再生核空间中集中信号的随机采样与重构
Random sampling and reconstruction of concentrated signals in a reproducing kernel space
摘要：
在本文中，我们考虑（随机）采样信号集中在一个有界的度量测度空间的Corkscrew域$Ω$，并重建集中的信号近似从他们的（未）损坏的采样数据上包含在$Ω$的采样集。本文在再生核空间中建立了集中信号集上的（随机）采样方案的双Lipschitz型加权稳定性。双Lipschitz型的加权稳定性提供了一个弱的鲁棒性的采样方案，但由于集中的信号集的非凸性，它并不意味着唯一的信号重建。从（未）损坏的样本上包含在$Ω$的有限采样集，我们提出了一个算法来找到一个有界的Corkscrew域$Ω$集中的信号的近似。随机抽样是根据概率分布随机选取抽样位置的抽样方案。接下来，我们表明，以高概率，信号集中在一个有界Corkscrew域$Ω$可以重建近似从他们的未损坏（或随机损坏）的样本在i.i.d.在$Ω$上绘制的随机位置，前提是采样大小至少为$μ(Ω) \ln (μ(Ω))$阶，其中$μ(Ω)$是集中域$Ω$的度量。最后，我们证明了所提出的逼近原始集中的信号时，采样过程中采取的小密度或随机与大尺寸的性能。
英文摘要：
In this paper, we consider (random) sampling of signals concentrated on a bounded Corkscrew domain $Ω$ of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in $Ω$. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel space. The weighted stability of bi-Lipschitz type provides a weak robustness to the sampling scheme, however due to the nonconvexity of the set of concentrated signals, it does not imply the unique signal reconstruction. From (un)corrupted samples taken on a finite sampling set contained in $Ω$, we propose an algorithm to find approximations to signals concentrated on a bounded Corkscrew domain $Ω$. Random sampling is a sampling scheme where sampling positions are randomly taken according to a probability distribution. Next we show that, with high probability, signals concentrated on a bounded Corkscrew domain $Ω$ can be reconstructed approximately from their uncorrupted (or randomly corrupted) samples taken at i.i.d. random positions drawn on $Ω$, provided that the sampling size is at least of the order $μ(Ω) \ln (μ(Ω))$, where $μ(Ω)$ is the measure of the concentrated domain $Ω$. Finally, we demonstrate the performance of proposed approximations to the original concentrated signal when the sampling procedure is taken either with small density or randomly with large size.
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