Normal Multiresolution Approximation of Curves

A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we study properties such as regularity, convergence, and stability of a normal multiresolution analysis. In particular, we show that these properties critically depend on the underlying subdivision scheme and that, in general, the convergence of normal multiresolution approximations equals the convergence of the underlying subdivision scheme.     

Beyond Besov Spaces,Part 2: Oscillation Spaces

We study several extensions of Besov spaces; these extensions include the oscillation spaces O_p^s,swhich take into account correlations between the positions of large wavelet coefficients through the scales and, more generally, spaces defined through the distributions of suprema of wavelet coefficients on wavelet subtrees. These spaces are independent of the particular wavelet basis chosen. Examples of applications will be taken from image processing and multifractal analysis.     

Hidden corrosion detection in aircraft aluminum structures using laser ultrasonics and wavelet transform signal analysis

Preliminary results of hidden corrosion detection in aircraft aluminum structures using a noncontact laser based ultrasonic technique are presented. A short laser pulse focused to a line spot is used as a broadband source of ultrasonic guided waves in an aluminum 2024 sample cut from an aircraft structure and prepared with artificially corroded circular areas on its back surface. The out of plane surface displacements produced by the propagating ultrasonic waves were detected with a heterodyne Mach-Zehnder interferometer. Time-frequency analysis of the signals using a continuous wavelet transform allowed the identification of the generated Lamb modes by comparison with the calculated dispersion curves. The presence of back surface corrosion was detected by noting the loss of the S(1) mode near its cutoff frequency. This method is applicable to fast scanning inspection techniques and it is particularly suited for early corrosion detection.     

Interpolation operators associated with sub-frame sets

Interpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the ``local commutant' space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the -congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same -congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set.

     

小波理论在VaR计算中的应用

本文根据小波理论研究金融市场中的资产组合收益的概率密度函数的估计问题 ,进而讨论金融市场风险的重要度量———VaR的计算问题 ,并给出相应结果 .     

An integral transform associated to the Poisson integral and inversion of Flett potentials

We introduce a new weighted wavelet-like transform, generated by the Poisson integral and a “wavelet measure.” By making use of the relevant Calderón-type reproducing formula, we obtain an explicit inversion formula for the Flett potentials which are interpreted as negative fractional powers of the operator (E+Λ), where Λ=(−Δ)^1/2, Δ is the Laplacian and E is the identity operator.     

Wavelet frames and shift-invariant subspaces of periodic functions

A general approach based on polyphase splines, with analysis in the frequency domain, is developed for studying wavelet frames of periodic functions of one or higher dimensions. Characterizations of frames for shift-invariant subspaces of periodic functions and results on the structure of these subspaces are obtained. Starting from any multiresolution analysis, a constructive proof is provided for the existence of a normalized tight wavelet frame. The construction gives the minimum number of wavelets required. As an illustration of the approach developed, the one-dimensional dyadic case is further discussed in detail, concluding with a concrete example of trigonometric polynomial wavelet frames.     

Sard-optimal prefilters for the Fast Wavelet Transform

A direct input of function samples into the Fast Wavelet Transform often gives inaccurate results. We use the refinement equation for the construction of prefiltering quadrature formulas which are optimal in Sard's sense, i.e.,in the standard class of functions with ‖ f ^(p)‖₂ < ∞. A detailed analysis of the error and several applications are given. This revised version was published online in August 2006 with corrections to the Cover Date.     

联用色谱数据的多尺度分析

引入基于正交小波基的多尺度分析，并将其用于色谱／光谱二维数据压缩和消除背景影响的研究，结果表明，该法不但成功地消除了复杂的色谱／光谱背景有同时大计压缩数据量，对模拟和实际体系的均获得满意的结果。     

一种自适应的盲水印方法

目前数字水印算法多局限于在图象中嵌入特定的标记,并通过改变空域值或频域系数值来达到目的.提出了一种新颖的自适应盲水印方法.首先基于小波变换的零树结构,找出重要系数,然后根据重要系数所在区域的特性构造出一个序列信息,再利用置乱变换最终得到水印信息,而不修改图象本身的任何特征.因此水印的不可见性从根本上得到了保证.其中,重要系数的选择是自适应于图象的,并且水印的检测是不需要原始图象的盲检测.实验证明该方法的鲁棒性较强,是一种行之有效的方法.