Applications, Modeling Theories Of PDEs And Wavelet In Image Processing

In recent years, wavelet multi-resolution analysis and partial differential equations (PDEs) play active roles in many image processing fields. They have become two basic tools for image processing and computer vision. This dissertation mainly studies the models and applications of wavelet multi-resolution and PDEs in image processing. The main work can be summarized as follows:1.Wavelet iterative regularization method and its application in image denoising.Firstly, we generalize the wavelet-based iterative regularization method and the wavelet-based inverse scale space to shift invariant wavelet-based cases. Then, a stepwise parameter is derived from wavelet-based iterative regularization. The wavelet-based iterative regularization with the new parameters, which controls the extent of denoising more precisely in the wavelet domain, leads to iterative global wavelet shrinkage. We provide a proof of the convergence and obtain a stopping criterion for the iterative procedure with the new scale parameter based on wavelet transform. Numerical examples show that the proposed iterative regularized method can well preserve the details of images.2.The modified iterative non-local means (NLM) for image denoising.We propose two iterative NLM algorithms for image denoising. The first one can be in-terpreted as a fixed point method for minimizing a weighted non-local regularization functional. The stability of the iterative procedures is guaranteed. The second one generalizes the first one. The computation of the similarity is based on the gray value of the last iterated image, and then the weighted averaging is computed over the last iterated image. Numerical experiments illustrate that the proposed algorithms can preserve the texture structures well, and the contrast of the processed images are much more clearer.3.Image zooming models based on wavelet and partial differential equations.We first give a noisy image zooming method which is based on wavelet and diffusion equations with scalar diffusivity. We use the original image as the low pass and estimate its high pass, and then reconstruct to obtain a zoomed image. Finally, to enhance the edges, forward and backward (FAB) diffusion filtering is used. The second model is based on the wavelet and diffusion equation with matrix valued diffusivity. The high pass are set to zeros. The diffusion equation with matrix-valued diffusivity is used to the zoomed image. Then we apply wavelet analysis to estimate high pass again. Numerical examples illustrate that with the proposed two models while the image is zoomed it is denoised and edges are preserved well.4. Image zooming and enhancing models based on coupled wavelet shrinkage.A new image zooming and enhancing method is proposed using the relationship between wavelet transform and diffusion equations. We use the wavelet method to obtain the initial zoomed image. We apply the shift invariant wavelet analysis to the initial zoomed image, use the coupled wavelet shrinkage to the high pass, and finally reconstruct to obtain the final zoomed image. The proposed model is easy to implement. Numerical experiments illustrate that it is a valid image zooming and enhancing algorithm.5. Image denoising based on wavelet shrinkage and inverse scale space method (ISS).The curvature term of the ISS can suppress the edge artifacts and preserve sharp edges. Based on this property, we apply inverse scale space to wavelet transform. A stopping criterion is obtained by Bregman distance on wavelet domain. Numerical examples show that the pro-posed model can improve the denoising ability. The details of the processed images are well preserved.In conclusion, based on the advantages of wavelet multi-resolution and partial differential equations (PDEs) in image denoising and image zooming, several variational regularization algorithms are proposed. The proposed image denoising and zooming algorithms are valuable for the study of computer vision, image processing and pattern recognition.