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.

This work explores performance vs. communication energy tradeoffs in wireless sensor networks that use the recently proposed cutset deployment strategy in which sensors are placed densely along a grid of intersecting lines. For a given number of sensors, intersensor spacing is less for a cutset network than for a conventional lattice deployment, so that cutset networks require less communication energy, albeit with some potential loss in network performance. Previous work analyzed the energy-performance tradeoffs for square-grid cutset networks in the context of specific decentralized algorithms for source localization based on received signal strength (RSS). The current work also considers the RSS based source localization problem. However, it takes a more fundamental approach to analyzing the tradeoff by considering a centralized task, minimum energy communication paths, Maximum Likelihood estimation algorithms and Cramér-Rao bounds. Moreover, it analyzes triangular and honeycomb cutset deployments, in addition to square-grid ones. The results indicate that cutset networks offer sizable decreases in energy with only modest losses of performance.

This paper presents new sufficient conditions under which a field (or image) can be perfectly reconstructed from its samples on a union of two lattices that share a common coarse lattice. In particular, if samples taken on the first lattice can be used to reconstruct a field bandlimited to some spectral support region, and likewise samples taken on the second lattice can reconstruct a field bandlimited to another spectral support region, then under certain conditions, a field bandlimited to the union of these two spectral regions can be reconstructed from its samples on the union of the two respective lattices. These results generalize a previous perfect reconstruction theorem for Manhattan sampling, where data is taken at high density along evenly spaced rows and columns of a rectangular grid. Additionally, a sufficient condition is given under which the Landau lower bound is achieved.

Cutset sampling is a new approach to image sampling where 2D data is recorded densely along intersecting lines. A special case of cutset sampling is Manhattan sampling, where data is sampled densely along evenly-spaced rows and columns. This paper presents a new method for interpolating pixels from their Manhattan samples called the orthogonal gradient (OG) algorithm, which exploits the fact that pixels tend to be correlated along the direction orthogonal to the image gradient. Such an approach is enhanced by Manhattan sampling, where dense sampling along straight lines allows for better reconstruction of both sharp and soft image edges. In particular, the OG algorithm alternates between solving a constrained optimization problem, and changing the weights of the optimization problem according to the direction of the gradient of each new image estimate. The proposed method improves upon previous Manhattan interpolation algorithms, both qualitatively as well as in mean-squared error. Furthermore, unlike the previous algorithms, the OG algorithm can easily be extended to any sampling scheme.

For wireless sensor networks, many decentralized algorithms have been developed to address the problem of locating a source that emits acoustic or electromagnetic waves based on received signal strength. Among the motivations for decentralized algorithms is that they reduce the number of transmissions between sensors, thereby increasing sensor battery life. Whereas most such algorithms are designed for arbitrary sensor placements, such as random placements, this paper focuses on applications that permit a choice of sensor placement. In particular, to make communications costs small, it is proposed to place sensors uniformly along evenly spaced rows and columns, i.e., a Manhattan grid. For such a placement, the Midpoint Algorithm is proposed, which is a simple noniterative decentralized algorithm. The results of this paper show that Manhattan grid networks offer improved accuracy vs. energy tradeoff over randomly distributed networks. Results also show the proposed Midpoint Algorithm offers further energy savings over the recent POCS algorithm.

This paper presents a sampling theorem for Manhattan-grid sampling, which is a sampling scheme in which data is taken along evenly spaced rows and columns. Given the spacing between the rows, the columns, and the samples along the rows and columns, the theorem shows that an image can be perfectly reconstructed from Manhattan-grid sampling if its spectrum is bandlimited to a cross-shaped region whose arm lengths and widths are determined by the aforementioned sample spacings. The nature of such cross-bandlimited images is demonstrated by filtering an image with a cross-bandlimiting filter for several choices of sampling parameters.

This paper builds upon previous work for image reconstruction problems in which samples are taken on evenly spaced rows and columns, i.e., a Manhattan grid. A new reconstruction method is proposed that uses three steps to interpolate the interior of each block under the model that an image can be decomposed into piecewise planar regions plus noise. First, the K-planes algorithm is developed in order to fit several planes to the observed pixel values on the border. Second, one of theK planes is assigned to each pixel of the block interior, by a process of partitioning the block with polygons, thereby creating a piecewise planar approximation. Third, the interior pixels are interpolated by modeling them as a Gauss Markov random field whose mean is the piecewise planar approximation just obtained. The new method is shown to improve significantly upon previous methods, especially in the preservation of “soft” image edges.

This paper presents a new approach to sampling images in which samples are taken on a cutset with respect to a graphical image model. The cutsets considered are Manhattan grids, for example every Nth row and column of the image. Cutset sampling is motivated mainly by applications with physical constraints, e.g. a ship taking water samples along its path, but also by the fact that dense sampling along lines might permit better reconstruction of edges than conventional sampling at the same density. The main challenge in cutset sampling lies in the reconstruction of the unsampled blocks. As a first investigation, this paper uses segmentation followed by linear estimation. First, the ACA method [1] is modified to segment the cutset, followed by a binary Markov random field (MRF) inspired segmentation of the unsampled blocks. Finally, block interiors are estimated from the pixels on their boundaries, as well as their segmentation, with methods that include a generalization of bilinear interpolation and linear MMSE methods based on Gaussian MRF models or separable autocorrelation models. The resulting reconstructions are comparable to those obtained with conventional sampling at higher sampling densities, but not generally as good as conventional sampling at lower rates.