On Variable Density Compressive Sampling

Puy, Gilles; Vandergheynst, Pierre; Wiaux, Yves

doi:10.1109/lsp.2011.2163712

Cited by 125 publications

(94 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For Fourier sampling, numerous empirical sampling strategies have been proposed to overcome this problem [61,85], and several other works have followed more principled approaches based on designing sampling strategies to match the underlying coherence pattern (see [10,55,70,71] and references therein). However, these works do not explain the key role played by asymptotic sparsity in the CS recovery.…”

Section: Relation To Other Workmentioning

confidence: 99%

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Adcock

Hansen

Poon³

et al. 2017

Forum of Mathematics, Sigma

269

461

View full text Add to dashboard Cite

This paper presents a framework for compressed sensing that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications. In doing so, it also introduces a new sampling method that yields substantially improved recovery over existing techniques. In many applications of compressed sensing, including medical imaging, the standard principles of incoherence and sparsity are lacking. Whilst compressed sensing is often used successfully in such applications, it is done largely without mathematical explanation. The framework introduced in this paper provides such a justification. It does so by replacing these standard principles with three more general concepts: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. Moreover, not only does this work provide such a theoretical justification, it explains several key phenomena witnessed in practice. In particular, and unlike the standard theory, this work demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on the structure of the signal to be recovered. Another key consequence of this framework is the introduction of a new structured sampling method that exploits these phenomena to achieve significant improvements over current state-of-the-art techniques.2010 Mathematics Subject Classification: 94A20, 94A08 (primary); 42C40, 65R32, 92C55 (secondary)

show abstract

Section: Relation To Other Workmentioning

confidence: 99%

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Adcock

Hansen

Poon³

et al. 2017

Forum of Mathematics, Sigma

269

461

View full text Add to dashboard Cite

show abstract

“…To generate visibilities, a u-v coverage is generated randomly through the variable density sampling profile (Puy et al 2011) in half the Fourier plane with 10% of Fourier coefficients of each ground truth image; see Figure 2 for an example of the sampling profile. The visibilities are then corrupted by zero mean complex Gaussian noise with standard deviation σ computed by σ = f ∞10 −SNR/20 , where · ∞ is the infinity norm (the maximum absolute value of components of f ), and SNR (signal to noise ratio) is set to 30 dB for all simulations.…”

Section: Simulationsmentioning

confidence: 99%

Uncertainty quantification for radio interferometric imaging – I. Proximal MCMC methods

Cai

Pereyra

McEwen

2018

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Since radio interferometric imaging requires solving a high-dimensional, ill-posed inverse problem, uncertainty quantification is difficult but also critical to the accurate scientific interpretation of radio observations. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, traditional high-dimensional sampling methods are generally limited to smooth (e.g. Gaussian) priors and cannot be used with sparsity-promoting priors. Sparse priors, motivated by the theory of compressive sensing, have been shown to be highly effective for radio interferometric imaging. In this article proximal MCMC methods are developed for radio interferometric imaging, leveraging proximal calculus to support non-differential priors, such as sparse priors, in a Bayesian framework. Furthermore, three strategies to quantify uncertainties using the recovered posterior distribution are developed: (i) local (pixel-wise) credible intervals to provide error bars for each individual pixel; (ii) highest posterior density credible regions; and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner.

show abstract

“…In each of these works, the proposed designs involve parameters that need to be learned or tuned, and we are not aware of any efficient principled approaches for doing so. An alternative approach is taken in [12] based on minimizing coherence (see also [13] for a related work with unstructured matrices), but the optimization is only based on the measurement and sparsity bases, as opposed to training data.…”

Section: B Related Workmentioning

confidence: 99%

Learning-Based Compressive Subsampling

Baldassarre

Scarlett

et al. 2016

IEEE J. Sel. Top. Signal Process.

View full text Add to dashboard Cite

The problem of recovering a structured signal x ∈ C p from a set of dimensionality-reduced linear measurements b = Ax arises in a variety of applications, such as medical imaging, spectroscopy, Fourier optics, and computerized tomography. Due to computational and storage complexity or physical constraints imposed by the problem, the measurement matrix A ∈ C n×p is often of the form A = PΩΨ for some orthonormal basis matrix Ψ ∈ C p×p and subsampling operator PΩ : C p → C n that selects the rows indexed by Ω. This raises the fundamental question of how best to choose the index set Ω in order to optimize the recovery performance. Previous approaches to addressing this question rely on non-uniform random subsampling using application-specific knowledge of the structure of x. In this paper, we instead take a principled learning-based approach in which a fixed index set is chosen based on a set of training signals x1, . . . , xm. We formulate combinatorial optimization problems seeking to maximize the energy captured in these signals in an average-case or worst-case sense, and we show that these can be efficiently solved either exactly or approximately via the identification of modularity and submodularity structures. We provide both deterministic and statistical theoretical guarantees showing how the resulting measurement matrices perform on signals differing from the training signals, and we provide numerical examples showing our approach to be effective on a variety of data sets.

show abstract

On Variable Density Compressive Sampling

Cited by 125 publications

References 6 publications

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Uncertainty quantification for radio interferometric imaging – I. Proximal MCMC methods

Learning-Based Compressive Subsampling

Contact Info

Product

Resources

About