Out-of-focus Blur: Image De-blurring

Image de-blurring is important in many cases of imaging a real scene or object by a camera. This project focuses on de-blurring an image distorted by an out-of-focus blur through a simulation study. A pseudo-inverse filter is first explored but it fails because of severe noise amplification. Then Tikhonov regularization methods are employed, which produce greatly improved results compared to the pseudo-inverse filter. In Tikhonov regularization, the choice of the regularization parameter plays a critical rule in obtaining a high-quality image, and the regularized solutions possess a semi-convergence property. The best result, with the relative restoration error of 8.49%, is achieved when the prescribed discrepancy principle is used to decide an optimal value. Furthermore, an iterative method, Conjugated Gradient, is employed for image de-blurring, which is fast in computation and leads to an even better result with the relative restoration error of 8.22%. The number of iteration in CG acts as a regularization parameter, and the iterates have a semi-convergence property as well.