Tailored Finite Point Methods For Image Processing Based On Variational Principle

Digital image processing is the process of converting the image signals into digital signals and using computers to process them,which has wide applications in many science and engineering fields.Traditional image processing methods include image coding,image enhancement,image restoration,image compression,image segmentation and image analysis.We focus on image restoration and image segmentation in this thesis.The essence of the image restoration is the inverse of image degradation process.It restores the obtained degraded images containing noises,blurring or loss of information to the original clean images,which is an optimal estimation of the original image under a certain sense,including many aspects,such as image denoising and deblurring,image restoration,and the super-resolution of the image,etc.For the image denoising and deblurring,we propose using the tailored-finite-point method?TFPM?to solve the resulting parabolic or elliptic equations when minimizing the Rician denoising model developed by Getreuer et al.in^[1]using augmented Lagrangian methods?ALM?.Different from traditional finite difference schemes,TFPM employs the method of weighted residuals with collocation technique,which helps get more accurate approximate solutions to the equations and thus reserve more details in restored images.Numerical experiments demonstrate that with the new schemes the quality of restored images has been improved.Besides these,the existence of the minimizer of the Rician denoising model have also been established in this article.For the image restoration,we propose using the tailored-finite-point method?TFPM?to solve the modified Chan-Hilliard equation when inpainting the binary and grayscale images developed by Bertozzi et al.in^[2]and Cherfils et al.^[3]using convexity splitting algorithm.Different from traditional finite difference schemes,TFPM employs method of weighted residuals with collocation technique,which helps get more accurate approximate solution to the equations and thus reserve more details in the restored images.Numerical experiments demonstrate that with the new schemes the quality of restored images has been improved.Besides these,the stability of the algorithm have also been established in this article.Image segmentation is the process of dividing an image into distinct regions and extracting the objects of interest^[4].Image segmentation is a classic problem in image processing.Since the seventies,the problem of image segmentation has attracted a lot of researchers to make great efforts.So far there is no universal method.For image segmentation,we propose a novel model for image segmentation by using the Cahn-Hilliard equation.An interesting feature of this model lies in its ability of interpolating missing contours along wide gaps in order to form meaningful object boundaries,which is often achieved by curvature dependent models in the literature.To solve the associated equation,we employ a recently developed technique,that is,the tailored-finite-point method?TFPM?,which helps preserve sharp jumps and thus helps locate segmentation contours.Numerical experiments are presented to demonstrate the effectiveness of the proposed model and its features.Beside these,analytical results on the existence and uniqueness of the associated equation are also provided.