MAP Estimation of Semi-Metric MRFs

1
MAP Estimation of Semi-Metric MRFs via
Hierarchical Graph Cuts
M. Pawan Kumar Daphne Koller
MAP Estimation
Semi-Metric Potentials
?b(k)
Bounds
Aim To obtain accurate, efficient maximum a
posteriori (MAP) estimation for Markov random
fields (MRF) with semi-metric pairwise potentials
lk
?a(i)
?ab(i,k) wab d(i,k)
For ?1 (Metric)
li
?ab(i,k)
d(i,i) 0, d(i,j) d(j,i) gt 0
Linear Program O(log H)
va
vb
d(i,j) - d(j,k) ? d(i,k)
Graph Cuts 2 dmax/dmin
minf Q(f)
Variables V, Labels L
f a,b, 1, , H
Our Method O(log H)
Q(f) ? ?a(f(a)) ? ?ab(f(a),f(b))
f(a)-f(b)
f(a)-f(b)
r-HST Metrics
r-HST Metric Labeling
Efficient Divide-and-Conquer Approach
Combine fi using ?-Expansion
A
A

• Initialize f0 f1

• At each iteration
• Choose an fi
• ft(a) ft-1(a) OR
• ft(a) fi(a)

B
B
C
C
Optimal move using graph cuts
l1
l2
l3
l4
l1
l2
l3
l4
l5
l6
Distance dT ? path length
f1 minf Q(f)
f2 minf Q(f)
f3 minf Q(f)

• Repeat

C A/r
B A/r
f(a) ? 1,2
f(a) ? 3,4
f(a) ? 5,6
Analysis
Overview
Bound of 1 for unary potentials, 2r/(r-1) for
pairwise potentials
Mathematical Induction
Unary potential bound follows from ?-Expansion
d ? ?1dT1 ?2dT2 .
A
A
minf Q(fdT1)
fT1
minf Q(fdT2)
fT2
B
B
.
C
C
.
va
vb
va
vb
va
vb
l1
l2
l3
l4
Combine fT1, fT2 .
Bound 2dmax/dmin 2r/(r-1)
Bound 1
Bound 1
True for children
Use ?-Expansion
Learning a Mixture of rHSTs (Hierarchical
Clustering)
Refinement (Hard EM)
??tdTt(i,k)
min maxi,k
d(i,k)

• Initial labeling f

l1
l3
l4
Cluster Cj
Derandomization

• Root ?1 cluster

Boosting-style descent

• yik contribution of (i,k)
• to current labeling

• Choose random p

• yik Residual

• For li in cluster Cj
• Find first lk in p
• s.t. d(i,k) T

• min ?yik dT(i,k)

l2
l3
l1
l4
Permutation p
yik ?wabf(a)if(b)k

• Update yik. Repeat.

• min ?yik dT(i,k)

l3
Bounds

• Decrease T by r

• New labeling f

• For ?1, O(log H)

Cluster Cj1

• Repeat

l4
l1

• For ??1, O((?log H)2)

• Approximate E and M

Fakcharoenphol et al., 2000
Synthetic Experiments
100 randomly generated 4-connected grid graphs of
size 100x100
Q Exp Swap TRW BP RSwp RExp Our EM
T-L1 48645 48721 47506 50942 48045 47998 47850 47823
T-L2 52094 51938 51318 60269 51842 51641 51587 51413
rHST 50221 51055 48132 52841 - - 48146 48146
Met 48112 48487 47355 48136 - - 47538 47382
SMet 47613 47579 46612 47402 - - 46651 46638
Time Exp Swap TRW BP RSwp RExp Our EM
T-L1 0.4 0.6 104.3 15.8 2.0 5.8 10.2 25.7
T-L2 0.4 0.9 179.0 45.6 10.7 30.7 12.8 64.1
rHST 0.3 0.5 713.7 150.4 - - 1.9 5.0
Met 0.3 0.5 703.8 129.7 - - 10.6 32.7
SMet 0.4 0.5 70.9.4 141.8 - - 12.2 57.5
Image Denoising
Clean up an image with noise and missing data
Exp TRW BP Our EM
Q 86163 73383 526969 81820 81820
Time 26.1 529.6 115.8 294.7 465.6
Exp TRW BP Our EM
Q 75641 68226 105845 72828 72332
Time 5.1 174.3 32.9 70.6 204.5
Stereo Reconstruction
Find correspondence between two epipolar
corrected images of a scene
Exp TRW BP Our EM
Q 78776 62777 126824 65116 65008
Time 12.1 263.3 50.4 152.8 361.8
Exp TRW BP Our EM
Q 15322 13257 56280 14135 14135
Time 4.5 169.1 29.6 72.1 203.1
Scene Registration
Find correspondence between two scenes with
common elements (building, fire)
Exp TRW BP Our EM
Q 82036 81118 84396 81315 81258
Time 1.7 1371.1 218.0 104.9 373.6
Exp TRW BP Our EM
Q 68572 67616 70239 67682 67676
Time 1.3 1058.2 160.0 73.6 240.5