Frame Theory, Phase Retrieval, and Approximation of Vector Bundle Sections

Frames share many characteristics with orthonormal bases, including continuous, linear, and stable reconstruction formulas, but their inherent redundancy provides greater flexibility for recovering lost information, making them strong candidates for applications. One such application is phase retrieval, where only the intensity of each linear measurement of a signal is available and the phase information is lost. We consider the setting of Parseval frames for Hilbert spaces which move continuously over a manifold to give a global coordinate system for a vector bundle. We show that for vector bundles over compact manifolds, one can use the frame coefficients at finitely many samples of a moving Parseval frame to approximate a section of the vector bundle.  Further, when using a moving Parseval frame which does phase retrieval for a vector bundle, one can similarly approximate a section (up to an equivalence relation) from only finitely many phaseless samples. The talk will begin with an introduction to frames and then discuss how these ideas extend to moving frames over manifolds and the vector bundle setting. This is joint work with Kieran Favazza, Olivia Frank, Daniel Freeman, Bernadette King, and N. Lovasoa Randrianarivony.