Linear Inversion of Band-Limited Reflection Seismograms

Santosa, Fadil; Symes, William W.

doi:10.1137/0907087

Cited by 639 publications

(453 citation statements)

References 6 publications

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“…The use of`1 minimization for the recovery of sparse spike trains from noisy band-limited measurements has a long history and was proposed in the 1980s by researchers in seismic prospecting [8,28,36]. For finite signals and under the rather restrictive assumption that the signal is real valued and nonnegative, [21] and [13] prove that k spikes can be recovered from 2k C 1 Fourier coefficients by this method.…”

Section: Comparison With Related Workmentioning

confidence: 99%

Towards a Mathematical Theory of Super‐resolution

Candès

Fernandez‐Granda

2013

Comm Pure Appl Math

1,060

1,574

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This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an objectthe high end of its spectrum-from coarse scale information only-from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in OE0; 1 and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff f c . We show that one can super-resolve these point sources with infinite precision-i.e., recover the exact locations and amplitudes-by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2=f c . This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary.

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Section: Comparison With Related Workmentioning

confidence: 99%

Towards a Mathematical Theory of Super‐resolution

Candès

Fernandez‐Granda

2013

Comm Pure Appl Math

1,060

1,574

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“…Because of this feature, (1) has been used for more than three decades in several signal processing problems where sparseness is sought; some early references are [12], [37], [50], [53]. In the 1990's, seminal work on the use of ℓ 1 sparseness-inducing penalties/log-priors appeared in the literature: the now famous basis pursuit denoising (BPDN, [11,Section 5]) criterion and the least absolute shrinkage and selection operator (LASSO, [54]).…”

Section: A Backgroundmentioning

confidence: 99%

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Figueiredo

Nowak

Wright

2007

IEEE J. Sel. Top. Signal Process.

3,056

1,847

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Abstract-Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a sparseness-inducing (ℓ1) regularization term.Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution, and compressed sensing are a few wellknown examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the BarzilaiBorwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.

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“…The L1 norm of a vector is simply the sum of the absolute values of the vector components. L1 norm minimization has been studied well and its history can be traced back to geophysics in the mideighties (Santosa & Symes 1986).…”

Section: Reconstructionmentioning

confidence: 99%

The application of compressive sampling to radio astronomy

Cornwell

Hoog

2011

A&A

103

114

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Compressive sampling is a new paradigm for sampling, based on sparseness of signals or signal representations. It is much less restrictive than Nyquist-Shannon sampling theory and thus explains and systematises the widespread experience that methods such as the Högbom CLEAN can violate the Nyquist-Shannon sampling requirements. In this paper, a CS-based deconvolution method for extended sources is introduced. This method can reconstruct both point sources and extended sources (using the isotropic undecimated wavelet transform as a basis function for the reconstruction step). We compare this CS-based deconvolution method with two CLEANbased deconvolution methods: the Högbom CLEAN and the multiscale CLEAN. This new method shows the best performance in deconvolving extended sources for both uniform and natural weighting of the sampled visibilities. Both visual and numerical results of the comparison are provided.

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Linear Inversion of Band-Limited Reflection Seismograms

Cited by 639 publications

References 6 publications

Towards a Mathematical Theory of Super‐resolution

Towards a Mathematical Theory of Super‐resolution

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

The application of compressive sampling to radio astronomy

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