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