Near-Optimal Compressed Sensing Guarantees for Total Variation Minimization

Needell, Deanna; Ward, Rachel

doi:10.1109/tip.2013.2264681

Cited by 107 publications

(67 citation statements)

References 46 publications

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“…For dimensions d ≥ 2, Needell and Ward [NW13a,NW13b] derived recovery results for measurement matrices having the restricted isometry property (RIP) when composed with the Haar wavelet transform. Here we say that a matrix Φ has the RIP of order k and level δ if for every k-sparse vector x it holds that…”

Section: Recovery From Haar-incoherent Measurementsmentioning

confidence: 99%

Total Variation Minimization in Compressed Sensing

Krahmer

Kruschel

Sandbichler

2017

Compressed Sensing and Its Applications

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This chapter gives an overview over recovery guarantees for total variation minimization in compressed sensing for different measurement scenarios. In addition to summarizing the results in the area, we illustrate why an approach that is common for synthesis sparse signals fails and different techniques are necessary. Lastly, we discuss a generalizations of recent results for Gaussian measurements to the subgaussian case.

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Section: Recovery From Haar-incoherent Measurementsmentioning

confidence: 99%

Total Variation Minimization in Compressed Sensing

Krahmer

Kruschel

Sandbichler

2017

Compressed Sensing and Its Applications

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“…In 2D imaging CS, in order to use the piecewise smoothness in spatial domain, the total variation (TV) is often used to reconstruct the image from incomplete measurements. The widely used form of TV is TV l 1 l 2 , denoted as

{TV}_{{normall}_{1} {normall}_{2}} ({boldx}_{k, j}) = \sum_{i} \sqrt{{({boldD}_{normalh} {boldI}_{k, j})}_{i}^{2} + {({boldD}_{normalv} {boldI}_{k, j})}_{i}^{2}}

, where I k , j is the matrix form of x k , j , with i corresponding to the pixel position, and D h and D v the horizontal and vertical gradient operators.…”

Section: Proposed Algorithmmentioning

confidence: 99%

“…In 2D imaging CS, in order to use the piecewise smoothness in spatial domain, the total variation (TV) [28,29] is often used to reconstruct the image from incomplete measurements. The widely used form of TV is TV l 1 l 2 , denoted as [30]…”

Section: Two-dimensional Total Variationmentioning

confidence: 99%

Compressed sampling reconstruction of hyperspectral images based on spectral prediction

Wang

Feng

2015

IEEJ Transactions Elec Engng

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With increasing amounts of hyperspectral images (HSI) and the limitations of the memory requirements, compressive techniques for hyperspectral images have attracted extensive research efforts in recent years. The main difficulty of applying compressed sampling (CS) theory to compression and reconstruction of hyperspectral images lies in using the spatial correlation and spectral correlation of hyperspectral images. In this paper, a reconstruction algorithm of hyperspectral images taking advantage of two‐dimensional compressed sampling (2DCS) and two‐dimensional total variation (2DTV) incorporating spectral prediction (SP) is investigated. In the sampling process, the hyperspectral images are divided into reference bands and common bands, and all bands are sampled using 2DCS independently. In the reconstruction process, the reference bands are reconstructed by 2DTV first. In order to improve the reconstruction quality of common bands, spectral prediction utilizing the spectral correlation between reference bands and common bands is conducted. Then the spectral compensation is computed by using a combination of the prediction value and the initial approximation for the common bands. The residual between the compensation value and the original value is obtained to revise the approximation for the common bands. The algorithm is implemented in an iterative manner to enhance the performance. Experimental results on popular hyperspectral datasets reveal that the proposed algorithm exploiting spectral prediction outperforms the algorithm 2DCS‐2DTV, which does not use spectral correlation, as well as the state‐of‐the‐art algorithms in terms of peak signal‐to‐noise ratio (PSNR). In particular, when the sampling rate of the reference bands is higher than that of the common bands, the proposed algorithm would improve the reconstruction quality dramatically. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

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“…The gradient operator ∇ is not an orthonormal basis or a tight frame, thus neither the standard theory of CS nor the theoretical extensions in [5] concerning the analysis model apply to (3.1), even for images with truly sparse gradient ∇u. The recent work [27,26] proves that stable recovery is possible via the convex program (3.1) and considers a general matrix A which is incoherent with the multidimensional Haar wavelet transform and satisfies a RIP condition. The Haar wavelet transform provides a sparsifying basis for 2D and 3D images and is closely related to the discrete gradient operator.…”

Section: Sparsity and Tv-based Reconstructionmentioning

confidence: 99%

“…6], and [26] and specializes them to the case of anisotropic TV (1.7) as considered in the present paper, see also the Remarks following [27,Thm. 6] and [26,Main Thm.]. Theorem 3.1.…”

Section: Sparsity and Tv-based Reconstructionmentioning

confidence: 99%

Phase Transitions and Cosparse Tomographic Recovery of Compound Solid Bodies from Few Projections

Deniţiu

Petra

Schnörr

2014

Fundamenta Informaticae

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We study unique recovery of cosparse signals from limited-view tomographic measurements of two-and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality with respect to the discrete gradient operator. We relate both to the number of measurements and to a nullspace condition with respect to the measurement matrix, so as to achieve unique recovery by linear programming. These results are supported by comprehensive numerical experiments that show a high correlation of performance in practice and theoretical predictions. Despite poor properties of the measurement matrix from the viewpoint of compressed sensing, the class of uniquely recoverable signals basically seems large enough to cover practical applications, like contactless quality inspection of compound solid bodies composed of few materials.

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Near-Optimal Compressed Sensing Guarantees for Total Variation Minimization

Cited by 107 publications

References 46 publications

Total Variation Minimization in Compressed Sensing

Total Variation Minimization in Compressed Sensing

Compressed sampling reconstruction of hyperspectral images based on spectral prediction

Phase Transitions and Cosparse Tomographic Recovery of Compound Solid Bodies from Few Projections

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