<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The recovery of structured signals from a few linear measurements is a central point in both compressed sensing (CS) and discrete tomography. In CS the signal structure is described by means of a low complexity model e.g. co-/sparsity. The CS theory shows that any signal/image can be undersampled at a rate dependent on its intrinsic complexity. Moreover, in such undersampling regimes, the signal can be recovered by sparsity promoting convex regularization like <italic>ℓ</italic><sub>1</sub>- or total variation (TV-) minimization. Precise relations between many low complexity measures and the sufficient number of random measurements are known for many sparsity promoting norms. However, a precise estimate of the undersampling rate for the TV seminorm is still lacking. We address this issue by: a) providing dual certificates testing uniqueness of a given cosparse signal with bounded signal values, b) approximating the undersampling rates via the statistical dimension of the TV descent cone and c) showing empirically that the provided rates also hold for tomographic measurements.</p></abstract>

The recovery of structured signals from a few linear measurements is a central point in both compressed sensing (CS) and discrete tomography. In CS the signal structure is described by means of a low complexity model e.g. co-/sparsity. The CS theory shows that any signal/image can be undersampled at a rate dependent on its intrinsic complexity. Moreover, in such undersampling regimes, the signal can be recovered by sparsity promoting convex regularization like ℓ1- or total variation (TV-) minimization. Precise relations between many low complexity measures and the sufficient number of random measurements are known for many sparsity promoting norms. However, a precise estimate of the undersampling rate for the TV seminorm is still lacking. We address this issue by: a) providing dual certificates testing uniqueness of a given cosparse signal with bounded signal values, b) approximating the undersampling rates via the statistical dimension of the TV descent cone and c) showing empirically that the provided rates also hold for tomographic measurements.

Performance Bounds For Co-/Sparse Box Constrained Signal Recovery

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Performance Bounds For Co-/Sparse Box Constrained Signal Recovery"}

The recovery of structured signals from a few linear measurements is a central point in both compressed sensing (CS) and discrete tomography. In CS the signal...

Performance Bounds For Co-/Sparse Box Constrained Signal Recovery

Publié en ligne: 02 mars 2019

Pages: 79 - 106

Reçu: 01 mai 2017

Accepté: 01 oct. 2017

DOI: https://doi.org/10.2478/auom-2019-0005

Mots clés
Compressed Sensing, Discrete Tomography, Conic Geometry, Linear Inverse Problems, Convex Programming

© 2019 Jan Kuske et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Performance Bounds For Co-/Sparse Box Constrained Signal Recovery

Publié en ligne: 02 mars 2019

Pages: 79 - 106

Reçu: 01 mai 2017

Accepté: 01 oct. 2017

DOI: https://doi.org/10.2478/auom-2019-0005

Mots clésCompressed Sensing, Discrete Tomography, Conic Geometry, Linear Inverse Problems, Convex Programming

© 2019 Jan Kuske et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
Compressed Sensing, Discrete Tomography, Conic Geometry, Linear Inverse Problems, Convex Programming