Transformation to Energy-Minimization
Factorization is not possible.
Hammersley - Clifford theorem [Clifford, 1990]:
p(x|z, θ) = 1
Z
exp(−E(x, z, θ))
MAP (Maximum a Posterior) solution is:
arg min
x
E(x, z, θ)
arg min
x
.
.
∑
vi∈V
Ei(xi, zi) + .
.
∑
eij∈E
Eij(xi, xj)
.
.
GMM Likelihood .
.
Smoothness Penalty
Equivalent to a Min-Cut on 2-terminal graph
s
x
2
x
6
x
7
x
8
x
3
t
x
4
x
5
x
0
x
1
s-t cut
E1(x1)+E2(x2)+E12(x1, x2)
E1(1)
E1(0)
E2(1)
E2(0)
E12(1,0)=E12(0,1)
v
2
v
1
s
t
.
Graph Theoretical Segmentation 10/41