Signal Phase Recovery and Unwrapping

The need to recover a signal from incomplete or corrupted measurements is a central challenge in signal processing. A particular problem of this type is recovery of a signal after its Fourier magnitude or its Fourier phase is lost. This problem has a rich history that originated in the field of x-ray crystallography and continues to be of substantial interest in molecular imaging and numerous other applications. It has been observed that Fourier phase is typically more important in representing recognizable features of one-dimensional signals (e.g., audio waveforms) and two-dimensional signals, such as images. Classical experiments illustrating this observation are reproduced in this thesis, and practical iterative algorithms for recovering a signal from either its phase or magnitude are demonstrated. Unsurprisingly, it is typically more difficult to compensate for the loss of phase information, and recovery of a signal from its Fourier magnitude is seen to be less effective than recovery from its Fourier phase. A partitioning method is introduced to improve image recovery from magnitude information, and the phase unwrapping problem for one-dimensional signals is discussed briefly.