The singular value decomposition of the dynamic ray transforms operators acting on 2-tensor fields in ℝ<sup>2</sup>

Polyakova, A. P.; Svetov, I. E.

doi:10.1088/1742-6596/1715/1/012040

Cited by 2 publications

(2 citation statements)

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“…A solution on general differential manifolds is presented in [41] providing an explicit inverse formula for reconstruction of the solenoidal component of a second rank tensor field from projections acquired about three axes. Different from these decompositions is thesingular value decomposition of a dynamic acquisition of 2-tensors in R 2 ued to solve the inverse of the dynamic tensor projections [58]. This is to our knowledge the first application of tensor tomography to a dynamic acquisition of tensor projections.…”

Section: Helmholtz Decompositionmentioning

confidence: 99%

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

Tao

Rohmer

Gullberg

et al. 2022

IEEE Trans. Med. Imaging

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Tensor fields are useful for modeling the structure of biological tissues. The challenge to measure tensor fields involves acquiring sufficient data of scalar measurements that are physically achievable and reconstructing tensors from as few projections as possible for efficient applications in medical imaging. In this paper, we present a filtered back-projection algorithm for the reconstruction of a symmetric second-rank tensor field from directional X-ray projections about three axes. The tensor field is decomposed into a solenoidal and irrotational component, each of three unknowns. Using the Fourier projection theorem, a filtered back-projection algorithm is derived to reconstruct the solenoidal and irrotational components from projections acquired around three axes. A simple illustrative phantom consisting of two spherical shells and a 3D digital cardiac diffusion image obtained from diffusion tensor MRI of an excised human heart are used to simulate directional X-ray projections. The simulations validate the mathematical derivations and demonstrate reasonable noise properties of the algorithm. The decomposition of the tensor field into solenoidal and irrotational components provides insight into the development of algorithms for reconstructing tensor fields with sufficient samples in terms of the type of directional projections and the necessary orbits for the acquisition of the projections of the tensor field. Index Terms-Filtered back-projection algorithm, solenoidal and irrotational components, tensor tomography, directional X-ray projections. I. INTRODUCTION T ENSOR tomography has found important applications in the physical sciences [1], [2], mathematics [3], and medicine [4]. Here we consider the tensor tomography problem as Manuscript

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Section: Helmholtz Decompositionmentioning

confidence: 99%

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

Tao

Rohmer

Gullberg

et al. 2022

IEEE Trans. Med. Imaging

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show abstract

“…In [8,47] algorithms based on the truncated SVdecomposition method were developed and numerically implemented for the approximate recovery of two-dimensional vector and symmetric two-tensor fields. In addition, the SVdecomposition method was used to recover vector [32,34] and tensor [33] fields in the dynamic case.…”

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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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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The singular value decomposition of the dynamic ray transforms operators acting on 2-tensor fields in ℝ²

Cited by 2 publications

References 27 publications

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

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

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The singular value decomposition of the dynamic ray transforms operators acting on 2-tensor fields in ℝ2

Cited by 2 publications

References 27 publications

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

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

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The singular value decomposition of the dynamic ray transforms operators acting on 2-tensor fields in ℝ²