Statistical image reconstruction Intro: SPECT, PET, CT MLEM (back) projection model OSEM MAP...
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![Page 1: Statistical image reconstruction Intro: SPECT, PET, CT MLEM (back) projection model OSEM MAP –uniform resolution –anatomical prior –lesion detection.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649d265503460f949fdb78/html5/thumbnails/1.jpg)
Statistical image reconstruction• Intro: SPECT, PET, CT• MLEM• (back) projection model• OSEM• MAP
– uniform resolution– anatomical prior– lesion detection
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CTCT
yT II0 e ( )dL
yE (x)L e
( )dx
d
dx
PETPET
yE (x)dxL e
( )dL
jj ijlii eby
j jiji ay
SPECTSPECT
j jiji ay
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Sinogram
position
projection angle
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sinogram
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MLEMmaximumlikelihood
expectationmaximisation
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computing p(recon | data) difficult inverse problem
computing p(data | recon) “easy” forward problem
one wishes to find recon that maximizes p(recon | data)
Bayes:
p(recon | data) = p(data | recon) p(recon)
p(data)
datarecon
~
Maximum Likelihood
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Maximum Likelihood
p(recon | data) ~
p(data | recon)
projection Poisson
j j
ijiji say
j = 1..Ji = 1..I
i i
yiy!y
ye
ii
i
ii )y|y(p
i
iiii )!yln(yylnyln(p(data | recon)) = L(data | recon) = ~
p(data | recon)recon data
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Maximum Likelihood
i
iii yylnyL(data | recon)
i i
iiij
j y
yya
)(Lfind recon:
J..1j,0sa
saya
i k ikik
k ikikiij
Iterative inversion needed
j
ijiji say j
ijiji say
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Expectation Maximisation
i
i
iij
i ijj
newj y
ya
a1
• produces non-negative solution
ML-EM algorithm:
ji ij
jj
newj
La
• can be written as additive gradient ascent:
• several useful alternative derivations exist
• only involves projections and backprojections (“easy” forward operations)
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Optimisation transferOptimisation transfer
i
iii yylnyL(data | recon)
j
ijiji say
ij jiji af
i j
jijijg
(data | recon)
In every iteration:
with L(data | current) = (data | current)
current
L
likelihoodlikelihood
new
Expectation Maximisation
current
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MEASUREMENT
REPROJECTION
COMPAREUPDATE RECON
likelihood
iteration
Iterative Reconstruction
iterationiteration
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h00189
FBPFBP
MLEMMLEM
FBP vs MLEM
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uniform Poisson
FBP vs MLEM
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uniformuniform
FBPFBPFBPFBP MLEMMLEMMLEMMLEM
Poisson
FBPFBPFBPFBP MLEMMLEMMLEMMLEM
Poisson
FBP vs MLEM
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8 iter8 iter 100 iter100 iter FBPFBPtrue imagetrue image
sinogramsinogramwithnoise
withnoise
smoothedsmoothed
MLEM: non-uniform convergence
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(back) projection model:model for image resolution
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resolution model: simulation
projection backprojection
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no noiseno noise
Poisson noisePoisson noise
mlemmlem
mlemmlem
resolution model: simulation
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no noise Poisson noise
resolution model: simulation
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no noise Poisson noise
compute: estimated sinogram – given sinogram = “unexplained part of the data”
compute: estimated sinogram – given sinogram = “unexplained part of the data”
resolution model: simulation
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compute sum of squared differences along vertical lines
resolution model: simulation
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MLEM withsingle rayprojector
MLEM withsingle rayprojector
MLEM withGaussiandiffusionprojector
MLEM withGaussiandiffusionprojector
(back)projection in SPECT
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3D-PET FDG: OSEM, no resolution model
3D-PET FDG: OSEM, with resolution model
(back)projection in PET
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after 8 iterations
• assume full convergence 0)|y(L
0)|yy(L
0)|y(L
yy
)|y(L
kj
22
can be used to estimate• impulse response• covariance matrixof ML-solution
first derivatives are zero
likelihood is maximized
yy
)|y(L)|y(L 21
kj
2
small change of the data...
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after 8 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
— Point 1— Point 2— Point 3— Point 4
target resolution
8 1612
gain in contrast to noise ratio due to better resolution model
gain in contrast to noise ratio due to better resolution model
4
2
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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...
(back) projection model
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OSEM
ordered subsetsexpectation maximisation
Hudson & Larkin, Sydney
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Reference
2 4 8 16 25 50 100 200
Subsets...
Filtered backprojection of the subsets.
OSEM
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01
2
3
410
40
1 iteration of 40 subsets(2 proj per subset)
1 iteration of 40 subsets(2 proj per subset)
OSEM
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MLEM-iterations
1 OSEM iteration with 40 subsets
0 1 2 3 4 10 40
Reference
0 1 2 3 4 10 40
OSEM
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s1s2
s3s4
no noise (and subset balance)
with noise
Convergence to limit cycleConvergence 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
ML
initialimage
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64x164x1 1x641x64truetrue differencedifference
OSEM
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MAPmaximum a posteriori
• short intro• MAP
• uniform resolution• anatomical priors• lesion detection
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MAP
computing p(recon | data) difficult inverse problem
computing p(data | recon) “easy” forward problem
one wishes to find recon that maximizes p(recon | data)
Bayes:
p(recon | data) = p(data | recon) p(recon)
p(data)
datarecon
~
![Page 35: Statistical image reconstruction Intro: SPECT, PET, CT MLEM (back) projection model OSEM MAP –uniform resolution –anatomical prior –lesion detection.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649d265503460f949fdb78/html5/thumbnails/35.jpg)
MAP
Bayes: p(recon | data) ~ p(data | recon) p(recon)
ln p(recon | data) ~ ln p(data | recon) + ln p(recon)
posteriorposterior likelihoodlikelihood priorprior
- penalty- penalty
local prior or Markov prior:
Gibbs distribution:
p(reconj | recon) = )N(EexpZ1
jjj
p(reconj | recon) = p(reconj | reconk, k is neighbor of j)
ln p(reconj | recon) = j Ej(Nj) + constant
j
k
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MAP
ln p(reconj | recon) = j Ej(Nj)
jNk
kj )(E
j – kj – k
E(j – k)E(j – k) quadratic
Huber
Geman
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MAP vs smoothed ML
MLEMMLEM smoothedMLEM
smoothedMLEM
MAP withquadratic prior
MAP withquadratic prior
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When postsmoothed-MLEM and MAPhave same resolutionsame resolution, they have same covariancesame covariance!When postsmoothed-MLEM and MAPhave same resolutionsame resolution, they have same covariancesame covariance!
Use non-uniform “prior” to smooth• more where likelihood is “strong”• less where likelihood is “weak”
Use non-uniform “prior” to smooth• more where likelihood is “strong”• less where likelihood is “weak”
Likelihood provides non-uniform information:• some information is destroyed by
• attenuation• Poisson noise• finite detector sensitivity and resolution• ...
Likelihood provides non-uniform information:• some information is destroyed by
• attenuation• Poisson noise• finite detector sensitivity and resolution• ...
MAP with uniform resolution
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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
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T1T1 GreyGrey WhiteWhite CSFCSF
prior knowledge,valid for severaltracer(FDG, ECD, ...)
prior knowledge,valid for severaltracer(FDG, ECD, ...)
• CSF: no tracer uptake• white: uniform, low tracer uptake• grey: higher tracer uptake,
possibly lesions
• CSF: no tracer uptake• white: uniform, low tracer uptake• grey: higher tracer uptake,
possibly lesions
MAP with anatomical prior
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• smoothing prior in gray matter (relative difference)• Intensity prior in white (with estimated mean)• Intensity prior in CSF (mean = 0)
MLEMMLEM MRIMRI MAPMAP
MAP with anatomical prior
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MAP with anatomical prior
Theoretical analysis indicates that
PV-correction with MAP-reconstruction is superiorsuperior to
PV-correction with post-processed MLEM
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phantomphantom
ml with resolutionmodeling
ml with resolutionmodeling
make anatomicalregions uniform
make anatomicalregions uniform ml-pml-p
map withanatomical priorand resolutionmodeling
map withanatomical priorand resolutionmodeling
mapmapsinogramsinogram
projection with finiteresolution(2 pixels FWHM)
MAP with anatomical prior
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ml-pml-p
mapmap
MAP with anatomical prior
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MAP with anatomical prior
MAP yields better noise characteristics
than post-processed MLEM
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MAP and lesion detection
human observer studyhuman observer study
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MAP and lesion detection
MLEM MAP
moresmoothing
moresmoothing
higher higher
observerscore
observerscore
moresmoothing
moresmoothing
higher higher
observerresponsetime
observerresponsetime
MLEM MAP
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MAP and lesion detection
(non-uniform quadratic) MAPseems better for lesion detection
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thanks