The image degradation produced by atmospheric turbulence and optical
aberrations is usually alleviated using post-facto image reconstruction
techniques, even when observing with adaptive optics systems. These techniques
rely on the development of the wavefront using Zernike functions and the
non-linear optimization of a certain metric. The resulting optimization
procedure is computationally heavy. Our aim is to alleviate this
computationally burden. To this aim, we generalize the recently developed
extended Zernike-Nijboer theory to carry out the analytical integration of the
Fresnel integral and present a natural basis set for the development of the
point spread function in case the wavefront is described using Zernike
functions. We present a linear expansion of the point spread function in terms
of analytic functions which, additionally, takes defocusing into account in a
natural way. This expansion is used to develop a very fast phase-diversity
reconstruction technique which is demonstrated through some applications. This
suggest that the linear expansion of the point spread function can be applied
to accelerate other reconstruction techniques in use presently and based on
blind deconvolution.Comment: 10 pages, 4 figures, accepted for publication in Astronomy &
Astrophysic
