【作者】 姜东焕；

【导师】 宋国乡；

【作者基本信息】 西安电子科技大学 ， 应用数学， 2007， 博士

【摘要】 小波分析是二十世纪九十年代出现的一门新的数学方法,由于它具有时-频局部化特点和多尺度特性,在图像处理领域得到了广泛应用。小波分析已成为新的静态图像压缩标准JPEG2000的核心技术。近年来,图像处理和分析中另一个最新的进展是偏微分方程图像处理理论。本论文主要围绕小波分析与偏微分方程在图像处理中的应用来进行研究。主要做了以下几个方面的工作:1.讨论了小波阈值与尺度空间的关系:一方面,可以用小波阈值建立新的尺度空间;另一方面,可以在尺度空间的基础上推导出新的阈值函数。提出一种基于Besov权的平移不变小波阈值函数。仿真实验的结果表明了新阈值函数的有效性。2.针对Perona-Malik各向异性扩散方程不能保留边缘细节的缺点,从小波和形态学的角度对扩散方程的扩散系数进行修改,分别得到两个新的各向异性扩散方程:基于非线性小波阈值和基于形态学算子的各向异性扩散方程。新的各向异性扩散方程在计算扩散系数时,不仅考虑了图像的梯度,而且考虑了二阶导数,并分别在作用非线性小波阈值和形态学闭开算子后的图像上估计各阶导数。这样,这两种新算子不但能够有效地去除噪声,还能保持图像中的尖峰和窄边缘,从而更好地保持边缘位置。实验结果表明用这两种新算子处理后的图像具有较高的清晰度和对比度。3.提出了两种新的图像边缘检测算法。一是基于小波耦合阈值的边缘检测算法。Canny边缘检测中使用高斯滤波平滑图像,由于高斯滤波使图像出现边缘过度平滑及边缘移位现象,提出用平移不变小波非线性耦合阈值来代替Canny边缘检测中的高斯平滑,得到基于小波耦合阈值的边缘检测算法;另外是小波尺度空间中的边缘检测算法。从尺度空间的观点看,Canny边缘检测方法的多尺度形式等价于在一个线性高斯尺度空间中进行Canny最优算子的边缘检测。用非线性的小波尺度空间代替高斯尺度空间做边缘检测能够获得更好的边缘检测结果,即小波尺度空间中的边缘检测算法。实验仿真表明新算法具有更好的边缘检测性能。4.提出了三种新的图像分解算法。一是基于修正小波阈值的图像分解算法。用具有更高正则性的修正小波阈值(分段n次多项式小波阈值和指数阈值)做图像分解,得到图像分解的变分泛函的近似最小值,其中n可以取任意正数。随着n的增大,图像分解的变分问题的近似最小值越逼近精确最小值。这样,得到了图像分解和修正小波阈值之间的联系。实验结果表明用修正小波阈值分解图像的有效性;另外是基于变分和小波变换的图像分解算法,该算法把图像分解成两部分更多还原

【Abstract】 Wavelets are a new mathematical technique, arising in 1980’s. Wavelets have found wide applications in image processing due to its time-frequency localization and multiscale analysis property. It has been adopted as a key technique in the new still image compression standard JPEG-2000. Recently, another active research is the application of Partial Differential Equations (PDEs) in image processing. In this dissertation, we focus on combining wavelets and PDEs for image processing. The main work can be summarized as follows:1. The relationship between wavelet shrinkage and scale space is discussed. On the one hand, wavelet shrinkage can be used to build a new image smoothing scale space; On the other hand, a new shrinkage function can be derived from the wavelet based scale space. A new translational invariant wavelet shrinkage function depending on Besov weight is further derived. Experiments show that the new shrinkage function is superior to the other wavelet shrinkages.2. As the Perona-Malik anisotropic diffusion cannot preserve edge details well, two improved diffusion models are proposed from the point of wavelet and morphology respectively, which are the anisotropic diffusion equation based on nonlinear wavelet shrinkage and on close-open operator. The modified diffusion coefficient is dependent not only on the gradient of the image, but also on the second derivatives estimated on the image after being applied to nonlinear wavelet shrinkage or close-open operator respectively. The most important advantage of the new models is that they can give a good trade-off between noise removal and edge details preservation. Finally, numerical results are given.3. Two edge detectors are proposed. One is the Canny edge detection based on wavelet coupled shrinkage. As Gaussian filter results in over-smoothness and edge movement of the original image in the Canny edge detection, the new algorithm is proposed in which the Gaussian filter is replaced with the nonlinear wavelet coupled shrinkage. The other is the edge detection in the wavelet-based scale space. From the scale space point of view, the multiscale version of the Canny edge detection is, in fact, the edge detection in a Gaussian scale space. The linear Gaussian scale space is substituted with a nonlinear wavelet-based one, obtaining the edge detection in the wavelet-based scale space. Experimental results demonstrate its superiority over the multiscale version of the Canny edge detection.4. Three algorithms of image decomposition are presented. Firstly, the image更多还原