MAP Estimation of Semi-Metric MRFs
via Hierarchical Graph Cuts
M. Pawan Kumar
Daphne Koller
b(k)
MAP Estimation
Aim: To obtain
Semi-Metric Potentials
lk
a(i)
ab(i,k) = wab d(i,k)
d(i,i) = 0, d(i,j) = d(j,i) > 0
d(i,j) - d(j,k) ≤  d(i,k)
accurate, efficient
li
ab(i,k)
maximum a posteriori
v
v
(MAP) estimation for
a
b
Markov random fields
Variables V, Labels L minf Q(f)
(MRF) with semi-metric
f : {a,b, …}
{1, …, H}
pairwise potentials
Q(f) = ∑ a(f(a)) + ∑ ab(f(a),f(b))
r-HST Metrics
r-HST Metric Labeling
Bounds
f(a)-f(b)
For =1 (Metric)
Linear Program:
O(log H)
Graph Cuts:
2 dmax/dmin
Our Method:
O(log H)
f(a)-f(b)
Efficient Divide-and-Conquer Approach
Combine
f
using
-Expansion
i
A
A
0
• Initialize f = f1
• At each iteration
B
B C
C
• Choose an fi
Optimal
move
l1
l2
l3
l4
l1
l2
l3
l4
l5
l6
• ft(a) = ft-1(a) OR using graph
t(a) = f (a)
•
f
Distance dT  path length f1 = minf Q(f)
i
f2 = minf Q(f) f3 = minf Q(f)
cuts
C ≤ A/r
B ≤ A/r
f(a)  {1,2}
f(a)  {3,4}
f(a)  {5,6} • Repeat
Analysis Bound of 1 for unary potentials, 2r/(r-1) for pairwise potentials
Overview
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
l1
Combine fT1, fT2 ….
Use -Expansion
C
l2
C
l3
va
l4
vb
Bound = 1
True for children
va
va
vb
Bound = 2dmax/dmin = 2r/(r-1)
Bound = 1
Learning a Mixture of rHSTs (Hierarchical Clustering) min maxi,k ∑tdTt(i,k)
l1
l2
l3
l3
l4
l1
l4
Cluster Cj
• Root 1 cluster
• Choose random π
• For li in cluster Cj
Permutation π • Find first lk in π
s.t. d(i,k) ≤ T
• Decrease T by r
Cluster Cj+1
• Repeat
l3
l4
l1
Fakcharoenphol et al., 2000
Synthetic Experiments
Q
Exp
Swap TRW
vb
Refinement (Hard EM)
d(i,k)
Derandomization
• Initial labeling f
Boosting-style descent • yik: contribution of (i,k)
• yik = Residual
to current labeling
• min ∑yik dT(i,k)
yik = ∑wab[f(a)=i][f(b)=k]
• Update yik. Repeat.
•
min
∑y
d
(i,k)
ik
T
Bounds
• For =1, O(log H)
2
• For 1, O((log H) )
• New labeling f’
• Approximate E and M
100 randomly generated 4-connected grid graphs of size 100x100
BP
RSwp RExp
Our
+EM
Time
Exp
Swap TRW
BP
RSwp RExp
Our
+EM
T-L1 48645 48721 47506 50942 48045 47998 47850 47823
T-L1
0.4
0.6
104.3
15.8
2.0
5.8
10.2
25.7
T-L2 52094 51938 51318 60269 51842 51641 51587 51413
T-L2
0.4
0.9
179.0
45.6
10.7
30.7
12.8
64.1
rHST 50221 51055 48132 52841
-
-
48146 48146
rHST
0.3
0.5
713.7 150.4
-
-
1.9
5.0
48112 48487 47355 48136
-
-
47538 47382
Met
0.3
0.5
703.8 129.7
-
-
10.6
32.7
SMet 47613 47579 46612 47402
-
-
46651 46638
SMet
0.4
0.5
70.9.4 141.8
-
-
12.2
57.5
Met
Image Denoising
Clean up an image with noise and missing data
Exp
Q
TRW
BP
Our
75641 68226 105845 72828 72332
Time
5.1
174.3
32.9
70.6
Exp
+ EM
204.5
Q
Time
TRW
BP
Our
+ EM
86163 73383 526969 81820 81820
26.1
529.6
115.8
294.7 465.6
Stereo Reconstruction Find correspondence between two epipolar corrected images of a scene
Exp
Q
Time
12.1
Our
+ EM
263.3
50.4
152.8 361.8
Exp
Q
Time
TRW
BP
Our
+ EM
15322 13257 56280 14135 14135
4.5
169.1
29.6
72.1
203.1
Find correspondence between two scenes with common elements (building, fire)
Exp
Time
BP
78776 62777 126824 65116 65008
Scene Registration
Q
TRW
TRW
82036 81118
1.7
BP
Our
+ EM
84396 81315 81258
1371.1 218.0
104.9 373.6
Exp
Q
Time
TRW
BP
Our
+ EM
68572 67616 70239 67682 67676
1.3
1058.2 160.0
73.6
240.5