Codes and Expansions (CodEx) Seminar

Daniel Freeman (St Louis University):
Recovering vectors from saturated measurements

A frame \((x_j)_{j\in J}\) for a Hilbert space \(H\) allows for a linear and stable reconstruction of any vector \(x\in H\) from the linear measurements \((\langle x,x_j\rangle)_{j\in J}\). However, there are many situations where some information of the frame coefficients is lost. In applications such as signal processing, electrical engineering, and digital photography one often uses sensors with an effective range and any measurement above that range is registered as the maximum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. Our goal is to develop and motivate a frame theoretic approach to this problem in a similar way to what Balan, Casazza, and Edidin did for phase retrieval. We will compare some of what is known about phase retrieval in frame theory with the corresponding results in saturation recovery. This perspective motivates many interesting open problems.