Chapter 14: Looking Ahead

Hansen, P. C.; Jørgensen, Jørgen S.; Lionheart, William R B

doi:10.1137/1.9781611976670.ch14

Cited by 12 publications

(25 citation statements)

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“…Underdetermined linear system. Consider a small scale CT problem [10] of the form b " Ax `e, where the true signal x P R 100ˆ100 is the Shepp-Logan phantom and the noise satisfies e " N p0, λ ´1I q with λ " 10. Furthermore, the forward operator A is a discretized Radon transform at 20 equally spaced angles from 0 to 180 degrees with 120 rays per angle and using parallel-beam geometry.…”

Section: Gibbs Samplermentioning

confidence: 99%

Sparse Bayesian Inference with Regularized Gaussian Distributions

Everink¹,

Dong²,

Andersen³

2023

Preprint

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Regularization is a common tool in variational inverse problems to impose assumptions on the parameters of the problem. One such assumption is sparsity, which is commonly promoted using lasso and total variation-like regularization. Although the solutions to many such regularized inverse problems can be considered as points of maximum probability of well-chosen posterior distributions, samples from these distributions are generally not sparse. In this paper, we present a framework for implicitly defining a probability distribution that combines the effects of sparsity imposing regularization with Gaussian distributions. Unlike continuous distributions, these implicit distributions can assign positive probability to sparse vectors. We study these regularized distributions for various regularization functions including total variation regularization and piecewise linear convex functions. We apply the developed theory to uncertainty quantification for Bayesian linear inverse problems and derive a Gibbs sampler for a Bayesian hierarchical model. To illustrate the difference between our sparsity-inducing framework and continuous distributions, we apply our framework to small-scale deblurring and computed tomography examples.

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Section: Gibbs Samplermentioning

confidence: 99%

Sparse Bayesian Inference with Regularized Gaussian Distributions

Everink¹,

Dong²,

Andersen³

2023

Preprint

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“…Projections are recorded at different angles around the object. By modelling attenuation using the Beer-Lambert law with line integrals [7], we can formulate a discretized linear inverse problem b = Au + e.…”

Section: The Inverse Problem Of Computed Tomographymentioning

confidence: 99%

“…A ∈ R n×m is the system matrix where the entry A i,j describes the intersection length of the i'th x-ray with the j'th pixel. For more details on the general CT inverse problem see [7], for example, for a motivation of the form of (1), which arises from the Beer-Lambert law of x-ray attenuation in matter combined with a postlogarithm Gaussian approximation to the general Poisson noise model for transmission CT.…”

Section: The Inverse Problem Of Computed Tomographymentioning

confidence: 99%

“…Likelihood. We assume that high-intensity x-rays are used for the data collection and therefore we model the additive data noise as being independent and identically distributed (IID) Gaussian with mean zero and variance λ −1 , i.e., e ∼ N 0, λ −1 I , (7) where I ∈ R n×n is an identity matrix. From this follows that the (joint) likelihood function for the inverse problem (6) given an observation of the data b obs is given by…”

Section: Bayesian Formulationmentioning

confidence: 99%

“…Traditionally CT problems are addressed by formulating and solving deterministic linear inverse problems [6,7]. In that framework a single reconstruction is obtained with no information about its accuracy or uncertainty, which can arise from e.g.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Bayesian approach for CT reconstruction with defect detection for subsea pipelines

Christensen,

Riis,

Pereyra

et al. 2023

Inverse Problems

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Subsea pipelines can be inspected via 2D cross-sectional x-ray computed tomography (CT). Traditional reconstruction methods produce an image of the pipe’s interior that can be post-processed for detection of possible defects. In this paper we propose a novel Bayesian CT reconstruction method with built-in defect detection. We decompose the reconstruction into a sum of two images; one containing the overall pipe structure, and one containing defects, and infer the images simultaneously in a Gibbs scheme. Our method requires that prior information about the two images is very distinct, i.e. the first image should contain the large-scale and layered pipe structure, and the second image should contain small, coherent defects. We demonstrate our methodology with numerical experiments using synthetic and real CT data from scans of subsea pipes in cases with full and limited data. Experiments demonstrate the effectiveness of the proposed method in various data settings, with reconstruction quality comparable to existing techniques, while also providing defect detection with uncertainty quantification.

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CUQIpy: I. Computational uncertainty quantification for inverse problems in Python

Riis,

Alghamdi,

Uribe

et al. 2024

Inverse Problems

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This paper introduces CUQIpy, a versatile open-source Python package for computational uncertainty quantification (UQ) in inverse problems, presented as Part I of a two-part series. CUQIpy employs a Bayesian framework, integrating prior knowledge with observed data to produce posterior probability distributions that characterize the uncertainty in computed solutions to inverse problems. The package offers a high-level modeling framework with concise syntax, allowing users to easily specify their inverse problems, prior information, and statistical assumptions. CUQIpy supports a range of efficient sampling strategies and is designed to handle large-scale problems. Notably, the automatic sampler selection feature analyzes the problem structure and chooses a suitable sampler without user intervention, streamlining the process. With a selection of probability distributions, test problems, computational methods, and visualization tools, CUQIpy serves as a powerful, flexible, and adaptable tool for UQ in a wide selection of inverse problems. Part II of the series focuses on the use of CUQIpy for UQ in inverse problems with partial differential equations (PDEs).

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Chapter 14: Looking Ahead

Cited by 12 publications

References 0 publications

Sparse Bayesian Inference with Regularized Gaussian Distributions

Sparse Bayesian Inference with Regularized Gaussian Distributions

A Bayesian approach for CT reconstruction with defect detection for subsea pipelines

CUQIpy: I. Computational uncertainty quantification for inverse problems in Python

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