Integral Geometry of Tensor Fields

Sharafutdinov, V. A.

doi:10.1515/9783110900095

Cited by 389 publications

(592 citation statements)

References 72 publications

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“…If we define the covector field v = (v 1 , v 2 ) on M by 16) then the previous formula can be written as follows:…”

Section: =ẋ(T) = E −µ(X(t)y(t)) Cos θ(T)γ 2 (T) =ẏ(T) = E −µ(X(t)ymentioning

confidence: 99%

“…In [21], the linear problem is considered under some assumption that is weaker than the simplicity. There is also a couple of results for manifolds with nonconvex boundary [16,6]. In the case of a simple manifold, it is known that the solenoidal part of any tensor field f ∈ Z 2 (S m τ M ) is smooth [4,17,20] and the kernel Z ∞ (S m τ M ) of the ray transform has a finite dimension modulo potential fields [15].…”

Section: Introductionmentioning

confidence: 99%

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Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Sharafutdinov

2007

J Geom Anal

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The Dirichlet-to-Neumann (DN) map Λ g : C ∞ (∂M ) → C ∞ (∂M ) on a compact Riemannian manifold (M, g) with boundary is defined by Λ g h = ∂u/∂ν| ∂M , where u is the solution to the Dirichlet problem ∆u = 0, u| ∂M = h and ν is the unit normal to the boundary. If g t = g + tf is a variation of the metric g by a symmetric tensor field f , then Λ g t = Λ g + tΛ f + o(t). We study the question: how do tensor fields f look like for whichΛ f = 0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: for the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.

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“…If we define the covector field v = (v 1 , v 2 ) on M by 16) then the previous formula can be written as follows:…”

Section: =ẋ(T) = E −µ(X(t)y(t)) Cos θ(T)γ 2 (T) =ẏ(T) = E −µ(X(t)ymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Sharafutdinov

2007

J Geom Anal

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“…(42b) and (42d) we learn that elements of the kernel of the ray transform can be characterized by the identical vanishing of a differential expression of the vector/tensor components w i , u ij . The differential operator acting in (42b) and (42d) is generically called the Saint-Venant operator [13]. Its vanishing can be understood as an integrability condition ensuring the existence of the scalar/vector fields φ and v occuring in eqs.…”

Section: Case Bmentioning

confidence: 99%

“…Finally, we might wonder whether there is any more information to be retrieved from the ray transforms Iw and Iu, or alternatively Ï and Í, as so far we could only reconstruct one tensor component from these data. The answer is no: As proven in [13], if any two stress tensors σ (1) and σ (2) satisfy (26, 27) and σ (1) 33 = σ (2) 33 then they produce the same ray transforms, Iw (1) = Iw (2) and Iu (1) = Iu (2) . As a consequence, the retrieval of one tensor component is the maximum information to be gleaned from a ray transform which is performed for horizontal lines only.…”

Section: Longitudinal and Transverse Tensor Componentsmentioning

confidence: 99%

“…In [13], Sharafutdinov has proposed a generalization of the Radon transform to the case of symmetric tensor fields of arbitrary degree m defined on a, possibly curved, Riemannian manifold of arbitrary dimension n; he has termed his construction the ray transform. Amongst many other aspects, his work encompasses the examination of the formal structure of the ray transform, its relation to the Fourier transform, its non-trivial kernel and the problem of inversion of the ray transform given the fact that the kernel is non-zero.…”

Section: Introductionmentioning

confidence: 99%

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Application of Sharafutdinov?s Ray Transform in Integrated Photoelasticity

Hammer¹,

Lionheart²

2004

J Elasticity

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Abstract. We explain the main concepts centered around Sharafutdinov's ray transform, its kernel, and the extent to which it can be inverted. It is shown how the ray transform emerges naturally in any attempt to reconstruct optical and stress tensors within a photoelastic medium from measurements on the state of polarization of light beams passing through the strained medium. The problem of reconstruction of stress tensors is crucially related to the fact that the ray transform has a nontrivial kernel; the latter is described by a theorem for which we provide a new proof which is simpler and shorter as in Sharafutdinov's original work, as we limit our scope to tensors which are relevant to Photoelasticity. We explain how the kernel of the ray transform is related to the decomposition of tensor fields into longitudinal and transverse components. The merits of the ray transform as a tool for tensor reconstruction are studied by walking through an explicit example of reconstructing the σ33-component of the stress tensor in a cylindrical photoelastic specimen. In order to make the paper self-contained we provide a derivation of the basic equations of Integrated Photoelasticity which describe how the presence of stress within a photoelastic medium influences the passage of polarized light through the material.

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Consistent Inversion of Noisy Non‐Abelian X‐Ray Transforms

Monard

Nickl

Paternain

2020

Comm Pure Appl Math

133

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For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field normalΦ:M→frakturso()n from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite‐dimensional MCMC methods. It is further shown that as the number N of measurements of point evaluations of CΦ increases, the statistical error in the recovery of Φ converges to 0 in L2(M)‐distance at a rate that is algebraic in 1/N and approaches 1/N for smooth matrix fields Φ. The proof relies, among other things, on a new stability estimate for the inverse map CΦ → Φ. Key applications of our results are discussed in the case n = 3 to polarimetric neutron tomography. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC

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Integral Geometry of Tensor Fields

Cited by 389 publications

References 72 publications

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

Application of Sharafutdinov?s Ray Transform in Integrated Photoelasticity

Consistent Inversion of Noisy Non‐Abelian X‐Ray Transforms

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