PHASE RECONSTRUCTION VIA NONLINEAR LEAST-SQUARES

Consider the problem of reconstructing the phase-phi of a complex-valued function fe(i-phi), given knowledge of the magnitude Absolute value of f and the magnitude of the Fourier transform \(fe(i-phi)and\. In this paper we consider formulation as a least-squares minimization problem. It is shown that the linearized problem is ill posed. Also, suprisingly, the gradient of the least-squares objective functional is not Frechet differentiable. A regularization is introduced which restores differentiability and also counteracts instability. It is shown how a certain implementation of Newton's method can be used to solve the regularized least-squares problem efficiently, and that the method converges locally, almost quadratically. Numerical examples are given with an application to diffractive optics.