Image%20reconstruction%20in%20PET

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Image reconstruction in PET
Luis Manuel Janeiro
Instituto de Biofísica e Eng. Biomédica - Fac.
Ciências Univ. Lisboa Lisboa Service
Hospitalier Frédéric Joliot C.E.A - Orsay
m26307_at_fc.ul.pt
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Aims

• To briefly show how data could be acquired and
  organized, in
• PET.

• To overview the different approaches used for
  the image reconstruction, in PET.

PET understood as the standard PET.
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Data acquisition in PET
LOR
y
Transaxial plane
Line of response
Oblique planes
y
? y
y
x
3D aquisition
x
y
Scanner axis
x
?
?
?2
?0 0
?1
? x
x
Values go to this line
Values go to this line
Values go to this line
x
x xcos? ysin? y -xsin? ycos?
Sinogram organization
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Data acquisition and image reconstruction in PET
Data Acquisition
Septa used between axial planes
No septa between axial planes
2D
3D
Reconstruction
2D Rec.
3D Rec.
Rebinning

• Concerning the scanner sensitivity, a 3D
  acquisition is better than 2D.
• The rebinning operation preserves the increased
  sensitivity of a 3D acquisition.

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Different approaches for image reconstruction (I)
3D data
Analytical
How to reconstruct?
y
y
The total number of coincidences between any
detector pair is, approximately, a line integral
through the source distribution.
g(x,y)
x
?
O
x
Reference frame
Search for an exact solution for the equation
P?(x)
(example for a transaxial plane)
Radon transform
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Analytical reconstruction FBP
Inversion of the Radon transform
The objective...
The tool...
Central slice theorem
Relates de 2D Fourier Transform of the object
with the 1D FT of its projection, along a certain
direction.
The algorithm...
Two steps
Different filters are used in practice Hanning,
Hamming, Butterworth, etc...
1st) Filtering
2nd) Integration
An integration along a sinusoid in the Radon
domain. Backprojection operation
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Analytical reconstruction 3DRP
3D FBP
Non truncated projections are needed...
The objective...
To recover g(x,y,z) from
Finite size
The tool...
Central section theorem
Truncated oblique proj.
Relates each section of the 3D Fourier transform
of the image through the origin with the 2D FT of
its projection, along a certain direction.
3D RP
Reconstruction is possible for projections
satisfying Orlovs sufficienty condition
The algorithm...
1st) Filtering step Colsher Filter
2nd) Backprojection
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Disadvantages of analytical reconstruction
The main disadvantages of an analytical
reconstruction, are

• The limitation imposed by the approximations
  implicit in the line integral model onto which
  the formulae are based.

It is not possible to model the detection and
acquisition process.

• Does not take in account the statistical
  variability inherent to the photon limited
  coincidence detection.

The noise is controlled at the expense of
resolution, varying the cut-off frequency of a
filter applied to the sinograms.

• The reconstruction problem is formulated on a
  continuous base.

Discretization is a contingence.
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Advantages of analytical reconstruction
The main advantages of an analytical
reconstruction, are

• Well known properties

• Easy to implement

There are many groups using this type of
reconstruction. Can be useful to compare results

• Fastest reconstruction

Differences are becoming less significant with
the use of faster computers
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Different approaches for image reconstruction (II)
3D data
Algebraic
How to reconstruct?
FBP (2D rec.) 3DRP (3D rec)
Analytical
Needed...

• Image parameterisation ? ?j, j 1, , n.
• Model for the acquisition process
• Objective function a distance to be minimized.
• Iterative algorithm

The object as a group of VOXELS(volume elements)
Relates the estimated image - f(A,?) - with the
acquired data y.
System matrix
Least-squares
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System matrix
Used to model the aquisition and detection
processes
Factorized representation...
A Adet.sen.Adet.blur.Aattn.Ageom.Apositron
Adet.sen matrix that contains the detection
efficiency of each detector pair.
Adet.blur local blurring function applied to the
sinogram, that accounts for the not exact
co-linearity of photons, the scattering of
photons from one crystal to another and the fact
that a photon may penetrate through one or more
crystals before being stopped.
Aattn matrix with the attenuation terms.
Ageom contains the geometrical mapping between
the source and data. The (i, j)th element is
equal to the probability that a photon pair
produced in voxel j reaches the front faces of
the detector pair i
Apositron includes the effect of the distance
travelled by the emitted positron.
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Different approaches for image reconstruction
(III)
3D data
How to reconstruct?
FBP (2D rec.) 3DRP (3D rec)
Algebraic
Analytical
Needed...
Statistical model for the data?

• Image parameterisation ? ?j, j 1, , n.
• Model for the acquisition process
• Objective function a distance that should be
  minimized
• Iterative algorithm (to minimize the objective
  function)

No
Yes
Non Statistical Reconstruction
It is better to go back to the objective function
!!!
ART
Algebraic Reconstruction Technique
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Objective function and likelihood
A statistical measure which is maximized when the
difference between estimated and measured
projections is minimized.
Likelihood
Obtain an estimate, , of ? (source activity)
which maximizes the probability p(y ?) of
observing the actual detector count data, y, over
all possible densities ?.
Assuming a distribution for the data, the maximum
of the likelihood function is the minimum of a
distance.
Least squares
Gaussian data
Likelihood function
Regularization term
Relates the image with the measured data.
Relates the image with a prior.
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Different approaches for image reconstruction (IV)
3D data
How to reconstruct?
FBP (2D rec.) 3DRP (3D rec)
Algebraic
Analytical
Needed...
Statistical model for the data?

• Image parameterisation ? ?j, j 1, , n.
• Model for the acquisition process
• Objective function a distance that should be
  minimized
• Iterative algorithm (to minimize the objective
  function)

Yes
No
Objective function
Non Statistical Reconstruction
Non Bayesian rec.
Iterative algorithms
ART
Bayesian reconstruction
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Bayesian reconstruction
In the objective function it is included a
regularization term
Inclusion of information a priori
Examples of Bayesian algorithms
GEM Generalized Expectiation Maximization OSL
One-step-late SAGE Space-alternating generalized
EM Gradient ascent methods
These algorithms could produce superior results
than analytical or non-Bayesian methods for image
reconstruction.
Advantages
High computational cost.
Disadvantages
The behavior of these nonlinear methods is not
well understood
Statistical Approaches in Quantitative Positron
Emission Tomography, R. Leahy and J. Qi,
Statistics and Computing, Vol. 10, April 2000, pp
147-165
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Non-Bayesian reconstruction ML-EM
ML-EM Maximum-Likelihood Expectation-Maximization
The iterative algorithm used to maximize the
likelihood
The objective function to maximize
Discretized object. Model for the acquisition and
detection process (system matrix). Poisson data.
Assumptions
Measured data
The algorithm includes two steps

• E-step calculation of the expectation

The estimated data, based on the model and the
source distiribution estimated in the previous
iteration.

• M-step maximization of the expectation

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Non-Bayesian reconstruction OS-EM
OS-EM Ordered Subsets Expectation-Maximization
In practice, understood as an accelerated version
of ML-EM
M subsets
1 iteration
? (M EM iterations)
1 subset (only part of the acquired projections)
1 sub-iteration
Steps involved...
1) Initialization of ? , n 0
2) To repeat in each iteration (update)
64 projections (64 samples in ?)
2 projections / subset
for subset s 1, ..., M
32 subsets / iteration
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Advantages and disadvantages of OS-EM over
analytical reconstruction
Advantages

• The inclusion of a model for the
  aquisition/detection process.

It is possible to account for the attenuation,
scatter, detectors efficiency, and other effects.

• Statistical properties of the data are considered
  (Poisson data)

• Better noise properties

Noise amplitude is lower in regions of low
counts. Absence of striking artifacts.
Disadvantages

• Less well known properties.

• Computational cost.

Something that has been improved with the use of
OS-EM instead of ML-EM
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Different approaches for image reconstruction (V)
3D data
How to reconstruct?
FBP (2D rec.) 3DRP (3D rec)
Algebraic
Analytical
Needed...
Statistical model for the data?

• Image parameterisation ? ?j, j 1, , n.
• Model for the acquisition process
• Objective function a distance that should be
  minimized
• Iterative algorithm (to minimize the objective
  function)

Yes
No
Objective function
Non Statistical Reconstruction
Rebinning?
Non Bayesian rec.
ML-EM OS-EM
ART
Bayesian reconstruction
GEM, OSL, SAGE, ...
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Rebining algorithm
An algorithm which sorts the 3D-data into an
ordinary 2D data set.
FORE1 Fourier Rebinning
SSRB FORE
B
N2 Oblique Sinograms
A
3D ACQUISITION
FORE - 2Rec.
REBINNING
3D Image 2N-1 Slices
Fourier Space
2D Rec.
2N-1 Direct Sinograms
(1) Exact and Approximate Rebinning Algorithms
for 3D PET Data, Michel Defrise, P. E. Kinahan,
D. Townsend, C. Michel, M. Sibomana, D,F Newport,
IEEE TMI 16(2), 1997, pp 145-158
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Different approaches for image reconstruction (VI)
3D data
How to reconstruct?
FBP (2D rec.) 3DRP (3D rec)
Algebraic
Analytical
Needed...
Statistical model for the data?

• Image parameterisation ? ?j, j 1, , n.
• Model for the acquisition process
• Objective function a distance that should be
  minimized
• Iterative algorithm (to minimize the objective
  function)

Yes
No
Objective function
Non Statistical Reconstruction
Rebinning?
Non Bayesian rec.
ML-EM OS-EM
ART
Yes
Bayesian reconstruction
GEM, OSL, SAGE, ...
2D Reconstruction
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Rebinning 2D reconstruction
3D data
Consistent projections, corrected for randoms,
attenuation, scatter, etc.
Needs...
FORE (Fourier Rebinning)
Rebinning
OS - EM (Ordered Subsets Expectation
Maximization)
2D Reconstruction
Assumes...
No more POISSON DATA
X
POISSON DATA
AW-OSEM NEC-OSEM
Decorrect data and include the corrections in the
matrix Aij
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Different approaches for image reconstruction
(VII)
3D data
How to reconstruct?
FBP (2D rec.) 3DRP (3D rec)
Algebraic
Analytical
Needed...
Statistical model for the data?

• Image parameterisation ? ?j, j 1, , n.
• Model for the acquisition process
• Objective function a distance that should be
  minimized
• Iterative algorithm (to minimize the objective
  function)

Yes
No
Objective function
Non Statistical Reconstruction
Rebinning?
Non Bayesian rec.
ML-EM OS-EM
ART
Yes
No
Bayesian reconstruction
GEM, OSL, SAGE, ...
Fully 3D Reconstruction
How to deal with corrections applied to the data?
2D Reconstruction
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CONCLUSIONS
In most of the situations, a 3D aquisition is
better than a 2D aquisition.
The properties of analytical algorithms are well
established and then these algorithms could be
useful, for example, to test the performance of a
new scanner.
The assumption of the statistical properties of
data and the inclusion of a model for the
acquisition/detection process support the better
results obtained with statistical algebraic
algorithms.
Among the statistical algorithms, OS-EM is
becoming widespread and assuming an important
role as an alternative to FBP.
Due the computational cost (timing constraints),
a solution to reconstruct 3D data is the use of a
rebinning algorithm followed by a 2D
reconstruction. There is no significant loss,
but a fully 3D reconstruction, if affordable,
would be preferable.
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S. Miguel - Azores