### Finite Frames and Phase Retrieval

Directed by Dr. Christopher Dock

###### Project Description

Frame theory provides a flexible extension to theory of bases, both for finite-dimensional spaces modeled by **C**^{n} and for infinite dimensional spaces like L^{2}(**R**). The redundancy provided by frames allows us to accomplish data recovery under adverse circumstances, such as the loss of phase information (as occurs, for example, in X-Ray crystallography and in reading out information from quantum computers).

You will be exploring several areas of Frame Theory research, with a particular emphasis on designing frames that admit phase retrieval and have good properties. We will also be looking at retrieval properties for structured frames, for example, Gabor Frames (which are well understood in the context of Phase Retrieval) and Wavelet Frames (which are not).

You will be tasked with (and guided towards) exploring these problems both theoretically and experimentally. In particular, I am interested in seeing what you come up with in terms of analyzing the difficult non-linear optimization problems that occur when trying to evaluate the “retrieval performance” of a given frame.

**Required Background**

Solid knowledge of linear algebra and MATLAB or other high-level computer languages. Some familiarity with functional analysis and signal processing would be helpful but is not required.

**About Me**

I am a Norbert Wiener Fellow at Tufts. My research is in extending frame theory and the concept of phase retrievability to a non-abelian framework, both in the finite-dimensional setting and the infinite-dimensional setting. I also perform research on statistical models in the context of generative machine learning, with an emphasis on models that have sound theoretical underpinning and high expressivity.

**References**(roughly in order of increasing technicality)

- Peter G Casazza. The art of frame theory. Taiwanese journal of mathematics 4.2 (2000): 129-201.
- Peter G Casazza, Gitta Kutyniok, Friedrich Philipp. Introduction to finite frame theory. Finite frames (2013): 1-53.
- Yoav Shechtman, Yonina C Eldar, Oren Cohen, Henry Nicholas Chapman, Jianwei Miao, and Mordechai Segev. Phase retrieval with application to optical imaging: a contemporary overview. IEEE signal processing magazine, 32(3):87-109, 201.
- Emmanuel Candes, Yanina C Eldar, Thomas Strohmer, and Vlad Voroninski. Phase retrieval via matrix completion problem. SIAM J. Imag. Sci., 6(1):199-225, 2013.
- Michael V Klibanov, Paul E Sacks, and Alexander V Tikhonravov. The phase retrieval problem. Inverse problems, 11(1):1, 1995.
- Radu Balan, Pete Casazza, and Dan Edidin. On signal reconstruction without phase. Applied and Computational Harmonic Analysis, 20(3):345-356, 2006.