Compressive phase retrieval

The theory of compressive sensing enables accurate and robust signal reconstruction from a number of measurements dictated by the signal's structure rather than its Fourier bandwidth. A key element of the theory is the role played by randomization. In particular, signals that are compressible in the time or space domain can be recovered from just a few randomly chosen Fourier coefficients. However, in some scenarios we can only observe the magnitude of the Fourier coefficients and not their phase. In this paper, we study the magnitude-only compressive sensing problem and in parallel with the existing theory derive sufficient conditions for accurate recovery. We also propose a new iterative recovery algorithm and study its performance. In the process, we develop a new algorithm for the phase retrieval problem that exploits a signal's compressibility rather than its support to recover it from Fourier transform magnitude measurements.