小波理论及其在图像、信号处理中的算法研究
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摘要
小波分析是国际上新兴的一个前沿研究领域，研究小波的新理论、新方法以及新应用具有重要的理论意义和实用价值。本文旨在完善小波的基本理论，设计新的小波应用算法，进一步拓宽小波的应用范围。主要工作包括：
较为详细地讨论了小波快速算法的矩阵实现形式和卷积实现形式；总结了小波基的数学特性，分析了它们对实际应用的影响和作用；提出了一种从紧框架出发构造规范正交小波基的方法；讨论了M进小波的构造与特性，构造了一组三进规范正交小波基对应的滤波器系数。
针对模极大值原理去噪过程中存在的重构小波系数难的问题，本文提出了一种分段三次样条插值(PCSI)新算法，可以快速高效地重构小波系数；在相关去噪的基础上，提出了一种基于区域相关的小波滤波算法，克服了通常相关算法中由于各尺度间小波系数的偏移导致的判断准确率低的缺点；针对硬阈值法不连续和软阈值法有偏差的缺点，提出了多项式插值法，软、硬阈值折衷法和模平方处理方法等三种改进方案。
推导出Poisson噪声在小波变换下随尺度变化的变化公式，提出了一种基于区间的小波局部域复合滤波算法；针对Film-grain型噪声的特性，通过最小化估计信号与真实信号之间的均方差计算并得到了一个变换域最佳滤波算子，使得阈值选取具有自适应性。
依次提出了基音周期检测与汉语声调识别的小波变换峰值检测算法、基于小波变换的语音数字水印嵌入与检测算法、图像噪声去除的小波相位滤波算法以及基于小波变换的遥感图像多尺度数据融合算法；首次把小波包变换的方法用于医学中的胃动力检测；首次将小波变换的方法用于太阳射电爆发中的网纹消除与图像增强。仿真试验结果表明了上述算法的有效性和可行性。
Wavelet analysis is a novel research field in the world. To study the new theory, methods and applications of wavelets is of great theoretical significance and practical value. This paper aims to consummate the wavelet theory, present some new algorithms and develop the new scopes of wavelet applications. The main results include:
Two fonns, the matrix form and the convolution form, for the realization of the wavelet fast algorithm are discussed in detail. An analysis is made on the influence of the wavelet bases on practical applications by studying their mathematical properties. A method for constructing orthonormal wavelet bases from tight frames is presented. A rank 3 wavelet basis and its corresponding filter coefficients are constructed by analyzing the properties of rank M wavelet.
To overcome the difficulty of reconstructing wavelet coefficients in the modulus maximum denoising, this paper presents a new piecewise cubic spline interpolating (PC SI) algorithm, with which the wavelet coefficients can be reconstructed fast and efficiently. A threshold filtering algorithm based on the region relativity of the wavelet coefficients is presented to overcome the disadvantage of the relativity-based algorithm available which is inaccurate in computing the relativities of the deflected wavelet coefficients. To avoid the discontinuity caused by using the hard-thresholding model and the biased estimation caused by using the sofi-thresholding model, we present three improved models of threshold estimation. These three models are: polynomial interpolating model, compromise model (in between the hard-thresholding and softthresholding models), and the modulus squared model.
A variation formula of Poisson noise with the decomposition scale in waveletdomain is derived and then a local wavelet-domain multiple filtering algorithm is presented. According to the property of Film-grain noise, we compute and obtain an optimal filtering operator in transform domain by minimizing the mean square error between the estimated signal and the original signal, which makes the threshold selfadaptive.
A peak-value detection algorithm with the wavelet transform is given, which can be used for exact pitch detection and accurate Chinese tone recognition. And the following three algorithms are presented respectively: an embedded and detection algorithm of audio digital water marking with the wavelet transform, a wavelet phase filtering algorithm for image noise removal, and a new method for multiscale image
data fusion based on the wavelet transfonn. Besides, the wavelet packet method is used, for the first time, to detect the gastric motility and the wavelet transform method is used, for the first time, to remove the grid texture and enhance the image in solar radio bursts. The experimental results show that all the methods presented in this paper are efficient and practical.
较为详细地讨论了小波快速算法的矩阵实现形式和卷积实现形式；总结了小波基的数学特性，分析了它们对实际应用的影响和作用；提出了一种从紧框架出发构造规范正交小波基的方法；讨论了M进小波的构造与特性，构造了一组三进规范正交小波基对应的滤波器系数。
针对模极大值原理去噪过程中存在的重构小波系数难的问题，本文提出了一种分段三次样条插值(PCSI)新算法，可以快速高效地重构小波系数；在相关去噪的基础上，提出了一种基于区域相关的小波滤波算法，克服了通常相关算法中由于各尺度间小波系数的偏移导致的判断准确率低的缺点；针对硬阈值法不连续和软阈值法有偏差的缺点，提出了多项式插值法，软、硬阈值折衷法和模平方处理方法等三种改进方案。
推导出Poisson噪声在小波变换下随尺度变化的变化公式，提出了一种基于区间的小波局部域复合滤波算法；针对Film-grain型噪声的特性，通过最小化估计信号与真实信号之间的均方差计算并得到了一个变换域最佳滤波算子，使得阈值选取具有自适应性。
依次提出了基音周期检测与汉语声调识别的小波变换峰值检测算法、基于小波变换的语音数字水印嵌入与检测算法、图像噪声去除的小波相位滤波算法以及基于小波变换的遥感图像多尺度数据融合算法；首次把小波包变换的方法用于医学中的胃动力检测；首次将小波变换的方法用于太阳射电爆发中的网纹消除与图像增强。仿真试验结果表明了上述算法的有效性和可行性。
Wavelet analysis is a novel research field in the world. To study the new theory, methods and applications of wavelets is of great theoretical significance and practical value. This paper aims to consummate the wavelet theory, present some new algorithms and develop the new scopes of wavelet applications. The main results include:
Two fonns, the matrix form and the convolution form, for the realization of the wavelet fast algorithm are discussed in detail. An analysis is made on the influence of the wavelet bases on practical applications by studying their mathematical properties. A method for constructing orthonormal wavelet bases from tight frames is presented. A rank 3 wavelet basis and its corresponding filter coefficients are constructed by analyzing the properties of rank M wavelet.
To overcome the difficulty of reconstructing wavelet coefficients in the modulus maximum denoising, this paper presents a new piecewise cubic spline interpolating (PC SI) algorithm, with which the wavelet coefficients can be reconstructed fast and efficiently. A threshold filtering algorithm based on the region relativity of the wavelet coefficients is presented to overcome the disadvantage of the relativity-based algorithm available which is inaccurate in computing the relativities of the deflected wavelet coefficients. To avoid the discontinuity caused by using the hard-thresholding model and the biased estimation caused by using the sofi-thresholding model, we present three improved models of threshold estimation. These three models are: polynomial interpolating model, compromise model (in between the hard-thresholding and softthresholding models), and the modulus squared model.
A variation formula of Poisson noise with the decomposition scale in waveletdomain is derived and then a local wavelet-domain multiple filtering algorithm is presented. According to the property of Film-grain noise, we compute and obtain an optimal filtering operator in transform domain by minimizing the mean square error between the estimated signal and the original signal, which makes the threshold selfadaptive.
A peak-value detection algorithm with the wavelet transform is given, which can be used for exact pitch detection and accurate Chinese tone recognition. And the following three algorithms are presented respectively: an embedded and detection algorithm of audio digital water marking with the wavelet transform, a wavelet phase filtering algorithm for image noise removal, and a new method for multiscale image
data fusion based on the wavelet transfonn. Besides, the wavelet packet method is used, for the first time, to detect the gastric motility and the wavelet transform method is used, for the first time, to remove the grid texture and enhance the image in solar radio bursts. The experimental results show that all the methods presented in this paper are efficient and practical.
引文
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