Adaptive Gradient Algorithm Image Restoration

Image processing has become an important approach to obtain complex information from external world ininformation age, and it has extended the vision of human beings in some degree. At present image processingtechnology has been developed quickly. It has been an important and useful technique in the fields of imageunderstanding and computer vision. Currently, image processing has been widely applied in lots of fields andacquired enormous social and economic benefits. In this thesis, we mainly study some image restoration prob-lems in image processing. The main work of this thesis is list as follows.(1) Images are often corrupted by salt-and-pepper noise. The principle of noise removal is to suppress the noisewhile preserving image details information. For this kind of noise, we firstly present a model to denoise, andthen apply a two-phase method to solve it. In the first phase, adaptive median filter is used to detect the con-taminated pixels. Then in the second phase, the candidate pixels will be restored by minimizing a regularizationfunctional. To end of this, a spectral conjugate gradient method is considered. Its global convergence result canbe established under some suitable conditions. Experimental results show that the proposed approach is efficientand practical.(2) Image restoration from a blurry and noisy observation is known to be ill-posed. We give a primal-dual-basedalternating direction iterative minimization method for image restoration. This method contains two steps. Wefirstly perform the deblurring by solving a linear system. Secondly, we apply a nonmonotone adaptive projectedgradient algorithm to the image generated by the previous step. Under some suitable conditions, we get theglobal convergence of this method. Numerical results and some comparisons indicate that the proposed primal-dual-based alternating minimization method is stable and efficient.(3) Finally, we propose a projected conjugate gradient method for recovering sparse signals. Global convergenceresult is established under some suitable conditions. In order to illustrate the effectiveness of the proposed algo-rithm, we apply it to solve the signal recovering model in compressive sensing. The construction of this methodconsists of two main phases:(1) reformulate a l1regularized least squares problem into an equivalent nonlinearsystem of monotone equations;(2) apply a conjugate gradient method with projection strategy to the resultingsystem. The derived method only needs matrix-vector products at each step and could be easily implemented.Numerical results demonstrate that the proposed method can improve the computation time while obtainingsimilar reconstructed quality.