去削波与从饱和测量中恢复向量
Declipping and the recovery of vectors from saturated measurements
AI总结：
本文针对饱和测量（削波）下的向量恢复问题，提出框架理论方法，刻画可行条件，证明最优框架对应射影空间最小多重填充，并改进经典框架算法实现重建。
摘要：
希尔伯特空间$H$中的框架$(x_j)_{j\in J}$允许从线性测量$(\langle x,x_j\rangle)_{j\in J}$中对任意向量$x\in H$进行线性且稳定的重建。然而，在许多情况下，框架系数中的某些信息会丢失。在使用具有固定动态范围的传感器的应用中，任何高于该范围的测量值被记录为最大值，任何低于该范围的测量值被记录为最小值。根据上下文，从这类测量中恢复向量被称为去削波或饱和恢复。我们以类似于[BCE06]对相位恢复所做的方式，开创了一种框架理论方法来解决饱和恢复问题。我们刻画了饱和恢复何时可行，证明了用于饱和恢复的最优框架对应于射影空间中的最小多重填充，并证明了经典框架算法可以适应这一非线性问题，从而提供一种重建算法。
英文摘要：
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.
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