Some Theory Studies And Applications Of Digital Image Deblurring

The main idea of this dissertation is to use analysis methods, including mathematical induction, variational method and PDE method, to study the problems, e.g., blurred image denoising, image deblurring, blind image deconvolution, image super-reconstruction and image segmentation. Firstly, we transform the problem to an energy minimization problem, establish the variational model and discuss the existence and uniqueness of the minimizer; Secondly, the corresponding Euler-Lagrange equation is derived; Thirdly, we use the steepest method to derive the heat flow equation and study the existence and uniqueness of the heat flow equation; Finally, we seek numerical solution of the equation and prove the validity of our model through numerical experiment results.The main research results are as follows:1. The study of blurred image using non-blind deblurringWe have studied two aspects of blurred image in the case of the known point spread function (PSF). One of the aspects is that we bring forward blurred image denoising model based on bilateral filter. Firstly, we make use of structuring method to derive the existence of minimizer of the energy, then use discrete functional analysis method and mathematical induction method to discuss the numerical solution of minimizer of the energy. The other is that we put forward an alternate iterative algorithm of combining denoising and deblurring. In the firstly, we utilize the first result of the TV-L~1 denoising model to act as the first initial value of the TV deblurring model, and then use the first result of the TV deblurring model to become the second initial value of the TV-L~1 denoising model again. In such a manner, we get the end results of denoising and deblurring. In this part, we give some experimental results about the above two aspects to verify the efficiency of denoising and alternate iterative algorithm.2. The study of blurred image using blind deconvolution We also have studied two aspects of blurred image when the point spread function is unknown. One of the aspects is that we study the theory of blindly deblurring model. Firstly, we use regularization method to the existence of minimizer of energy functional; Secondly, using the theory of subdifferential and BV function, we derive the Euler-Lagrange equation; Finally, we get the system of heat flow equations by the steepest method. In order to prove the solution of the system of heat flow equations, we firstly get some estimates by discussing the system of discrete heat flow equations according to numerical calculation mode; and then prove the existence of the system of continuous heat flow equations by using these estimates. The other is that we propose two blind deconvolution model based on bilateral filter for gray and color images, respectively. In experimental part, we compare our results with TV blind deconvolution model and non-blind deblurring model.3. The study of combining image super-reconstruction with segmentationIn most cases, image super-reconstruction and segmentation are separated. Based on the ideal of super-reconstruction model by Michael et al. and relaxing Mumford-Shah functional, we recommend a new variational model to image super-reconstruction and image segmentation. Because of the influencing of regularization term, this model can play a role in inpainting.