论文标题
基于POCS的信号重建框架从广义非均匀样品中
POCS-based framework of signal reconstruction from generalized non-uniform samples
论文作者
论文摘要
我们将投影在凸集集合（POC）上正式使用，以重建来自非均匀样品的信号，以其最高的通用性。这涵盖了任何Hilbert Space $ \ MATHSCR H $中的信号，包括多维和多渠道信号，以及最通常是信号中具有给定内核函数的信号的样品，均以$ \ mathscr h $中的给定。 POCS方法的一个有吸引力的特征是其迭代的无条件收敛到与输入样品一致的估计值，即使这些样品在其不均匀性的基础上具有非常异构性的性质，并且/或采样不足。此外，迭代的误差在系统上单调下降，并且只要样品是该信号的唯一特征，它们的极限就会检索输入信号。在本文的第二部分中，我们重点介绍了$ \ Mathscr h $中的采样内核函数正交的情况，而输入可以限制在较小的封闭空间$ \ mathscr $ a $（例如bandlimitation）中。这涵盖了通过集成编码的时间越来越流行的应用，包括多通道编码。在这种情况下，我们通过提供特殊的并行化版本来推动对POCS方法的分析，显示了它与样品定义的线性运算符的伪内的联系，并提供了无乘数的离散时间实现，以矛盾地加速了迭代的互换。
We formalize the use of projections onto convex sets (POCS) for the reconstruction of signals from non-uniform samples in their highest generality. This covers signals in any Hilbert space $\mathscr H$, including multi-dimensional and multi-channel signals, and samples that are most generally inner products of the signals with given kernel functions in $\mathscr H$. An attractive feature of the POCS method is the unconditional convergence of its iterates to an estimate that is consistent with the samples of the input, even when these samples are of very heterogeneous nature on top of their non-uniformity, and/or under insufficient sampling. Moreover, the error of the iterates is systematically monotonically decreasing, and their limit retrieves the input signal whenever the samples are uniquely characteristic of this signal. In the second part of the paper, we focus on the case where the sampling kernel functions are orthogonal in $\mathscr H$, while the input may be confined in a smaller closed space $\mathscr A$ (of bandlimitation for example). This covers the increasingly popular application of time encoding by integration, including multi-channel encoding. We push the analysis of the POCS method in this case by giving a special parallelized version of it, showing its connection with the pseudo-inversion of the linear operator defined by the samples, and giving a multiplierless discrete-time implementation of it that paradoxically accelerates the convergence of the iteration.
