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Fast wave-front reconstruction by solving the Sylvester equation with the alternating direction implicit method.

Large degree-of-freedom real-time adaptive optics (AO) control requires reconstruction algorithms that are computationally efficient and readily parallelized for hardware implementation. In particular, we find the wave-front reconstruction for the Hudgin and Fried geometry can be cast into a form of the well-known Sylvester equation using the Kronecker product properties of matrices. We derive the filters and inverse filtering formulas for wave-front reconstruction in two-dimensional (2-D) Discrete Cosine Transform (DCT) domain for these two geometries using the Hadamard product concept of matrices and the principle of separable variables. We introduce a recursive filtering (RF) method for the wave-front reconstruction on an annular aperture, in which, an imbedding step is used to convert an annular-aperture wave-front reconstruction into a squareaperture wave-front reconstruction, and then solving the Hudgin geometry problem on the square aperture. We apply the Alternating Direction Implicit (ADI) method to this imbedding step of the RF algorithm, to efficiently solve the annular-aperture wave-front reconstruction problem at cost of order of the number of degrees of freedom, O(n). Moreover, the ADI method is better suited for parallel implementation and we describe a practical real-time implementation for AO systems of order 3,000 actuators.