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 mod f mod and the magnitude of the Fourier transform mod (fe i phi ) V-product mod . The author considers the formulation as a least-squares minimization problem. It is shown that the linearized problem is ill posed. Also, surprisingly, 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.