Study Of Wavelet And Partial Differential Equations For Image Processing

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...