Sparse Tomography

Hämäläinen, K.; Kallonen, Aki; Kolehmainen, Ville; Lassas, Matti; Niinimäki, Kati; Siltanen, Samuli

doi:10.1137/120876277

Cited by 60 publications

(72 citation statements)

References 59 publications

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“…The data are sampled, i.e., only finitely many projections along finitely many lines (through the region of interest) are taken. The case that projections are taken from only very few directions [29,34,48] is not considered in the present work.…”

Section: Numerical Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Wavelet methods for a weighted sparsity penalty for region of interest tomography

2015

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We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓ ppenalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the regularized solutions converge to the exact piecewise constant solution if the noise tends to zero. Tests on phantoms demonstrate the effectiveness of the method.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Stopping criteria such as Morozov's discrepancy principle or the S-curve method of [29] are still to be examined. 6.1.…”

Section: Numerical Resultsmentioning

confidence: 99%

Wavelet methods for a weighted sparsity penalty for region of interest tomography

2015

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“…Several recent studies apply the approach to a variety of applications, including medical imaging, signal analysis, and compressed sensing (see, e.g., Hämäläinen et al, 2013;Knopp and Weber, 2013;Candès et al, 2011). All of these applications make use of the fact that the underlying process can be described as a localized signal in a suitable representation system.…”

Section: N Hase Et Al: Atmospheric Inverse Modeling Via Sparse Recomentioning

confidence: 99%

Atmospheric inverse modeling via sparse reconstruction

et al. 2017

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Abstract. Many applications in atmospheric science involve ill-posed inverse problems. A crucial component of many inverse problems is the proper formulation of a priori knowledge about the unknown parameters. In most cases, this knowledge is expressed as a Gaussian prior. This formulation often performs well at capturing smoothed, large-scale processes but is often ill equipped to capture localized structures like large point sources or localized hot spots.Over the last decade, scientists from a diverse array of applied mathematics and engineering fields have developed sparse reconstruction techniques to identify localized structures. In this study, we present a new regularization approach for ill-posed inverse problems in atmospheric science. It is based on Tikhonov regularization with sparsity constraint and allows bounds on the parameters. We enforce sparsity using a dictionary representation system. We analyze its performance in an atmospheric inverse modeling scenario by estimating anthropogenic US methane (CH 4 ) emissions from simulated atmospheric measurements.Different measures indicate that our sparse reconstruction approach is better able to capture large point sources or localized hot spots than other methods commonly used in atmospheric inversions. It captures the overall signal equally well but adds details on the grid scale. This feature can be of value for any inverse problem with point or spatially discrete sources. We show an example for source estimation of synthetic methane emissions from the Barnett shale formation.

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“…Sparsity promoting algorithms have been used as a solution to the ill-posed inverse problem of tomography with limited projections [1]. Coded aperture X-ray tomography is a novel scheme constructed from concepts in coded spectral imaging and reference structure tomography introduced in [2], which can be used as a different alternative to sense the object and reconstruct images from less projections, thus reducing the radiation dose.…”

Section: Introductionmentioning

confidence: 99%