Reconstruction of strain –elds from tomographic data · residual elastic strain throughout...
Transcript of Reconstruction of strain –elds from tomographic data · residual elastic strain throughout...
Reconstruction of strain �elds from tomographic data
Victor PalamodovTel Aviv University
Ban¤ International Research StationHybrid methods in Imaging
June 18, 2015
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 1 / 31
The objective is to reconstruct three-dimensional maps of variation inresidual elastic strain throughout samples using one-dimensionalprojected data.
Evaluation of the strain requires imaging a six-component tensorquantity in 3D.
An application of tomographic ideas to reconstruction of smallresidual strain �elds from data of di¤raction pattern under penetratedX-ray or neutron radiation was proposed by A. Korsunsky et al:
I Korsunsky A M et al 2006 The principle of strain reconstructiontomography: Determination of quench strain distribution fromdi¤raction measurements Acta Materialia 54
I Korsunsky A M et al 2009 Feasibility study of neutron straintomography Procedia Engineering 1
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 2 / 31
The objective is to reconstruct three-dimensional maps of variation inresidual elastic strain throughout samples using one-dimensionalprojected data.
Evaluation of the strain requires imaging a six-component tensorquantity in 3D.
An application of tomographic ideas to reconstruction of smallresidual strain �elds from data of di¤raction pattern under penetratedX-ray or neutron radiation was proposed by A. Korsunsky et al:
I Korsunsky A M et al 2006 The principle of strain reconstructiontomography: Determination of quench strain distribution fromdi¤raction measurements Acta Materialia 54
I Korsunsky A M et al 2009 Feasibility study of neutron straintomography Procedia Engineering 1
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 2 / 31
The objective is to reconstruct three-dimensional maps of variation inresidual elastic strain throughout samples using one-dimensionalprojected data.
Evaluation of the strain requires imaging a six-component tensorquantity in 3D.
An application of tomographic ideas to reconstruction of smallresidual strain �elds from data of di¤raction pattern under penetratedX-ray or neutron radiation was proposed by A. Korsunsky et al:
I Korsunsky A M et al 2006 The principle of strain reconstructiontomography: Determination of quench strain distribution fromdi¤raction measurements Acta Materialia 54
I Korsunsky A M et al 2009 Feasibility study of neutron straintomography Procedia Engineering 1
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 2 / 31
The objective is to reconstruct three-dimensional maps of variation inresidual elastic strain throughout samples using one-dimensionalprojected data.
Evaluation of the strain requires imaging a six-component tensorquantity in 3D.
An application of tomographic ideas to reconstruction of smallresidual strain �elds from data of di¤raction pattern under penetratedX-ray or neutron radiation was proposed by A. Korsunsky et al:
I Korsunsky A M et al 2006 The principle of strain reconstructiontomography: Determination of quench strain distribution fromdi¤raction measurements Acta Materialia 54
I Korsunsky A M et al 2009 Feasibility study of neutron straintomography Procedia Engineering 1
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 2 / 31
The objective is to reconstruct three-dimensional maps of variation inresidual elastic strain throughout samples using one-dimensionalprojected data.
Evaluation of the strain requires imaging a six-component tensorquantity in 3D.
An application of tomographic ideas to reconstruction of smallresidual strain �elds from data of di¤raction pattern under penetratedX-ray or neutron radiation was proposed by A. Korsunsky et al:
I Korsunsky A M et al 2006 The principle of strain reconstructiontomography: Determination of quench strain distribution fromdi¤raction measurements Acta Materialia 54
I Korsunsky A M et al 2009 Feasibility study of neutron straintomography Procedia Engineering 1
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 2 / 31
Stress tensor
σ = ∑ σijdxidxj
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 3 / 31
Strain tensor
Constitutive equation σ = cε. where ε is the strain tensor, c is thesti¤ness tensor.
Coaxial case
σij = λδij trε+ 2µεij , trε = ε11 + ε22 + ε33,
where λ, µ are Lamé parameters.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 4 / 31
Strain tensor
Constitutive equation σ = cε. where ε is the strain tensor, c is thesti¤ness tensor.
Coaxial case
σij = λδij trε+ 2µεij , trε = ε11 + ε22 + ε33,
where λ, µ are Lamé parameters.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 4 / 31
Strain tensor
Constitutive equation σ = cε. where ε is the strain tensor, c is thesti¤ness tensor.
Coaxial case
σij = λδij trε+ 2µεij , trε = ε11 + ε22 + ε33,
where λ, µ are Lamé parameters.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 4 / 31
Mathematical model
Let E be an Euclidean space of dimension 3. Symmetric tensor �eldsof rank 1 and 2 are
f = ∑ fidx i , g = ∑ gijdx i � dx j ,
where the product of forms is symmetric.
The coe¢ cients fi and gij = gji transform as vectors and bivectors,respectively.
The symmetric di¤erential reads
Df = g , gii = ∂i fi , gij =12(∂i fj + ∂j fi ) , ∂i = ∂/∂x i .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 5 / 31
Mathematical model
Let E be an Euclidean space of dimension 3. Symmetric tensor �eldsof rank 1 and 2 are
f = ∑ fidx i , g = ∑ gijdx i � dx j ,
where the product of forms is symmetric.
The coe¢ cients fi and gij = gji transform as vectors and bivectors,respectively.
The symmetric di¤erential reads
Df = g , gii = ∂i fi , gij =12(∂i fj + ∂j fi ) , ∂i = ∂/∂x i .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 5 / 31
Mathematical model
Let E be an Euclidean space of dimension 3. Symmetric tensor �eldsof rank 1 and 2 are
f = ∑ fidx i , g = ∑ gijdx i � dx j ,
where the product of forms is symmetric.
The coe¢ cients fi and gij = gji transform as vectors and bivectors,respectively.
The symmetric di¤erential reads
Df = g , gii = ∂i fi , gij =12(∂i fj + ∂j fi ) , ∂i = ∂/∂x i .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 5 / 31
Axial (longitudinal) integrals
For a 2-tensor �eld g with compact support, the axial (longitudinal)ray integrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
where y , v 2 E , v 6= 0 and
g (x ; u, v) = ∑ gij (x) uivj , u, v 2 E .Xg (y , v) = 0 if g is potential, that is g = Df , and f (y) = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 6 / 31
Axial (longitudinal) integrals
For a 2-tensor �eld g with compact support, the axial (longitudinal)ray integrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
where y , v 2 E , v 6= 0 and
g (x ; u, v) = ∑ gij (x) uivj , u, v 2 E .
Xg (y , v) = 0 if g is potential, that is g = Df , and f (y) = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 6 / 31
Axial (longitudinal) integrals
For a 2-tensor �eld g with compact support, the axial (longitudinal)ray integrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
where y , v 2 E , v 6= 0 and
g (x ; u, v) = ∑ gij (x) uivj , u, v 2 E .Xg (y , v) = 0 if g is potential, that is g = Df , and f (y) = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 6 / 31
Bragg scattering
Bragg�s law: 2d sin θ = Nλ :
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 7 / 31
Bragg scattering
Bragg�s law: 2d sin θ = Nλ :
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 7 / 31
Debye-Scherrer rings
Di¤raction patterns (D�S rings) are collected by employing aposition-sensitive detector
Incident Xray beam
2D detector
sample
DebyeScherrer rings
E=87KeV, d=0.1mm
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 8 / 31
Determination of the axial integrals
Strain measurementdirection Diffracted
beam
Diffractedbeam
Incident X or neutron ray20
The model for the Bragg di¤raction is the axial (longitudinal) rayintegrals of the strain tensor.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 9 / 31
Determination of the axial integrals
Strain measurementdirection Diffracted
beam
Diffractedbeam
Incident X or neutron ray20
The model for the Bragg di¤raction is the axial (longitudinal) rayintegrals of the strain tensor.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 9 / 31
Saint-Venant tensor
Adhemar de Saint-Venant (1860) introduced the di¤erentialconsistency equation Vε = 0 for a small potential tensor �eld ε.Boussinesq, Beltrami, Cesaro proved its su¢ ciency for a 2-tensor tobe potential in a simply connected domain. Volterra...
Sharafutdinov: a tensor �eld g in Rn: of rank m with compactsupport is potential if (and only if) the axial line integrals of gvanishes for all lines.
Paternain, Salo, Uhlmann: this property holds for the geodesictransform in simple surfaces with boundary
I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47
I Monard F 2014 On reconstruction formulas for the ray transform actingon symmetric di¤erentials on surfaces Inverse Probl. 30
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 10 / 31
Saint-Venant tensor
Adhemar de Saint-Venant (1860) introduced the di¤erentialconsistency equation Vε = 0 for a small potential tensor �eld ε.Boussinesq, Beltrami, Cesaro proved its su¢ ciency for a 2-tensor tobe potential in a simply connected domain. Volterra...
Sharafutdinov: a tensor �eld g in Rn: of rank m with compactsupport is potential if (and only if) the axial line integrals of gvanishes for all lines.
Paternain, Salo, Uhlmann: this property holds for the geodesictransform in simple surfaces with boundary
I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47
I Monard F 2014 On reconstruction formulas for the ray transform actingon symmetric di¤erentials on surfaces Inverse Probl. 30
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 10 / 31
Saint-Venant tensor
Adhemar de Saint-Venant (1860) introduced the di¤erentialconsistency equation Vε = 0 for a small potential tensor �eld ε.Boussinesq, Beltrami, Cesaro proved its su¢ ciency for a 2-tensor tobe potential in a simply connected domain. Volterra...
Sharafutdinov: a tensor �eld g in Rn: of rank m with compactsupport is potential if (and only if) the axial line integrals of gvanishes for all lines.
Paternain, Salo, Uhlmann: this property holds for the geodesictransform in simple surfaces with boundary
I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47
I Monard F 2014 On reconstruction formulas for the ray transform actingon symmetric di¤erentials on surfaces Inverse Probl. 30
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 10 / 31
Saint-Venant tensor
Adhemar de Saint-Venant (1860) introduced the di¤erentialconsistency equation Vε = 0 for a small potential tensor �eld ε.Boussinesq, Beltrami, Cesaro proved its su¢ ciency for a 2-tensor tobe potential in a simply connected domain. Volterra...
Sharafutdinov: a tensor �eld g in Rn: of rank m with compactsupport is potential if (and only if) the axial line integrals of gvanishes for all lines.
Paternain, Salo, Uhlmann: this property holds for the geodesictransform in simple surfaces with boundary
I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)
I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47
I Monard F 2014 On reconstruction formulas for the ray transform actingon symmetric di¤erentials on surfaces Inverse Probl. 30
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 10 / 31
Saint-Venant tensor
Adhemar de Saint-Venant (1860) introduced the di¤erentialconsistency equation Vε = 0 for a small potential tensor �eld ε.Boussinesq, Beltrami, Cesaro proved its su¢ ciency for a 2-tensor tobe potential in a simply connected domain. Volterra...
Sharafutdinov: a tensor �eld g in Rn: of rank m with compactsupport is potential if (and only if) the axial line integrals of gvanishes for all lines.
Paternain, Salo, Uhlmann: this property holds for the geodesictransform in simple surfaces with boundary
I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47
I Monard F 2014 On reconstruction formulas for the ray transform actingon symmetric di¤erentials on surfaces Inverse Probl. 30
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 10 / 31
Saint-Venant tensor
Adhemar de Saint-Venant (1860) introduced the di¤erentialconsistency equation Vε = 0 for a small potential tensor �eld ε.Boussinesq, Beltrami, Cesaro proved its su¢ ciency for a 2-tensor tobe potential in a simply connected domain. Volterra...
Sharafutdinov: a tensor �eld g in Rn: of rank m with compactsupport is potential if (and only if) the axial line integrals of gvanishes for all lines.
Paternain, Salo, Uhlmann: this property holds for the geodesictransform in simple surfaces with boundary
I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47
I Monard F 2014 On reconstruction formulas for the ray transform actingon symmetric di¤erentials on surfaces Inverse Probl. 30
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 10 / 31
Saint-Venant operator
Let Σ2 and Λ2 be the bundles of symmetric and skew symmetricdi¤erential forms of degree 2.
B4 + Λ2 S Λ2 be the symmetric square.
Any b 2 B4 is a tensor �eld whose components bij ,kl are functionssuch that bkl ,ij = bij ,kl , bji ,kl = �bij ,kl , bij ,lk = �...Saint-Venant operator V : Σ2 ! B4 is de�ned for a �eld g 2 Σ2 by
(Vg)ij ,kl + ∂i∂kgjl � ∂i∂lgjk � ∂j∂kgil + ∂j∂lgik .
The �elds ∂i , ∂j , ∂k , ∂l can be replaced here by arbitrary tangentvectors α, β,γ, δ in E :
(Vg)αβ,γδ = ∂α∂γg (β, δ)� ∂α∂δg (β,γ)� ∂β∂γg (α, δ)+ ∂β∂δg (α,γ) .
Tensor Vg vanishes for any potential �eld g = Df , since VD = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 11 / 31
Saint-Venant operator
Let Σ2 and Λ2 be the bundles of symmetric and skew symmetricdi¤erential forms of degree 2.
B4 + Λ2 S Λ2 be the symmetric square.
Any b 2 B4 is a tensor �eld whose components bij ,kl are functionssuch that bkl ,ij = bij ,kl , bji ,kl = �bij ,kl , bij ,lk = �...Saint-Venant operator V : Σ2 ! B4 is de�ned for a �eld g 2 Σ2 by
(Vg)ij ,kl + ∂i∂kgjl � ∂i∂lgjk � ∂j∂kgil + ∂j∂lgik .
The �elds ∂i , ∂j , ∂k , ∂l can be replaced here by arbitrary tangentvectors α, β,γ, δ in E :
(Vg)αβ,γδ = ∂α∂γg (β, δ)� ∂α∂δg (β,γ)� ∂β∂γg (α, δ)+ ∂β∂δg (α,γ) .
Tensor Vg vanishes for any potential �eld g = Df , since VD = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 11 / 31
Saint-Venant operator
Let Σ2 and Λ2 be the bundles of symmetric and skew symmetricdi¤erential forms of degree 2.
B4 + Λ2 S Λ2 be the symmetric square.
Any b 2 B4 is a tensor �eld whose components bij ,kl are functionssuch that bkl ,ij = bij ,kl , bji ,kl = �bij ,kl , bij ,lk = �...
Saint-Venant operator V : Σ2 ! B4 is de�ned for a �eld g 2 Σ2 by
(Vg)ij ,kl + ∂i∂kgjl � ∂i∂lgjk � ∂j∂kgil + ∂j∂lgik .
The �elds ∂i , ∂j , ∂k , ∂l can be replaced here by arbitrary tangentvectors α, β,γ, δ in E :
(Vg)αβ,γδ = ∂α∂γg (β, δ)� ∂α∂δg (β,γ)� ∂β∂γg (α, δ)+ ∂β∂δg (α,γ) .
Tensor Vg vanishes for any potential �eld g = Df , since VD = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 11 / 31
Saint-Venant operator
Let Σ2 and Λ2 be the bundles of symmetric and skew symmetricdi¤erential forms of degree 2.
B4 + Λ2 S Λ2 be the symmetric square.
Any b 2 B4 is a tensor �eld whose components bij ,kl are functionssuch that bkl ,ij = bij ,kl , bji ,kl = �bij ,kl , bij ,lk = �...Saint-Venant operator V : Σ2 ! B4 is de�ned for a �eld g 2 Σ2 by
(Vg)ij ,kl + ∂i∂kgjl � ∂i∂lgjk � ∂j∂kgil + ∂j∂lgik .
The �elds ∂i , ∂j , ∂k , ∂l can be replaced here by arbitrary tangentvectors α, β,γ, δ in E :
(Vg)αβ,γδ = ∂α∂γg (β, δ)� ∂α∂δg (β,γ)� ∂β∂γg (α, δ)+ ∂β∂δg (α,γ) .
Tensor Vg vanishes for any potential �eld g = Df , since VD = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 11 / 31
Reconstruction of the Saint-Venant tensor
Theorem
Let K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .
Notations. For a 2-tensor �eld g with compact support, the rayintegrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
We set
∂αXg (y ; v) = hα,ry iXg (y ; v) , ∂;αXg (y ; v) =∂
∂tXg (y ; v + tα)jt=0 .
for any α 2 E .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 12 / 31
Reconstruction of the Saint-Venant tensor
TheoremLet K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .
S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .
Notations. For a 2-tensor �eld g with compact support, the rayintegrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
We set
∂αXg (y ; v) = hα,ry iXg (y ; v) , ∂;αXg (y ; v) =∂
∂tXg (y ; v + tα)jt=0 .
for any α 2 E .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 12 / 31
Reconstruction of the Saint-Venant tensor
TheoremLet K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .
Notations. For a 2-tensor �eld g with compact support, the rayintegrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
We set
∂αXg (y ; v) = hα,ry iXg (y ; v) , ∂;αXg (y ; v) =∂
∂tXg (y ; v + tα)jt=0 .
for any α 2 E .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 12 / 31
Reconstruction of the Saint-Venant tensor
TheoremLet K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .
Notations. For a 2-tensor �eld g with compact support, the rayintegrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
We set
∂αXg (y ; v) = hα,ry iXg (y ; v) , ∂;αXg (y ; v) =∂
∂tXg (y ; v + tα)jt=0 .
for any α 2 E .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 12 / 31
Reconstruction of the Saint-Venant tensor
TheoremLet K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .
Notations. For a 2-tensor �eld g with compact support, the rayintegrals are
Xg (y ; v) =Z ∞
0g (y + tv ; v , v)dt.
We set
∂αXg (y ; v) = hα,ry iXg (y ; v) , ∂;αXg (y ; v) =∂
∂tXg (y ; v + tα)jt=0 .
for any α 2 E .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 12 / 31
Reconstruction from ray integrals
Problem: reconstruct the Saint-Venant tensor Vε from data of rayintegrals Xε.
First step: for a plane H � E , a point y 2 H and arbitrary vectorsα, β, compute
RH (α,ω; β,ω) =12
Z∂α∂β∂3;ωXε (y ; v)dθ
where ω is a normal vector orthogonal to H and v = v (θ) runs overthe unit circle in H � y .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 13 / 31
Reconstruction from ray integrals
Problem: reconstruct the Saint-Venant tensor Vε from data of rayintegrals Xε.
First step: for a plane H � E , a point y 2 H and arbitrary vectorsα, β, compute
RH (α,ω; β,ω) =12
Z∂α∂β∂3;ωXε (y ; v)dθ
where ω is a normal vector orthogonal to H and v = v (θ) runs overthe unit circle in H � y .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 13 / 31
Reconstruction (cont.)
Second step: for any plane H and arbitrary vectors α, β,γ, δ, set
RH (α, β;γ, δ) = RH (hβ,ωi α,ω; hδ,ωi γ,ω)
� RH (hα,ωi β,ω; hδ,ωi γ,ω)
� RH (hβ,ωi α,ω; hγ,ωi δ,ω)
+ RH (hα,ωi β,ω; hγ,ωi δ,ω) .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 14 / 31
Reconstruction (cc.)
Third step: for arbitrary x 2 K and arbitrary α, β,γ, δ, by H.Lorentz�s formula
(Vε)αβ;γδ (x) = �18π2
Zω2S2
∂pRH (p,ω) jp=hx ,ωi (α, β;γ, δ) ˙,
where ˙ is the area form in unit sphere S2 in Eand H (p,ω) = fy 2 E , hω, yi = pg.
Remark. The �eld ε need not to be smooth e.g. it can hasdiscontinuity on the boundary of the specimen. In this case thecomponents of Vε are singular (generalized) functions in E .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 15 / 31
Reconstruction (cc.)
Third step: for arbitrary x 2 K and arbitrary α, β,γ, δ, by H.Lorentz�s formula
(Vε)αβ;γδ (x) = �18π2
Zω2S2
∂pRH (p,ω) jp=hx ,ωi (α, β;γ, δ) ˙,
where ˙ is the area form in unit sphere S2 in Eand H (p,ω) = fy 2 E , hω, yi = pg.Remark. The �eld ε need not to be smooth e.g. it can hasdiscontinuity on the boundary of the specimen. In this case thecomponents of Vε are singular (generalized) functions in E .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 15 / 31
Gauge �eld
Let ε be an unknown 2-tensor �eld with compact support such thattensor Vε is known.
Any 2-tensor �eld g such that Vg = Vε is called gauge of ε.
Theorem. For any 2-tensor �eld ε supported in a simply-connectedcompact K in E , a gauge �eld g supported in K can be analyticalyconstructed from Vε.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 16 / 31
Gauge �eld
Let ε be an unknown 2-tensor �eld with compact support such thattensor Vε is known.
Any 2-tensor �eld g such that Vg = Vε is called gauge of ε.
Theorem. For any 2-tensor �eld ε supported in a simply-connectedcompact K in E , a gauge �eld g supported in K can be analyticalyconstructed from Vε.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 16 / 31
Gauge �eld
Let ε be an unknown 2-tensor �eld with compact support such thattensor Vε is known.
Any 2-tensor �eld g such that Vg = Vε is called gauge of ε.
Theorem. For any 2-tensor �eld ε supported in a simply-connectedcompact K in E , a gauge �eld g supported in K can be analyticalyconstructed from Vε.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 16 / 31
Gauge �eld (cont.)
Set h = fhijg + Vε and write the S-V system for a gauge g = fgijg :
∂22g11 � 2∂12g12 + ∂11g22 = h33, (1)
∂33g11 � 2∂13g13 + ∂11g33 = h22,
�∂23g11 + ∂12g13 + ∂13g12 � ∂11g23 = h23,
∂33g22 � 2∂23g23 + ∂22g33 = h11,
�∂12g33 + ∂13g23 + ∂23g13 � ∂33g12 = h12,
�∂13g22 + ∂12g23 + ∂23g12 � ∂22g13 = h13,
where ∂ij = ∂i∂j , i , j = 1, 2, 3 and gij = (Vε)ki ,kj .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 17 / 31
Gauge �eld (cc.)
Main step. Find functions gij , i , j = 1, 2 such that
h33 = ∂22g11 � 2∂12g12 + ∂11g22.
The �rst line of (1) with g replaced by ε impliesZh33 (x)dx1dx2 =
Zx1h33 (x)dx1dx2 =
Zx2h33 (x)dx1dx2 = 0
for any x3 2 R, since all εij have compact support.
LetaRe0dt = 1 for a smooth function e0 with support in [0, a1]. The
function
f0 (x1, x2, x3) = h33 (x1, x2, x3)� e0 (x1)Z a1
0h33 (t, x2, x3)dt
ful�ls Z a1
0f0 (t, x2, x3)dt = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 18 / 31
Gauge �eld (cc.)
Main step. Find functions gij , i , j = 1, 2 such that
h33 = ∂22g11 � 2∂12g12 + ∂11g22.
The �rst line of (1) with g replaced by ε impliesZh33 (x)dx1dx2 =
Zx1h33 (x)dx1dx2 =
Zx2h33 (x)dx1dx2 = 0
for any x3 2 R, since all εij have compact support.Leta
Re0dt = 1 for a smooth function e0 with support in [0, a1]. The
function
f0 (x1, x2, x3) = h33 (x1, x2, x3)� e0 (x1)Z a1
0h33 (t, x2, x3)dt
ful�ls Z a1
0f0 (t, x2, x3)dt = 0.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 18 / 31
Gauge �eld (ccc.)
Functions
f1 (x) =Z x1
0f0 (t, x2, x3)dt, f2 (x) = e0 (x1)
Z x2
0
Z a1
0h33 (t, s, x3)dtds
are supported in K and satisfy h33 = ∂1f1 + ∂2f2.
Apply this method to f1 and f2 :
fi = ∂i1f1 + ∂i2f2, i = 1, 2,
and set g11 = f11, g22 = f22, g12 = 1/2 (f12 + f21) .Next steps...
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 19 / 31
Gauge �eld (ccc.)
Functions
f1 (x) =Z x1
0f0 (t, x2, x3)dt, f2 (x) = e0 (x1)
Z x2
0
Z a1
0h33 (t, s, x3)dtds
are supported in K and satisfy h33 = ∂1f1 + ∂2f2.
Apply this method to f1 and f2 :
fi = ∂i1f1 + ∂i2f2, i = 1, 2,
and set g11 = f11, g22 = f22, g12 = 1/2 (f12 + f21) .
Next steps...
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 19 / 31
Gauge �eld (ccc.)
Functions
f1 (x) =Z x1
0f0 (t, x2, x3)dt, f2 (x) = e0 (x1)
Z x2
0
Z a1
0h33 (t, s, x3)dtds
are supported in K and satisfy h33 = ∂1f1 + ∂2f2.
Apply this method to f1 and f2 :
fi = ∂i1f1 + ∂i2f2, i = 1, 2,
and set g11 = f11, g22 = f22, g12 = 1/2 (f12 + f21) .Next steps...
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 19 / 31
Photoelastic tomography
It is an experimental method to determine the stress distribution in anoptically transparent material.
Measurements of the motion of the polarization ellipse
I Aben H, Errapart A, Ainola L, Anton J 2005 Photoelastic tomographyfor residual stress measurement in glass Opt. Eng. 44 093601
I Puro A 1998 Magnetophotoelasticity as parametric tensor �eldtomography Inverse Probl. 14 1315-1330
I Lionheart W and Sharafutdinov V 2009 Reconstruction algorithm forthe linearized polarization tomography problem with incomplete data inimaging microstructures Contemp. Math. 49 137�159 AMS
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 20 / 31
Photoelastic tomography
It is an experimental method to determine the stress distribution in anoptically transparent material.Measurements of the motion of the polarization ellipse
I Aben H, Errapart A, Ainola L, Anton J 2005 Photoelastic tomographyfor residual stress measurement in glass Opt. Eng. 44 093601
I Puro A 1998 Magnetophotoelasticity as parametric tensor �eldtomography Inverse Probl. 14 1315-1330
I Lionheart W and Sharafutdinov V 2009 Reconstruction algorithm forthe linearized polarization tomography problem with incomplete data inimaging microstructures Contemp. Math. 49 137�159 AMS
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 20 / 31
Photoelastic tomography
It is an experimental method to determine the stress distribution in anoptically transparent material.Measurements of the motion of the polarization ellipse
I Aben H, Errapart A, Ainola L, Anton J 2005 Photoelastic tomographyfor residual stress measurement in glass Opt. Eng. 44 093601
I Puro A 1998 Magnetophotoelasticity as parametric tensor �eldtomography Inverse Probl. 14 1315-1330
I Lionheart W and Sharafutdinov V 2009 Reconstruction algorithm forthe linearized polarization tomography problem with incomplete data inimaging microstructures Contemp. Math. 49 137�159 AMS
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 20 / 31
Photoelastic tomography
It is an experimental method to determine the stress distribution in anoptically transparent material.Measurements of the motion of the polarization ellipse
I Aben H, Errapart A, Ainola L, Anton J 2005 Photoelastic tomographyfor residual stress measurement in glass Opt. Eng. 44 093601
I Puro A 1998 Magnetophotoelasticity as parametric tensor �eldtomography Inverse Probl. 14 1315-1330
I Lionheart W and Sharafutdinov V 2009 Reconstruction algorithm forthe linearized polarization tomography problem with incomplete data inimaging microstructures Contemp. Math. 49 137�159 AMS
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 20 / 31
Photoelastic tomography
It is an experimental method to determine the stress distribution in anoptically transparent material.Measurements of the motion of the polarization ellipse
I Aben H, Errapart A, Ainola L, Anton J 2005 Photoelastic tomographyfor residual stress measurement in glass Opt. Eng. 44 093601
I Puro A 1998 Magnetophotoelasticity as parametric tensor �eldtomography Inverse Probl. 14 1315-1330
I Lionheart W and Sharafutdinov V 2009 Reconstruction algorithm forthe linearized polarization tomography problem with incomplete data inimaging microstructures Contemp. Math. 49 137�159 AMS
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 20 / 31
Photoelastic tomography (cont.)
Figure: Strain in a plastic protractor seen under polarized light
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 21 / 31
Normal traceless integrals
For a small strain tensor ε, the motion of the polarization ellipse ismodelled by the line integral of the traceless normal part of ε.
For a vector v 2 E , the traceless normal part of ε is the 2-tensor Qv ε
Qv ε jP = ε jP �12
tr (ε jP ) id jP
de�ned in any plane P orthogonal to v .
If v = (0, 0, 1)
Qv ε =12(ε11 � ε22) (dx1)
2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)
2 .
The traceless normal ray integral of ε is
Tv ε (y ; u,w) =Z ∞
0Qv ε (y + tv ; u,w)dt, u,w 2 P
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 22 / 31
Normal traceless integrals
For a small strain tensor ε, the motion of the polarization ellipse ismodelled by the line integral of the traceless normal part of ε.
For a vector v 2 E , the traceless normal part of ε is the 2-tensor Qv ε
Qv ε jP = ε jP �12
tr (ε jP ) id jP
de�ned in any plane P orthogonal to v .
If v = (0, 0, 1)
Qv ε =12(ε11 � ε22) (dx1)
2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)
2 .
The traceless normal ray integral of ε is
Tv ε (y ; u,w) =Z ∞
0Qv ε (y + tv ; u,w)dt, u,w 2 P
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 22 / 31
Normal traceless integrals
For a small strain tensor ε, the motion of the polarization ellipse ismodelled by the line integral of the traceless normal part of ε.
For a vector v 2 E , the traceless normal part of ε is the 2-tensor Qv ε
Qv ε jP = ε jP �12
tr (ε jP ) id jP
de�ned in any plane P orthogonal to v .
If v = (0, 0, 1)
Qv ε =12(ε11 � ε22) (dx1)
2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)
2 .
The traceless normal ray integral of ε is
Tv ε (y ; u,w) =Z ∞
0Qv ε (y + tv ; u,w)dt, u,w 2 P
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 22 / 31
Normal traceless integrals
For a small strain tensor ε, the motion of the polarization ellipse ismodelled by the line integral of the traceless normal part of ε.
For a vector v 2 E , the traceless normal part of ε is the 2-tensor Qv ε
Qv ε jP = ε jP �12
tr (ε jP ) id jP
de�ned in any plane P orthogonal to v .
If v = (0, 0, 1)
Qv ε =12(ε11 � ε22) (dx1)
2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)
2 .
The traceless normal ray integral of ε is
Tv ε (y ; u,w) =Z ∞
0Qv ε (y + tv ; u,w)dt, u,w 2 P
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 22 / 31
Recovering of the displacement �eld
Theorem
Let K be a compact and Γ � EnK be a curve satisfying Tuy�scondition for K . For any 2-tensor ε with support in K such thatVε = 0, the displacement �eld ϕ such that Dϕ = ε can bereconstructed from data of the second derivatives of Tv ε (y)for y 2 Γ, v 2 S2.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 23 / 31
Recovering of the displacement �eld
TheoremLet K be a compact and Γ � EnK be a curve satisfying Tuy�scondition for K . For any 2-tensor ε with support in K such thatVε = 0, the displacement �eld ϕ such that Dϕ = ε can bereconstructed from data of the second derivatives of Tv ε (y)for y 2 Γ, v 2 S2.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 23 / 31
Reconstruction from normal integrals
For any plane H and a point y 2 H, the rays R = R (y , v) withdirection vectors v = v (θ) parallel to H cover H.
For any v , a vector u parallel to H and the normal ω to H form anorthogonal basis in any plane P orthogonal to v .
v uw
H
y
The circle of integration
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 24 / 31
Reconstruction from normal integrals
For any plane H and a point y 2 H, the rays R = R (y , v) withdirection vectors v = v (θ) parallel to H cover H.For any v , a vector u parallel to H and the normal ω to H form anorthogonal basis in any plane P orthogonal to v .
v uw
H
y
The circle of integrationVictor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 24 / 31
Reconstruction (cont.)
Tensor Tv ε (y) is known for y 2 Γ and unit v parallel to H. It hastwo independent components in this basis:
Tv ε (y ;ω,ω) + 12
Z ∞
0(εωω (y + tv)� εuu (y + tv))dt,
Tv ε (y ;ω, u) +Z ∞
0εωu (y + tv)dt.
Calculate the integrals
I1 (y , p,ω) +Z 2π
0∂2;ωTv (θ)ε (y ,ω,ω)dθ,
I2 (y , p,ω) +Z 2π
0∂2;ωTv (θ)ε (y ,ω, u)dθ.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 25 / 31
Reconstruction (cc.)
This yields
∂
∂y3I2 (y , p,ω) = ∂3p
ZH
ϕ2dH,∂
∂y2I2 (y , p,ω) = �∂3p
ZH (p,ω)
ϕ3dH.
Integrating in p we get
∂
∂y3J2 (p,ω) = ∂2p
ZH (p,ω)
ϕ2dH,∂
∂y2J2 (p,ω) = �∂2p
ZH (p,ω)
ϕ3dH,
whereJ2 (p,ω) +
Z p
�∞I2 (y , q,ω)dq.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 26 / 31
Reconstruction (cc.)
This yields
∂
∂y3I2 (y , p,ω) = ∂3p
ZH
ϕ2dH,∂
∂y2I2 (y , p,ω) = �∂3p
ZH (p,ω)
ϕ3dH.
Integrating in p we get
∂
∂y3J2 (p,ω) = ∂2p
ZH (p,ω)
ϕ2dH,∂
∂y2J2 (p,ω) = �∂2p
ZH (p,ω)
ϕ3dH,
whereJ2 (p,ω) +
Z p
�∞I2 (y , q,ω)dq.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 26 / 31
Reconstruction (ccc.)
Compute
J1 (p,ω) + ∂2p
ZH (p,ω)
ϕ1dH =Z p
�∞
∂2
∂y22I1 (y , q,ω)dq.
Finally,
ϕ (x) = � 18π2
ZS2
J (hx ,ωi ,ω) ˙,
where
hJ (p,ω) , ei = he,ωi J1 (p,ω) + [ω, e, ∂/∂y ] J2 (p,ω) .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 27 / 31
Reconstruction (ccc.)
Compute
J1 (p,ω) + ∂2p
ZH (p,ω)
ϕ1dH =Z p
�∞
∂2
∂y22I1 (y , q,ω)dq.
Finally,
ϕ (x) = � 18π2
ZS2
J (hx ,ωi ,ω) ˙,
where
hJ (p,ω) , ei = he,ωi J1 (p,ω) + [ω, e, ∂/∂y ] J2 (p,ω) .
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 27 / 31
Algorithm of full reconstruction of a strain tensor
Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.
Step 2. Find a gauge �eld g with compact support such thatVg = Vε.
Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �
I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 28 / 31
Algorithm of full reconstruction of a strain tensor
Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.Step 2. Find a gauge �eld g with compact support such thatVg = Vε.
Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �
I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 28 / 31
Algorithm of full reconstruction of a strain tensor
Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.Step 2. Find a gauge �eld g with compact support such thatVg = Vε.
Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.
Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �
I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 28 / 31
Algorithm of full reconstruction of a strain tensor
Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.Step 2. Find a gauge �eld g with compact support such thatVg = Vε.
Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.
Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �
I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 28 / 31
Algorithm of full reconstruction of a strain tensor
Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.Step 2. Find a gauge �eld g with compact support such thatVg = Vε.
Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .
The reconstruction reads ε = g +Dϕ. �
I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 28 / 31
Algorithm of full reconstruction of a strain tensor
Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.Step 2. Find a gauge �eld g with compact support such thatVg = Vε.
Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �
I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 28 / 31
Algorithm of full reconstruction of a strain tensor
Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.Step 2. Find a gauge �eld g with compact support such thatVg = Vε.
Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �
I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 28 / 31
More pictures
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 29 / 31
Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 30 / 31