IGL Project Proposal
Webtools Form 2331998 submitted by:
Joe Rosenblatt, rosnbltt@illinois.edu
College of Liberal Arts & Sciences (KV), Department of Mathematics (257), Staff
07/20/2022 10:54 A.M.
Phase retrieval is a fascinating topic that connects different aspects of mathematics with scientific areas in physics, chemistry, astronomy, and electrical engineering. Phase retrieval became a topic of interest in the early years of the 20th century when we began to use Xrays to determine the structure of chemical crystals. When Xrays are passed through a crystal, the intensity of the diffraction of the Xray can be determined. But the phase of this diffraction cannot be so easily determined. Without knowing the phase, the structure of the crystal cannot be determined unambiguously.
Using the physics of Xray diffraction by crystals, this becomes a mathematical problem. You know the modulus of the complexvalued function F (which is the Fourier transform FT(f) of the unknown atomic distribution f of the crystal). But you do not know the phase of F . So you cannot determine F or f . But what are the possible functions f that will work here? Simply put, imagine you know the length of a vector v in the plane, but you do not know the angle it makes with the xaxis. Then how can you determine v?! This is impossible without more information gathered from the applied setting in which you are working.
Phase retrieval as an area of applied mathematics has developed over the years in many ways to solve the inverse problem: use F  to find all the distributions f which could reasonably include the one you want. Then determine which f to use from various physical or chemical aspects of the structure.
We will consider lots of interesting and fun questions that remain unsolved in the mathematics of phase retrieval. For example:
Question: Suppose you have a set of points P on a circle, Let D(P ) be the set of distances along the circle between the points. How can we find all other finite sets of points Q on the circle such that the set of distances along the circle D(Q) between the points in Q is the same as D(P )? That is, given D(Q) = D(P ), what is Q?
Question: Suppose we have a polynomial P (x) with positive coefficients. There are methods for finding all the complexcoefficient polynomials Q such that P (x)P (−x) = Q(x)Q(−x). But how can we find all the polynomials Q with positive coefficients that do this?
You can read about some of the background and the basics of phase retrieval in the article J. Rosenblatt, Phase Retrieval, Communications in Mathematical Physics 95 (1984) 317343. As a start to our project, we can work through the parts of this article and the questions raised there that are the most interesting to you. We can find out what work has been done in the almost 40 years since this article was published. And then from these starting points, we can jump off into new directions of research in phase retrieval!
