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.
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