Frames and Phase Retrieval for Vector Bundles

Kieran Favazza, Olivia Frank, Daniel Freeman, Dorsa Ghoreishi, Bernadette King, Nirina Lovasoa Randrianarivony

Session: General

Keywords: Frame theory, phase retrieval for vector bundles

Abstract:

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),
one must use a frame because 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, we have that 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.

Submission Number: 119