Performance Bounds for Expander-Based Compressed Sensing in Poisson Noise

Raginsky, Maxim; Jafarpour, Sina; Harmany, Zachary T.; Marcia, Roummel F.; Willett, Rebecca; Calderbank, Robert

doi:10.1109/tsp.2011.2157913

Cited by 48 publications

(48 citation statements)

References 36 publications

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“…Some of the major theoretical challenges associated with the application of CS to linear optical systems in the presence of Poisson noise have been addressed in the recent literature [38,39]. These works considered two novel sensing paradigms, based on either pseudo-random dense sensing matrices (akin to the shifted and scaled dense sensing matrix described above) or expander graph constructions, both of which satisfy the nonnegativity and flux preservation constraints.…”

Section: Sidebar] Sparse Recovery: Methods and Guaranteesmentioning

confidence: 99%

Sparsity and Structure in Hyperspectral Imaging : Sensing, Reconstruction, and Target Detection

Willett

Duarte

Davenport

et al. 2014

IEEE Signal Process. Mag.

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179

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Section: Sidebar] Sparse Recovery: Methods and Guaranteesmentioning

confidence: 99%

Sparsity and Structure in Hyperspectral Imaging : Sensing, Reconstruction, and Target Detection

Willett

Duarte

Davenport

et al. 2014

IEEE Signal Process. Mag.

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179

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“…The case of a Poisson single-measurement model (without perturbation on the sensing matrix) has been considered in [22,24]. However, it is worth noting that even though the multiple-measurement case can be formulated concisely as y ∼ Pois(Af + λ), akin to the single-measurement situation [22,24], a fundamental difference is that the sensing matrix A in this case is limited to a block-diagonal structure, rather than being arbitrary, as in the single-measurement case. The block-diagonal measurement matrix poses a more challenging (and practical) problem.…”

Section: Concentration-of-measure Inequalitiesmentioning

confidence: 99%

“…The block-diagonal measurement matrix poses a more challenging (and practical) problem. Hence, the proof techniques from [22,24] cannot be applied to the multiple-measurements case, and the nonnegativity constraint on the sensing matrix also invalidates adaptation of the results in [8,21].…”

Section: Concentration-of-measure Inequalitiesmentioning

confidence: 99%

“…The properties of the Poisson measurement model have been studied from various perspectives. Algorithms for recovering the (sparse) Poisson rate function (i.e., the associated parameter of the Poisson distribution) have been studied in [10], and performance bounds for inversion algorithms have been developed in [22,24]. However, these algorithms assume perfect knowledge of the sensing matrix.…”

Section: Introduction and Related Workmentioning

confidence: 99%

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Signal Recovery and System Calibration from Multiple Compressive Poisson Measurements

Wang

Huang

Yuan

et al. 2015

SIAM J. Imaging Sci.

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Abstract. The measurement matrix employed in compressive sensing typically cannot be known precisely a priori, and must be estimated via calibration. One may take multiple compressive measurements, from which the measurement matrix and underlying signals may be estimated jointly. This is of interest as well when the measurement matrix may change as a function of the details of what is measured. This problem has been considered recently for Gaussian measurement noise, and here we develop this idea with application to Poisson systems. A collaborative maximum likelihood algorithm and alternating proximal gradient algorithm are proposed, and associated theoretical performance guarantees are established based on newly derived concentration-of-measure results. A Bayesian model is then introduced, to improve flexibility and generality. Connections between the maximum likelihood methods and the Bayesian model are developed, and example results are presented for a real compressive X-ray imaging system.

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“…In previous work, we evaluated CS approaches for generating high resolution images and video from photon-limited data. 59,60 In particular, we showed how a feasible positivity-and flux-preserving sensing matrix can be constructed, and analyzed the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures image sparsity. We showed that for a fixed image intensity, the error bound actually grows with the number of measurements or sensors.…”

Section: Non-negativity and Photon Noisementioning

confidence: 99%

Compressed sensing for practical optical imaging systems: a tutorial

2011

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Abstract. The emerging field of compressed sensing has potentially powerful implications for the design of optical imaging devices. In particular, compressed sensing theory suggests that one can recover a scene at a higher resolution than is dictated by the pitch of the focal plane array. This rather remarkable result comes with some important caveats however, especially when practical issues associated with physical implementation are taken into account. This tutorial discusses compressed sensing in the context of optical imaging devices, emphasizing the practical hurdles related to building such devices, and offering suggestions for overcoming these hurdles. Examples and analysis specifically related to infrared imaging highlight the challenges associated with large format focal plane arrays and how these challenges can be mitigated using compressed sensing ideas. C 2011 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: 10.1117/1.3596602] Subject terms: compressed sensing; sampling; image reconstruction; inverse problems; computational imaging; infrared; coded aperture imaging; optimization.Paper 100978TR received Nov. 29, 2010; revised manuscript received May 8, 2011; accepted for publication May 12, 2011; published online Jul. 6, 2011. IntroductionThis tutorial describes new methods and computational imagers for increasing system resolution based on recently developed compressed sensing (CS, also referred to as compressive sampling) 1, 2 techniques. CS is a mathematical framework with several powerful theorems that provide insight into how a high resolution image can be inferred from a relatively small number of measurements using sophisticated computational methods. For example, in theory a 1 mega-pixel array could potentially be used to reconstruct a 4 mega-pixel image by projecting the desired high resolution image onto a set of low resolution measurements (via spatial light modulators, for instance) and then recovering the 4 mega-pixel scene through sparse signal reconstruction software. However, it is not immediately clear how to build a practical system that incorporates these theoretical concepts. This paper provides a tutorial on CS for optical engineers which focuses on 1. a brief overview of the main theoretical tenets of CS, 2. physical systems designed with CS theory in mind and the various tradeoffs associated with these systems, and 3. an overview of the state-of-the-art in sparse reconstruction algorithms used for CS image formation. There are several other tutorials on CS available in the literature which we highly recommend; 3-7 however, these papers do not address important technical issues related to optical systems, including a discussion of the tradeoffs associated with non-negativity, photon noise, and the practicality of implementation in real imaging systems.Although the CS theory and methods we describe in this paper can be applied to many general imaging systems, we concentrate on infrared (IR) technology as a specific example to highlight the challenges associated...

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Performance Bounds for Expander-Based Compressed Sensing in Poisson Noise

Cited by 48 publications

References 36 publications

Sparsity and Structure in Hyperspectral Imaging : Sensing, Reconstruction, and Target Detection

Sparsity and Structure in Hyperspectral Imaging : Sensing, Reconstruction, and Target Detection

Signal Recovery and System Calibration from Multiple Compressive Poisson Measurements

Compressed sensing for practical optical imaging systems: a tutorial

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