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.

翻译：设Hilbert空间$H$中的框架$(x_j)_{j\in J}$能够从线性测量值$(\langle x,x_j\rangle)_{j\in J}$中对任意向量$x\in H$实现线性且稳定的重建。然而在许多情况下，框架系数中的部分信息会丢失。当使用具有固定动态范围的传感器时，任何超出该范围的测量值均被记录为最大值，低于该范围的测量值则被记录为最小值。根据具体应用场景，从这类测量中恢复向量的过程被称为削波恢复或饱和重建。本文借鉴[BCE06]处理相位恢复问题的思路，首次从框架理论角度研究饱和重建问题。我们刻画了饱和重建可行的条件，证明最优饱和重建框架对应于射影空间中的多重极小填充，并论证经典框架算法可经改进后应用于该非线性问题，从而提供有效的重建算法。