Abstract

Many physical problems in adaptive optics and imaging application result in the optimization of a Hermitian form, ${\overline{\mathbf{x}}}^{T}H\mathbf{x},$ where $H$ is an $(n\times n)$ Hermitian matrix and x is an $n$-component complex vector whose elements are constrained to have unit magnitude. In this work the technique of Lagrange multiplers is used to derive the governing nonlinear equations. An efficient numerical algorithm is constructed to solve the nonlinear equations. Of particular interest are applications that admit of nonunique solutions (e.g., problems arising from phase difference measurements). Newton’s method is applied to an inflated system of equations. This significantly improves the region inside which Newton’s method converges quadratically. A practical example of phasor reconstruction from phase difference measurements is given to illustrate the developed theory. In the example it is shown how the Lagrange multipliers can be used to give an a posteriori estimate of the measurement noise. For general Hermitian matrices, practical considerations of the developed algorithm are discussed.

© 1997 Optical Society of America

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