Multidimensional Manhattan Sampling and Reconstruction

TL;DR

This paper develops a theory for multidimensional Manhattan sampling, proving sampling theorems and providing efficient reconstruction methods for images bandlimited to unions of Nyquist regions, applicable in both continuous and discrete spaces.

Contribution

It introduces multidimensional Manhattan sampling, proves new sampling theorems, and offers efficient reconstruction algorithms for images sampled on Manhattan grids, extending previous work to higher dimensions.

Findings

Sampling theorems for 2D and higher dimensions established

Efficient onion-peeling reconstruction procedures developed

Maximality of reconstructable image set demonstrated

Abstract

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.