Treffer: Performance modeling of adaptive-optics imaging systems using fast Hankel transforms

Real-time adaptive-optics is a means for enhancing the resolution of ground based, optical telescopes beyond the limits previously imposed by the turbulent atmosphere. One approach for linear performance modeling of closed-loop adaptive-optics systems involves calculating very large covariance matrices whose components can be represented by sums of Hankel transform based integrals. In this paper we investigate approximate matrix factorizations of discretizations of such integrals. Two different approximate factorizations based upon representations of the underlying Bessel function are given, the first using a series representation due to Ellerbroek and the second an integral representation. The factorizations enable fast methods for both computing and applying the covariance matrices. For example, in the case of an equally spaced grid, it is shown that applying the approximated covariance matrix to a vector can be accomplished using the derived integral-based factorization involving a 2-D fast cosine transform and a 2-D separable fast multipole method. The total work is then O(N log N) where N is the dimension of the covariance matrix in contrast to the usual O(N2) matrix-vector multiplication complexity. Error bounds exist for the matrix factorizations. We provide some simple computations to illustrate the ideas developed in the paper.