Microlocal Analysis in Tomography

Krishnan, Venkateswaran P.; Quinto, Eric Todd

doi:10.1007/978-3-642-27795-5_36-2

Cited by 10 publications

(8 citation statements)

References 100 publications

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“…Singularities which are invisible in the microlocal sense are related to the instability of inverting f from its limited X-ray data [22,29,30,31]. See also [21,32] for discussion of which part of the wave front set is visible in limited data tomography. Even though our theorem loses stability it gives uniqueness which is relevant for applications.…”

Section: 2mentioning

confidence: 99%

Unique continuation of the normal operator of the x-ray transform and applications in geophysics

Ilmavirta

Mönkkönen²

2020

Inverse Problems

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We show that the normal operator of the X-ray transform in R d , d ≥ 2, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes earlier results to lower regularity. Our proof also gives a unique continuation property for certain Riesz potentials. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.

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Section: 2mentioning

confidence: 99%

Unique continuation of the normal operator of the x-ray transform and applications in geophysics

Ilmavirta

Mönkkönen²

2020

Inverse Problems

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“…Generally cotangent spaces are used to describe microlocal ideas, but they would complicate this exposition, so we will identify a covector (x, ξdx) with the associated ordered pair of vectors (x, ξ). The book chapter [26] provides some basic microlocal ideas and a more elementary exposition adapted for tomography. The concept of the wavefront set is a central notion of microlocal analysis.…”

Section: Wavefront Setsmentioning

confidence: 99%

“…We adapt the standard terminology of microlocal analysis and consider wavefront sets as subsets of cotangent spaces [57]. Elementary presentations of microlocal analysis for tomography are in [26,27]. Standard references include [14,55].…”

Section: Appendix a Proofsmentioning

confidence: 99%

Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data

Borg

Frikel²,

Jørgensen

et al. 2018

SIAM J. Imaging Sci.

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This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray CT with arbitrary incomplete data. We prove that these singular artifacts arise from points at the boundary of the data set. Our results show that, depending on the geometry of this boundary, two types of artifacts can arise: object-dependent and object-independent artifacts. Object-dependent artifacts are generated by singularities of the object being scanned and these artifacts can extend along lines. They generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts, on the other hand, are essentially independent of the object and take one of two forms: streaks on lines if the boundary of the data set is not smooth at a point and curved artifacts if the boundary is smooth locally. We prove that these streak and curve artifacts are the only singular artifacts that can occur for FBP in the continuous case. In addition to the geometric description of artifacts, the article provides characterizations of their strength in Sobolev scale in certain cases. The results of this article apply to the well-known incomplete data problems, including limitedangle and region-of-interest tomography, as well as to unconventional X-ray CT imaging setups that arise in new practical applications. Reconstructions from simulated and real data are analyzed to illustrate our theorems, including the reconstruction that motivated this work-a synchrotron data set in which artifacts appear on lines that have no relation to the object.

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“…Basic discussions of microlocal analysis in tomography are in [27,42] and microlocal analysis is used in more general settings (e.g., [17]), seismics (e.g., [6]) radar (e.g., [26,48]) and X-ray CT (e.g., [11,25,28,37]). …”

Section: Basic Microlocal Analysismentioning

confidence: 99%

Artifacts and Visible Singularities in Limited Data X-Ray Tomography

Quinto

2017

Sens Imaging

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We describe a principle to determine which features of an object will be easy to reconstruct from limited X-ray CT data and which will be difficult. The principle depends on the geometry of the data set, and it applies to any limited data set. We also describe a characterization of Frikel and the author explaining artifacts that can be added to limited angle reconstructions, and we provide an easy-to-implement method to decrease them. These ideas are justified using microlocal analysis, deep mathematics that involves Fourier theory. Reconstructions from simulated and real limited data are given to illustrate our ideas.

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Microlocal Analysis in Tomography

Cited by 10 publications

References 100 publications

Unique continuation of the normal operator of the x-ray transform and applications in geophysics

Unique continuation of the normal operator of the x-ray transform and applications in geophysics

Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data

Artifacts and Visible Singularities in Limited Data X-Ray Tomography

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