Restoration of Atmospherically Blurred Images by Symmetric Indefinite Conjugate Gradient Techniques

M. Hanke and J. Nagy

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 selfadjoint. 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 selfadjoint operator fails to be positive definite, a symmetric indefinite conjugate gradient technique, called MRII 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.