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
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,
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.