequential improvement criterion as well as with the novel radial basis function implementation, compared to the runs with conventional sequential improvement criteria and kriging interpolation.     

Meshless method to solve nonstationary heat conduction problems using atomic radial basis functions

The authors consider a meshless method to solve 3D nonstationary boundary-value heat conduction problems. It is implemented through an iterative scheme based on a combination of the double substitution method and the method of fundamental solutions with the use of atomic radial basis functions. The approaches to the visualization of the desired solution are considered.     

Multifidelity surrogate modeling based on radial basis functions

Multiple models of a physical phenomenon are sometimes available with different levels of approximation. The high fidelity model is more computationally demanding than the coarse approximation. In this context, including information from the lower fidelity model to build a surrogate model is desirable. Here, the study focuses on the design of a miniaturized photoacoustic gas sensor which involves two numerical models. First, a multifidelity metamodeling method based on Radial Basis Function, the co-RBF, is proposed. This surrogate model is compared with the classical co-kriging method on two analytical benchmarks and on the photoacoustic gas sensor. Then an extension to the multifidelity framework of an already existing RBF-based optimization algorithm is applied to optimize the sensor efficiency. The co-RBF method does not bring better results than co-kriging but can be considered as an alternative for multifidelity metamodeling.