小波变换用于重叠化学信号的分辨研究

【作者】 聂磊；

【导师】 林祥钦； 吴守国；

【作者基本信息】 中国科学技术大学 ， 分析化学， 2002， 博士

【摘要】 本论文工作在综述了小波变换在化学领域应用的基础上, 对小波变换用于分辨重叠化学信号进行了研究, 解决了相关的理论问题, 并在此基础上细致研究了小波变换用于重叠信号的定性分析和定量分析. 其中创新性的工作主要包括以下几个方面: 1. 以特殊小波(Marr, Haar 和DOG 小波和双正交小波), 分别利用连续和离散小波变换对不同类型(Gaussian, Lorentzian和sech2-function型)重叠信号的峰位置提取进行了详细研究, 给出了相应的理论依据. 以数字模拟信号及方波伏安实验信号验证表明: (1) 在合适的尺度和分解次数下, 连续和离散小波变换均能够有效地提取重叠化学信号各组分的峰位置, 可以用于重叠信号的定性分析; (2) Marr, Haar 和DOG 三种小波函数用于连续小波变换提取峰位置的结果相同; (3) 连续小波变换比离散小波变换更具有简便快捷, 尺度选择精细, 对信噪比较低的信号也可直接应用等优越性, 具有广阔的应用前景. 2. 基于小波变换是线性变换, 提出了一种小波变换用于双组分重叠方波伏安信号定量分析的新的基线选取方法, (1) 通过该基线建立的定量测量方法在不同尺度或不同分解次数, 不同分离度以及不同峰参数(峰高和峰宽)的影响下, 均可得到比较满意的峰高定量结果. 与其它基线方法相比, 系统误差绝对值最小, 可直接用于重叠方波伏安信号的定量分析. (2) 考察了几种小波函数, 结果证明: Marr, Haar 和DOG 小波以及双正交小波均可用于定量分析, 且所得结果基本相同. 3. 基于Marr 小波函数, 对信号连续小波变换的实质进行了讨论. 在此基础上, (1)结合连续小波变换的特点和卷积的微分性质, 提出了使用Gaussian 函数的一阶和二阶导数, Haar和三次样条函数的一阶导数作为小波函数的连续小波变换计算信号近似导数的一般性方法, 与其他导数计算方法(包括数字微分法, 多项式滤波法, Fourier 变换法和离散小波变换法)相比, 本法简单便捷, 计算速度快, 对于噪声含量较高的信号(S/N 为5), 只要适当调节尺度即可获得比较满意的结果. (2) 对离散小波变换法进行了改进, 克服了原法的不足. (3) 从理论上分析了连续(以Gaussian函数的一阶和二阶导数, Haar, 三次样条函数的一阶导数为小波函数)和离散(改进)小波变换法(以Haar和三次样条函数的一阶导数为小波)提高分辨率的最大限度. 4. 在连续小波变换提取峰位置的基础上, (1) 建立了贯穿于原始信号及其连续小波变换的交叉迭代算法(CIACWTOS), 获得了更加精确的峰位置. (2) 进而提出了分离双组分重叠方波伏安信号的新方法?反摺平移加减法(FSSM), 并运用模拟信号和实验信号进行了验证, 结果证明该方法比较简单, 能够有效地进行双峰分离, 引入的误差也比较小.更多还原

【Abstract】 On the basis of the development and process of application of wavelet transform (WT) in chemistry, a detailed investigation of resolving and quantifying the overlapped chemical signals by using wavelet transform (WT) was carried out, and the relevant theory was also provided. The innovative research work mainly focused on the four aspects as follows: 1. Using the special wavelets, which included Marr, Haar and DOG for the continuous wavelet transform (CWT) and biorthogonal wavelet for the discrete wavelet transform (DWT), the peak positions of the component’s responses, described by Gaussian, Lorentzian and sech2-function, could be determined via WT from the overlapped signals. The corresponding theoretical bases were also presented in this work. According to the validation of synthetic and real data, the results manifested that (1) the CWT and DWT could both be employed to find the peak positions of the components from the unresolved signals under the suitable dilations or the appropriate resolutions. Hence, the WT could be used to perform the qualitative analysis. (2) For CWT, the different wavelets did not exert the effect on the determination of the peak positions. (3) In term of extracting the peak positions, the CWT had some superior aspects as compared with the DWT, such as simplicity, refined dilations and straightforward application to very noisy signals, so the application of CWT would be very promising. 2. Depending on the linear property of WT, a new method to construct the suitable baseline for quantitative analysis of the overlapped square wave voltammogram (SWV) of two components in WT domain was proposed. (1) The relative satisfactory results of evaluating the peak height could be achieved by using the graphical measures established on this appropriate baseline in the case of different dilations or resolutions, degrees of separation and parameters of the peak (peak width and peak height). Since the proposed measure yielded the least absolute values of systematic errors as compared with other measures, it could be applied to performing the quantitative analysis straightforward for the overlapped SWV without separating each component. (2) The comparison of quantification between the different wavelets was conducted, and demonstrated that Marr, Haar, DOG and biorthogonal wavelet could be used for quantitative analysis and results had little correlation with the wavelets. 3. Using Marr wavelet, the essence of CWT of the signal was discussed. (1) Combining the definition of CWT and the derivative property of convolution, we constructed a general method to calculate the approximate derivative of signal through CWT by using the first and second derivative of Gaussian function, Haar, and the first derivative of three-order-Spline function as wavelets. As compared with the other approaches of calculating derivative, which include the numerical differentiation, polynomial filters, Fourier transform, and the recently proposed DWT method, fast calculation and simple mathematical operation were remarkable advantages of CWT method. For the signal corrupted by severe noise (Signal-to-Noise Ratio = 5), the satisfactory results could also obtained via CWT method through appropriately adjusting the dilations. (2) The improved DWT method after updating the original one was able to cope with the limitations. (3) The limit of resolution enhancement obtained by using CWT method (the first and second derivative of Gaussian function, Haar, and the first derivative of three-order-Spline function as wavelets) and the improved DWT method (Haar and the first derivative of three-order-Spline function as wavelets) was also analyzed from the theoretical angle. 4. (1) An algorithm, cross-iterative algorithm of continuous wavelet transform and original signal (CIACWTOS), based on finding the peak positions by using the Marr wavelet through the CWT, was established on to locate the refined peak positions. (2) On the basis of CIACWTOS, a new method, flip shift subtraction method (FSSM), was proposed to separate the overlapped SWV of the two components. Depending on the validation of simulated signals and real data, the results showed that FSSM was a simple method, and could be utilized to separate the two peaks efficiently. Meanwhile, the introduced errors of FSSM were relatively small.更多还原