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
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
t
θ
Transform.Radon)(),)((
)).sin(),(cos(),sin,(cos0
dssD θt
θt
Transmission Tomography (Scalar Case)Transmission Tomography (Scalar Case)
Helmholtz DecompositionHelmholtz Decomposition
2-D Vector Field2-D Vector Field Solenoidal PartSolenoidal Part Potential PartPotential Part
= +
= +),(grad),(curl),( yxyxyx a
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
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
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
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.
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
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
knknkn
knkn
knkn
ZZDD
ZcFD
FcZD
,2,,
*
,1,*
,2,
:
:
:
TransformRadon Adjoint
TransformRadon
)()(:
)()(:
211
2*
1122
LSSLD
SSLLD
basisFourier standard)12()1(, kinikn eeF
ValuesSingular 01
,
nOkn
SVD of the Radon Transform (scalar case)SVD of the Radon Transform (scalar case)
spolynomial Zernike, knZ
Singular ValuesSingular Values
nOn
1 nOn
nOn
n
On1
Radon TransformRadon Transform Inverse Radon TransformInverse Radon Transform
Integration OperatorIntegration Operator Differentiation OperatorDifferentiation Operator
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 !
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)
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
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
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
.
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
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:
)2,0[),(),,)((:),(
)2,0[),(),(),()(),(),(
ttaDtfu
zzazuzezuezu iz
iz
Inversion formula (sketch)Inversion formula (sketch)
),(:)(),)((),(
)(
zudeazaDz
z
dz
:equation transport theosolution t a is function The ),( zu
equation transportfor the problem inversean as
problemomography emission t theFormulate 1)
) planecomplex with the(identify
riablescomplex va of language the Utilize
2 CR
formcomplex in sformRadon tran attenuated theRewrite
z
),( z
)2,0[),(),,)((:),(
)2,0[),(),(),()(),(),(
ttaDtfu
zzazuzezuezu iz
iz
Zn
inn ezuzu
zu
)(),(
),( seriesFourier theinto function theExpand 2)
., :Notationzz
Inversion formula (sketch)Inversion formula (sketch)
equation. transport theintoexpansion thisSubstitute
Znezaeuuu i
Zn
innnn
,)()( 12
1),(
}1{\,0
011
12
nzauuu
Znuuu nnn
Zn
inn ezuzu
zu
)(),(
),( seriesFourier theinto function theExpand 2)
Inversion formula (sketch)Inversion formula (sketch)
Znezaeuuu i
Zn
innnn
,)()( 12
., :Notationzz
equation. transport theintoexpansion thisSubstitute
1),(
}1{\,0
011
12
nzauuu
Znuuu nnn
Zn
inn ezuzu
zu
)(),(
),( seriesFourier theinto function theExpand 2)
Inversion formula (sketch)Inversion formula (sketch)
FormulaInversion )()())(Re(2)( 01 zuzzuza
., :Notationzz
equation. transport theintoexpansion thisSubstitute
1),(
}1{\,0
011
12
nzauuu
Znuuu nnn
fu
uuu
in0)( *2* UU
fu
uu
in0*UA
),0(,...),,(: 2210
notationoperator in equations Rewrite 3)
luuuu
),0( ,...),,(,...),,(:
,...),,,0(,...),,(:
2321210*
210210
inAdjoint its
OperatorShift
luuuuuuU
uuuuuuU
AUA A :,)(: 2*:Notation
Inversion formula (sketch)Inversion formula (sketch)
FormulaInversion )()())(Re(2)( 01 zuzzuza
eUA )( * A
zzez z on dependent smoothly is
) variablesof change(by n attenuatio of ridGet 4)
),()( )( vu
])([)( ** vvv
UeeU AAA
then If ,,)()( * zUzzA
gfv
0v
:
in
e
A
Inversion formula (sketch)Inversion formula (sketch)
e
FormulaInversion )()())(Re(2)( 01 zuzzuza
fu
uu
in0*UA
e ! OperatorsSimilar
Inversion formula (scalar case)Inversion formula (scalar case)
,),(*Im)( ),(),()(2),)((2
0
),(
deveeei
zza zieDzDi zi
),(),(2
1),()(),(* ),( tevviItv i
),,(),( ),()(2 tfetv tD
Sinogram.
Operator,Identity
Transform,Hilbert Angular
)f(t,
I
.,)(sin||122
),( 222 iii ezzeezz
z
R
Rtsdtts
thsHh ,,
)(1))((
2
0
ˆ)ˆ(2
ˆctg
2
1))(( dvv
Equivalence of the Inversion FormulaeEquivalence of the Inversion Formulae
Hilbert TransformAngular Hilbert Transform
Inversion formula (vector case)Inversion formula (vector case)
,
)(
...Im2)()(
),)((2
021
z
deei
zz
ziaza
zDi
- components of the vector field being reconstructed,
- a known attenuation function:
)(),( 21 zaza
)(z
• 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.
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
360 degree Medium Attenuation No Noise
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
360 degree Medium Attenuation Large NoiseLarge Noise
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
360 degree XXL Attenuation [6,14]XXL Attenuation [6,14] No Noise
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
270 degree270 degree Medium Attenuation No Noise
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
180 degree180 degree Medium Attenuation No Noise
Emission Tomography: Numerical Examples (Scalar Case)Emission Tomography: Numerical Examples (Scalar Case)
90 degree !90 degree ! Medium Attenuation No Noise
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
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
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
),( 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)
),( 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)
),( 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)
),( 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)
),( 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)
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.
Top Related