Correspondence Transfer for the Registration of Multimodal Images(8)

2.5.Probabilisticinterpretation

Informalterms,ourgoalcanbestatedasseekingthemaximuma-posterioriprobabilityofadiffemorphicwarping,thatis

w =.arginfw

logp(w|I1,I2)=

=arginfw

logp(I1,I2|w)p(w).

(8)

Thesecondterm,logp(w),canbeeasilyrecog-nized, inthemodel(5),asthegenericregularizer

12likelihoodD| w(x)|dx.Soweconcentrateonthelog-termlogp(I1,I2|w).Thiscanbeobtainedviap(I1,I2|w)=p(I1|I2,w)p(I2).Tothisend,themod-elsproposedby[18]couldbeemployed,inprinciple,sowhatweneedtocomputeisp(I1|I2,w).Now,againinpurelyformalterms,wecouldrepresenttheprobabil-ityofmatchingbilateralregionsasP(B1 B2);thenwhatwewishtocomputeis

p(I1|I2,w)=p(I1|I2,w,B1 B2)P(B1 B2)+

+p(I1|I2,w,U )(1 P(B1 B2)).(9)Now,thisisjustformalnotation.Thedif cultycomesinwhenwetrytowriteexplicitlytheprobabilitiesabove,becausetheconditionB1 B2isspeci ctoeachpointx∈D,soagainwehavetospecifythespatialstatisticsoftheimage,whichwouldleadtoaninferenceproblemwhereallpossiblecombinationsofstatesarepossibleandtime-consumingMarkov-ChainMonteCarlometh-odsbecomenecessaryratherthansimplelocaldescentalgorithms.

So,insteadofattemptingtocomputetheabovelike-lihood,wewillapproximateitbyassumingthatallpix-elsareindependent,andcomputinganaverage(expec-tation) overpixelsoftheprobabilitye 1|I1(w(x)) I2(x)|2 e 1

)) φ2(x)|2|φ1(w(xB1 B2

U

(10)

which,modulotechnicalities,shouldconvergeto(4),the“stiffversion”ofourfunctional(5).

3.Extensions

Whiletheapproachdescribedintheprevioussec-tioncanbeapplieddirectlytosimplesyntheticphantomssuchasthoseinFig.1,applicationtorealmulti-modalimagesrequiresamorepowerfulmodel.

Speci cally,ratherthanL2,weusemutualinforma-tion[21]betweenthedeformedtemplateI1 wandthetargetI2,denotedby

Φp(I1,I2|w)

MI(w)=

log

p(I(IdP(I1,I2|w).(11)

1|w)p2)

Estimationforthejointimageintensitydistribution

p(I1,I2|w)iscarriedontheregionofoverlapAofbothimagesbyusing2-DParzenwindowingwithGaussiankernelGσ:

p(I1

1(w(x)),I2(x)|w)=|A|Gσ(I1(w(x))

A

I1(w(y)),I2(x) I2(y))dy.(12)Itisequivalenttoconvolvingthejointintensityhis-togramwithadiscreteapproximationofGσ.ThejointhistogramsofI1 wandI2withintheirregionofoverlapareconstructedbybinningthecorrespondingintensitypairs(I1(w(x)),I2(x)),andthemarginalhistogramsareobtainedbyintegratingoverrowsandcolumns,re-spectively.

Substituting(12)into(11)andrearrangingfollowing[4,9],yieldsthe rstvariationofΦMI:δΦMI

=1 δw |A|·I1,I ·G Lσ w

2

I(I1(w(x)),I2(x)) I1(w(x))

(13)

1

with theconvolutionoperator,andLIw1,I2

givenby

LI(I1,I2|w)

w1,I2

=1+log

pp(Iw)p(I.

(14)

1|2)

InplaceofanL2regularizer,weusea uidmodel[5]wherethedeformationvelocityv(x,t)isgovernedbythesimpli edNavier-Stokesequations

µ 2v+(µ+λ) ( ·v)+f=0

(15)

withµandλviscosityconstants.Herefistheforce eldwhichdrivesthewarpingwintheappropriatedi-rection.Itisderivedfromimageinformation,usuallyset

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