Performance of Phase Retrieval via Phaselift and Quadratic Inversion in Circular Scanning Case

The reconstruction of the field radiated by a source from square amplitude-only data falls into the realm of phase retrieval. In this paper, we tackle the phase retrieval with two different approaches. The first one is based on a convex optimization problem called PhaseLift. The latter exploits the lifting technique to recast phase retrieval as a linear problem with an increased number of unknowns and then, because the linear problem is highly undetermined, it adds further constraints (based on the mathematical properties of the solution) to estimate it. The second approach formulates phase retrieval as a least squares problem and, therefore, it requires to tackle the minimization of a quartic functional which will be carried out by applying a gradient descent method. In the second part of this paper, in order to corroborate the effectiveness of both approaches, we present the numerical results. Afterward, we provide a comparison between the two methods and finally, we emphasize how the ratio between the number of independent data and the number of unknowns impacts on the performance in both approaches.