小波阈值去噪的性能分析及基于能量元的小波阈值去噪方法研究
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摘要
在数据处理中，通常数据中都存在着各种不易消除的噪声。小波分析由于在时域频域同时具有良好的局部化性质和多分辨率分析的特点，因此不仅能满足各种去噪要求如低通、高通、陷波、随机噪音的去除等，而且与传统的去噪方法相比较，有着无可比拟的优点，成为信号分析的一个强有力的工具，被誉为分析信号的数学显微镜。
小波分析由于能同时在时域和频域中对信号进行分析，所以它能有效地实现对信号的去噪。这也是小波分析的一个重要的应用领域。但是，不同的信号小波去噪的结果也不尽相同。由于可用于去噪的小波母函数是一个集合，在小波去噪的实际应用中采用哪一种小波函数才能得到最好的去噪效果，是一个有待解决的、同时很有实际价值的研究课题。
小波去噪的另一个重要问题就是阈值的选取问题。采用同一种小波对同一个信号进行去噪处理的时候，阈值的选取直接关系到去噪效果的优劣。如果阈值选取过小，那么有一部分的噪声小波系数将不能被置零，从而在去噪后的信号中保留了部分噪声信息，使去噪的效果变差。如果阈值选的偏大，则会将一部分有用信号去掉，使得去噪后的信号丢失信息。因此，在去噪的过程中如何更有效地进行阈值选取，使得噪声被去除的同时尽可能的避免有用信号的丢失，也是一个值得研究的问题。
本文尝试对上述两个问题进行了一些探讨和研究，主要内容如下：
第一章绪论部分介绍了本课题的研究目的。并介绍了目前常用的去噪方法及其之间的比较，接下来针对小波去噪理论和方法着重进行了介绍，包括小波去噪的原理、方法和阈值去噪处理方面的内容；最后介绍了本课题的研究工具。
第二章以用于测定汽油辛烷值的红外吸收光谱分析为背景，构造了一个理想的原始光谱信号，分析评估采用小波去噪方法时各种现有小波和阈值组合的去噪能力。首先讨论了这种方法的可行性。并考虑到小波去噪后信噪比以及原始光谱信号保留率这两者之间的协调关系，基于信噪比(SNR)定义了一个评价去噪优劣的评估系数η。然后对实验数据进行了去噪处理实验。以评价各种小波函数和阈值选取方法的优劣。
第三章以“rigrsure”阈值选取规则为基础，提出了一种基于能量元的小波阈值去噪处理方法，它不仅可以在去噪的过程中将信号增强，使得在有效地去除噪声噪声的同时，减少有用信息的丢失，而且还可以将保留下来的小波系数还原，避免失真。该方法较适用于信噪比较高且含有部分较高频信息的信号。首先介绍了该方法的原理和步骤。然后以叠加了白噪声的理想光谱信号来对这种去噪方法进行仿真分析。仿真实验结果显示，对于不同的小波母函数，采用该方法去噪后的结果优于未经能量元转换的原方法，而且对于含有较高频信息的信号，该方法的结果比原方法明显更优越。
第四章对本文的工作进行了总结，并对小波去噪的理论研究和小波去噪在实际应用
l！摘 要
中存在的问题进行了讨论，并对其进行了展望。
The noise exists inevitably in date processing. Wavelet has good localizing quality at time domain and frequency domain simultaneously and the characteristic of multi-resolution ratio analysis, so it can fulfill all kinds of wave-filtering needs such as low-pass, high-pass, sink wave, random noise denoising. Compare with readitional wave-filtering methods, wavelet has incomparable advantage. Wavelet has become an effective means of signal analysis and is intituled as math microscope of signal analysis.
Wavelet can analyze signal at time domain and frequency simultaneously, so it can denoise effectively. This is an important application area of wavelet. Because the mother wavelets is a gather, that which mother wavelet can get best denoising effect at practical application is an await-resolving research subject which has practiceal value.
Another important problem is threshold selection. The threshold selection has immediacy relation to the result of denosing. Partial wavelet coefficients can't be set zero when the threshold is undersize so parts of noises are retained and some useful signals will be taked off when the threshold is iusto major so parts of useful signals are lost. These cases result in dissatisfactory denoising. So how to select threshold effectively to avoid lose useful signal is problem worth to research.
Some research is processed here on the problems above as follow:
(1). In exordium the research purpose and means is introduced. The denoising methods in common use are presented and compared with each other, then wavelet denoising method and threshold selection is introduced emphatically.
(2). An ideal spectrum signal prototype is constructed based on the infrared ray spectrum of octane level measure to evaluate the performances of wavelet based threshold denoising approaches via different combinations of mother wavelet functions and thresholds. The feasibility of this metnod is discussed first. A performance index (η) is defined to assess the signal to noise ratios (SNR) of denoising results, in consideration of the trade-off between the SNR and the distortion of the original signal after wavelet denoising, the better. Four families of mother wavelets, four threshold selection rules and three threshold rescaling methods are tested in a series of experiments to estimate the functioning of those wavelets and thresholding parameters.
(3). A novel energy cell based wavelet-thresholding denoising method is presented based
on the 'rigrsure' threshold selection scheme. The energy cell is used not only on threshold selection but also on coefficients disposing after threshold selecting. The wavelet coefficients of real signals are strengthened in the process of wavelet denoising to distinguish them from those of noise more efficiently, and, as result, the significant loss of information in the signal content is reduced. The retained wavelet coefficients are restored after thresholding to avoid anamorphosis. This method is propitious to signals with high SNR and part relative high frequentcy informations. The element and the step of this method are presented. The simulation analysis using the ideal spectrum signal prototype which is constructed based on the infrared ray spectrum of octane level measure indicates that this method gives better signal to noise ratios and performance index (g) than conventional wavelet thresholding method via different kinds of mother wavelets.
At the end, a summary is carried out on the research, and the problems existing at wavelet denoising method researching and practical application are discussed.
小波分析由于能同时在时域和频域中对信号进行分析，所以它能有效地实现对信号的去噪。这也是小波分析的一个重要的应用领域。但是，不同的信号小波去噪的结果也不尽相同。由于可用于去噪的小波母函数是一个集合，在小波去噪的实际应用中采用哪一种小波函数才能得到最好的去噪效果，是一个有待解决的、同时很有实际价值的研究课题。
小波去噪的另一个重要问题就是阈值的选取问题。采用同一种小波对同一个信号进行去噪处理的时候，阈值的选取直接关系到去噪效果的优劣。如果阈值选取过小，那么有一部分的噪声小波系数将不能被置零，从而在去噪后的信号中保留了部分噪声信息，使去噪的效果变差。如果阈值选的偏大，则会将一部分有用信号去掉，使得去噪后的信号丢失信息。因此，在去噪的过程中如何更有效地进行阈值选取，使得噪声被去除的同时尽可能的避免有用信号的丢失，也是一个值得研究的问题。
本文尝试对上述两个问题进行了一些探讨和研究，主要内容如下：
第一章绪论部分介绍了本课题的研究目的。并介绍了目前常用的去噪方法及其之间的比较，接下来针对小波去噪理论和方法着重进行了介绍，包括小波去噪的原理、方法和阈值去噪处理方面的内容；最后介绍了本课题的研究工具。
第二章以用于测定汽油辛烷值的红外吸收光谱分析为背景，构造了一个理想的原始光谱信号，分析评估采用小波去噪方法时各种现有小波和阈值组合的去噪能力。首先讨论了这种方法的可行性。并考虑到小波去噪后信噪比以及原始光谱信号保留率这两者之间的协调关系，基于信噪比(SNR)定义了一个评价去噪优劣的评估系数η。然后对实验数据进行了去噪处理实验。以评价各种小波函数和阈值选取方法的优劣。
第三章以“rigrsure”阈值选取规则为基础，提出了一种基于能量元的小波阈值去噪处理方法，它不仅可以在去噪的过程中将信号增强，使得在有效地去除噪声噪声的同时，减少有用信息的丢失，而且还可以将保留下来的小波系数还原，避免失真。该方法较适用于信噪比较高且含有部分较高频信息的信号。首先介绍了该方法的原理和步骤。然后以叠加了白噪声的理想光谱信号来对这种去噪方法进行仿真分析。仿真实验结果显示，对于不同的小波母函数，采用该方法去噪后的结果优于未经能量元转换的原方法，而且对于含有较高频信息的信号，该方法的结果比原方法明显更优越。
第四章对本文的工作进行了总结，并对小波去噪的理论研究和小波去噪在实际应用
l！摘 要
中存在的问题进行了讨论，并对其进行了展望。
The noise exists inevitably in date processing. Wavelet has good localizing quality at time domain and frequency domain simultaneously and the characteristic of multi-resolution ratio analysis, so it can fulfill all kinds of wave-filtering needs such as low-pass, high-pass, sink wave, random noise denoising. Compare with readitional wave-filtering methods, wavelet has incomparable advantage. Wavelet has become an effective means of signal analysis and is intituled as math microscope of signal analysis.
Wavelet can analyze signal at time domain and frequency simultaneously, so it can denoise effectively. This is an important application area of wavelet. Because the mother wavelets is a gather, that which mother wavelet can get best denoising effect at practical application is an await-resolving research subject which has practiceal value.
Another important problem is threshold selection. The threshold selection has immediacy relation to the result of denosing. Partial wavelet coefficients can't be set zero when the threshold is undersize so parts of noises are retained and some useful signals will be taked off when the threshold is iusto major so parts of useful signals are lost. These cases result in dissatisfactory denoising. So how to select threshold effectively to avoid lose useful signal is problem worth to research.
Some research is processed here on the problems above as follow:
(1). In exordium the research purpose and means is introduced. The denoising methods in common use are presented and compared with each other, then wavelet denoising method and threshold selection is introduced emphatically.
(2). An ideal spectrum signal prototype is constructed based on the infrared ray spectrum of octane level measure to evaluate the performances of wavelet based threshold denoising approaches via different combinations of mother wavelet functions and thresholds. The feasibility of this metnod is discussed first. A performance index (η) is defined to assess the signal to noise ratios (SNR) of denoising results, in consideration of the trade-off between the SNR and the distortion of the original signal after wavelet denoising, the better. Four families of mother wavelets, four threshold selection rules and three threshold rescaling methods are tested in a series of experiments to estimate the functioning of those wavelets and thresholding parameters.
(3). A novel energy cell based wavelet-thresholding denoising method is presented based
on the 'rigrsure' threshold selection scheme. The energy cell is used not only on threshold selection but also on coefficients disposing after threshold selecting. The wavelet coefficients of real signals are strengthened in the process of wavelet denoising to distinguish them from those of noise more efficiently, and, as result, the significant loss of information in the signal content is reduced. The retained wavelet coefficients are restored after thresholding to avoid anamorphosis. This method is propitious to signals with high SNR and part relative high frequentcy informations. The element and the step of this method are presented. The simulation analysis using the ideal spectrum signal prototype which is constructed based on the infrared ray spectrum of octane level measure indicates that this method gives better signal to noise ratios and performance index (g) than conventional wavelet thresholding method via different kinds of mother wavelets.
At the end, a summary is carried out on the research, and the problems existing at wavelet denoising method researching and practical application are discussed.
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