# Space-Varying Restoration of Optical Images

J. Nagy, P. Pauca, R. Plemmons, T. Torgersen

The improvement in optical image quality is now generally attempted in two stages. The first stage involves techniques in adaptive optics and occurs as the observed image is initially formed. The second stage of enhancing the quality of optical images generally occurs off--line, and consists of the postprocessing step of image restoration. Image restoration is an ill--posed inverse problem which involves the removal or minimization of degradations caused by noise and blur in an image, resulting from, in this case, imaging through a medium. Our work here concerns a new space--varying regularization approach, and associated techniques for accelerating the convergence of iterative image postprocessing computations. Denoising methods, including total variation minimization, followed by segmentation--based preconditioning methods for minimum residual conjugate gradient iterations, are investigated. Regularization is accomplished by segmenting the image into (smooth) segments and varying the preconditioners across the segments. The method appears to work especially well on images that are piecewise smooth. Our algorithm has computational complexity of only $O(\ell n^2 \log n)$, where $n^2$ is the number of pixels in the image and $\ell$ is the number of segments used. Also, parallelization is straightforward. Numerical tests are reported on both simulated and practical atmospheric imaging problems. Comparisons are made with the case where segmentation is not used. It is found that our approach is especially attractive for restoring images with high noise levels, and that magnification of noise is effectively suppressed in the iterations, leading to a robust regularized iterative restoration algorithm.