Parametric Level Set Methods for Inverse Problems

Aghasi, Alireza; Kilmer, Misha E.; Miller, Eric L.

doi:10.1137/100800208

Cited by 112 publications

(181 citation statements)

References 86 publications

(160 reference statements)

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“…In applications like FWI, it is not obvious how to update the level-set away from the boundary of the domain, because h 0 e (x) quickly tends to zero. To address this issue, we follow Aghasi et al (2011) and parametrize the level-set function using n terms as…”

Section: Level-set Methodsmentioning

confidence: 99%

A parametric level-set approach for seismic full-waveform inversion

Kadu

Leeuwen

Mulder

2016

SEG Technical Program Expanded Abstracts 2016

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SUMMARYIn seismic exploration, the delineation of large bodies with hard exterior contrasts but nearly constant interior properties is a challenge. Examples include salt diapirs, salt slabs, anhydrite or basalt layers. Salt geometries are of particular interest because they often have hydrocarbon reservoirs on their sides or underneath. This papers introduces a parametric level-set method for the reconstruction of such geometries in seismic full-waveform inversion (FWI). The level-set determines the outline of the salt geometry and evolves during the inversion in terms of its underlying parameters. For the latter, we employ Gaussian radial basis functions that can represent a large class of shapes with a small number of parameters. This keeps the dimensionality of the inverse problem small, which makes it easier to solve. First tests on a simple 2-D square box model show dramatic improvements over classic FWI.

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Section: Level-set Methodsmentioning

confidence: 99%

A parametric level-set approach for seismic full-waveform inversion

Kadu

Leeuwen

Mulder

2016

SEG Technical Program Expanded Abstracts 2016

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“…However, one can recover regions of large inhomogeneities in the absorption and/or diffusion coefficient, which may reveal the presence of some physical disorder (e.g., cancer). Therefore, we often use a parametric imaging model to target these regions for recovery [18,20,1], reducing the number of unknowns from the (very large) number of pixels/voxels in the image to the number of parameters. In this paper, we aim to recover images that are almost piecewise constant, looking for anomalous regions of diffusion and absorption on a possibly unknown constant background.…”

Section: Image Parameterizationmentioning

confidence: 99%

“…In this paper, we are interested in models h(p) that are nonlinear in p. In particular, we use regularization by parametric level sets [18,1] (see section 2), which reduces the imaging problem to the nonlinear optimization of a regularized (well-posed) problem for the vector of nonlinear parameters p.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Regularized Gauss–Newton Trust Region Approach to Imaging in Diffuse Optical Tomography

Sturler¹,

Kilmer²

2011

SIAM J. Sci. Comput.

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Abstract. We present a new algorithm for the solution of nonlinear least squares problems arising from parameterized imaging problems with diffuse optical tomographic data [D. Boas et al., IEEE Signal Process. Mag., 18 (2001), pp. 57-75]. The parameterization arises from the use of parametric level sets for regularization [M. E. Kilmer et al., Proc. SPIE, 5559 (2004), pp. 381-391], [A. Aghasi, M. E. Kilmer, and E. L. Miller, SIAM J. Imaging Sci., 4 (2011), pp. 618-650]. Such problems lead to Jacobians that have relatively few columns, a relatively modest number of rows, and are ill-conditioned. Moreover, such problems have function and Jacobian evaluations that are computationally expensive. Our optimization algorithm is appropriate for any inverse or imaging problem with those characteristics. In fact, we expect our algorithm to be effective for more general problems with ill-conditioned Jacobians. The algorithm aims to minimize the total number of function and Jacobian evaluations by analyzing which spectral components of the Gauss-Newton direction should be discarded or damped. The analysis considers for each component the reduction of the objective function and the contribution to the search direction, restricting the computed search direction to be within a trust region. The result is a truncated SVD-like approach to choosing the search direction. However, we do not necessarily discard components in order of decreasing singular value, and some components may be scaled to maintain fidelity to the trust region model. Our algorithm uses the Basic Trust Region Algorithm from [A. R. Conn, N. I. M. Gould, and Ph. L. Toint, Trust-Region Methods, SIAM, Philadelphia, 2000]. We prove that our algorithm is globally convergent to a critical point. Our numerical results show that the new algorithm generally outperforms competing methods applied to the DOT imaging problem with parametric level sets, and regularly does so by a significant factor.

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“…One is a statistical method, e.g., a Monte Carlo [26,27] or Bayesian [28,29] method. The other is optimization using a least-squares criterion [18,19,30,31], a regularization method [18][19][20][21][22][23], a level set method [32,33], etc. As a new molecular modality, the methods used for CLT reconstruction are mainly focused on optimization.…”

Section: Introductionmentioning

confidence: 99%

Probability method for Cerenkov luminescence tomography based on conformance error minimization

Ding¹,

Wang²,

Jie³

et al. 2014

Biomed. Opt. Express

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Cerenkov luminescence tomography (CLT) was developed to reconstruct a three-dimensional (3D) distribution of radioactive probes inside a living animal. Reconstruction methods are generally performed within a unique framework by searching for the optimum solution. However, the ill-posed aspect of the inverse problem usually results in the reconstruction being non-robust. In addition, the reconstructed result may not match reality since the difference between the highest and lowest uptakes of the resulting radiotracers may be considerably large, therefore the biological significance is lost. In this paper, based on the minimization of a conformance error, a probability method is proposed that consists of qualitative and quantitative modules. The proposed method first pinpoints the organ that contains the light source. Next, we developed a 0-1 linear optimization subject to a space constraint to model the CLT inverse problem, which was transformed into a forward problem by employing a region growing method to solve the optimization. After running through all of the elements used to grow the sources, a source sequence was obtained. Finally, the probability of each discrete node being the light source inside the organ was reconstructed. One numerical study and two in vivo experiments were conducted to verify the performance of the proposed algorithm, and comparisons were carried out using the hp-finite element method (hp-FEM). The results suggested that our proposed probability method was more robust and reasonable than hp-FEM.

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Parametric Level Set Methods for Inverse Problems

Cited by 112 publications

References 86 publications

A parametric level-set approach for seismic full-waveform inversion

A parametric level-set approach for seismic full-waveform inversion

A Regularized Gauss–Newton Trust Region Approach to Imaging in Diffuse Optical Tomography

Probability method for Cerenkov luminescence tomography based on conformance error minimization

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