Primal-dual algorithm and ADMM for digital image processing

This thesis introduce two algorithms to remove the noise and blur from the image. In the First section, we will talk about the primal-dual algorithms, which is efficient to solve the non-smooth convex problem. For the general problem, this method will converge to the saddle point with rate O(1/N) in finite dimension Hilbert space. Furthermore, when either the primal object or dual object is uniformly convex, we can deduce that the convergence rate can achieve O(W N/2). When both the primal object and dual object are uniformly convex, we can deduce that the convergence rate can achieve O(1/N 2 ). Since the primal-dual algorithm is sensitive to the regularization parameter and it depends on that the dual problem is solvable, in the second section, we introduce a method using the alternating direction method of multipliers (ADMM) strategy by just adding a new variable. This method will converge to the solution when the data obtained is exact. When the data is noisy, we should add an additional stop criterion to obtain a solution