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Introduction to PET reconstruction
Maurizio Conti, Siemens Molecular Imaging,
Knoxville, Tennessee, USA
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Maurizio Conti: Introduction to PET
Maurizio Conti: Introduction to PET
- reproduction authorized for educational purposes only
Summary
Emission data corrections
Analytical methods (FBP)
Iterative methods (MLEM, OSEM)
Point Spread Function (PSF)
Time-of-Flight (TOF)
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Emission Sinogram
0°
45°
90°
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A
Need to correct the data
C
B
Normalization
A
D
C
A
C
B
Attenuation
B
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Scatter
D reconstruction
Maurizio Conti: PET
D
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Emission Sinogram
Normalization
Scatter simulation
Scatter
correction
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m-map
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Attenuation
correction
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E
N
NxE
recon
X
=
A
A x (N x E)
recon
=
X
S
_
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A x (NxE) - S
=
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recon
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Reconstruction
Analytical: “exact” solution to a system of equations. For example:
Filtered Back Projection, in image or Fourier space.
Iteratives: iterative process in which an intermediate image converges
toward the “true” image. For example:
- Maximum Likelihood Expectation Maximization (MLEM),
- Ordered Subsets Expectation Maximization (OSEM),
- Row Action Maximum Likelihood Algorithm (RAMLA) .
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Reconstruction from projections
Radon (1917):
p(xr, )
xr
p(xr, ) =  dyr f(x,y)
= R {f(x,y)}
f(x,y) = R -1{p(xr, )} = d b(xr, )
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Backprojection operator
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Back projection in image space
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Filtered back projection in image space
Apply a filter function to the projections before back projection
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Filtered back projection in Fourier space
Central slice (or central section) theorem
2 D ( f ( x, y )) u yr 0  1D ( p( xr ,  ))
f ( x, y )  21D ( fˆ (u x , u y ))
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Filtered back projection in Fourier space
Fourier transform (1D) of a projection
filter the projection in frequency space
pˆ (ur ,  )  ur  H (ur )
fˆ (ur , )  pˆ (ur , )  ur  H (ur )
repeat for all angles 
Interpolate on the rectangular grid
Inverse Fourier transform (2D)
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pˆ (ur ,  )  1D ( p( xr ,  ))
fˆ (ux , u y )  resample( pˆ (ur ,  )  ur  H (ur ))
f ( x, y)  1 [ fˆ (u , u )]
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2D
x
y
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Filtered back projection in Fourier space : filters
pˆ (ur ,  )  ur  H (ur )
Ramp filter
Additional filter
The ramp filter is used to eliminate the star artifact and improve spatial
resolution, but it also amplifies the noise (high frequencies).
To compensate for these effects, low-pass smoothing filters are applied to
cutoff frequencies higher than a certain limit.
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Filtered back projection in Fourier space
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Iterative methods
The Maximum Likelihood Expectation Maximization (MLEM) is based on the
maximization of the logarithm of a Poisson-likelihood probability function.
It updates the image during each iteration by using a multiplicative factor assessed as
the ratio between the original acquired projections and the newly estimated ones.
Advantages of this iterative method are low noise, good spatial resolution, and the fact
that all reconstructed values will be positive because a nonnegativity condition is imposed
on the original data.
The main disadvantage is the large number of iterations required to converge to an
optimal solution.
To overcome the problem of slow convergence rate, the ordered-subsets expectation
maximization (OSEM) algorithm was proposed in 1994.
The OSEM is a modified version of MLEM with the main difference being that projections
are grouped into subsets of angles. Within each iteration the image is updated as many
times as the number of subsets, proportionally accelerating convergence.
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MLEM and OSEM
Yj
i
i
Yj
MLEM

n 1
i

p(Y j | i )  ei
Yj!
Yj

 

L( )  P(Y |  )   ei i
Yj!
i, j
in
 cij
Yj
c
ij
j
j
n
c

 kj k
k
Which  maximizes the L() ?
OSEM
n1
i


in
c
ij jS s
j
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c
ij
Yj
c
n
kj k

Y j   cij i
i
k
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MLEM and OSEM, references
MLEM: L.A.Shepp, Y.Vardi, Maximum Likelihood reconstruction for
emission tomography (maximum likelihood expectation maximization)
IEEE Trans.Med. Imaging, vol. 1, no.2, 1982, 113-122.
MLEM: K.Lange, R.Carson, EM reconstruction algorithms for emission
and transmission tomography, Journal of Computer Assisted
Tomography, vol.8, no.2, 1984, 306-316.
OSEM: H.Malcolm and R.Larkin, Accelerated Image reconstruction using
Ordered Subsets of projection data (ordered subsets expectation
maximization) IEEE Trans. Nucl. Sci., vol. 13, no.8, 1994, 601-609.
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MLEM, with physics corrections
BackProjector

n 1
i

in
c
ij
Yj
c
ij
j
j

n 1
i

c

k

n
i
 cij
c
A j ( N j (Y j  R j )  S j )
scatter correction
n
c

 kj k
ij
j
j
k
attenuation correction
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ForwardProjector
n
kj k
randoms correction
normalization correction
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2D or 3D reconstruction
2D reconstruction is faster
Rebinning techniques
move 3D data into 2D data
New fast algorithms and
computer make rebinning
less necessary!
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Recent reconstruction improvements
Iterative algorithm (OSEM) :
I.
rebin 3D2D + OSEM2D
II.
OSEM3D
III.
OSEM3D + Point Spread Function (PSF)
IV. OSEM3D + Point Spread Function + Time-of-Flight (TOF)
OSEM-3D + PSF + TOF
OSEM-3D + PSF
OSEM-3D
rebin 3D2D + OSEM-2D
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Maurizio Conti: Introduction to PET
Maurizio Conti: Introduction to PET
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Point Spread Function (PSF)
When a photon strikes a crystal, it travels
a certain distance before its energy is
converted into light. If the photon comes
from the center of the field of view (FOV),
the line of response (LOR) is likely to be
correctly localized in the crystal in which
the photon entered.
The further away from the
center of the FOV, the less
likely the LOR will be
calculated correctly
because the photon will hit
the crystal on an angle and
continue traveling to
another crystal before it
lights up.
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Point Spread Function (PSF)
A Point Spread Function (PSF)
describes the response of an
imaging system to a point
source or point object. A
system that knows the
response of a point source
from everywhere in its field of
view can use this information
to recover the original shape
and form of imaged objects.
PSFs are used in precision
imaging instruments, such as
microscopy, ophthalmology,
and astronomy (e.g. the
Hubble telescope) to make
geometric corrections to the
final image.
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Time-of-Flight (TOF)
Conventional PET detects
coincidence photons and
records individual lines of
response (LOR) between the
crystals. The actual location
where the annihilation
occurred along the LOR is not
measured.
Siemens ultraHD•PET with
Time of Flight (TOF)
measures the actual time
difference between the
detection of each
coincidence photon. This
timing information is used to
better localize the event
along each LOR.
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Time-of-Flight (TOF)
Time of Flight (TOF) systems measure the time between each
coincidence photon to determine the event location along the
line of response. The event location accuracy can be measured
proportionally to the system’s time resolution.
TOF systems are able to record segments of response instead
of lines of response. The time resolution defines the size of the
segment of response (“time bin”).
Conventional line of response
T2
T1
0
x
Segments of response (“time bins”)
s
x
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(T 2  T 1)  c
2
c = speed of light
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s => directly proportional to
the system’s time resolution
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Future:
faster implementations of the reconstruction algorithms
other algorithms development (other than ML based)
listmode instead of sinogram based
dynamic or 4D reconstructions
application oriented reconstruction methods (oncology, brain
functions, cardiac,..)
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