Multidimensional Manhattan Sampling and Reconstruction


This paper introduces Manhattan sampling in two and better dimensions, and proves sampling theorems for them. In two-D, Manhattan sampling, which takes samples densely along a Manhattan grid of lines, will be viewed as sampling on the union of 2 rectangular lattices, one dense horizontally and the other vertically, with the coarse spacing of every being a multiple of the fine spacing of the opposite. The sampling theorem shows that the images bandlimited to the union of the Nyquist regions of the 2 rectangular lattices can be recovered from their Manhattan samples, and an economical procedure for doing so is given. Such recovery is attainable while there's an overlap among the spectral replicas induced by Manhattan sampling. In 3 and better dimensions, there are various possible configurations for Manhattan sampling, each consisting of the union of special rectangular lattices referred to as bi-step lattices. This paper identifies them, proves a sampling theorem showing that the photographs bandlimited to the union of the Nyquist regions of the bi-step rectangular lattices are recoverable from Manhattan samples, presents an economical onion-peeling procedure for doing so, and shows that the union of Nyquist regions is as large as any bandlimited region, such that each one images supported by such will be stably reconstructed from samples taken at the speed of the Manhattan sampling. It also develops a special illustration for the bi-step lattices with a variety of useful properties. While most of this paper deals with continuous-area images, Manhattan sampling of discrete-house images is additionally considered, for infinite, furthermore finite, support pictures.

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