SOME TOPICS ON WAVELET ANALYSIS

A review of the advance in the theory of wavelet analysis in recent years is given.     

Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L ₂(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X _H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? ^N of Hausdorff dimension D. This rate turns out to be of order n ^?H/D (log?n)^1/2. In the case T=[0,1] ^N we present a concrete wavelet representation of X _H leading to an approximation of X _H with the optimal rate n ^?H/N (log?n)^1/2.     

Duality and interpolation of anisotropic Triebel–Lizorkin spaces

We study properties of anisotropic Triebel–Lizorkin spaces associated with general expansive dilations and doubling measures on using wavelet transforms. This paper is a continuation of (Bownik in J Geom Anal 2007, to appear, Trans Am Math Soc 358:1469–1510, 2006), where we generalized the isotropic methods of dyadic -transforms of Frazier and Jawerth (J Funct Anal 93:34–170, 1990) to non-isotropic settings. By working at the level of sequence spaces, we identify the duals of anisotropic Triebel–Lizorkin spaces. We also obtain several real and complex interpolation results for these spaces. The author was partially supported by the NSF grants DMS-0441817 and DMS-0653881. The author wishes to thank Michael Frazier and Dachun Yang for valuable comments and discussions on this work.     

Sub-optimal wavelet denoising of coaveraged spectra employing statistics from individual scans

This paper proposes a novel wavelet denoising method, which exploits the statistics of individual scans acquired in the course of a coaveraging process. The proposed method consists of shrinking the wavelet coefficients of the noisy signal by a factor that minimizes the expected square error with respect to the true signal. Since the true signal is not known, a sub-optimal estimate of the shrinking factor is calculated by using the sample statistics of the acquired scans. It is shown that such an estimate can be generated as the limit value of a recursive formulation. In a simulated example, the performance of the proposed method is seen to be equivalent to the best choice between hard and soft thresholding for different signal-to-noise ratios. Such a conclusion is also supported by an experimental investigation involving near-infrared (NIR) scans of a diesel sample. It is worth emphasizing that this experimental example concerns the removal of actual instrumental noise, in contrast to other case studies in the denoising literature, which usually present simulations with artificial noise. The simulated and experimental cases indicate that, in classic denoising based on wavelet coefficient thresholding, choosing between the hard and soft options is not straightforward and may lead to considerably different outcomes. By resorting to the proposed method, the analyst is not required to make such a critical decision in order to achieve appropriate results.     

Wavelet Transformation and Wigner-Husimi Distribution Function

We find that the optical wavelet transformation can be used to study the Husimi distribution function in phase space theory of quantum optics. We prove that the Husimi distribution function of a quantum state |ψ〉 is just the modulus square of the wavelet transform of with ψ(x) being the mother wavelet up to a Gaussian function. Thus a convenient approach for calculating various Husimi distribution functions of miscellaneous quantum states is presented.     

Wavelet Bases for Confinements of the Infrared Divergence

We propose the construction of wavelet bases with pseudo-polynomials adapted to the homogeneous Sobolev spaces , s−n/2∈ℕ. They provide a confinement of the infrared divergence by decomposing as a direct sum X ⊕ Y where X is a “small” space which carries the divergence and Y can be embedded in . In the case of we also construct such an orthonormal basis, which provides a confinement of the Mumford process.     

Wavelet expansions and asymptotic behavior of distributions

We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S₀(R)⊂S(R) and its dual space , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in . A characterization of boundedness and convergence in is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.     

基于双目立体视觉实现物体三维测量的研究

以双目立体视觉理论为基础,提取立体图像的Harris角点作为特征点,经过基于小波变换的子线段匹配方法,采用由粗到细的匹配策略匹配端点,盒滤波技术加速法,解决了匹配准确度和匹配速度之间的平衡问题.采用该方法对水泥冷却机内熟料进行测量,实验表明该方法能够较精确的对物体实现三维测量.     

非Kolmogorov大气湍流温度谱标度指数的测量与分析

 阐述了非Kolmogorov湍流谱理论以及湍流谱标度指数的测量与计算方法。在近地面多个地点对大气湍流温度起伏进行了多次的实验观测，结果表明：实际大气湍流温度谱标度指数多数不等于-5/3，并且通常在-2到-1之间变化。分析了湍流温度谱标度指数与湍流发展程度的相关性，利用小波分析方法展现了不同湍流强度下湍流温度脉动能量在各尺度之间的分配状态，发现湍流温度谱标度指数的绝对值在一定程度上随湍流强度的增加而增大。     

高温拉曼光谱提高信噪比的途径分析

结合累积时间分辨和空间分辨建成了上海大学高温拉曼谱仪SU-HTRS(T/S).由于高温噪声的不可避免,采用了移动平均方法,傅里叶变换滤波和小波分析的方法对其进行了数据降噪处理.在不同噪声水平上分析了不同处理方式对谱峰中心位置、强度、面积的影响及处理后数据的光滑度的影响.三种方法比较起来,移动平均法快而简单,小波分析的去噪效果最好.