Alternating projection (AP) of various forms, including the Parallel AP (PAP), Real-constrained AP (RAP) and the Serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition.To compensate for the local nature of convergence, the null initialization is proposed for initial guess and proved to produce asymptotically accurate initialization for the case of Gaussian random measurement. Numerical experiments show that the null initialization produces more accurate initial guess than the spectral initialization and that AP converges faster to the true object than other iterative schemes for non-convex optimization such as the Wirtinger Flow. In numerical experiments AP with the null initialization converges globally to the true object.1.1. Set-up. Let us recall the measurement schemes of [28]. Let x 0 (n) be a discrete object function with n = (n 1 , n 2 , · · · , n d ) ∈ Z d . Consider the object space consisting of all functions supported in M = {0 ≤ m 1 ≤ M 1 , 0 ≤ m 2 ≤ M 2 , · · · , 0 ≤ m d ≤ M d }.We assume d ≥ 2.

The null vector method, based on a simple linear algebraic concept, is proposed as a solution to the phase retrieval problem. In the case with complex Gaussian random measurement matrices, a non-asymptotic error bound is derived, yielding an asymptotic regime of accurate approximation comparable to that for the spectral vector method.

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