On sampling a high-dimensional bandlimited field on a union of shifted lattices

Unnikrishnan, Jayakrishnan; Vetterli, Martin

doi:10.1109/isit.2012.6283507

Cited by 7 publications

(7 citation statements)

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“…We find new sufficient conditions under which a two-dimensional field can be perfectly reconstructed from its samples on a slanted Manhattan grid. We also generalize the limiting case result of [18] and [19] to arbitrary non-convex spectral regions. We are particularly concerned with conditions where the image spectrum explicitly satisfies the Landau bound.…”

Section: Introductionmentioning

confidence: 83%

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Sampling 2-D signals on a union of lattices that intersect on a lattice

Unnikrishnan

Prelee

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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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.

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Section: Introductionmentioning

confidence: 83%

“…Sampling theorems and reconstruction methods for this limiting scenario were studied in [18] and [19]. The primary application addressed in these works was the sampling of spatially bandlimited fields using mobile sensors [20].…”

Section: Introductionmentioning

confidence: 99%

Sampling 2-D signals on a union of lattices that intersect on a lattice

Unnikrishnan

Prelee

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…Closer to the line of our work, the rate distortion function has been characterized when multiple Gaussian signals from a random field are sampled and quantized (centralized or distributed) in [24], [22], [23]. Also, in a series [28], [29], [30] (see also [15], [25], [8]), various aspects of random field-sampling and reconstruction for special models are considered. In a setting of distributed acoustic sensing and reconstruction, centralized as well as distributed coding schemes and sampling lattices are studied, and their performance is compared with corresponding rate distortion bounds [20].…”

Section: Introductionmentioning

confidence: 99%

Sampling Rate Distortion

Boda

Narayan

2017

IEEE Trans. Inform. Theory

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Consider a discrete memoryless multiple source with m components of which k ≤ m possibly different sources are sampled at each time instant and jointly compressed in order to reconstruct all the m sources under a given distortion criterion. A new notion of sampling rate distortion function is introduced, and is characterized first for the case of fixed-set sampling. Next, for independent random sampling performed without knowledge of the source outputs, it is shown that the sampling rate distortion function is the same regardless of whether or not the decoder is informed of the sequence of sampled sets. Furthermore, memoryless random sampling is considered with the sampler depending on the source outputs and with an informed decoder. It is shown that deterministic sampling, characterized by a conditional point-mass, is optimal and suffices to achieve the sampling rate distortion function. For memoryless random sampling with an uninformed decoder, an upper bound for the sampling rate distortion function is seen to possess a similar property of conditional point-mass optimality. It is shown by example that memoryless sampling with an informed decoder can outperform strictly any independent random sampler, and that memoryless sampling can do strictly better with an informed decoder than without. Index TermsDiscrete memoryless multiple source, independent random sampler, memoryless random sampler, random sampling, rate distortion, sampling rate distortion function. † V.P. Boda and P. Narayan are with the

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“…We conclude the introduction by relating the present work to previous work. Multidimensional sampling theorems, showing that images with certain spectral support regions can be reconstructed from certain samplings sets, appeared first for lattice sampling sets in Peterson and Middleton [11], then later for unions of shifted lattices, i.e., lattice cosets, [16]- [27], although they were not always described as such.…”

Section: Introductionmentioning

confidence: 99%

“…Nonoverlapping spectral replicas were not required in later work [17]- [27], and more complex reconstruction procedures were proposed. Though not specifically intended for images, a seminal contribution stimulating a number of advances in image sampling was the multichannel, generalized sampling introduced by Papoulis [28].…”

Section: Introductionmentioning

confidence: 99%

Multidimensional Manhattan Sampling and Reconstruction

Prelee

Neuhoff

2016

IEEE Trans. Inform. Theory

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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, the other vertically, with the coarse spacing of each 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. Index Termsimage sampling, lattice sampling, Landau sampling rate, nonuniform periodic sampling.

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On sampling a high-dimensional bandlimited field on a union of shifted lattices

Cited by 7 publications

References 31 publications

Sampling 2-D signals on a union of lattices that intersect on a lattice

Sampling 2-D signals on a union of lattices that intersect on a lattice

Sampling Rate Distortion

Multidimensional Manhattan Sampling and Reconstruction

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