Codes and Expansions (CodEx) Seminar

Dorsa Ghoreishi (Saint Louis University):
Frames and Phase retrieval for Vector Bundles

A frame for a Hilbert space \(H\), like an orthonormal basis, gives a continuous, linear, and stable reconstruction formula for any vector \(x \in H\). However, the redundancy of frames allows for more adaptability to different applications. For example, in order to do phase retrieval to recover a vector from only the magnitudes of a collection of linear measurements (such as in coherent diffraction imaging), we must use a frame instead of a basis as a basis cannot recover any loss of information. Frames are also necessary when working with a coordinate system for a vector bundle which moves continuously over a manifold. Although topological restrictions often prevent the existence of a continuously moving basis for a vector bundle, every vector bundle over a paracompact manifold has a moving redundant frame. We consider a combination of these two situations where one must recover a section of a vector bundle (up to an equivalence relation) from only the magnitudes of a collection of linear measurements on each fiber. Furthermore, we consider how to approximate a section from only finitely many samples. This is a joint work with Kieran Favazza, Olivia Frank, Daniel Freeman, Bernadette King and N. Lovasoa Randrianarivony.