Our research in this program is aimed at developing new methods for what is generically known as the phase retrieval problem. In its classical formulation, phase retrieval amounts to reconstruction of an unknown waveform from the magnitude of its Fourier transform. Currently, the most widely used algorithm for such problems is Hybrid Input-Output (HIO), introduced in 1978 [1]. Notwithstanding many attempts, no better alternative has been established. There is, however, a great interest in developing reconstruction algorithms based on convex optimization techniques, which have been developed extensively in recent decades, both theoretically and practically. Unfortunately, due to the strong non-convexity of the problem, such methods have not met with success. Nevertheless, in our previous research in the framework of the IMG4 MAGNET we showed that incorporating some additional information on the phase can change this situation dramatically. In particular, we developed a convex optimization method for phase retrieval when phase uncertainty is limited by π radians. Our method is several orders of magnitude faster than the HIO algorithm in this case, and it demonstrates a much better reconstruction quality when measurements contain noise [5]. In this year we continued our research on the phase retrieval with alternative additional information. This additional information can of course be used together with partial phase information if available. In addition to combining the Fourier magnitude with other sources of information, we developed several methods for reconstruction from defocused images or from interferograms. Initially, we developed a method for image reconstruction from the Fourier magnitude and a single interferometric pattern. Interferometry is widely used in optics, and is probably the most widely used method for phase estimation. The method and results are described in Section 2. (More details are available in the Y4 annual report.) This method was later
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