Abstract

A new algorithm is given for inverting an autocorrelation function R = R₁,…,R_n for its inputs a. The autocorrelation of a function a may be regarded as the cross correlation between two functions b and a, where b = a. This is made the basis for a cyclic algorithm. In the cross-correlation operation b ⊗ a, make b a numerical guess at a, and leave a as a set of unknowns. The resulting n equations R = b ⊗ a are linear in a and are in echelon form, allowing immediate solution (without recourse to a matrix inversion scheme). If output a obeys a = b, then a is the final solution, and the algorithm is stopped. Otherwise, continue by now making a the new guess solution b. This algorithm is cycled until a bistable solution state is reached: An input guess b gives an output a; then a used as input again gives b, etc. If the geometric mean $\sqrt{\mathbf{a}\mathbf{b}}$ between these bistable solutions is taken, the result is very close to solving the problem. At this point, each new guess solution b is formed as the geometric mean between the previous input-output pair. Convergence is then attained at an exponential rate. Application to 1-D and 2-D problems is discussed.

© 1989 Optical Society of America