Phase Retrieval of Sparse Signals from Magnitude Information

The ability to recover the phase information of a signal of interest from a measurement process plays an important role in many practical applications. When only the Fourier transform magnitude of the signal is recorded, recovering the complete signal from these nonlinear measurements turns into a problem of phase retrieval.

Many practical algorithms exist to handle the phase retrieval problem. However, they present the drawback of convergence to a local minimum because of the non-convex Fourier magnitude constraints. Recent approaches formulating the problem in a higher dimensional space overcome this drawback but require a sufficiently large number of measurements. By using compressive sensing (CS) techniques, the number of measurements required for phase retrieval can be reduced with the additional information pertaining to the signal structure.

With the aim of reducing the number of measurements, this dissertation focuses on the problem of signal recovery by exploiting the sparsity information present in the signal samples. In this thesis, two approaches are proposed to accomplish sparse signal recovery from fewer magnitude measurements, modified Phase Cut and improved Phase Lift. In these approaches, we combine the phase retrieval methods, both Phase Cut and Phase Lift, which formulate the problem in a higher dimensional space, with l_(1)-norm minimization idea in CS by exploiting the sparse structure of the signals. The minimum number of measurements required for signal recovery by the proposed approaches is less than the number that Phase Cut and Phase Lift methods require. Both the modified Phase Cut and the improved Phase Lift approaches outperform another variation of the Phase Lift method, Compressive Phase Retrieval via Lifting; namely, better signal reconstruction rate is obtained for different sparsity degrees. However, in terms of computation time, Phase Lift based methods are faster than the Phase Cut based methods.

Ultimately, combining the phase retrieval methods with the l_(1)-norm minimization enables the usage of the sparse structure of the signal for the exact recovery up to a sparsity degree from fewer magnitude measurements. However, challenges remain, particularly those related with computation time of methods and the sparsity degree of the signal which the methods could recover up to by fewer measurements.