Improved Moves for Truncated Convex Models

1
Improved Moves for Truncated Convex Models
Aim To efficiently obtain accurate maximum a
posteriori estimate of random fields with
truncated convex model pairwise potentials
Our Approach
Im im1, jm
jm-im L
d(L) M
Analysis
Local vs. Global optimum
fm1(a) fm(a) OR fm1(a) ? Im
for all Va
Graph Construction
Maximum a Posteriori (MAP) Estimation
?bf(b)
Va
Vb
Va
Vb
V Va, Vb,
l l0, l1,
?af(a)
Unary
?abf(a)f(b)
Current fC
Previous fP
f a,b, 0,1,
Va
Vb
Pairwise
minf Q(f) ? ?af(a) ? ?abf(a)f(b)
Modelled as convex using the method of Veksler,
CVPR07
fm(b) ? Im
fm(a) ? Im
Truncated Convex Models
Va
Vb
Va
Vb
?abf(a)f(b) wab mind(f(a)-f(b)), M
Global fG
Partial fO
M
M
M
(ak,ak1)
Q(fc) Q(fc) Q(fO) ?
fm(b) ? Im
fm(a) ? Im
wab (d(L-kim) d(k-im))
Q(fP) - Q(fc) Q(fP) -Q(fO)
Potts
2
Trunc. Linear
Trunc. Quad.
If fP was the local optimum of the method
(ak,bl)
f(a)-f(b)
f(a)-f(b)
f(a)-f(b)
Q(fP) - Q(fc) 0 ? Q(fP) -Q(fO) 0
Linear Programming (LP) Relaxation
wab (d(k-l1) - 2d(k-l) d(k-l-1))
Truncated Linear
L v2 M
2 v2
fm(a) ? Im
fm(b) ? Im
Best possible integrality gap (most accurate in
worst case).
Truncated Quad.
L vM
O(vM)
2
Potts Model - 2, Trunc. Linear - 2v2, Trunc.
Quad. - O(vM)
Results
Properties of the Graph
Unary potential represented exactly
Move-Making Algorithms using Graph Cuts
Initialize the labelling f0.
Construct an st-MINCUT graph.
S
Current labelling fm
Cut induces labelling.
Trunc. Linear
Trunc. Quad.
Find st-MINCUT. fm1
?abfm(a)fm(b)
Repeat
t
Overestimation Q(f) of Gibbs energy due
to pairwise potentials
Potts Model - 2, Trunc. Linear - 2M, Trunc.
Quad. - 2M
Teddy Stereo Pair
wabd(f(a)-f(b))
wab(d(f(a)-im-1) M)
New move-making method as accurate as LP and
faster.
More results in tech report.