Hermite Radial Basis Functions Implicits

The Hermite radial basis functions (HRBF) implicits reconstruct an implicit function which interpolates or approximates scattered multivariate Hermite data (i.e. unstructured points and their corresponding normals). Experiments suggest that HRBF implicits allow the reconstruction of surfaces rich in details and behave better than previous related methods under coarse and/or non‐uniform samplings, even in the presence of close sheets. HRBF implicits theory unifies a recently introduced class of surface reconstruction methods based on radial basis functions (RBF), which incorporate normals directly in their problem formulation. Such class has the advantage of not depending on manufactured offset‐points to ensure existence of a non‐trivial implicit surface RBF interpolant. In fact, we show that HRBF implicits constitute a particular case of Hermite–Birkhoff interpolation with radial basis functions, whose main results we present here. This framework not only allows us to show connections between the present method and others but also enable us to enhance the flexibility of our method by ensuring well‐posedness of an interesting combined interpolation/regularization approach.     

Evolutionary Radial Basis Functions for Credit Assessment

Credit analysts generally assess the risk of credit applications based on their previous experience. They frequently employ quantitative methods to this end. Among the methods used, Artificial Neural Networks have been particularly successful and have been incorporated into several computational tools. However, the design of efficient Artificial Neural Networks is largely affected by the definition of adequate values for their free parameters. This article discusses a new approach to the design of a particular Artificial Neural Networks model, RBF networks, through Genetic Algorithms. It presents an overall view of the problems involved and the different approaches employed to optimize Artificial Neural Networks genetically. For such, several methods proposed in the literature for optimizing RBF networks using Genetic Algorithms are discussed. Finally, the model proposed by the authors is described and experimental results using this model for a credit risk assessment problem are presented.     

基于径向基函数的图像修复技术

图像修复是指恢复图像中破损区域的颜色信息或者去除图像中的多余物体。本文提出了一种新的基于径向基函数的图像修复算法，由用户交互地指定需要修复的区域，算法自动地计算破损区域的轮廓并沿轮廓法向扩张，确定合适的径向基函数重构区域，将该区域内图像的颜色值看作规则采样点上的高度场，把二维图像修复问题转化为三维散乱点重建问题，利用径向基函数曲面重建的优势来修补破损的图像。实验表明，该算法能正确、稳定地处理各种破损区域。     

All-Frequency Lighting with Multiscale Spherical Radial Basis Functions

This paper proposes a novel multiscale spherical radial basis function (MSRBF) representation for all-frequency lighting. It supports the illumination of distant environment as well as the local illumination commonly used in practical applications, such as games. The key is to define a multiscale and hierarchical structure of spherical radial basis functions (SRBFs) with basis functions uniformly distributed over the sphere. The basis functions are divided into multiple levels according to their coverage (widths). Within the same level, SRBFs have the same width. Larger width SRBFs are responsible for lower frequency lighting while the smaller width ones are responsible for the higher frequency lighting. Hence, our approach can achieve the true all-frequency lighting that is not achievable by the single-scale SRBF approach. Besides, the MSRBF approach is scalable as coarser rendering quality can be achieved without reestimating the coefficients from the raw data. With the homogeneous form of basis functions, the rendering is highly efficient. The practicability of the proposed method is demonstrated with real-time rendering and effective compression for tractable storage.     

Numerical Cubature on Scattered Data by Radial Basis Functions

We study cubature formulas on relatively small scattered samples in the unit square, obtained by integrating radial basis function (RBF) interpolants. Numerical tests show that, due to the small size of the corresponding weights (which are not all positive in general), thin-plate splines and Wendland's compactly supported radial functions give the most reliable RBF cubature methods.     

非对称径向基函数与稳定边界图像变形算法

提出了一种基于非对称径向基函数的图像变形算法,该算法克服了基于对称径向基函数算法中由于对称径向基函数的全局性导致的图像边界变形过大的不合理变形现象．实验结果表明,文中算法简单、有效,得到的变形图像既具有基于对称径向基函数算法所得的变形图像的光滑性,又具有良好的局部变形效果,更为重要的是稳定了变形图像的边界．     

Surface Reconstruction Based on Hierarchical Floating Radial Basis Functions

In this paper we address the problem of optimal centre placement for scattered data approximation using radial basis functions (RBFs) by introducing the concept of floating centres. Given an initial least‐squares solution, we optimize the positions and the weights of the RBF centres by minimizing a non‐linear error function. By optimizing the centre positions, we obtain better approximations with a lower number of centres, which improves the numerical stability of the fitting procedure. We combine the non‐linear RBF fitting with a hierarchical domain decomposition technique. This provides a powerful tool for surface reconstruction from oriented point samples. By directly incorporating point normal vectors into the optimization process, we avoid the use of off‐surface points which results in less computational overhead and reduces undesired surface artefacts. We demonstrate that the proposed surface reconstruction technique is as robust as recent methods, which compute the indicator function of the solid described by the point samples. In contrast to indicator function based methods, our method computes a global distance field that can directly be used for shape registration.     

A Parallel Multivariate Interpolation Algorithm With Radial Basis Functions

This paper presents an efficient and highly scalable parallel version of the Modified RBF Shepard's method presented in [5]. This method maintains the "metric" nature and the advantages of Shepard's method and, at the same time, improves its accuracy by exploiting the characteristics of flexibility and accuracy which have made the radial basis functions a well-established tool for multivariate interpolation. Due to its locality, this method can be easily and efficiently parallelized on a distributed memory parallel architecture. The performance of the parallel algorithm has been studied theoretically and the experimental results obtained by running its implementation on a Cray T3E parallel machine, using the MPI interface, confirm the theoretical efficiency.     

Large Scattered Data Fitting Based on Radial Basis Functions

Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.     

Edge-driven Image Interpolation using Adaptive Anisotropic Radial Basis Functions

This paper investigates the image interpolation problem, where the objective is to improve the resolution of an image by dilating it according to a given enlargement factor. We present a novel interpolation method based on Radial Basis Functions (RBF) which recovers a continuous intensity function from discrete image data samples. The proposed anisotropic RBF interpolant is designed to easily deal with the local anisotropy in the data, such as edge-structures in the image. Considering the underlying geometry of the image, this algorithm allows us to remove the artifacts that may arise when performing interpolation, such as blocking and blurring. Computed examples demonstrate the effectiveness of the method proposed by visual comparisons and quantitative measures.     

Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint

Implicit planar curve and surface fitting to a set of scattered points plays an important role in solving a wide variety of problems occurring in computer graphics modelling, computer graphics animation, and computer assisted surgery. The fitted implicit surfaces can be either algebraic or non‐algebraic. The main problem with most algebraic surface fitting algorithms is that the surface fitted to a given data set is often unbounded, multiple sheeted, and disconnected when a high degree polynomial is used, whereas a low degree polynomial is too simple to represent general shapes. Recently, there has been increasing interest in non‐algebraic implicit surface fitting. In these techniques, one popular way of representing an implicit surface has been the use of radial basis functions. This type of implicit surface can represent various shapes to a high level of accuracy. In this paper, we present an implicit surface fitting algorithm using radial basis functions with an ellipsoid constraint. This method does not need to build interior and exterior layers for the given data set or to use information on surface normal but still can fit the data accurately. Furthermore, the fitted shape can still capture the main features of the object when the data sets are extremely sparse. The algorithm involves solving a simple general eigen‐system and a computation of the inverse or psedo‐inverse of a matrix, which is straightforward to implement.     

Prediction of Critical Desalination Parameters Using Radial Basis Functions Networks

Prediction of critical desalination parameters (recovery and salt rejection) of two distinct processes based on real operational data is presented. The proposed method utilizes the radial basis function network using data clustering and histogram equalization. The scheme involves center selection and shape adjustment of the localized receptive fields. This algorithm causes each group of radial basis functions to adapt to regions of the clustered input space. Networks produced by the proposed algorithm have good generalization performance on prediction of non-linear input–output mappings and require a small number of connecting weights. The proposed method was used for the prediction of two different critical parameters for two distinct Reverse Osmosis (RO) plants. The simulation results indeed confirm the effectiveness of the proposed prediction method.     

Image Warping by Radial Basis Functions: Application to Facial Expressions

The human face is an elastic object. A natural paradigm for representing facial expressions is to form a complete 3D model of facial muscles and tissues. However, determining the actual parameter values for synthesizing and animating facial expressions is tedious; evaluating these parameters for facial expression analysis out of gray-level images is ahead of the state of the art in computer vision. Using only 2D face images and a small number of anchor points, we show that the method of radial basis functions provides a powerful mechanism for processing facial expressions. Although constructed specifically for facial expressions, our method is applicable to other elastic objects as well.     

构造径向基函数的一般方法及其在图象处理中的应用

gi．引言对于函数J：＊”＋R如果存在函数h：B＋R使得对VxE＊”;都有人对二周卜DD）,其中卜D为X的欧几里得范数,则称函数人X）为一个径向函数．径向函数具有本质上的一维性,它可以将任意高继问题映射为一维问题处理．当一个径向函数的平移和伸缩能够构成某函数空间的基底时,利用该基底的线性组合便可以设计径向基函数网络．径向基函数网络的降维特性为多层前馈网络的学习提供了一条新颖而简捷的途径山,现已广泛应用于非线性函数逼近、信号及图象处理等领域［‘,‘,‘］．它不仅具有良好的推广能力,而且避免了象反传算法那样繁…     

Weighted Radial Basis Functions for improved pattern recognition and signal processing

The paper describes an improved version of the Radial Basis Function algorithm, which integrates the advantages of Multi-Layer Perceptrons and Radial Basis Functions alone. The proposed paradigm is more general in nature, since it has the other two as particular subcases. It finds applications in several pattern recognition and classification tasks. Furthermore it can also be used as a method to map Fuzzy Inference Systems on Artificial Neural Networks.     

基于广义径向基函数的神经网络分类预测

径向基函数网络是神经网络中一种广泛使用的设计方法.它把神经网络的设计看作是一个高维空间的曲线逼近问题.相对于其他的神经网络方法.径向基函数神经网络除了具有一般神经网络的优点,如多维非线性映射能力、泛化能力、并行信息处理能力等,还具有很强的聚类分析能力,学习算法简单方便等优点.针对一个实际分类问题,利用广义径向基函数网络的思想训练一个网络并实现对测试数据集的分类预测.本算法采用k-均值聚类算法训练广义径向基函数网络中心,使用奇异值分解计算输出层权值.对该网络的实现细节及待改进之处进行简要分析.实验表明广义径向基函数神经网络的思想具有很强的聚类分析能力,学习算法简单方便等优点.     

使用局部支撑径向基函数的隐式曲线曲面几何迭代算法

由散乱数据稳定重构曲线曲面,在变分拟插值方法的基础之上,提出了使用局部支撑径向基函数的隐式几何迭代算法.首先,根据给定数据点的法向构造隐式函数的非零约束,构造计算隐函数系数的迭代格式,并讨论其收敛性;然后,在此基础上引入加速因子,对隐式迭代算法进行加速,同时讨论了加速算法的收敛性;最后,为了降低迭代过程空间和时间的复杂度,给出了一种加速算法的改进版本.数值实验表明,使用局部支撑径向基函数的隐式几何迭代算法对曲线曲面重构是有效的,并对部分信息缺失、非均匀分布、带噪声采样数据的重构也达到了较好的效果,且实现简单,易于并行.