Phase Retrieval from Incomplete Magnitude Information via Total Variation Regularization

The phase retrieval problem has drawn considerable attention, as many optical detection devices can only measure magnitudes of the Fourier transform of the underlying object (signal or image). This paper addresses the phase retrieval problem from incomplete data, where only partial magnitudes of Fourier transform are obtained. In particular, we consider structured illuminated patterns in holography and find that noninteger values used in designing such patterns often yield better reconstruction than the conventional integer-valued ones. Furthermore, we demonstrate theoretically and numerically that three diffracted sets of (complete) magnitude data are sufficient to recover the object. To compensate for incomplete information, we incorporate a total variation regularization a priori to guarantee that the reconstructed image satisfies some desirable properties. The proposed model can be solved efficiently by an alternative directional multiplier method with provable convergence. Numerical experiments validate the theoretical finding and demonstrate the effectiveness of the proposed method in recovering objects from noisy and incomplete data.