An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data

Desai, Naeem; Lionheart, William R B

doi:10.1088/0266-5611/32/11/115009

Cited by 15 publications

(23 citation statements)

References 12 publications

(28 reference statements)

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“…See [117] for a proof. To see how this relates to the Lens Rigidity Problem (see Problem 1), fix two metrics g andg on M and suppose they have same lens data 19 . Both metrics give rise to geodesic vector fields V and V on T * M , each of which uniquely characterizes g andg.…”

Section: From Problem 2 To Problemmentioning

confidence: 99%

“…Nowadays this field forms the basis of several non-invasive approaches to imaging internal properties of materials: seismology [42,123], or how to reconstruct the density inside the Earth from first arrival times of seismic wavefronts; medical imaging since the development of X-ray Computerized Tomography [75, 119,25]; Single-Photon Emission Computerized Tomography using the attenuated X-ray transform [76,74,78]; vector tomography in helio-seismology [51,94,52]; ocean imaging [73]; X-ray diffraction strain tomography [59,19] and tomography in elastic media [103,Ch. 7][108]; neutron imaging, as applied to the imaging of vertebrate remains [102] and shales [11].…”

Section: Introductionmentioning

confidence: 99%

“…Such problems, namely travel-time tomography and boundary rigidity as later formulated in [72,62], are concerned with recovering a Riemannian metric from the shortest length between any two boundary points. Such problems and their cousins (described below), now make the field of integral geometry, or how to reconstruct geometric features of a manifold from integral functionals defined over that manifold.Nowadays this field forms the basis of several non-invasive approaches to imaging internal properties of materials: seismology [42,123], or how to reconstruct the density inside the Earth from first arrival times of seismic wavefronts; medical imaging since the development of X-ray Computerized Tomography [75,119,25]; Single-Photon Emission Computerized Tomography using the attenuated X-ray transform [76,74,78]; vector tomography in helio-seismology [51,94,52]; ocean imaging [73]; X-ray diffraction strain tomography [59,19] and tomography in elastic media [103, Ch. 7][108]; neutron imaging, as applied to the imaging of vertebrate remains [102] and shales [11].…”

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confidence: 99%

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4. Integral geometry on manifolds with boundary and applications

Ilmavirta¹,

Monard²

2019

The Radon Transform

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We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.Johann Radon and his contemporaries formulated several integral geometric problems, not only in linear but also in non-linear settings [42,123]. Such problems, namely travel-time tomography and boundary rigidity as later formulated in [72,62], are concerned with recovering a Riemannian metric from the shortest length between any two boundary points. Such problems and their cousins (described below), now make the field of integral geometry, or how to reconstruct geometric features of a manifold from integral functionals defined over that manifold.Nowadays this field forms the basis of several non-invasive approaches to imaging internal properties of materials: seismology [42,123], or how to reconstruct the density inside the Earth from first arrival times of seismic wavefronts; medical imaging since the development of X-ray Computerized Tomography [75,119,25]; Single-Photon Emission Computerized Tomography using the attenuated X-ray transform [76,74,78]; vector tomography in helio-seismology [51,94,52]; ocean imaging [73]; X-ray diffraction strain tomography [59,19] and tomography in elastic media [103, Ch. 7][108]; neutron imaging, as applied to the imaging of vertebrate remains [102] and shales [11]. Non-linear integral geometric problems also continue to find new applications: recently, Neutron Spin Tomography [99] as a means to measure magnetic fields in materials, has arised as a novel method which can be of use in electrical engineering, superconductivity, etc. The transform to invert in this case is a non-linear operator, the so-called "non-abelian X-ray transform" of the magnetic field, see Problem 3 below.Recent breakthroughs have fuelled the field, exploiting a combination of old and new methods. Examples of such methods are: the systematic use of analysis on the unit sphere bundle combining energy methods (also coined "Pestov identities"), initiated by Mukhometov [71] and generalized in [91,103], and harmonic analysis on the tangent fibers [15,83,86]; in dimensions three and higher, the discovery in [120] that the existence of a foliation of the domain by strictly convex hyperfsurfaces, local or global, yields a powerful and robust approach to integral geometric inversions [116,118,125, 127], via a successful use of Melrose's scattering calculus [60]; the systematic use of analytic microlocal analysis to produce 'generic' results, implying the unique identifiability of unknown parameters in an open and dense subset of all cases [112,45, 128]; finally, recent results in the context of Anosov flows, leading to positive results for certain geometries with trapped sets [33,34,38].This review article aims at giving an overview of the arsenal of these methods, and to describe to what extent they help coping with various geometric settings, whose complexity is mainly governed by two features ...

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Section: From Problem 2 To Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

4. Integral geometry on manifolds with boundary and applications

Ilmavirta¹,

Monard²

2019

The Radon Transform

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“…This decomposition, Lemma 5.4, is very powerful for understanding the tensor tomography problem from an analytical stand-point. Theorem 5.3 lifts invertibility results from [30,47] to demonstrates when (or how much) the non-symmetric TRT is invertible. On the other hand, results from [47] also compute the (pseudo) inverses of the LRT and symmetric TRT which could be lifted to a filtered-back projection-like pseudo-inverse of the non-symmetric TRT.…”

Section: Invertibility Of Tensor Ray Transformsmentioning

confidence: 86%

Scanning electron diffraction tomography of strain

et al. 2020

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Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain components perpendicular to the incident electron beam direction. Extension of this method to recover the full three-dimensional strain tensor field has been restricted though by the absence of a formal framework for tensor tomography using such data. Here, we show that it is possible to reconstruct the full non-symmetric strain tensor field as the solution to an ill-posed tensor tomography inverse problem. We then demonstrate the properties of this tomography problem both analytically and computationally, highlighting why incorporating precession to perform scanning precession electron diffraction may be important. We establish a general framework for non-symmetric tensor tomography and demonstrate computationally its applicability for achieving strain tomography with scanning precession electron diffraction data.

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“…This work is extended in Desai and Lionheart [4] through the presentation of an explicit inversion formula for the transverse ray transform. In general, it was shown that this formula allows for the reconstruction of strain from high-energy X-ray measurements made around three axes of rotation.…”

Section: Introductionmentioning

confidence: 99%

A Bayesian approach to triaxial strain tomography from high‐energy X‐ray diffraction

Hendriks

Wensrich

Wills

2020

Strain

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Diffraction of high-energy X-rays produced at synchrotron sources can provide rapid strain measurements, with high spatial resolution, and good penetrating power. With an uncollimated diffracted beam, through-thickness averages of strain can be measured using this technique, which poses an associated rich tomography problem. This paper proposes a Gaussian process (GP) model for three-dimensional strain fields satisfying static equilibrium and an accompanying algorithm for tomographic reconstruction of strain fields from high-energy X-ray diffraction. This method can achieve triaxial strain tomography in three dimensions using only a single axis of rotation. The method builds upon recent work where the GP approach was used to reconstruct two-dimensional strain fields from neutron-based measurements. A demonstration is provided from simulated data, showing the method is capable of rejecting realistic levels of Gaussian noise.

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An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data

Cited by 15 publications

References 12 publications

4. Integral geometry on manifolds with boundary and applications

4. Integral geometry on manifolds with boundary and applications

Scanning electron diffraction tomography of strain

A Bayesian approach to triaxial strain tomography from high‐energy X‐ray diffraction

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