A Generative Perspective on MRFs in Low-Level Vision Supplem(2)

Byusingthewell-knownpropertyitfollowsthat

x=WZW

T 1

y～N(0,I) Ay～N(0,AIAT),√

√ √ T T 1WZIWZWWZ

(6)

WZy～Nx;0,WZW

T 1

～Nx;0,WZW

T 1

(7)

isindeedavalidsamplefromtheconditionaldistributionasderivedinEq.(3).Sincethescalesareconditionallyindependent

giventheimagebyconstruction,theconditionaldistributionp(z|x;Θ)isreadilygivenas

2

p(zic|x;Θ)∝p(zic)·N(JTix(c);0,σi/szic).

(8)

1.2.ConditionalSampling

Inordertoavoidextremevaluesatthelessconstrainedboundarypixels[5]duringlearningandmodelanalysis,orto

performinpaintingofmissingpixelsgiventheknownones,werelyonconditionalsampling.Inparticular,wesamplethepixelsxAgiven xedxBandscaleszaccordingtotheconditionalGaussiandistribution

p(xA|xB,z;Θ),

whereAandBdenotetheindexsetsoftherespectivepixels.Withoutlossofgenerality,weassumethat

xA

x=,

xB

Σ=WZW

T 1

A

=

CT

CB 1

,

(10)(9)

wherethesquaresub-matrixAhasasmanyrowsandcolumnsasthevectorxAhaselements,etc.Theconditionaldistributionofinterestcannowbederivedas

T

1xAACxA

p(xA|xB,z;Θ)∝exp

CTBxB2xB

(11) T 1 1 1

∝exp xA+ACxBAxA+ACxB

2 ∝NxA; A 1CxB,A 1.ThematricesAandCaregivenbytheappropriatesub-matricesofWiandZi,andallowforthesameef cientsamplingscheme.Themeanµ= A 1CxBcanalsobecomputedbysolvingaleastsquaresproblem.Samplingtheconditionaldistributionofscalesp(z|xA,xB;Θ)=p(z|x;Θ)remainsasbefore.

1.3.SamplingthePosteriorforImageDenoising

Assumingadditivei.i.d.Gaussiannoisewithknownstandardderivationσ,theposteriorgivenscaleszcanbewrittenas

p(x|y,z;Θ)∝p(y|x)·p(x|z;Θ)

1 1

T 12

∝exp 2 y x ·exp xΣx

2σ 2

1I 1TyT

∝exp 2x2+x+Σx

2σσ2

2 ∝Nx;Σy/σ,Σ,

(12)

=I/σ2+Σ 1 1andΣasinEq.(4).Theconditionaldistributionofthescalesp(z|x,y;Θ)=p(z|x;Θ)whereΣ

remainsasbefore.