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 in the frame coefficients is lost. We are interested in applications which employ sensors with a fixed dynamic range, where measurements above that range are registered as the maximum and measurements below that range are registered as the minimum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. We initiate a frame theoretic approach to saturation recovery in finite dimensions, analogous to what Balan, Casazza, and Edidin did for phase retrieval. We characterize when saturation recovery is possible, show optimal frames for use with saturation recovery correspond to minimal multi-fold packings in projective space, and prove that the classical frame algorithm may be adapted to this non-linear problem to provide a reconstruction algorithm.
