1. MAP Estimation of Semi-Metric MRFs via Hierarchical Graph Cuts
M. Pawan Kumar Daphne Koller
MAP Estimation
b(k)
Semi-Metric Potentials
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) > 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 l1 l2 l3 l4
Optimal move
using graph
cuts l1 l2 l3 l4 l5 l6
Distance dT  path length
f1 = minf Q(f)
f2 = minf Q(f)
f3 = minf Q(f)
Repeat
B ≤ A/r
C ≤ A/r
f(a)  {1,2}
f(a)  {3,4}
f(a)  {5,6}
Overview
Analysis
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(f;dT1)
fT1
minf Q(f;dT2)
fT2 . B B C C . va vb va vb va vb l1 l2 l3 l4
Combine fT1, fT2 ….
Bound = 1
Bound = 1
Bound = 2dmax/dmin = 2r/(r-1)
Use -Expansion
True for children
Learning a Mixture of rHSTs (Hierarchical Clustering)
Refinement (Hard EM)
∑tdTt(i,k)
min maxi,k
d(i,k) l1 l3 l4
Initial labeling f
Cluster Cj
Root 1 cluster
Derandomization
Boosting-style descent
yik: contribution of (i,k)
to current labeling
Choose random π
yik = Residual
For li in cluster Cj
Find first lk in π
s.t. d(i,k) ≤ T
min ∑yik dT(i,k) l2 l3 l1 l4
Permutation π
yik = ∑wab[f(a)=i][f(b)=k]
Update yik. Repeat.
min ∑yik dT(i,k) l3
Bounds
Decrease T by r
For =1, O(log H)
New labeling f’
Cluster Cj+1
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
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
0.9
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
703.8
129.7
10.6
32.7
SMet
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
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
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