Numerical Computation Of Inverse Problems In Image Processing

Image decomposition and image denoising are two very important problems in image processing that have seen many recent developments. One of the most popular models is the total variation minimizing model proposed by Rudin, Osher, and Fatemi (ROF). The ROF model is well known for its ability to remove noise while preserving sharp edges. However, a particular caveat of TV regularization is staircasing phenomenon in recovered images. Besides, many details can be found in removed noise. First, the thesis did some analysis on the total variation minimizing models, especially considering staircasing phenomenon and oscillating term. Based on above analysis, the thesis proposed a new model by adding a constrained condition, which combined higher derivatives. Then, the thesis introduced negative Sobolev norm to measure the fidelity term, and used iterative regularization in order to improve recovered image results. Finally, the dual method and Newton method were used to solve the minimization problems, and proofs of convergence were proved. The numerical experiments showed that the proposed new model and iterative procedure preserved more details and reduced staircasing phenomenon. In addition, it can be claimed that the dual method is faster and more stable than time marching algorithms.