Consistent Sampling and Signal Recovery
Consistent Sampling and Signal Recovery
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DOI：
10.1109/tsp.2007.895996
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发表时间：
2007-08
影响因子：
5.4
通讯作者：
A. Hirabayashi;M. Unser
中科院分区：
文献类型：
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作者：
A. Hirabayashi;M. Unser
采样问题的一个有吸引力的表述是基于一致信号重建的原理。要求是重构信号与输入无法区分，因为它产生完全相同的测量结果。这样的系统可以解释为到给定重建空间的倾斜投影。标准公式要求输入测量值和重建模型之间存在一对一的关系。不幸的是，当分析基函数和重构基函数之间的互相关矩阵不可逆时，这个条件就失效了。特别是当测量值少于重建函数的数量时。在本文中，我们提出了一致采样的扩展，它也适用于这些奇异情况，并产生独特且定义明确的解决方案。该解决方案还利用投影算子并具有几何解释。关键思想是从重建空间中排除采样算子的零空间，并强制其补集的一致性。我们指定了一类与互补重建空间的不同选择相对应的一致重建算法。该公式包括摩尔-彭罗斯广义逆，以及保留某些优先信号的其他可能更有趣的重建。特别是，我们展示了保留多项式或正弦曲线的解决方案，因此在实际应用中表现良好。
An attractive formulation of the sampling problem is based on the principle of a consistent signal reconstruction. The requirement is that the reconstructed signal is indistinguishable from the input in the sense that it yields the exact same measurements. Such a system can be interpreted as an oblique projection onto a given reconstruction space. The standard formulation requires a one-to-one relationship between the input measurements and the reconstructed model. Unfortunately, this condition fails when the cross-correlation matrix between the analysis and reconstruction basis functions is not invertible; in particular, when there are less measurements than the number of reconstruction functions. In this paper, we propose an extension of consistent sampling that is applicable to those singular cases as well, and that yields a unique and well-defined solution. This solution also makes use of projection operators and has a geometric interpretation. The key idea is to exclude the null space of the sampling operator from the reconstruction space and to enforce consistency on its complement. We specify a class of consistent reconstruction algorithms corresponding to different choices of complementary reconstruction spaces. The formulation includes the Moore-Penrose generalized inverse, as well as other potentially more interesting reconstructions that preserve certain preferential signals. In particular, we display solutions that preserve polynomials or sinusoids, and therefore perform well in practical applications.