Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data

Borg, Leise; Frikel, Jürgen; Jørgensen, Jakob Sauer; Quinto, Eric Todd

doi:10.1137/18m1166833

Cited by 33 publications

(69 citation statements)

References 55 publications

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“…Test 2. In this case, we set Ω = (−1, 1) × (3,5). The distance between the source line Γ d and the domain Ω is 3, which is three times larger than the distance in test 1.…”

Section: 3mentioning

confidence: 99%

“…The missing data leads to the instability of the reconstruction. We cite some important papers [1,5,13,30,31] and references therein that characterize and (or) introduce the strategies to reduce the resulting artifacts, which appear in the reconstructed image. To treat the case of incompleteness, one might non-rigorously fill the missing data by the number 0, see figures 2-4 in [5].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

PDE-based numerical method for a limited angle x-ray tomography

Klibanov¹,

Nguyen²

2019

Inverse Problems

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A new numerical method for X-ray tomography for a specific case of incomplete Radon data is proposed. Potential applications are in checking out bulky luggage in airports. This method is based on the analysis of the transport PDE governing the X-ray tomography rather than on the conventional integral formulation. The quasi-reversibility method is applied. Convergence analysis is performed using a new Carleman estimate. Numerical results are presented and compared with the inversion of the Radon transform using the well-known filtered back projection algorithm. In addition, it is shown how to use our method to study the inversion of the attenuated X-ray transform for the same case of incomplete data.

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“…Test 2. In this case, we set Ω = (−1, 1) × (3,5). The distance between the source line Γ d and the domain Ω is 3, which is three times larger than the distance in test 1.…”

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

PDE-based numerical method for a limited angle x-ray tomography

Klibanov¹,

Nguyen²

2019

Inverse Problems

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show abstract

“…We hope that our framework for limited angle CT will spark further interest in applying similar ideas to other inverse problems. We anticipate that this might be particularly fruitful for problems, where larger amounts of data are not acquired in the measurements, as it is for instance the case in region of interest and exterior tomography [6]. Furthermore, we have observed that we could improve the learning phase by optimizing over a more sophisticated loss function defined over the image domain or by adding additional regularizers.…”

Section: Resultsmentioning

confidence: 99%

Learning the invisible: a hybrid deep learning-shearlet framework for limited angle computed tomography

et al. 2019

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The high complexity of various inverse problems poses a significant challenge to modelbased reconstruction schemes, which in such situations often reach their limits. At the same time, we witness an exceptional success of data-based methodologies such as deep learning. However, in the context of inverse problems, deep neural networks mostly act as black box routines, used for instance for a somewhat unspecified removal of artifacts in classical image reconstructions. In this paper, we will focus on the severely ill-posed inverse problem of limited angle computed tomography, in which entire boundary sections are not captured in the measurements. We will develop a hybrid reconstruction framework that fuses model-based sparse regularization with data-driven deep learning. Our method is reliable in the sense that we only learn the part that can provably not be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to the remaining parts. Such a decomposition into visible and invisible segments is achieved by means of the shearlet transform that allows to resolve wavefront sets in the phase space. Furthermore, this split enables us to assign the clear task of inferring unknown shearlet coefficients to the neural network and thereby offering an interpretation of its performance in the context of limited angle computed tomography.Our numerical experiments show that our algorithm significantly surpasses both pure model-and more data-based reconstruction methods.

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“…This is for example the basis of geometrical tomography [12] where the singularities of an attenuation are reconstructed from the singularities of its Radon transform. Furthermore this provides a framework for the description of artifacts that arise from limited data in transmission tomography [3,5].…”

Section: Tomographymentioning

confidence: 99%

Heuristic imaging from generic projections: backprojection outside the range of the Radon transform

Bellet¹,

Berginc²

2020

MathematicS in Action

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Reflective tomography is an efficient method for optical imaging in the visible and near infrared ranges. It computes empirical reconstructions based on algorithms from X-ray tomography. This subject introduces mathematical gaps to be filled, about the meaning of the reconstructions, and about their artifacts. To tackle these questions, we study more generally the filtered backprojection on projections outside the range of the Radon transform. We consider generic projections that can involve any kind of physical and geometric parameters. We claim that the backprojection contains partially the geometry of the original scene. More precisely, we compare the singularities of the backprojection with the singularities of a representation of the scene. This comparison of wavefront sets, inspired by studies of the artifacts in X-ray tomography, is based on microlocal analysis. It gives a precise meaning to the well-reconstructed geometry, describes the invisible parts, and the artifacts. We illustrate the heuristic and the analysis principle on canonical cases that belong to various fields: shape from silhouettes, constructible tomography, cloaking, reconstruction from cartoon images, imaging of occluded lambertian objects. Numerical results show the relevance of the heuristic and its analysis. In a word, this study provides a mathematical framework that covers the solver of reflective tomography, and exhibits an imaging method whose range of application is wide.

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Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data

Cited by 33 publications

References 55 publications

PDE-based numerical method for a limited angle x-ray tomography

PDE-based numerical method for a limited angle x-ray tomography

Learning the invisible: a hybrid deep learning-shearlet framework for limited angle computed tomography

Heuristic imaging from generic projections: backprojection outside the range of the Radon transform

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