Declipping and the recovery of vectors from saturated measurements
<|reference_start|>Declipping and the recovery of 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 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.<|reference_end|>
@article{alharbi2024declipping,
    title={Declipping and the recovery of vectors from saturated measurements},
    author={Wedad Alharbi, Daniel Freeman, Dorsa Ghoreishi, Brody Johnson, and N.
  Lovasoa Randrianarivony},
    journal={arXiv preprint arXiv:2402.03237},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03237},
    primaryClass={math.FA cs.NA math.MG math.NA}
}