A Generative Perspective on MRFs in Low-Level Vision Supplem

AGenerativePerspectiveonMRFsinLow-LevelVision

SupplementalMaterial

UweSchmidt QiGao StefanRothDepartmentofComputerScience,TUDarmstadt

1.Derivations

1.1.SamplingthePrior

We rstrewritethemodeldensityfromEqs.(1)and(2)ofthemainpaperas

p(x;Θ)=

z

1 x 2/22

ep(zic)·N(JTix(c);0,σi/szic),Z(Θ)i=1

c∈C

N

(1)

wherewetreatthescalesz∈{1,...,J}N×|C|foreachexpertandcliqueasrandomvariableswithp(zic)=αizic(i.e.,

theGSMmixtureweights).Insteadofmarginalizingoutthescales,wecanalsoretainthemexplicitlyandde nethejointdistribution(cf.[13])

N 212

e x /2p(zic)·N(JT(2)p(x,z;Θ)=ix(c);0,σi/szic).Z(Θ)i=1

c∈C

Theconditionaldistributionp(x|z;Θ)canbederivedasthemultivariateGaussian

s2z

p(x|z;Θ)∝e x /2exp icJTix(c)2σic∈Ci=1

N 1szic

T

∝exp xT I+wicwicx2σii=1c∈C

N 1

T

∝Nx;0, I+WiZiWi,

2

N

(3)

i=1

T2

wherethewicarede nedsuchthatwicxistheresultofapplying lterJitocliquecoftheimagex.Zi=diag{szic/σi}arediagonalmatriceswithentriesforeachclique,andWiare ltermatricesthatcorrespondtoaconvolutionoftheimage

TTTTTTT

with lterJi,i.e.Wix=[wiC1x,...,wiC|C|x]=[Jix(C1),...,Jix(C|C|)].

FollowingLeviandWeiss[4,11],wefurtherrewritethecovarianceasthematrixproduct

1

TZ1···0W1

1 ..N . 1... .. TT. Σ= I+WiZiWi= [W1,...,WN,I] .(4) =WZW T . WN i=1.ZN

I0··· I

andsampley～N(0,I)toobtainasamplexfromp(x|z;Θ)bysolvingtheleast-squaresproblem

√

WZWTx=WZy.

The

(5)

rsttwoauthorscontributedequallytothiswork.