Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim...

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


.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


.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%)



(noise level: 20%)



(noise level: 20%)



(noise level: 20%)



(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


reconstruction from


reconstruction from


reconstruction from


reconstruction from


reconstruction from


reconstruction from




(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


equation. transport theintoexpansion thisSubstitute

Znezaeuuu i

Zn

innnn

,)()( 12

1),(

}1{\,0

011

12

nzauuu

Znuuu nnn

Zn

inn ezuzu

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)(),(



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,)()( 12

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1),(

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011

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nzauuu

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Zn

inn ezuzu

zu

)(),(



FormulaInversion )()())(Re(2)( 01 zuzzuza

., :Notationzz


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*

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inAdjoint its

OperatorShift

luuuuuuU

uuuuuuU

AUA A :,)(: 2*:Notation



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),()( )( vu

])([)( ** vvv

UeeU AAA

then If ,,)()( * zUzzA

gfv

0v

:

in

e

A


e


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


360 degree Medium Attenuation Large NoiseLarge Noise


360 degree XXL Attenuation [6,14]XXL Attenuation [6,14] No Noise


270 degree270 degree Medium Attenuation No Noise


180 degree180 degree Medium Attenuation No Noise


90 degree !90 degree ! Medium Attenuation No Noise


180 degree180 degree Large Attenuation [4,7]Large Attenuation [4,7] No Noise


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


SinogramSinogram




SinogramSinogram




SinogramSinogram




SinogramSinogram




SinogramSinogram



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.

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Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim...

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