Recent Advances in Phase Retrieval [Lecture Notes]

In many applications in science and engineering, one is given the modulus squared of the Fourier transform of an unknown signal and then tasked with solving the corresponding inverse problem, known as phase retrieval. Solutions to this problem have led to some noteworthy accomplishments, such as identifying the double helix structure of DNA from diffraction patterns, as well as characterizing aberrations in the Hubble Space Telescope from point spread functions. Recently, phase retrieval has found interesting connections with algebraic geometry, low-rank matrix recovery, and compressed sensing. These connections, together with various new imaging techniques developed in optics, have spurred a surge of research into the theory, algorithms, and applications of phase retrieval. In this lecture note, we outline these recent connections and highlight some of the main results in contemporary phase retrieval.