基于Legendre正交多项式非参数混合模型的图像分割

针对有限混合模型中参数估计方法对先验假设存在过分依赖的问题,提出了一种非参数的Legendre正交多项式图像混合模型分割方法。首先,设计了一种基于Legendre正交多项式的图像非参数混合模型,并用最小均方差(MISE)估计每一个模型的平滑参数;其次,利用EM算法求解正交多项式系数和模型的混合比。此方法不需要对模型作任何假设,可以有效克服模型失配问题。通过图像的分割实验表明,该方法比其他非参数混合模型分割效果更好。     

多元Chebyshev正交多项式混合模型及其在医学图像分割中的应用

针对原有一元正交多项式混合模型只能根据灰度特征分割图像的问题,提出一种基于多元Chebyshev正交 多项式混合模型的多维特征的医学图像分割方法。首先,根据Fouricr分析方法与张量积理论推导出图像的多元 Chcbyshcv正交多项式,并构建多元正交多项式的非参数混合模型,用最小均方差(MISE)估计每一个模型的平滑参 数;然后,用EM算法求解正交多项式系数和模型的混合比。此方法不需要对模型作任何假设,可以有效克服“模型失 配”问题。通过实验,表明了该分割方法的有效性。     

基于抽样的正交多项式混合模型图像分割

针对有限混合模型中参数估计方法对先验假设存在过分依赖和图像数据量大的问题,提出了一种基于抽样的非参数余弦正交序列的图像混合模型分割方法.首先,基于图像的直方图进行分层随机抽样得到样本数据,根据样本数据构建非参数正交多项式混合模型,对于模型的平滑参数采用最小均方差方法进行估计;其次,采用NEM(Nonparametric Expectation Maximum)算法求解混合模型中正交多项式系数和模型的混合比;最后,根据贝叶斯准则进行图像分割.此方法能够克服参数模型的基本假设与实际的物理模型之间存在的差异,实验表明该方法比GMM和Hermite混合模型分割方法分割质量高,而且分割速度快.     

基于邻域信息的B样条密度模型的医学图像分割研究

针对有参混合模型的聚类算法需要假设模型为某种已知的参数模型而存在模型不匹配及应用于图像分割时对噪声比较敏感的问题,提出了一种基于空间邻域信息的B样条密度模型的图像分割方法。首先,通过构建基于规范化的B样条密度函数的非参数混合模型,定义空间信息函数,使得分割模型具有空间邻域信息;其次,利用非参数B样条期望最大(NNBEM)算法估计密度模型的未知参数;最后根据贝叶斯准则实现图像的分割。该图像分割方法不需要假设图像符合某种模型,就可以克服实际数据分布与假设图像模型不一致的问题。此方法有效克服了"模型失配"问题,而且有力抑制了噪声点,同时很好地保留了边界的特性。分别对模拟图像进行仿真,验证了基于空间邻域信息的B样条密度模型的分割方法的有效性。     

基于多分辨率的非参数正交多项式医学图像融合方法

基于估计理论的图像融合方法都是假设图像偏移或噪声服从高斯混合分布,容易造成模型不匹配和丢失局部细节等问题。文中提出一种基于小波的多分辨率的非参数正交多项式医学图像融合方法。首先,对图像进行多分辨率分解。对低频部分,根据图像信息模型和非参数正交多项式混合模型,采用非参数期望最大法估计模型参数,获得低频融合结果。对高频部分,采用系数绝对值选大法进行融合。然后,将高频和低频部分结果进行反变换,得到最终融合图像。实验结果表明,该方法融合质量优于其它方法,融合时间大为缩短。     

切比雪夫正交基神经网络的权值直接确定法

经典的BP神经网络学习算法是基于误差回传的思想.而对于特定的网络模型,采用伪逆思想可以直接确定权值进而避免以往的反复迭代修正的过程.根据多项式插值和逼近理论构造一个切比雪夫正交基神经网络,其模型采用三层结构并以一组切比雪夫正交多项式函数作为隐层神经元的激励函数.依据误差回传(BP)思想可以推导出该网络模型的权值修正迭代公式,利用该公式迭代训练可得到网络的最优权值.区别于这种经典的做法,针对切比雪夫正交基神经网络模型,提出了一种基于伪逆的权值直接确定法,从而避免了传统方法通过反复迭代才能得到网络权值的冗长训练过程.仿真结果表明该方法具有更快的计算速度和至少相同的工作精度,从而验证了其优越性.     

基于正交多项式核函数方法

针对实际工程中小样本数据的稀疏性、分布特征不明显等问题,分析了现有的一些方法并指出了现有方法存在的问题,重点讨论了一类基于切比雪夫多项式的核方法.由于切比雪夫多项式的正交性,使得这些核函数在高维特征空间能得到更优的超平面.通过实验测试了这一类核函数的泛化性能以及学习效率.证明它们比其它的核函数需要更少的支持向量并能保证更好的学习性能.最后论文讨论了这类核函数方法存在的问题,并指出切比雪夫多项式核函数在解决小样本回归问题时具有很大的潜力,值得进一步研究     

多元切比雪夫神经网络及其快速权值确定算法

与传统的多层感知器模型相比,切比雪夫神经网络具有收敛速度快,复杂度低,泛化能力强等优点,但是,其研究最为广泛的一元切比雪夫神经网络在解决实际应用中的多元问题时存在着很大局限。鉴于此,对一元切比雪夫神经网络进行扩展,提出了多元切比雪夫神经网络模型,并在切比雪夫多项式正交性的基础上给出了快速权值确定算法。仿真实验证明,相对于传统多层感知器神经网络,该方法在计算精度和计算速度等方面都存在明显优势。     

初始轨道的确定方法

卫星初轨确定在卫星轨道改进中起着重要作用.随着测量技术的发展,测量数据不断增多,为初轨确定提供了良好的基础.常见的方法有多项式逼近,切比雪夫多项式逼近.然而在实际初轨确定过程中,它们存在很大的缺点:主要是逼近精度不高.根据实际测量数据,提出了用样条函数逼近的方法来获取初始轨道,这种方法具有逼近精度高,实际容易操作的优点.并且通过计算结果进行了比较,指出了多项式逼近和切比雪夫多项式逼近存在的不足.     

基于正交多项式的核函数性质研究

核函数及其参数的选择是支持向量机(SVM)研究中的一个核心问题.正交多项式的正交性和可变性使其可以构造通用核函数以代替多项式核、高斯核等常用核函数.基于正交多项式构造核函数的参数仅在自然数中取值,因而能较大地简化核参数的选择.分析基于切比雪夫多项式、埃尔米特多项式、勒让德多项式及拉盖尔多项式构造的6类正交多项式核函数的性质,并在多个数据集上对比这些核函数的鲁棒性和泛化性,所得结论可为选择这些核函数进行支持向量分类提供理论依据和技术支持.     

New set of adapted Gegenbauer–Chebyshev invariant moments for image recognition and classification

Orthogonal moments and their invariants to geometric transformations for gray-scale images are widely used in many pattern recognition and image processing applications. In this paper, we propose a new set of orthogonal polynomials called adapted Gegenbauer–Chebyshev polynomials (AGC). This new set is used as a basic function to define the orthogonal adapted Gegenbauer–Chebyshev moments (AGCMs). The rotation, scaling, and translation invariant property of (AGCMs) is derived and analyzed. We provide a novel series of feature vectors of images based on the adapted Gegenbauer–Chebyshev orthogonal moments invariants (AGCMIs). We practice a novel image classification system using the proposed feature vectors and the fuzzy k-means classifier. A series of experiments is performed to validate this new set of orthogonal moments and compare its performance with the existing orthogonal moments as Legendre invariants moments, the Gegenbauer and Tchebichef invariant moments using three different image databases: the MPEG7-CE Shape database, the Columbia Object Image Library (COIL-20) database and the ORL-faces database. The obtained results ensure the superiority of the proposed AGCMs over all existing moments in representation and recognition of the images.

     

Analysis of linear time-varying systems and bilinear systems via shifted Chebyshev polynomials of the second kind

Linear time-varying systems and bilinear systems are analysed via shifted Chebyshev polynomials of the second kind. Using the operational matrix for integration and the product operational matrix, the dynamical equation of a linear time-varying system (or bilinear system) is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of shifted Chebyshev polynomials of the second kind can be determined by using the least-squares method. Illustrative examples show that shifted Chebyshev polynomials of the second kind having a finite number of terms are more accurate than either the Legendre or Laguerre methods.     

Image representation using separable two-dimensional continuous and discrete orthogonal moments

This paper addresses bivariate orthogonal polynomials, which are a tensor product of two different orthogonal polynomials in one variable. These bivariate orthogonal polynomials are used to define several new types of continuous and discrete orthogonal moments. Some elementary properties of the proposed continuous Chebyshev–Gegenbauer moments (CGM), Gegenbauer–Legendre moments (GLM), and Chebyshev–Legendre moments (CLM), as well as the discrete Tchebichef–Krawtchouk moments (TKM), Tchebichef–Hahn moments (THM), Krawtchouk–Hahn moments (KHM) are presented. We also detail the application of the corresponding moments describing the noise-free and noisy images. Specifically, the local information of an image can be flexibly emphasized by adjusting parameters in bivariate orthogonal polynomials. The global extraction capability is also demonstrated by reconstructing an image using these bivariate polynomials as the kernels for a reversible image transform. Comparisons with the known moments are performed, and the results show that the proposed moments are useful in the field of image analysis. Furthermore, the study investigates invariant pattern recognition using the proposed three moment invariants that are independent of rotation, scale and translation, and an example is given of using the proposed moment invariants as pattern features for a texture classification application.     

Parameter estimation of time-varying systems via Chebyshev polynomials of the second kind

A new method for the parameter estimation of linear time-varying systems using Chebyshev polynomials of the second kind is presented. The systems are characterized by linear differential equations with time-varying coefficients that are in the form of polynomials in the time variable. The operational matrices of integration and time-variable multiplication of Chebyshev polynomials of the second kind play key roles in the derivation of the algorithm. Least-squares estimation of overdetermined linear algebraic equations obtained from polynomial approximations of the systems is used to estimate the unknown parameters. Illustrative examples give satisfactory results     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

Properties and performance of orthogonal neural network in function approximation

Backpropagation neural network has been applied successfully to solving uncertain problems in many fields. However, unsolved drawbacks still exist such as the problems of local minimum, slow convergence speed, and the determination of initial weights and the number of processing elements. In this paper, we introduce a single‐layer orthogonal neural network (ONN) that is developed based on orthogonal functions. Since the processing elements are orthogonal to one another and there is no local minimum of the error function, the orthogonal neural network is able to avoid the above problems. Among the five existing orthogonal functions, Legendre polynomials and Chebyshev polynomials of the first kind have the properties of recursion and completeness. They are the most suitable to generate the neural network. Some typical examples are given to show their performance in function approximation. The results show that ONN has excellent convergence performance. Moreover, ONN is capable of approximating the mathematic model of backpropagation neural network. Therefore, it should be able to be applied to various applications that backpropagation neural network is suitable to solve. © 2001 John Wiley & Sons, Inc.     

基于Chebyshev神经网络的图像复原算法

退化图像的点扩散函数难以准确确定,为此,提出一种基于Chebyshev正交基函数的前向神经网络图像复原算法。该算法以一组Chebyshev正交基为隐层神经元的激励函数,采用BP算法对权值进行修正,达到收敛目标。给出2类Chebyshev神经网络的实现步骤及其相应衍生算法的图像恢复实现步骤。实验结果表明,该算法能较好地实现图像复原。