A Note on Compressed Sensing of Structured Sparse Wavelet Coefficients From Subsampled Fourier Measurements

@article{Adcock2016ANO,
  title={A Note on Compressed Sensing of Structured Sparse Wavelet Coefficients From Subsampled Fourier Measurements},
  author={Ben Adcock and Anders C. Hansen and Bogdan Roman},
  journal={IEEE Signal Processing Letters},
  year={2016},
  volume={23},
  pages={732-736}
}
We consider signal recovery from Fourier measurements using compressed sensing (CS) with wavelets. For discrete signals with structured sparse Haar wavelet coefficients, we give the first proof of near-optimal recovery from discrete Fourier samples taken according to an appropriate variable density sampling scheme. Crucially, in taking into account such structured sparsity-known as sparsity in levels-as opposed to just sparsity, this result yields recovery guarantees that agree with the… 
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38 Citations

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On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels
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It is demonstrated that for natural compressed sensing matrices involving a level based reconstruction basis (e.g., wavelets), the number of measurements required to recover all $s-sparse signals for reasonable $s$ is excessive.
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References

SHOWING 1-10 OF 22 REFERENCES
Stable and Robust Sampling Strategies for Compressive Imaging
TLDR
The local coherence framework developed in this paper implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis-as opposed to bounded maximal coherence-as long as the sampling strategy is adapted accordingly.
An Analysis of Block Sampling Strategies in Compressed Sensing
TLDR
A new random sampling approach that consists of projecting the signal over blocks of sensing vectors, which provides a good insight on the possibilities and limits of block compressed sensing in imaging devices, such as magnetic resonance imaging, radio-interferometry, or ultra-sound imaging.
A Probabilistic and RIPless Theory of Compressed Sensing
TLDR
It is proved that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements.
BREAKING THE COHERENCE BARRIER: A NEW THEORY FOR COMPRESSED SENSING
TLDR
A framework for compressed sensing is presented that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications and explains several key phenomena witnessed in practice, and demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on theructure of the signal to be recovered.
On asymptotic structure in compressed sensing
TLDR
By using multilevel sampling, which exploits the structure of the signal, the paper explains how one can outperform random Gaussian/Bernoulli sampling even when the classical $l^1$ recovery algorithm is replaced by modified algorithms which aim to exploit structure such as model based or Bayesian compressed sensing or approximate message passaging.
Variable density compressed image sampling
TLDR
This paper addresses the design of a novel variable density sampling strategy, where the “a priori” information about the statistical distributions that natural images exhibit in the wavelet domain is exploited.
Sparse MRI: The application of compressed sensing for rapid MR imaging
TLDR
Practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference and demonstrate improved spatial resolution and accelerated acquisition for multislice fast spin‐echo brain imaging and 3D contrast enhanced angiography.
On the absence of the RIP in real-world applications of compressed sensing and the RIP in levels
TLDR
A new framework based on a generalised RIP-like definition is introduced that fits the applications where compressed sensing is used and it is shown that the shortcomings that show that uniform recovery is unreasonable no longer apply if the authors instead ask for structured recovery that is uniform only within each of the levels.
Sparsity and incoherence in compressive sampling
TLDR
It is shown that ℓ1 minimization recovers x0 exactly when the number of measurements exceeds S, and μ is the largest entry in U properly normalized: .
Generalized Sampling and Infinite-Dimensional Compressed Sensing
TLDR
A framework and corresponding method for compressed sensing in infinite dimensions is introduced and the introduction of two novel concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize an infinite-dimensional problem.
...
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