On image restoration from random sampling noisy frequency data with regularization

Consider the image restoration using random sampling noisy frequency data by total variation regularization. By exploring image sparsity property under wavelet expansion, weestablish an optimization model with two regularizing terms specifying image sparsity and edge preservation on the restored image. The choice strategy for the regularizing parameters is rigorously set up together with corresponding error estimate on the restored image. The cost functional with data-fitting in the frequency domain is minimized using the Bregman iteration scheme. By deriving the gradient of the cost functional explicitly, the minimizer of the cost functional at each Bregman step is also generated by an inner iteration process with Tikhonov regularization, which is implemented stably and efficiently due to the special structure of the regularizing iterative matrix. Numerical tests are given to show the validity of the proposed scheme.