Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques

We consider an ill-posed deconvolution problem from astronomical imaging with a given noise-contaminated observation, and an approximately known convolution kernel. The limitations of the mathematical model and the shape of the kernel function motivate and legitimate a further approximation of the convolution operator by one that is self-adjoint. This simplifies the reconstruction problem substantially because the efficient conjugate gradient method can now be used for an iterative computation of a (regularized) approximation of the true unblurred image. Since the constructed self-adjoint operator fails to be positive definite, a symmetric indefinite conjugate gradient technique, called MR-II is used to avoid a breakdown of the iteration. We illustrate how the L-curve method can be used to stop the iterations, and suggest a preconditioner for further reducing the computations.