Statistical image reconstruction• Intro: SPECT, PET, CT• MLEM• (back) projection model• OSEM• MAP

– uniform resolution– anatomical prior– lesion detection

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

Sinogram

position

projection angle

sinogram

MLEMmaximumlikelihood

expectationmaximisation

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

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

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

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)

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

MEASUREMENT

REPROJECTION

COMPAREUPDATE RECON

likelihood

iteration

Iterative Reconstruction

iterationiteration

h00189

FBPFBP

MLEMMLEM

FBP vs MLEM

uniform Poisson

FBP vs MLEM

uniformuniform

FBPFBPFBPFBP MLEMMLEMMLEMMLEM

Poisson

FBPFBPFBPFBP MLEMMLEMMLEMMLEM

Poisson

FBP vs MLEM

8 iter8 iter 100 iter100 iter FBPFBPtrue imagetrue image

sinogramsinogramwithnoise

withnoise

smoothedsmoothed

MLEM: non-uniform convergence

(back) projection model:model for image resolution

resolution model: simulation

projection backprojection

no noiseno noise

Poisson noisePoisson noise

mlemmlem

mlemmlem

resolution model: simulation

no noise Poisson noise

resolution model: simulation

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

compute sum of squared differences along vertical lines

resolution model: simulation

MLEM withsingle rayprojector

MLEM withsingle rayprojector

MLEM withGaussiandiffusionprojector

MLEM withGaussiandiffusionprojector

(back)projection in SPECT

3D-PET FDG: OSEM, no resolution model

3D-PET FDG: OSEM, with resolution model

(back)projection in PET

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...

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

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

OSEM

ordered subsetsexpectation maximisation

Hudson & Larkin, Sydney

Reference

2 4 8 16 25 50 100 200

Subsets...

Filtered backprojection of the subsets.

OSEM

01

2

3

410

40

1 iteration of 40 subsets(2 proj per subset)

1 iteration of 40 subsets(2 proj per subset)

OSEM

MLEM-iterations

1 OSEM iteration with 40 subsets

0 1 2 3 4 10 40

Reference

0 1 2 3 4 10 40

OSEM

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

64x164x1 1x641x64truetrue differencedifference

OSEM

MAPmaximum a posteriori

• short intro• MAP

• uniform resolution• anatomical priors• lesion detection

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

~

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

MAP

ln p(reconj | recon) = j Ej(Nj)

jNk

kj )(E

j – kj – k

E(j – k)E(j – k) quadratic

Huber

Geman

MAP vs smoothed ML

MLEMMLEM smoothedMLEM

smoothedMLEM

MAP withquadratic prior

MAP withquadratic prior

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

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

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

• 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

MAP with anatomical prior

Theoretical analysis indicates that

PV-correction with MAP-reconstruction is superiorsuperior to

PV-correction with post-processed MLEM

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

ml-pml-p

mapmap

MAP with anatomical prior

MAP with anatomical prior

MAP yields better noise characteristics

than post-processed MLEM

MAP and lesion detection

human observer studyhuman observer study

MAP and lesion detection

MLEM MAP

moresmoothing

moresmoothing

higher higher

observerscore

observerscore

moresmoothing

moresmoothing

higher higher

observerresponsetime

observerresponsetime

MLEM MAP

MAP and lesion detection

(non-uniform quadratic) MAPseems better for lesion detection

thanks