The Geometry of Ambiguity in One-dimensional Phase Retrieval
The phase retrieval problem is the problem of reconstructing an unknown signal from its Fourier intensity function. This problem has a long history in physics and engineering and occurs in contexts such as X-ray crystallography, speech processing and computational biology. As stated, the phase retrieval problem is ill-posed as there may be up to $2^N$ non-equivalent signals (called ambiguities) with the same Fourier intensity function. To enforce uniqueness additional constraints must be imposed.
In this talk we discuss the geometry of the space of ambiguities obtained by varying the signal. By understanding this geometry we prove a general result characterizing constraints that enforce a unique solution to the phase retrieval problem. This result was applied in work with Tamir Bendory and Yonina Eldar on blind phaseless short-time Fourier transform recovery.