多维曼哈顿采样与重构
Multidimensional Manhattan Sampling and Reconstruction
摘要：
本文介绍了二维及更高维的曼哈顿抽样，并证明了抽样定理。在二维空间中，曼哈顿采样（Manhattan sampling），即沿着曼哈顿网格线密集采样，可以被看作是在两个矩形网格的并集上采样，一个是水平密集的，是另一个的精细间距的倍数。采样定理表明，图像的带限的联合的奈奎斯特区域的两个矩形格可以恢复从他们的曼哈顿样本，并给出了一个有效的程序，这样做。这样的恢复是可能的，即使有曼哈顿采样引起的光谱副本之间的重叠。在三维和更高的维度中，曼哈顿采样有许多可能的配置，每个配置都由称为双阶晶格的特殊矩形晶格的联合组成。本文识别它们，证明了一个采样定理，表明图像的带限的联合的Nyquist区域的双步骤矩形格子是可恢复的曼哈顿样本，并提出了一个有效的洋葱剥离过程这样做。此外，它还发展了双步格的一种特殊表示和一个具有良好性质的代数。它还表明，在朗道意义下的可重构图像的集合是最大的。虽然大部分的文件处理连续空间图像，曼哈顿采样的离散空间图像也被认为是无限的，以及有限的，支持图像。
英文摘要：
This paper introduces Manhattan sampling in two and higher dimensions, and proves sampling theorems. In two dimensions, Manhattan sampling, which takes samples densely along a Manhattan grid of lines, can be viewed as sampling on the union of two rectangular lattices, one dense horizontally, being a multiple of the fine spacing of the other. The sampling theorem shows that images bandlimited to the union of the Nyquist regions of the two rectangular lattices can be recovered from their Manhattan samples, and an efficient procedure for doing so is given. Such recovery is possible even though there is overlap among the spectral replicas induced by Manhattan sampling. In three and higher dimensions, there are many possible configurations for Manhattan sampling, each consisting of the union of special rectangular lattices called bi-step lattices. This paper identifies them, proves a sampling theorem showing that images bandlimited to the union of the Nyquist regions of the bi-step rectangular lattices are recoverable from Manhattan samples, and presents an efficient onion-peeling procedure for doing so. Furthermore, it develops a special representation for the bi-step lattices and an algebra with nice properties. It is also shown that the set of reconstructable images is maximal in the Landau sense. While most of the paper deals with continuous-space images, Manhattan sampling of discrete-space images is also considered, for infinite, as well as finite, support images.
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