Tomography of Tensor Fields in the Plain

Derevtsov, E.Yu.; Svetov, I. E.

doi:10.32523/2306-6172-2015-3-2-25-69

Cited by 15 publications

(24 citation statements)

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“…Such decomposition is unique. Using it, one can obtain a more detailed decomposition of a symmetric m-tensor field (see [11] for details).…”

Section: Differential Operatorsmentioning

confidence: 99%

“…For example, longitudinal, transverse and mixed ray transforms appear in the framework of 2D tensor tomography. In [11] tensor fields of arbitrary degree are analyzed, their representation through scalar potentials and a detailed decomposition are obtained. Also the paper presents the definition and classification of ray transforms acting on such fields and describes their kernels and images.…”

Section: Introductionmentioning

confidence: 99%

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Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

Polyakova,

Svetov

2023

Inverse Problems

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The operators of longitudinal, transverse and mixed ray transforms acting on two-dimensional symmetric tensor fields of arbitrary degree m in an unit disk are considered in the article. The singular value decompositions of the operators for a parallel scheme of data acquisition are constructed. Orthogonal bases in original spaces and image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions the polynomial expressions for the (pseudo)inverse and adjoint operators are obtained.

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“…Such decomposition is unique. Using it, one can obtain a more detailed decomposition of a symmetric m-tensor field (see [11] for details).…”

Section: Differential Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

Polyakova,

Svetov

2023

Inverse Problems

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“…It is easy to note that D 0 has a large kernel containing all the gradient fields vanishing at the boundary of the support. In response, a vast literature in tensor tomography concerns recovery questions (uniqueness, stability, reconstruction) on the solenoidal part of the tensor field; see the surveys [23,17,8,4,16,22] and reference therein. The problem is originally motivated by engineering practices [29,14,2,24].…”

Section: Introductionmentioning

confidence: 99%

Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc

Fujiwara¹,

Sadiq²,

Tamasan³

2022

IPI

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<p style='text-indent:20px;'>In two dimensions, we consider the problem of inversion of the attenuated <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula>-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula>-analytic functions in the sense of Bukhgeim.</p>

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“…Experimental and numerical details can be found in [31] and [32] respectively. In two dimensions, the work of [33] considers symmetric strain tomography, [34] considers symmetric tensors of arbitrary rank, and [35] also considers the non-symmetric strain A c c e p t e d M a n u s c r i p t case. Symmetric strain tomography of polycrystalline materials in three dimensions is explored by [36].…”

Section: Introductionmentioning

confidence: 99%

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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Tomography of Tensor Fields in the Plain

Cited by 15 publications

References 0 publications

Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc

Scanning electron diffraction tomography of strain

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