Haar Wavelet and Its Application in Optimal Control of Linear Time-invariant Systems

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整数Haar变换结合DPCM的无失真图像压缩

讨论了整数Ｈａａｒ 小波的构造方法,在此基础上提出了一种整数Ｈａａｒ 小波与ＤＰＣＭ 相结合的无失真图像编码方法,该方法运算速度快,硬件实现简单,与ＡＲＪ和ＪＰＥＧ 中使用的无失真编码方法相比,压缩比平均提高了１２ ％ 左右     

边Haar子波的光学实现及图像边特征提取

提出一种应用４ｆ 系统制作边 Ｈａａｒ 子波匹配滤波器从而实现光学子波变换的新方法．在傅里叶平面引入正弦光栅,使输入平面两个方孔的正负一级衍射像边缘密接,且位相相反,达到子波要求的极性．给出了理论分析和提取图像边特征的实验结果．     

归一化Haar变换谱技术在逻辑函数对称性检测中的应用

基于归一化Haar变换谱技术,讨论了检测逻辑函数对称性的一种新方法.实例表明该方法具有直观、简便和准确的特点.     

Topological equivalence of Haar measures on compact groups

The existence of homeomorphisms establishign an isometry of normalized Haar measures on (metrizable) compact groups is studied. In the case of 0-dimensional groups, a complete answer is given in terms of the indices of open normal subgroups. For example, for the countable powers of the groups ℤ/(m) and ℤ/(n), the answer is affirmative if and only ifm andn have the same prime divisors. A certain class of extensions of 0-dimensional groups is also studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 188–194, August, 2000.     

模拟尾流泡幕及实际尾流光学性质对比研究

对模拟尾流泡幕及实际尾流光学性质进行了对比研究.结果表明,实验室模拟尾流气泡幕的散射光平均照度均值随着压强变化梯度大,可据此定性区别不同压强的气泡幕.实际尾流随着航速增大,散射光照度减小,但这种减小不是线性的.散射光照度一定周期性的规律,说明随着航速的增大,尾流气泡幕逐渐增强,气泡幕中气泡密度、不同直径气泡分布等均随之变化,这可能预示着气泡幕的增强存在饱和度.实验室模拟尾流气泡幕与实际尾流的散射光信号在轮廓方面是相似的,都随着压强或航速的增加而减小.通过比较实验室模拟气泡幕和实际尾流的差值信号,可以有效地消除杂散光或自然光的影响,利用Haar小波对差值信号的分析也得到相同的结论.     

Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data

In this paper, we study the problem of estimating a multivariate normal covariance matrix with staircase pattern data. Two kinds of parameterizations in terms of the covariance matrix are used. One is Cholesky decomposition and another is Bartlett decomposition. Based on Cholesky decomposition of the covariance matrix, the closed form of the maximum likelihood estimator (MLE) of the covariance matrix is given. Using Bayesian method, we prove that the best equivariant estimator of the covariance matrix with respect to the special group related to Cholesky decomposition uniquely exists under the Stein loss. Consequently, the MLE of the covariance matrix is inadmissible under the Stein loss. Our method can also be applied to other invariant loss functions like the entropy loss and the symmetric loss. In addition, based on Bartlett decomposition of the covariance matrix, the Jeffreys prior and the reference prior of the covariance matrix with staircase pattern data are also obtained. Our reference prior is different from Berger and Yang’s reference prior. Interestingly, the Jeffreys prior with staircase pattern data is the same as that with complete data. The posterior properties are also investigated. Some simulation results are given for illustration.     

Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via rationalized Haar functions

Rationalized Haar functions are developed to approximate of the nonlinear Volterra–Fredholm–Hammerstein integral equations. The properties of rationalized Haar functions are first presented, and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.     

Estimation of the multivariate normal precision and covariance matrices in a star-shape model

In this paper, we introduce the star-shape models, where the precision matrix Ω (the inverse of the covariance matrix) is structured by the special conditional independence. We want to estimate the precision matrix under entropy loss and symmetric loss. We show that the maximal likelihood estimator (MLE) of the precision matrix is biased. Based on the MLE, an unbiased estimate is obtained. We consider a type of Cholesky decomposition of Ω, in the sense that Ω=Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements. A special group , which is a subgroup of the group consisting all lower triangular matrices, is introduced. General forms of equivariant estimates of the covariance matrix and precision matrix are obtained. The invariant Haar measures on , the reference prior, and the Jeffreys prior of Ψ are also discussed. We also introduce a class of priors of Ψ, which includes all the priors described above. The posterior properties are discussed and the closed forms of Bayesian estimators are derived under either the entropy loss or the symmetric loss. We also show that the best equivariant estimators with respect to is the special case of Bayesian estimators. Consequently, the MLE of the precision matrix is inadmissible under either entropy or symmetric loss. The closed form of risks of equivariant estimators are obtained. Some numerical results are given for illustration. The project is supported by the National Science Foundation grants DMS-9972598, SES-0095919, and SES-0351523, and a grant from Federal Aid in Wildlife Restoration Project W-13-R through Missouri Department of Conservation.     

Greedy Algorithm for General Biorthogonal Systems

We consider biorthogonal systems in quasi-Banach spaces such that the greedy algorithm converges for each xX (quasi-greedy systems). We construct quasi-greedy conditional bases in a wide range of Banach spaces. We also compare the greedy algorithm for the multidimensional Haar system with the optimal m-term approximation for this system. This substantiates a conjecture by Temlyakov.