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

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

Sinogram position projection angle

sinogram

MLEM maximum likelihood expectation maximisation

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)

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) = ~

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

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

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)

Iterative Reconstruction likelihood iteration MEASUREMENT iteration COMPARE UPDATE RECON REPROJECTION

FBP vs MLEM h00189 FBP MLEM

FBP vs MLEM uniform Poisson

FBP vs MLEM FBP MLEM Poisson FBP MLEM Poisson uniform uniform

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

(back) projection model: model for image resolution

resolution model: simulation projection backprojection

resolution model: simulation no noise mlem mlem Poisson noise

resolution model: simulation no noise Poisson noise

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

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

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

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

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

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

(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...

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

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

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

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

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)

OSEM 64x1 1x64 true difference

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

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)

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

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

MAP vs smoothed ML MLEM smoothed MAP with quadratic prior

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!

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

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

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)

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

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

MAP with anatomical prior ml-p

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

MAP and lesion detection human observer study

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

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

thanks