Phase retrieval: Approaching the theoretical limits in practice
In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. Today, very little is known about how to design injective intensity measurements, let alone stable measurements with efficient reconstruction algorithms. In fact, the state of the art applies certain routines called PhaseLift or PhaseCut, but for these, performance guarantees are only available when the measurement vectors are Gaussian random, which cannot be used in many applications. This talk will help fill the void - I will discuss a wide variety of recent results in phase retrieval, including various conditions for injectivity and stability (joint work with Afonso S. Bandeira (Princeton), Jameson Cahill (U Missouri) and Aaron A. Nelson (AFIT)) as well as measurement designs based on spectral graph theory which allow for efficient reconstruction (joint work with Boris Alexeev (Princeton), Afonso S. Bandeira (Princeton) and Matthew Fickus (AFIT)). In particular, I will show how Fourier-type tricks can be leveraged in concert with this graph-theoretic design to produce pseudorandom apertures for coherent diffractive imaging (joint work with Afonso S. Bandeira (Princeton) and Yutong Chen (Princeton)).