1. Maurizio Conti, Siemens Molecular Imaging, Knoxville, Tennessee, USA
Introduction to PET reconstruction
Maurizio Conti, Siemens Molecular Imaging, Knoxville, Tennessee, USA

2. Summary
Emission data corrections
Analytical methods (FBP)
Iterative methods (MLEM, OSEM)
Point Spread Function (PSF)
Time-of-Flight (TOF)

3. Emission Sinogram 0°
45°
90°

4. Need to correct the dataB
Attenuation A B C D
Normalization D
Scatter A B C D

5. Emission Sinogram
m-map
Normalization
Scatter simulation
Scatter
correction
Attenuation
correction

6. E X = N
N x E
recon X =
A x (N x E) A
recon = _
A x (NxE) - S S
recon

7. 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) .

8. Reconstruction from projections
Radon (1917): xr
p(xr, )
p(xr, ) = dyr f(x,y)
= R {f(x,y)} 
f(x,y) = R -1{p(xr, )} = d b(xr, ) 
Backprojection operator

9. Back projection in image space

10. Filtered back projection in image space
Apply a filter function to the projections before back projection

11. Filtered back projection in Fourier space
Central slice (or central section) theorem

12. Filtered back projection in Fourier space
Fourier transform (1D) of a projection
filter the projection in frequency space
repeat for all angles j
Interpolate on the rectangular grid
Inverse Fourier transform (2D)

13. Filtered back projection in Fourier space : filters
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.

14. Filtered back projection in Fourier space

15. 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.

16. MLEM and OSEM Yj li
MLEM
Which l maximizes the L(l) ?
OSEM

17. 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,
MLEM: K.Lange, R.Carson, EM reconstruction algorithms for emission and transmission tomography, Journal of Computer Assisted Tomography, vol.8, no.2, 1984,
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,

18. MLEM, with physics corrections
BackProjector
ForwardProjector
scatter correction
normalization correction
randoms correction
attenuation correction

19. 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!

20. Recent reconstruction improvements
Iterative algorithm (OSEM) :
rebin 3D2D + OSEM2D
OSEM3D
OSEM3D + Point Spread Function (PSF)
OSEM3D + Point Spread Function + Time-of-Flight (TOF)
OSEM-3D + PSF + TOF
OSEM-3D + PSF
OSEM-3D
rebin 3D2D + OSEM-2D

21. 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.

22. 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.

23. 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.

24. x Time-of-Flight (TOF) s c = speed of light
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
Segments of response (“time bins”) s
s => directly proportional to
the system’s time resolution T2 x T1
c = speed of light

25. 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,..)