Tomographic reconstruction techniques

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Summer school Clermont - Irène Buvat - july 2004 - 1

To m og ra ph ic re co nstru ctio n te ch niq ue s

Irène Buvat U494 INSERM, Paris

buvat@imed.jussieu.frhttp://www.guillemet.org/irene

New Technologies in Medical Imaging and Surgery New Technologies in Medical Imaging and Surgery

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Outline

• Introduction: the issue of tomographic reconstruction

• Basics

• Tomographic reconstruction methods

• Regularization

• New challenges

• Conclusion

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Introduction

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What is tomography ?

! An indirect measurement of aparameter of interest, using a detectorsensitive to some sort of radiations À Algorithms to recover the cartography of the parameters the measurements

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Direct (forward) problem

The tomographic system measures a set of “projection” data: integralsof a signal along certain specific directions.

unknown parameters measurements = projection

The mathematical formulation of the relationship between the unknownparameters and the measurements is the direct problem. direct problem

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Inverse problem

Tomographic reconstruction is the inversion of the direct problem.

unknown parameters measurements = projections

inverse problem

Inverse problem: estimating the 3D cartography of unknownparameters from the measured data.

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Ill-posed inverse problem

Limited spatial sampling

Ill-posed: several solutions compatible with the measurements 1 projection

projectiondirection 2 projections

Noisy Noisy measurements

No noise signal intensity

spatial direction

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Different types of tomography

^!

Transmission tomography

- External radiation source- Measurement of radiationstransmitted through the patient- Parameters related to the interactionsof radiations within the body

e.g., Computed Tomography (CT)e.g., SPECT and PET - Internal radiation source- Measurement of radiation emittedfrom the patient- Parameters related to the radiationsources within the body À Emission tomography

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Different types of tomography

PET

SPECT CT

Optical TomographySynchrotron Tomography

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Summer school Clermont - Irène Buvat - july 2004 -

B asics

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Summer school Clermont - Irène Buvat - july 2004 - 11

Factorization of the reconstruction problem

_3D

images from a set of 2D measurements

3D cartography of aparameter 2D projections

3D distribution

If real 3D: “Fully 3D reconstruction” 1D projections2D cartography of aparameter 2D images from a set of 1D measurements

transaxial slice

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K ey notion 1: projection

p(u,q) = f(x,y)dv

-• +• Ú

u projectionq

Modelling the direct problem

u = x cosq + y sinqv = -x sinq + y cosq qx yvf(x,y)u

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Projection: mathematical expression

u projectionq

p(u,q) = f(x,y)dv

-• +• Ú The 2D Radon transform

set of projections for q = [0, p ] = Radon transform of f(x,y)

R[ f(x,y) ] = p(u,

q

)d

q0 p

Ú

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Key notion 2: sinogram

All detected signal concerning 1 slice

q q2q3 q1

t

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Key notion 3: backprojection

f*(x,y) = p(u,q)dq0 p

Ú Tackling the inverse problem

u = x cosq + y sinqv = -x sinq + y cosq qx yvf(x,y)u u projectionq

Beware: backprojection is not the inverse of projection !

p(u,q)

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Methods of tomographic reconstruction

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Two approaches

^!

Analytical approaches

- Continuous formulation- Explicit solution using inversionformulae or successive transformations- Direct calculation of the solution- Fast- Discretization for numericalimplementation only

f*(x,y) = p’(u, q ) d q

0 p

Ú

À Discrete approaches

- Discrete formulation- Resolution of a system of linearequations or probabilistic estimation- Iterative algorithms- Slow convergence- Intrinsic discretization

p

i

= S r

ij

f

jj

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Analytical approach: central slice theorem

p(u,q) = f(x,y) dv-• +•

Ú

-• P(r,q) = p(u,q)e -i2pru du

Ú

+• Fourier transform

qx yvu u = x cosq + y sinqv = -x sinq + y cosqrx =rcosqry =rsinqdu.dv= dx.dy

1D FT of p with respect to u = 2D FT of f in a specific direction P(r,q) = f(x,y) e -i2p(xrx +yr

-•-• +•+•

Ú Ú

p(u,q)

P(r,q) = f(x,y) e -i2pru du.dv-• +•

-• +•

Ú Ú

P(r,q) = F(rx ,ry )

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Analytical approach: filtered backprojection (FBP)

P(r,q) = F(rx ,ry ) f(x,y) = F(rx ,ry ) e i2p(xrx +yry ) drx .dry-• +•

-• +•

Ú Ú

= P(r,q) e i2p(xrx +yry ) drx .dry-• +•

-• +•

Ú Ú

= P(r,q)|r|e i2pru dr. dq0 p+•

Ú Ú

= p’(u,q)dq with p’(u,q)= P(r,q)|r|e i2pru dr -•

0 p

Ú

-• +•

Ú

Ramp filter qx yvu u = x cosq + y sinqrx =rcosqry =rsinqr= (rx 2 + ry 2) 1/2rdrx dry = rdrdq

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Filtered backprojection: algorithm

P(r,q) 1D-FT

p’(u,q) FT -1 f(x,y)backprojection reconstructed image f(x,y)= p’(u,q)dq with p’(u,q)= P(r,q)|r|e i2pru d0 p

Ú

sinogram

p(u,q)

|r| P(r,q) filtering

Ramp filter

Ú

-• +•

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Filtered backprojection: beyond the Ramp filter

f(x,y)= p’(u,q)dq with p’(u,q) = P(r,q)|r|e i2pru dr0 p

Ú Ú

-• +•

r

0 0.8

| r |w (r)

⁰ 1

r | r |

0 0.8 10 |r|w(r)

high frequency amplificationnoisenoise control

loss of spatial resolution

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Two approaches

^!

Analytical approaches

- Continuous formulation- Explicit solution using inversionformulae or successive transformations- Direct calculation of the solution- Fast- Discretization for numericalimplementation only

f*(x,y) = p’(u, q ) d q

0 p

Ú

À Discrete approaches

- Discrete formulation- Resolution of a system of linearequations or probabilistic estimation- Iterative algorithms- Slow convergence- Intrinsic discretization

p

i

= S r

ij

f

jj

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Discrete approach: model

p

i

f

j

r11 r14

r41 r44 p1p2p3p4 f1f2f3f4

= p = R f

f1 f2

f3 f4 p1 p2p3p4 p1 = r11 f1 + r12 f2 + r13 f3 + r14 f4p2 = r21 f1 + r22 f2 + r23 f3 + r24 f4p3 = r31 f1 + r32 f2 + r33 f3 + r34 f4p4 = r41 f1 + r42 f2 + r43 f3 + r44 f4

projector

Given p and R, estimate f

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Discrete approach: calculation of R

• Geometric modelling- intersection between pixel and projection rays

f

j

p = R f R

models the direct problem

• Physics modelling- spatial resolution of the detector- physical interactions of radiations

r

ij

=1 r

i’j

=0 r

i

=0 r

i

f

j

f

j’

p

i

p

i’ parallel

p

i fan beam

spatial resolution

p

i

r

ij

≠ r

ij’

=0, f

j

f

j’

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Two classes of discrete methods

! Algebraic methods

- Generalized inverse methods À Statistical approaches

- Bayesian estimates- Optimization of functionals

- Account for noise properties

p

i

= S r

ij

f

jj

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Iterative algorithm used in discrete methods

f

0

f

1

correction p = R f

p = R f

p ^ comparison p ^ versus p

defines the iterative method:

additive if

f

n+1 =

f

n +

c

n

multiplicative if

f

n+1 =

f

n .

c

n

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A lgebraic m ethods

Minimisation of || p - R f ||

2

p = R f

Several minimisation algorithms are possible to estimate a solution:

e.g., SIRT (Simultaneous Iterative Reconstruction Technique)Conjugate GradientART (Algrebraic Reconstruction Technique)

e.g., additive ART:

f

j n+1

= f

j n i

+ ( p - p

k i nijik

) r / S r

2

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Statistical methods Probabilistic formulation (Bayes’ equation): proba ( f | p ) = proba ( p | f ) proba ( f ) / proba ( p ) p = R f

probability of obtaining fwhen p is measured likelihood of p Find a solution f maximizing proba(p|f) given a probabilistic model for prior on fprior on p

f

n+1

= f

n .

R

t

( p /p

n

)

e.g., if p follows a Poisson law: proba(p|f) =

P

exp(-pk ).pk pk/p

MLEM (Maximum Likelihood Expectation Maximisation):

and OSEM (accelerated version of MLEM) k

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Regularization

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Regularization

Set constraints on the solution f

Solution f: trade-off betw een the agreement with the observed data and the agreement with the constraints

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Regularization for analytical methods

Filtering

f(x,y)= P(r,q)w(r)|r|e i2pru dr0 p

Ú Ú

-• +•

Ramp filterButterworth filter

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Regularization for discrete methods Minimisation of || p - R f ||

2

+ l K( f )

l controls the trade-off betweenagreement with the projections and agreement with the constraints

proba ( f | p ) = proba ( p | f ) proba ( f ) / proba ( p )

prior on f, i.e. proba(f) non uniform

Examples of priors:f smoothf having discontinuities

Conjugate Gradient gives MAP-Conjugate Gradient (Maximum A Posteriori)MLEM gives MAP-EM

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New challenges

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Analytical reconstruction

• Fully 3D reconstruction for CT: - suited to the new designs of CT detectors (increased number of rows, increased gantry rotation speed, helical cone beam geometry) - real time

• Reconstruction algorithms managing source deformations (motions)

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Iterative reconstruction: fully 3D Monte Carlo reconstruction p = R f

modelling R using numerical (Monte Carlo) simulationsof the imaging procedure in emission tomography

tissue densityand composition+cross-section tables ofradiation interaction

All propagation and detection physics can be accurately modelled3D propagation of radiation is taken into account stochastic modellingof physical interactions probability that an“event” in j be detectedin i j i