# 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.