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Transcript of Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim...
![Page 1: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/1.jpg)
Tensor and Emission Tomography ProblemsTensor and Emission Tomography Problemson the Planeon the Plane
A. L. Bukhgeim
S. G. Kazantsev
A. A. Bukhgeim
Sobolev Institute of Mathematics
Novosibirsk, RUSSIA
![Page 2: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/2.jpg)
OverviewOverview
• 2D problems, in a unit disc on the plane• Isotropic case• Fan-beam scanning geometry
1) Transmission tomography• inversion formula
on the basis of SVD of the Radon transform• scalar, vector and tensor cases
2) Emission tomography• the first explicit inversion formula
(A.L.Bukhgeim, S.G.Kazantsev, 1997) • recent results that follow from it• scalar and vector cases
![Page 3: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/3.jpg)
t
θ
Transform.Radon)(),)((
)).sin(),(cos(),sin,(cos0
dssD θt
θt
Transmission Tomography (Scalar Case)Transmission Tomography (Scalar Case)
![Page 4: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/4.jpg)
Helmholtz DecompositionHelmholtz Decomposition
2-D Vector Field2-D Vector Field Solenoidal PartSolenoidal Part Potential PartPotential Part
= +
= +),(grad),(curl),( yxyxyx a
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Vectorial Radon TransformVectorial Radon Transform
θ
θ
t
t
0
,)(,),)(( dssD θtaθθta
.sincos, 21 aa aθ
0
,)(,),)(( dssD θtaθθta
.cossin, 21 aa aθ
Normal Flow Radon TransformNormal Flow Radon Transform
D
D
![Page 6: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/6.jpg)
Vectorial Radon TransformVectorial Radon Transform D
θt
θt
Normal Flow Radon TransformNormal Flow Radon TransformD
0
,)(,),)(( dssD θtaθθta
.cossin, 21 aa aθ
Solenoidal Part of the Vector Field
![Page 7: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/7.jpg)
Vectorial Radon TransformVectorial Radon Transform D
θt
θt
Normal Flow Radon TransformNormal Flow Radon TransformD
Solenoidal Part of the Vector Field
Potential Part of the Vector Field
![Page 8: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/8.jpg)
Tensorial Radon TransformTensorial Radon Transform
}.1:),{( 222 yxRyxConsider a unit disk on the plane:
Covariant symmetric tensor field of rank m:
.,...,1,2,1)),,((),( ...1msiyxayxa sii m
Due to symmetry it has m+1 independent components.
By analogy with the vector case:• similar decomposition into the solenoidal and potential parts,• define tensorial Radon transform.
Refer to:V. A. Sharafutdinov “Integral Geometry of Tensor Fields”Utrecht: VSP, 1994.
![Page 9: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/9.jpg)
Consider two Hilbert spaces: H with O.N.B and
SVD is one of the methods for solving ill-posed problems:
1}{ kku.}v{ 1kkK with O.N.S.
Singular value decomposition of an operator
1
.v,v,k
kkkkHkk AuuuAu
Then its generalized inverse operator will look like:
1
1 v,vvk
kKkk uA - can be unbounded.
/1
1 v,vvk
kKkk uT - truncated SVD.
KHA :
SVD of the Radon TransformSVD of the Radon Transform
![Page 10: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/10.jpg)
The presence of a singular value decomposition allows to:
• describe the image of the operator,
• invert the operator,
• estimate its level of incorrectness.
Bukhgeim A. A., Kazantsev S. G. “Singular-value decomposition of the fan-beam Radon transformof tensor fields in a disc” // Preprint of Russian Academy of Sciences, Siberian Branch. No. 86. Novosibirsk: Institute of Mathematics Press, October 2001. 34 pages.
The first SVD of the Radon transform for the parallel-beam geometry was derived by Herlitz in 1963 and Cormack in 1964 (scalar case only).
SVD of the Radon TransformSVD of the Radon Transform
![Page 11: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/11.jpg)
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:
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basisFourier standard)12()1(, kinikn eeF
ValuesSingular 01
,
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SVD of the Radon Transform (scalar case)SVD of the Radon Transform (scalar case)
spolynomial Zernike, knZ
![Page 12: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/12.jpg)
Singular ValuesSingular Values
nOn
1 nOn
nOn
n
On1
Radon TransformRadon Transform Inverse Radon TransformInverse Radon Transform
Integration OperatorIntegration Operator Differentiation OperatorDifferentiation Operator
![Page 13: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/13.jpg)
Transmission Tomography: Numerical Examples (Scalar Case)Transmission Tomography: Numerical Examples (Scalar Case)
original image reconstruction from 300
fan-projections; N=298
reconstruction from 512
noisy fan-projections; N=510
(noise level: 20%)
reconstruction from 512
noisy fan-projections; N=446
(noise level: 20%)
reconstruction from 512
noisy fan-projections; N=382
(noise level: 20%)
reconstruction from 512
noisy fan-projections; N=318
(noise level: 20%)
reconstruction from 512
noisy fan-projections; N=254
(noise level: 20%)
Compare with the talk of Emmanuel Candes !Compare with the talk of Emmanuel Candes !
![Page 14: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/14.jpg)
Transmission Tomography: Numerical Examples (Scalar Case)Transmission Tomography: Numerical Examples (Scalar Case)
original image reconstruction from
8 fan-projections; N=6
reconstruction from
16 fan-projections; N=14
reconstruction from
32 fan-projections; N=30
reconstruction from
64 fan-projections; N=62
reconstruction from
128 fan-projections; N=126
reconstruction from
256 fan-projections; N=254
reconstruction from
512 fan-projections; N=510
reconstruction from
1024 fan-projections; N=1022
reconstruction from 2048
noisy fan-projections; N=2046
(noise level: 5% in L2-norm)
reconstruction from 2048
noisy fan-projections; N=1022
(noise level: 5% in L2-norm)
reconstruction from 2048
noisy fan-projections; N=510
(noise level: 5% in L2-norm)
![Page 15: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/15.jpg)
Transmission Tomography: Numerical Examples (Vector Case)Transmission Tomography: Numerical Examples (Vector Case)
original (solenoidal) vector field
reconstruction from noisy (3%) projections
reconstruction from non-uniform projections
![Page 16: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/16.jpg)
AttenuatedRadon
Transform
Emission TomographyEmission Tomography
• Inject a radioactive solution into the patient, it is then spread all over the body with the blood
• Assume, that the attenuation map of the object is known
• Place detectors around and measure how many radioactive particles go through it in the given directions
• Reconstruct the Emission Map
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Emission tomography problem: reconstruct from its known attenuated Radon transform provided that the attenuation map is known.
Let represent an attenuation map and represent an emission map, both given in
Formulation of the emission tomography problemFormulation of the emission tomography problem
}.1:),{( 222 yxRyxConsider a unit disc on the plane:
The fan-beam Radon transform
The fan-beam attenuated Radon transform
.,)(),)((0
))((
tθtθt θθt dsesaaD sD
.),sin,(cos,)(),)((0
xθθxθx dssD
),( yx),( yxa
D
D
),( yxa
),( yx
.
![Page 18: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/18.jpg)
Attenuated Vectorial Radon TransformAttenuated Vectorial Radon Transform
θ
θ
t
t
.sincos, 21 aa aθ
.cossin, 21 aa aθ
Attenuated Normal Flow Radon TransformAttenuated Normal Flow Radon Transform
0
),)(( ,)(,),)(( dsesD sD θθtθtaθθta
0
),)(( ,)(,),)(( dsesD sD θθtθtaθθta
![Page 19: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/19.jpg)
Servey of the Results in Emission TomographyServey of the Results in Emission Tomography
• 1980, O.J. Tretiak, C. Metz. The first inversion formula for emission tomography with constant attenuation.
• 1997, K. Stråhlén. Inversion formula for full reconstruction of a vector field from both Exponential Vectorial Radon Transform and Exponential Normal Flow Transform, attenuation coefficient is constant.
• 1997, A.L. Bukhgeim, S.G. Kazantsev. The first explicit inversion formula for emission tomography (in the fan-beam formulation) with arbitrary non-constant attenuation (based on the theory of A-analytic functions).
• 2000, R.G. Novikov (and then F.Natterer in 2001). Inversion formula for emission tomography in the parallel-beam formulation which then was numerically implemented by L.A. Kunyansky in 2001.
• 2002, A.A. Bukhgeim, S.G. Kazantsev. Full reconstruction of a vector field only from its Attenuated Vectorial Radon Transform, arbitrary non-constant attenuation function is allowed.
SCALAR CASE:
VECTOR CASE:
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)2,0[),(),,)((:),(
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![Page 28: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/28.jpg)
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)(),( 21 zaza
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• For the full reconstruction of a vector field it’s sufficient to know only one transform: either Vectorial Attenuated Radon Transform or the Normal Flow Attenuated Radon Transform;
• Arbitrary non-constant attenuation is allowed.
![Page 29: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/29.jpg)
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
360 degree Medium Attenuation No Noise
![Page 30: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/30.jpg)
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
360 degree Medium Attenuation Large NoiseLarge Noise
![Page 31: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/31.jpg)
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
360 degree XXL Attenuation [6,14]XXL Attenuation [6,14] No Noise
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Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
270 degree270 degree Medium Attenuation No Noise
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Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
180 degree180 degree Medium Attenuation No Noise
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Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
90 degree !90 degree ! Medium Attenuation No Noise
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Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
180 degree180 degree Large Attenuation [4,7]Large Attenuation [4,7] No Noise
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Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
180 degree180 degree Large Attenuation [4,7]Large Attenuation [4,7] With NoiseWith Noise
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Emission Tomography: Numerical Examples (Vector Case)Emission Tomography: Numerical Examples (Vector Case)
),( 21 aaaOriginal Vector FieldOriginal Vector Field
SinogramSinogram
Reconstruction fromReconstruction from128 fan-projections128 fan-projections
![Page 38: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/38.jpg)
),( 21 aaaOriginal Vector FieldOriginal Vector Field
SinogramSinogram
Reconstruction fromReconstruction from256 fan-projections256 fan-projections
Emission Tomography: Numerical Examples (Vector Case)Emission Tomography: Numerical Examples (Vector Case)
![Page 39: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/39.jpg)
),( 21 aaaOriginal Vector FieldOriginal Vector Field
SinogramSinogram
Reconstruction fromReconstruction from512 fan-projections512 fan-projections
Emission Tomography: Numerical Examples (Vector Case)Emission Tomography: Numerical Examples (Vector Case)
![Page 40: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/40.jpg)
),( 21 aaaOriginal Vector FieldOriginal Vector Field
SinogramSinogram
Reconstruction fromReconstruction from256 fan-projections256 fan-projections
Emission Tomography: Numerical Examples (Vector Case)Emission Tomography: Numerical Examples (Vector Case)
![Page 41: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/41.jpg)
),( 21 aaaOriginal Vector FieldOriginal Vector Field
SinogramSinogram
Reconstruction fromReconstruction from256 fan-projections256 fan-projections
Emission Tomography: Numerical Examples (Vector Case)Emission Tomography: Numerical Examples (Vector Case)
![Page 42: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/42.jpg)
),( 21 aaaOriginal Vector FieldOriginal Vector Field
SinogramSinogram
Reconstruction fromReconstruction from256 fan-projections256 fan-projections
Emission Tomography: Numerical Examples (Vector Case)Emission Tomography: Numerical Examples (Vector Case)
![Page 43: Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e8f5503460f94b93c49/html5/thumbnails/43.jpg)
ConclusionConclusion
1) SVD of the Radon transform of tensor fields• description of the image of the operator,• inversion formula,• estimation of incorrectness of the inverse problem,• unified formula (for reconstruction of scalar, vector and tensor fieds),• numerical implementation;
2) The very first inversion formula (by A.L.Bukhgeim, S.G. Kazantsev)was re-derived• shows equivalence of the first inversion formula to the formulae
obtained later by Novikov and Natterer,• yields a pathbreaking inversion formula for the vectorial attenuated
Radon transfom,• numerical implementation.