Declipping and the recovery of vectors from saturated measurements

TL;DR

This paper develops a frame-theoretic approach to recover vectors from saturated measurements caused by sensor limitations, characterizing conditions for successful recovery and proposing an adapted reconstruction algorithm.

Contribution

It introduces a novel frame-theoretic framework for declipping and saturation recovery, linking optimal frames to minimal multi-fold packings and adapting classical algorithms for this non-linear problem.

Findings

Characterizes when saturation recovery is possible.

Identifies optimal frames as minimal multi-fold packings.

Adapts classical frame algorithms for non-linear saturation recovery.

Abstract

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. In applications where one is using sensors with a fixed dynamic range, any measurement above that range is registered as the maximum, and any measurement below that range is 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 a similar way to what [BCE06] 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.