Tomographic inversion using ℓ<sub>1</sub>-norm regularization of wavelet coefficients

Loris, Ignace; Nolet, Guust; Daubechies, Ingrid; Dahlen, F. A.

doi:10.1111/j.1365-246x.2007.03409.x

Cited by 151 publications

(126 citation statements)

References 26 publications

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“…There are also variety of works on Xray CT based on discrete transformations or dictionary bases [8], [15], [16]. In this paper, the Dual Tree-Complex Wavelet Transformation (DT-CWT) [17] is used.…”

Section: The Hierarchical Bayesian Methodsmentioning

confidence: 99%

“…Many methods have been studied in order to solve this l 1 norm optimization problem, for example the Newton's method [7] and the Split Bregman method [6]. Another class of the regularization method, called "Synthesis", considers a linear sparse transformation f = Dz and minimizes a criterion: J (z) = g − HDz 2 2 + λR(z) for example R(z) = z 1 [8]. When z obtained the object is reconstructed by f = D z.…”

Section: Introductionmentioning

confidence: 99%

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Bayesian method with sparsity enforcing prior of dual-tree complex wavelet transform coefficients for X-ray CT image reconstruction

Wang

Mohammad-Djafari

Gac

2017

2017 25th European Signal Processing Conference (EUSIPCO)

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Section: The Hierarchical Bayesian Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bayesian method with sparsity enforcing prior of dual-tree complex wavelet transform coefficients for X-ray CT image reconstruction

Wang

Mohammad-Djafari

Gac

2017

2017 25th European Signal Processing Conference (EUSIPCO)

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“…Seismic tomography, which estimates subsurface geologic facies, also has exploited sparsity for reconstruction. This has been demonstrated with wavelet representations of the subsurface and nonlinear forward models (Loris et al, 2007;Simons et al, 2011). Gholami and Siahkoohi (2010) used a split Bregman iteration (Goldstein and Osher, 2009) to solve a seismic tomography problem, imposing sparsity via soft thresholding (Donoho, 1995).…”

Section: Wavelets and Sparsity In Inverse Problemsmentioning

confidence: 99%

A multiresolution spatial parameterization for the estimation of fossil-fuel carbon dioxide emissions via atmospheric inversions

et al. 2014

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Abstract. The characterization of fossil-fuel CO 2 (ffCO 2 ) emissions is paramount to carbon cycle studies, but the use of atmospheric inverse modeling approaches for this purpose has been limited by the highly heterogeneous and nonGaussian spatiotemporal variability of emissions. Here we explore the feasibility of capturing this variability using a low-dimensional parameterization that can be implemented within the context of atmospheric CO 2 inverse problems aimed at constraining regional-scale emissions. We construct a multiresolution (i.e., wavelet-based) spatial parameterization for ffCO 2 emissions using the Vulcan inventory, and examine whether such a parameterization can capture a realistic representation of the expected spatial variability of actual emissions. We then explore whether sub-selecting wavelets using two easily available proxies of human activity (images of lights at night and maps of built-up areas) yields a low-dimensional alternative. We finally implement this low-dimensional parameterization within an idealized inversion, where a sparse reconstruction algorithm, an extension of stagewise orthogonal matching pursuit (StOMP), is used to identify the wavelet coefficients. We find that (i) the spatial variability of fossil-fuel emission can indeed be represented using a low-dimensional wavelet-based parameterization, (ii) that images of lights at night can be used as a proxy for sub-selecting wavelets for such analysis, and (iii) that implementing this parameterization within the described inversion framework makes it possible to quantify fossil-fuel emissions at regional scales if fossil-fuel-only CO 2 observations are available.

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“…Most of these inverse problems have been in the estimation of log-transformed permeability fields (Li and Jafarpour, 2010;Jafarpour, 2013), seismic tomography (Loris et al, 2007;Simons et al, 2011;Gholami and Siahkoohi, 2010) and estimation of point and distributed emissions (Hirst et al, 2013;Martinez-Camara et al, 2013). A more detailed review of the sparse reconstruction methods can be found in our previous paper .…”

Section: Introductionmentioning

confidence: 99%

A sparse reconstruction method for the estimation of multi-resolution emission fields via atmospheric inversion

et al. 2015

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Abstract. Atmospheric inversions are frequently used to estimate fluxes of atmospheric greenhouse gases (e.g., biospheric CO 2 flux fields) at Earth's surface. These inversions typically assume that flux departures from a prior model are spatially smoothly varying, which are then modeled using a multi-variate Gaussian. When the field being estimated is spatially rough, multi-variate Gaussian models are difficult to construct and a wavelet-based field model may be more suitable. Unfortunately, such models are very high dimensional and are most conveniently used when the estimation method can simultaneously perform data-driven model simplification (removal of model parameters that cannot be reliably estimated) and fitting. Such sparse reconstruction methods are typically not used in atmospheric inversions. In this work, we devise a sparse reconstruction method, and illustrate it in an idealized atmospheric inversion problem for the estimation of fossil fuel CO 2 (ffCO 2 ) emissions in the lower 48 states of the USA.Our new method is based on stagewise orthogonal matching pursuit (StOMP), a method used to reconstruct compressively sensed images. Our adaptations bestow three properties to the sparse reconstruction procedure which are useful in atmospheric inversions. We have modified StOMP to incorporate prior information on the emission field being estimated and to enforce non-negativity on the estimated field. Finally, though based on wavelets, our method allows for the estimation of fields in non-rectangular geometries, e.g., emission fields inside geographical and political boundaries.Our idealized inversions use a recently developed multiresolution (i.e., wavelet-based) random field model developed for ffCO 2 emissions and synthetic observations of ffCO 2 concentrations from a limited set of measurement sites. We find that our method for limiting the estimated field within an irregularly shaped region is about a factor of 10 faster than conventional approaches. It also reduces the overall computational cost by a factor of 2. Further, the sparse reconstruction scheme imposes non-negativity without introducing strong nonlinearities, such as those introduced by employing log-transformed fields, and thus reaps the benefits of simplicity and computational speed that are characteristic of linear inverse problems.

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Tomographic inversion using ℓ₁-norm regularization of wavelet coefficients

Cited by 151 publications

References 26 publications

Bayesian method with sparsity enforcing prior of dual-tree complex wavelet transform coefficients for X-ray CT image reconstruction

Bayesian method with sparsity enforcing prior of dual-tree complex wavelet transform coefficients for X-ray CT image reconstruction

A multiresolution spatial parameterization for the estimation of fossil-fuel carbon dioxide emissions via atmospheric inversions

A sparse reconstruction method for the estimation of multi-resolution emission fields via atmospheric inversion

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Tomographic inversion using ℓ1-norm regularization of wavelet coefficients

Cited by 151 publications

References 26 publications

Bayesian method with sparsity enforcing prior of dual-tree complex wavelet transform coefficients for X-ray CT image reconstruction

Bayesian method with sparsity enforcing prior of dual-tree complex wavelet transform coefficients for X-ray CT image reconstruction

A multiresolution spatial parameterization for the estimation of fossil-fuel carbon dioxide emissions via atmospheric inversions

A sparse reconstruction method for the estimation of multi-resolution emission fields via atmospheric inversion

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Product

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Tomographic inversion using ℓ₁-norm regularization of wavelet coefficients