Study On Image Denoising And Image Restoring Based On Partial Differential Equation

PDE method has been extensively applied to many fields, such as image processing and signal processing. However, its application is restricted by its computational complexity. Therefore, the PDE and multi-scale image analysis are combined to form a new image processing method. The new method can be applied in image denoising and image inpainting, and can greatly reduce the computation time spent on PDE. The new method has theoretical and practical value. At present, PDE has been applied to many fields, such as edge and feature extracton, pattern recognition, image segmentation as well as computer vision, and showed good prospects. But many PDE methods suffer from the staircase effect easily, which restricts its application in some fields such as image denoising and image inpainting. So, the fundamental theory and developments of PDE were systematically described in this dissertation. The characteristics of PDE and multi-scale image analysis were analyzed for image denoising and image inpainting in pixel domain and transform domain in detail. The principle and implementation of these methods were deeply studied in this paper.The adaptive P-Laplace diffusion method for images denoising is proposed in this dissertation. An adaptive factor p is proposed based on the local geometry characteristic of curvature and gradient feature of images. The adaptive factor p can control the diffusion direction and diffusion intensity. The Euler-Lagrange equations of the P-Laplace diffusion method are deduced. The diffusion performance of the adaptive P-Laplace diffusion equation is analyzed. The adaptive P-Laplace diffusion has strong diffusion coefficient in the edge direction and has small diffusion coefficient in the gradient direction when diffusion was performed at edge region. But the adaptive P-Laplace diffusion has the same diffusion coefficient at smooth region. The image inpainting algorithm of finite difference is proposed with the half point differential scheme based on the analysis of the adaptive P-Laplace diffusion. Theoretic analysis and experimental results show that the new method has better performances both in vision effect and image quality than the TV method and the constant P-Laplace method.An adaptive de-noising algorithm was proposed based on the nonsubsampled Contourlet transform. Firstly the coefficients in different scales and different directions are obtained by image decomposition using the nonsubsampled Contourlet transform. The texture of the image information is introduced by using the mean of decomposition scale and the variance of region. The threshold should be set lowly when the image has many textures at each decomposition scale. On the contrary, the threshold should be set large. Threshold functions are set with these coefficients adaptively. After the de-noising and reconstruction of these coefficients, image de-noising is implemented. Comparing to the wavelet transform threshold and Contourlet transform threshold, the nonsubsampled Contourlet transform picks up the image detail better and improves signal-to-noise ratio of the peak.We proposed a nonsubsampled Contourlet transform (NSCT) formulation combined with the adaptive P-Laplace variation method for images denoising. Our method aims at reducing Gibbs-type artifacts. The diffusion factor has different diffusional intensity at different region. Firstly, the denoised image which by threshold method of nonsubsampled Contourlet transform is obtained. Secondly, the NSCT coefficients which have been set to zero by the threshold procedure are retained, and the P-Laplace diffusion directly to the reconstruction image from the retained coefficients. Finally, the diffusion image is fused with the denoised image with the thresholds method, and the final denoising image is obtained. The experimental results indicate that our method can improve image quality and maintain much more detail information.We proposed a novel image inpainting method. The new method is anisotropic diffusion. It can control and restore the missing or damaged regions in the nonsubsampled Contourlet transform (NSCT) domain, instead of the pixel domain in which traditional inpainting problems are defined. In the wireless communication of these images, it could happen that certain wavelet packets are randomly lost or damaged during the transmission process. Our method can remedy the lost coefficients in the transform domain. Experimental results show that the proposed algorithm can remedy images effectively and improve image quality significantly, even with relatively large number of lost coefficients.A blind restoration algorithm of P-Laplace diffusion based on anisotropic and nonlinear regularizations is proposed for restoring degraded images, in which the anisotropic and adaptive regularizations are adopted according to the character of image curvature and gradient. The nonlinear and spatial anisotropic regularization functions are suggested to smooth adaptively in the process of recovering the object image. Finally, the cost functions are minimized by alternate minimization scheme. The nonlinear equations are linear by fixed-point iteration scheme. The images can be recovered quickly.