Research On Image Restoration With Combined Bases

Image restoration is a typical inverse problem, whose modeling and solution is one of the key issues in the field of image processing. Recently, sparsity is introduced into image processing and is widely utilized. On support of the National Natural Science Foundation of China under grant NSFC-607720191, a project named "sparse representation of images by combined bases and its applications", aiming at image restoration, concentrating on modeling and solutions of image recovery by sparsity constraint in wavelets domain, we have done a piece of probing research.First is an introduction to wavelet transform, including the continuous version, the discrete version and its manifest properties, such as multi-resolution. Following that, it is a discussion of the modeling of image restoration, solutions and difficulty of linear inverse problems, regularization and its commonly used methods, such as Wiener filtering, quadratic regularization and total variation.Then, with the fundamentally thorough research on non-quadratic regularization method by sparsity constraint, a mixed two step iterative shrinkage thresholding algorithm for image restoration, abbreviated as MixIST, is proposed. This algorithm employs coefficients from wavelet transform as regularization item, turning the restoration in wavelet domain into a majorization problem. MixIST absorbs the benefits of Iterative Re-weighted Shrinkage algorithm as well as Iterative Shrinkage Thresholding algorithm, and in consequence it bears a nice performance on whether image denoising, image deblurring or image recovery under ill-posed conditions. Meanwhile, the adoption of two step iterative policy enables a high efficiency of solution for image restoration.Finally, in the interest of inspection of real effect of MixIST for image restoration, we have done plenty analysis and experiments on its convergence and restoration performance. Through experiments of MixIST on degraded images, we verify its convergence, the Signal to Noise Ratio effect, the visual effect by comparison with other algorithms, the influences of redundant wavelets and so on. A great number of experiments show that MixIST restores original image faster with better performances, compared with either Iterative Shrinkage Thresholding algorithm or Wiener filters. The proposed algorithm, MixIST makes much progress in running efficiency while assuming good performance, and in consequence it not only serves as a good reference for further study on image restoration but also preliminarily takes on application value.