Multidimensional Manhattan Sampling and Reconstruction

This paper introduces Manhattan sampling in two and higher dimensions, and proves sampling theorems for them. In 2-D, 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 and the other vertically, with the coarse spacing of each being a multiple of the fine spacing of the other. The sampling theorem shows that the 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 an 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 the images bandlimited to the union of the Nyquist regions of the bi-step rectangular lattices are recoverable from Manhattan samples, presents an efficient onion-peeling procedure for doing so, and shows that the union of Nyquist regions is as large as any bandlimited region, such that all images supported by such can be stably reconstructed from samples taken at the rate of the Manhattan sampling. It also develops a special representation for the bi-step lattices with a number of useful properties. While most of this paper deals with continuous-space images, Manhattan sampling of discrete-space images is also considered, for infinite, as well as finite, support images.