1. Statistical image reconstruction
Intro: SPECT, PET, CT
MLEM
(back) projection model
OSEM
MAP
uniform resolution
anatomical prior
lesion detection

2. PET CT SPECT  y  I e y   ( x ) dx e  y   ( x )  e dx T   ( e   (  ) d L  y E   ( x ) dx L  e    d y E   ( x ) L  e    d dx

3. Sinogram
position
projection angle

4. sinogram

5. MLEM
maximum
likelihood
expectation
maximisation

6. Maximum Likelihood
one wishes to find recon that maximizes p(recon | data)
recon
data
computing p(recon | data)
difficult inverse problem
computing p(data | recon)
“easy” forward problem
Bayes:
p(data | recon) p(recon)
p(recon | data) = ~
p(data)

7. Maximum Likelihood lj p(recon | data) ~ p(data | recon) data recon
projection
Poisson lj
p(data | recon)
j = 1..J
i = 1..I
ln(p(data | recon)) = L(data | recon) = ~

8. Maximum Likelihood L(data | recon) find recon:
Iterative inversion needed

9. Expectation Maximisation
ML-EM algorithm:
produces non-negative solution
can be written as additive gradient ascent:
only involves projections and backprojections (“easy” forward operations)
several useful alternative derivations exist

10. Expectation Maximisation
Optimisation transfer
L(data | recon) l
likelihood L F
In every iteration:
= F(data | recon)
lcurrent
l current
lnew
with L(data | lcurrent) = F(data | lcurrent)

11. Iterative Reconstruction
likelihood
iteration
MEASUREMENT
iteration
COMPARE
UPDATE
RECON
REPROJECTION

12. FBP vs MLEM
h00189
FBP
MLEM

13. FBP vs MLEM
uniform
Poisson

14. FBP vs MLEM
FBP
MLEM
Poisson
FBP
MLEM
Poisson
uniform
uniform

15. MLEM: non-uniform convergence
true image
8 iter
100 iter
FBP
sinogram
with
noise
smoothed

16. (back) projection model: model for image resolution

17. resolution model: simulation
projection
backprojection

18. resolution model: simulation
no noise
mlem
mlem
Poisson noise

19. resolution model: simulation
no noise
Poisson noise

20. resolution model: simulation
compute: estimated sinogram – given sinogram
= “unexplained part of the data”
no noise
Poisson noise

21. resolution model: simulation
compute sum of squared differences along vertical lines

22. (back)projection in SPECT
MLEM with
single ray
projector
MLEM with
Gaussian
diffusion
projector

23. (back)projection in PET
3D-PET FDG: OSEM, no resolution model
3D-PET FDG: OSEM, with resolution model

24. 8 after iterations assume full convergence  likelihood is maximized
 first derivatives are zero
small change of the data...
can be used to estimate
impulse response
covariance matrix
of ML-solution

25. 8 after iterations Simulation:
SPECT system with blurring (detector and collimator): about 8 mm.
reconstructed with and without resolution modelling
post-filter to have same target resolution
compare CNR in 4 points
gain in contrast to noise ratio due to better resolution model
— Point 1
— Point 2
— Point 3
— Point 4 4 2 8 12 16
target resolution

26. (back) projection model
accurate modeling of the physics:
larger fraction of the data becomes consistent
 better resolution
larger fraction of the noise becomes inconsistent
 less noise
 we gain twice!
 but computation time goes up...

27. expectation maximisation
OSEM
ordered subsets
expectation maximisation
Hudson & Larkin, Sydney

28. OSEM Reference Subsets... 2 4 8 16 25 50 100 200
Hudson and Larkin 1994
Filtered backprojection of the subsets.

29. OSEM 1 2 3 4 10 40
1 iteration of 40 subsets
(2 proj per subset)

30. OSEM Reference 1 OSEM iteration with 40 subsets 1 2 3 4 10 40 1 2 3 41 2 3 4 10 40 1 2 3 4 10 40
MLEM-iterations

31. OSEM s1 ML s2 no noise (and subset balance) s3 s4 initial image
with noise
Convergence to limit cycle
Solutions:
apply converging block-iterative algorithm:
sacrifize some speed for guaranteed convergence
gradually decrease the number of subsets
ignore the problem (you may not want convergence anyway)

32. OSEM
64x1
1x64
true
difference

33. MAP maximum a posteriori short intro MAP uniform resolution
anatomical priors
lesion detection

34. MAP one wishes to find recon that maximizes p(recon | data) recon data
computing p(recon | data)
difficult inverse problem
computing p(data | recon)
“easy” forward problem
Bayes:
p(data | recon) p(recon)
p(recon | data) = ~
p(data)

35. MAP j k Bayes: p(recon | data) ~ p(data | recon) p(recon)
ln p(recon | data) ~ ln p(data | recon) + ln p(recon)
posterior
likelihood
prior
- penalty
local prior or Markov prior:
p(reconj | recon) = p(reconj | reconk, k is neighbor of j) j k
Gibbs distribution:
p(reconj | recon) =
ln p(reconj | recon) =
-bj Ej(Nj) constant

36. MAP ln p(reconj | recon) = -bj Ej(Nj) E(lj – lk) quadratic Huber Geman

37. MAP vs smoothed ML
MLEM
smoothed
MAP with
quadratic prior

38. MAP with uniform resolution
Likelihood provides non-uniform information:
some information is destroyed by
attenuation
Poisson noise
finite detector sensitivity and resolution
...
Use non-uniform “prior” to smooth
more where likelihood is “strong”
less where likelihood is “weak”
When postsmoothed-MLEM and MAP
have same resolution, they have same covariance!

39. MAP with uniform resolution
equivalent to post-smoothed MLEM
prior improves condition number:
MAP converges faster than MLEM:
fewer iterations required!
but more work per iteration

40. MAP with anatomical prior
Grey
White
CSF
prior knowledge,
valid for several
tracer
(FDG, ECD, ...)
CSF: no tracer uptake
white: uniform, low tracer uptake
grey: higher tracer uptake,
possibly lesions

41. MAP with anatomical prior
This is an example of using anatomical priors for MAP-reconstruction. This approach is taken as a case to evaluate the use of Lx as a substitute of Ly in a post-processing method, using a simple simulation.
In that simulation, we used the same prior as in MAP, and combined it with Sx and Lx. Our experiment indicates that ignoring the covariances causes measureable loss of information [Nuyts et al, M5-2, MIC2003].
MLEM
MRI
MAP
smoothing prior in gray matter (relative difference)
Intensity prior in white (with estimated mean)
Intensity prior in CSF (mean = 0)

42. MAP with anatomical prior
Theoretical analysis indicates that
PV-correction with MAP-reconstruction is superior to PV-correction with post-processed MLEM

43. MAP with anatomical prior
and resolution
modeling
map
sinogram
projection with finite
resolution
(2 pixels FWHM)
phantom
ml with resolution
modeling
make anatomical
regions uniform
ml-p

44. MAP with anatomical prior
ml-p

45. MAP with anatomical prior
MAP yields better noise characteristics than post-processed MLEM

46. MAP and lesion detection
human observer study

47. MAP and lesion detection
observer
score
MLEM
MAP
observer
response
time
MLEM
MAP
more
smoothing
higher b
more
smoothing
higher b

48. MAP and lesion detection
(non-uniform quadratic) MAP
seems better for lesion detection

49. thanks