  Context. The knowledge of the point-spread function compensated by adaptive
optics is of prime importance in several image restoration techniques such as
deconvolution and astrometric/photometric algorithms. Wavefront-related data
from the adaptive optics real-time computer can be used to accurately estimate
the point-spread function in adaptive optics observations. The only
point-spread function reconstruction algorithm implemented on astronomical
adaptive optics system makes use of particular functions, named $U\_{ij}$.
These $U\_{ij}$ functions are derived from the mirror modes, and their number
is proportional to the square number of these mirror modes. Aims. We present
here two new algorithms for point-spread function reconstruction that aim at
suppressing the use of these $U\_{ij}$ functions to avoid the storage of a
large amount of data and to shorten the computation time of this PSF
reconstruction. Methods. Both algorithms take advantage of the eigen
decomposition of the residual parallel phase covariance matrix. In the first
algorithm, the use of a basis in which the latter matrix is diagonal reduces
the number of $U\_{ij}$ functions to the number of mirror modes. In the second
algorithm, this eigen decomposition is used to compute phase screens that
follow the same statistics as the residual parallel phase covariance matrix,
and thus suppress the need for these $U\_{ij}$ functions. Results. Our
algorithms dramatically reduce the number of $U\_{ij}$ functions to be computed
for the point-spread function reconstruction. Adaptive optics simulations show
the good accuracy of both algorithms to reconstruct the point-spread function.

Astrophysics