Role of support information and zero locations in phase retrieval by a quadratic approach

A recently introduced approach to phase-retrieval problems is applied to present a unified discussion of support information and zero locations in the reconstruction of a discrete complex image from Fourier-transform phaseless data. The choice of the square-modulus function of the Fourier transform of the unknown as the problem datum results in a quadratic operator that has to be inverted, i.e., a simple nonlinearity. This circumstance makes it possible to consider and to point out some relevant factors that affect the local minima problem that arises in the solution procedure (which amounts to minimizing a quartic functional). Simple modifications of the basic procedure help to explain the role of support information and zeros in the data and to develop suitable strategies for avoiding the local minima problem. All results can be summarized by reference to the ratio between the effective dimensions of the data space and the space of unknowns. Numerical results identify the approach’s considerable robustness against false solutions, starting from completely random first guesses, if the above ratio is larger than 3. The algorithm also ensures robust performance in the presence of noise in the data.

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