Jacobi-like method for a control algorithm in adaptive-optics imaging
A study is made of a non-smooth optimization problem arising in adaptive-optics, which involves the real-time control of a deformable mirror designed to compensate for atmospheric turbulence and other dynamic image degradation factors. One formulation of this problem yields a functional f(U) = summation i=1n maxj{(UTMjU)ii} to be maximized over orthogonal matrices U for a fixed collection of n × n symmetric matrices Mj. We consider first the situation which can arise in practical applications where the matrices Mj are "nearly" pairwise commutative. Besides giving useful bounds, results for this case lead to a simple corollary providing a theoretical closed-form solution for globally maximizing f if the Mj are simultaneously diagonalizable. However, even here conventional optimization methods for maximizing f are not practical in a real-time environment. The general optimization problem is quite difficult and is approached using a heuristic Jacobi-like algorithm. Numerical tests indicate that the algorithm provides an effective means to optimize performance for some important adaptive-optics systems.
