Local reconstruction for sampling in shift-invariant spaces
Local reconstruction for sampling in shift-invariant spaces
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DOI：
10.1007/s10444-008-9109-0
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发表时间：
2010-04
影响因子：
1.7
通讯作者：
Qiyu Sun
中科院分区：
文献类型：
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作者：
Qiyu Sun
在信号处理的许多应用中，样本的局部重构是一个非常理想的性质，但它并没有得到足够的重视。在本文中，我们将考虑在移位不变空间中的信号的局部重建问题。特别是，我们考虑寻找采样setsX，使得移位不变空间中的信号可以从它们在X上的样本局部重构。对于一个局部有限维的平移不变空间V，我们表明，信号在V可以局部重建从它的样本在任何采样集具有足够大的密度。对于平移不变空间V（λ 1，...，... N，我们表征所有周期性非均匀采样集X，使得在该移位不变空间V（λ 1，...，N）可以从从X取得的样本局部重建。对于一个由紧支撑的可加细函数生成的可加细平移不变空间V（λ），我们证明了对于几乎所有的可加细平移不变空间V（λ），V（λ）中的任何信号都可以从其过采样率为2的样本中局部重构.我们关于可加细位移不变空间V（n）中局部采样与重构结果的证明主要依赖于可加细函数在具有正Lebesgue测度的可测集上的线性无关位移和可加细函数的几乎涟漪性质，这些性质本身是新的和有趣的.
The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling setsXsuch that signals in a shift-invariant space can be locally reconstructed from their samples onX. For a locally finite-dimensional shift-invariant spaceVwe show that signals inVcan be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant spaceV(ϕ1, ...,ϕN) generated by finitely many compactly supported functionsϕ1, ...,ϕN, we characterize all periodic nonuniform sampling setsXsuch that signals in that shift-invariant spaceV(ϕ1, ...,ϕN) can be locally reconstructed from the samples taken fromX. For a refinable shift-invariant spaceV(ϕ) generated by a compactly supported refinable functionϕ, we prove that for almost all, any signal inV(ϕ) can be locally reconstructed from its samples fromwith oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant spaceV(ϕ) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.