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StatisticsinImagingWorkshop

July8,2004

ABayesianDeformationModel

forImages

SiningChen

PostdoctoralFellow

BiostatisticsDivision,Dept.ofOncology,SchoolofMedicine

Outline

1.Introducingtheproblem

2.Modeldescription

3.Applicationtomousebrainsegmentation

4.Inter-subjectregistrationofhumanbrains

ProblemstoSolve

•Identifyingabnormaltissuefromaseriesofgatedcardiac

images

•Markingaregionofinterestinalargenumberofhumanbrain

MRIs

•Repeatingthesameoperationonsimilarimages

ImageRegistration

Tobringtwosimilarimagesintospatialalignment,suchthat

“correspondingpointsoftheimagedsceneappearinthesame

positionontheregisteredimages”.

Alsoreferredtoas”spatialnormalization”incertaincontexts.

RegistrationMethods

•Curvedtransformationsusingbasisfunctions(Ashburnerand

Friston1999HumanBrainMapping),implementedinthe

softwarepackageSPM.

•Modelsderivedfromthephysicsofelasticobjects(Bajcsyand

Kovacic1989Comp.Vision,Graphics&ImageProcessing)

•Spatialmodelingindeformabletemplates...

Facets–generalizedlandmarks

Deformationmodelmechanism:placelargenumberoffacetsinthe

volumeoftemplateimage.Thenlocateeachinthetargetimage.

templatetarget

ModelHeuristic

Abalancebetween

•“preservingspatialrelationships”–expressedintheprior

•“matchingimagefeatures”–expressedinthelikelihood

Notation

TemplateTarget

facetlocationµx

facetfeatureφf

Thefeatureofafacetisafunctionoftheimageevaluatedatthat

location:canbeintensity(brightness),gradience(edgeness),

laplacian(medialness),etc.

(a)(b)

(c)(d)

(e)(f)

Figure1:a)Originalimage;b)localrankofintensity;

c)d)directionalderivatives;e)gradientmagnitude;f)

Laplacian

Prior–presevingspatialrelationship

Markovrandomfieldpriorspecifiedonafirst-degreeneighborhood

system.

Thespatialrelationshipamongianditsneighbors∂iinthetarget

imageremainssimilarbeforeandafterdeformation.

ij

PSfragreplacements

γij

µ1

µ2

µ3

µ4

µ5

x1

x2

µ3|x∂3

x4

x5

PSfragreplacements

γij

µ1

µ2µ3µ4

µ5

x1

x2µ3|x∂3

x4

x5

Prior–continued

p(x)=1

Z(γ)exp{−

1

2

∑

〈i,j〉

γij((xi−µi)−(xj−µj))a}

Fullconditional(leta=2):

p(xi|.)∝exp{−1

2

∑

j∈∂i

γij((xi−µi)−(xj−µj))2}

∝exp{−1

2(∑

j∈∂i

γij)(xi−µi−

∑

j∈∂iγij(xj−µj)∑

j∈∂iγij

)2}

Likelihood–matchingfeatures

Foreachfaceti,featureafterdeformationfishouldbesimilarto

templatefeatureφi

p(xi|T,φi)=1

Ci(T,φi)exp{sim(fi,φi)}

where

fi=QT(xi)

MeasureofFeatureSimilarity

sim(fi,φi)=b(ri·ρi)α·cos(ui−νi),α>0

ri,ρi:quantilesofgradientmagnitude,

ui,νi:directionsofgradients.

PSfragreplacements(u,r)

(u,r)

(ν,ρ)(ν,ρ)

θ

MeasureofFeatureSimilarity–continued

Advantages:

•robustunderweakassumptionsonintensity

•abilitytodistinguishbetween“meaningful”and“meaningless”

features

•accomodates“no-match”situations

PosteriorInference

MAP(maximumaposteriori)estimatecanbeobtainedbyICM

(iterativeconditionalmodes,Besag1986).

Toobtainestimatethatisclosertoglobalmaximum,webuilda

hierarchyoffacetsinscale-space.

FacetTreeinScale-Space

AugmentedFull-Conditional

Spatialpartonly

p(xi|xs,s∈∂i,xp)∝exp{−γp((xi−µi)−(xp−µp))2

+∑

s∈∂i

γis((xi−µi)−(xs−µs))2}

Faceti,

facettargetlocationxi,

facettemplatelocationµi,

i’ssiblingss∈∂i,

i’sparentp,

pre-selectedparametersγpandγis

AutomatedSegmentationofMouseBrainImages

ManualSegmentationofMouseHippocampus

Goal:toinvestigatetherelationshipofAlzheimer’sdiseaseto

changesinthevolumeofthehippocampus.

AutomaticsegmentationofanothermouseMRI

TemplateImage

5010

015

020

025

0

50 100

150

200

250

Figure2:Threeregionsofinterestaremarkedonthetemplateimage

withrectangles.

12

01

40

16

01

80

12

0

13

0

14

0

15

0

16

0

17

0

18

0

12

01

40

16

01

80

12

0

13

0

14

0

15

0

16

0

17

0

18

0

80

100

120

140

60

70

80

90

100

110

80

100

120

140

60

70

80

90

100

110

160

180

200

30

40

50

60

70

80

150

160

170

180

190

200

30

40

50

60

70

80

Figure3:Facetsbeforeandafterdeformation.

QuantitativeAssessmentofAuto-segmentation

PSfragreplacementsmi

ticiI

tI

logm(k)i=logti+ε

(k)i,ε∼N(0,∆

2)

logc(k)iI=logtI+(logm

(k)i−logti)+e

(k)iI,e∼N(0,δ

2)

00.010.020.030.040.050.060.070

2000

4000

6000

8000

10000

12000

00.010.020.030.040.050.060.070

2000

4000

6000

8000

10000

12000

Figure4:a)∆2,b)δ

2

Inourdataset,theinconsistencyofauto-segmentationis

approximatelythesameashumansegmentation.

Inter-subjectRegistrationofHumanBrainMRIs

Areallbrainsalike?

Issuesinevaluatinginter-subjectregistration

Problematicduetolackofinformationongroundtruth.

Whatwasdone:

•Qualitativeassessment:visualinspection,mosaic,facet

movement;

•Quantitativeevaluation:segmentationbased,cross-correlation;

•DoneincomparisontotheindustrialstandardSPM.

Mosaic

beforeregistrationafterfacetreg.afterSPMreg.

Facetmovement

Figure5:Locatingfacetsonsubject3

HumanbrainMRIsegmentation

originalMRIwhitemattergraymatter(a)(b)(b)

(a)(b)(c)

Overlap,misclassification

Figure6:Comparisonofgraymattersegmentationof“warpedtem-

plate”andmanuallysegmentedtarget

Segmentationbasedevaluation

misclassify(%)facetSPM

subject117.420.7

subject224.424.4

subject316.515.8

Table1:Percentmisclassifiedforwhitematter

overlap(%)facetSPM

subject187.881.0

subject279.070.1

subject379.773.0

Table2:Percentageoverlapforwhitematter

misclassify(%)facetSPM

subject116.223.3

subject237.054.0

subject326.947.3

Table3:Percentmisclassifiedforgraymatter

overlap(%)facetSPM

subject272.767.8

subject565.960.2

subject666.262.7

Table4:Percentageoverlapforgraymatter

NormalizedCross-covarianceMap

C(f,g)=n·

∑

Afij·gij−∑

Afij·∑

Agij

n√

(∑

Af2+n−2n

2(∑

Af)2)(∑

Ag2+n−2n

2(∑

Ag)2)

(1)

facetSPM

Conclusion

•UsedMarkovrandomfieldtomodelpriorbeliefoffacet

locations;

•proposedanew,robustmeasureofsimilarity;

•successfullyperformedauto-segmentationofmousebrain

hippocampigivenasegmentedtemplate;

•registeredbrainMRIofdifferentindividuals;

•quantitativeevaluatedperformanceofregistration;

•themodelperformsmarginallybetterthanSPM.

Thankyou!