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

Ren, Hongwu and Dekany, Richard (2004) Fast wave-front reconstruction by solving the Sylvester equation with the alternating direction implicit method. Optics Express, 12 (14). pp. 3279-3296. ISSN 1094-4087.

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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.

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Additional Information:© 2004 Optical Society of America. Received 2 June 2004; revised 6 July 2004; accepted 8 July 2004. We appreciate Marcia Brown for improving the presentation of this paper. We acknowledge the financial support of the National Science Foundation through the Center for Adaptive Optics.
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Subject Keywords:OCIS codes: (010.1080) Adaptive Optics; (010.7350) Wave-front Sensing
Issue or Number:14
Record Number:CaltechAUTHORS:RENoe04
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:301
Deposited By: Archive Administrator
Deposited On:18 May 2005
Last Modified:02 Oct 2019 22:32

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