Wavelet SVM in Reproducing Kernel Hilbert Space for hyperspectral remote sensing image classification

Combining Support Vector Machine (SVM) with wavelet analysis, we constructed wavelet SVM (WSVM) classifier based on wavelet kernel functions in Reproducing Kernel Hilbert Space (RKHS). In conventional kernel theory, SVM is faced with the bottleneck of kernel parameter selection which further results in time-consuming and low classification accuracy. The wavelet kernel in RKHS is a kind of multidimensional wavelet function that can approximate arbitrary nonlinear functions. Implications on semiparametric estimation are proposed in this paper. Airborne Operational Modular Imaging Spectrometer II (OMIS II) hyperspectral remote sensing image with 64 bands and Reflective Optics System Imaging Spectrometer (ROSIS) data with 115 bands were used to experiment the performance and accuracy of the proposed WSVM classifier. The experimental results indicate that the WSVM classifier can obtain the highest accuracy when using the Coiflet Kernel function in wavelet transform. In contrast with some traditional classifiers, including Spectral Angle Mapping (SAM) and Minimum Distance Classification (MDC), and SVM classifier using Radial Basis Function kernel, the proposed wavelet SVM classifier using the wavelet kernel function in Reproducing Kernel Hilbert Space is capable of improving classification accuracy obviously.     

Large Deviations for the Fermion Point Process Associated with the Exponential Kernel

For the fermion point process on the whole complex plane associated with the exponential kernel , we show the central limit theorem for the random variable ξ(D _r , the number of points inside the ball D _r of radius r, as r → ∞ and we establish the large deviation principle for the random variables {r ⁻²ξ (D _r ), r > 0}.     

Group-Valued Measures on the Lattice of Closed Subspaces of a Hilbert Space

We show there are no non-trivial finite Abelian group-valued measures on the lattice of closed subspaces of an infinite-dimensional Hilbert space, and we use this to establish that the unigroup of the lattice of closed subspaces of an infinite-dimensional Hilbert space is divisible. The main technique is a combinatorial construction of a set of vectors in R²ⁿgeneralizing properties of those used in various treatments of the Kochen–Specker theorem in R⁴.     

Quantic Superpositions and the Geometry of Complex Hilbert Spaces

The concept of a superposition is a revolutionary novelty introduced by Quantum Mechanics. If a system may be in any one of two pure states x and y, we must consider that it may also be in any one of many superpositions of x and y. An in-depth analysis of superpositions is proposed, in which states are represented by one-dimensional subspaces, not by unit vectors as in Dirac’s notation. Superpositions must be considered when one cannot distinguish between possible paths, i.e., histories, leading to the current state of the system. In such a case the resulting state is some compound of the states that result from each of the possible paths. States can be compounded, i.e., superposed in such a way only if they are not orthogonal. Since different classical states are orthogonal, the claim implies no non-trivial superpositions can be observed in classical systems. The parameter that defines such compounds is a proportion defining the mix of the different states entering the compound. Two quantities, p and θ, both geometrical in nature, relate one-dimensional subspaces in complex Hilbert spaces: the first one is a measure of proximity relating two rays, the second one is an angle relating three rays. The properties of superpositions with respect to those two quantities are studied. The algebraic properties of the operation of superposition are very different from those that govern linear combination of vectors. This work was partially supported by the Jean and Helene Alfassa fund for research in Artificial Intelligence, by the Israel Science Foundation grant 183/03 on “Quantum and other cumulative logics” and by EPSRC Visiting Fellowship GR/T 24562 on “Quantum Logic”.     

A Finitely Additive State Criterion for the Completeness of Inner Product Spaces

We show that an inner product space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subspaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subspaces of S. Although it gives only a partial solution to the problem formulated by Pták on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subspaces of S.     

希尔伯特空间有限区域成像积分算子的谱

本文讨论了希尔伯特空间中有限区域成像积分算子的本征值和本征函数,研究了非对称核的情形,并用微扰和有限秩方法计算了衍射受限和离焦情况下的本征值和本征函数。     

有限维q非谐振子广义相干态振幅N次方压缩

利用Zhang等人提出的单模辐射场的振幅N次方压缩理论,研究了有限维希尔伯特空间中单参数q变形非简谐振子广义相干态的振幅N次方压缩效应.结果发现:该量子光场的确存在场的振幅N次方压缩效应,其压缩条件分别与相位角Φ、维度参数s、压缩幂次N、平均光子数的方根r和变形参数q相关;这一结果与无限维希尔伯特空间单参数q变形广义相干态的情形截然不同.     

Phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions

One of the main problems of statistical physics is to describe all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this article is to investigate phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions.     

Gibbs Measures for Brownian Paths Under the Effect of an External and a Small Pair Potential

We consider Brownian motion in the presence of an external and a weakly coupled pair interaction potential and show that its stationary measure is a Gibbs measure. Uniqueness of the Gibbs measure for two cases is shown. Also the typical path behaviour, the degree of mixing and some further properties are derived. We use cluster expansion in the small coupling parameter.     

Amplitude-squared squeezing of the generalized odd-even coherent states of the anharmonic oscillator in a finite-dimensional Hilbert space

This paper discusses the properties of amplitude-squared squeezing of the generalized odd-even coherent states of anharmonic oscillator in finite-dimensional Hilbert space. It demonstrates that the generalized odd coherent states do exhibit strong amplitude-squared squeezing effects in comparison with the generalized even coherent states.     

希尔伯特变换条纹分析法及其在非球面镜测量上的应用

提出了利用离散的希尔伯特变换法解调干涉条纹相位的方法,叙述了希尔伯特变换法解调相位的原理,并从单一的干涉图样中提取出条纹信号的正弦和余弦部分,正弦与余弦比值的反正切即为干涉相位;然后利用泽尼克多项式拟合,实现波前重建;最后讨论了条纹相位的算法,误差修正和测量面形的应用,从而实现对非球面镜的检测。实验表明,该方法可获得均方根(RMS)的精度为λ／10。     

Orthogonality and Disjointness in Spaces of Measures

The disjointness of measures and their Hahn–Jordan decomposition are instances of the general notion of minimal decomposition in base normed spaces. The mixing distance, a specification of a novel concept of angle in real normed vector spaces, is applied to provide a geometric interpretation of disjointness as orthogonality.     

有限维Hilbert空间Roy型奇偶非线性相干态的振幅平方压缩

构造出了有限维Hilbert空间Roy型奇偶非线性相干态, 讨论了它们的正交归一完备性和振幅平方压缩效应. 研究表明, 在此空间中Roy型奇偶非线性相干态是归一完备的, 但不具有正交性; 当复参数相位角θ满足一定条件时它们存在振幅平方压缩效应, 同时导出了压缩条件与参数s,r以及函数f(n)之间的关系. 最后借助于数值计算, 发现对于5维(或7维)Hilbert空间中Roy型偶(或奇)非线性相干态, 当参数θ和Lamb-Dike参数η取某一给定值时, 在参数r变化的不同取值范围内, 它们均可以呈现振幅平方 关键词： 有限维Hilbert空间 Roy型非线性相干态 奇偶非线性相干态 振幅平方压缩     

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.     

有限维Hilbert空间一种新的奇偶非线性相干态

利用数值计算方法,研究了有限维Hilbert空间一种新的奇偶非线性相干态的反聚束效应.研究结果表明:各有限维Hilbert空间新的偶非线性相干态均不出现反聚束效应.但各有限维Hilbert空间新的奇非线性相干态均可出现反聚束效应.     

Resonances in the Rigged Hilbert Space and Lax-Phillips Scattering Theory

The rigged Hilbert space formalism of quantum mechanics provides a framework in which one can identify resonance states and obtain the typical exponential decay law. However, there remain questions of the interpretation and extraction of physical information through the calculation of expectation values of observables. The Lax-Phillips scattering theory provides a mathematical construction in which resonances are assigned with states in a Hilbert space, thus no such difficulties arise. The original Lax-Phillips structure is inapplicable within standard nonrelativistic quantum theory. Through the powerful theory of H ^p spaces certain relations between the two theories are uncovered, which suggest that a search for a unifying framework might prove useful.     

On the (Non)Existence of States on Orthogonally Closed Subspaces in an Inner Product Space

Suppose that S is an incomplete inner product space. In (Dvurečenskij, 1992, Gleason's Theorem and Its Applications, Ister Science Press, Bratislava, Kluwer Academic Publishers, Dordrecht), A. Dvurečenskij shows that there are no finitely additive states on orthogonally closed subspaces, F(S), of S that are regular with respect to finitely dimensional spaces. In this note we show that the most important special case of the former result—the case of the evaluations given by vectors in the “Gleason manner”—allows for a relatively simple proof. This result further reinforces the conjecture that there are no finitely additive states on F(S) at all.     

双平衡式相干接收原理及其在外差干涉仪中的应用研究

利用琼斯矩阵理论推导了光学双平衡式相干接收的原理,得到了具有正交相关性的包含全部光学信息的IQ信号,给出了基于IQ正交信号解调待测光信号信息的信号处理算法.基于该原理设计了一种高速微机电系统扫描式激光外差干涉仪,得到了待测玻璃的厚度信息分布图.该干涉仪的波前相位差提取准确度可高达0.01 rad、测量速度可达40 frame/s、最大测量直径可达到300mm,在实时光学检测方面有广阔的应用价值.     

SPH核函数光滑长度最优选取和自适应准则研究

基于小波分析理论和RKPM再生核函数研究无网格方法SPH中多尺度诊断工具,多尺度再生核函数使得数值计算在不同尺度上的响应分离,并通过动态伸缩窗函数给出计算域不同位置的时频特性,实现在无网格体系下构造网格计算方法的“自适应网格”,从而达到对不同流场位置多分辨率分析的目的.利用多尺度诊断工具中的小波分解算法给出SPH核函数在频域内能量残差估计,发展一种核函数光滑长度最优选取准则.最后,基于可压缩流场激波稀疏波共存的现象,针对传统的光滑长度自适应的缺陷,构造一种避免数值计算“拖尾”现象的自适应准则.