Discrete OptimizationLecture 5 – Part 2

M. Pawan Kumar

pawan.kumar@ecp.fr

Slides available online http://mpawankumar.info

Interactive Binary Segmentation

Foreground histogram of RGB values FG

Background histogram of RGB values BG

‘1’ indicates foreground and ‘0’ indicates background

Interactive Binary Segmentation

More likely to be foreground than background

Interactive Binary Segmentation

More likely to be background than foreground

θa(0) proportional to -log(BG(da))

θa(1) proportional to -log(FG(da))

Interactive Binary Segmentation

More likely to belong to same label

Interactive Binary Segmentation

Less likely to belong to same label

θab(i,k) proportional to exp(-(da-db)2) if i ≠ k

θab(i,k) = 0 if i = k

Outline

• Minimum Cut Problem

• Submodular Energy Functions

Directed Graph

n1 n2

n3 n4

10

5

3 2

Important restriction

Positive arc lengths

D = (N, A)

Cut

n1 n2

n3 n4

10

5

3 2

Let N1 and N2 such that

• N1 “union” N2 = N

• N1 “intersection” N2 = Φ

C is a set of arcs such that• (n1,n2) A• n1 N1

• n2 N2

D = (N, A)

C is a cut in the digraph D

Cut

n1 n2

n3 n4

10

5

3 2

What is C?

D = (N, A)

N1

N2

{(n1,n2),(n1,n4)} ?

{(n1,n4),(n3,n2)} ?

{(n1,n4)} ?✓

Cut

n1 n2

n3 n4

10

5

3 2

What is C?

D = (N, A)N1N2

{(n1,n2),(n1,n4),(n3,n2)} ?

{(n1,n4),(n3,n2)} ?

{(n4,n3)} ?✓

Cut

n1 n2

n3 n4

10

5

3 2

What is C?

D = (N, A)N2N1

{(n1,n2),(n1,n4),(n3,n2)} ?

{(n1,n4),(n3,n2)} ?

{(n3,n2)} ?

✓

Cut

n1 n2

n3 n4

10

5

3 2

Let N1 and N2 such that

• N1 “union” N2 = N

• N1 “intersection” N2 = Φ

C is a set of arcs such that• (n1,n2) A• n1 N1

• n2 N2

D = (N, A)

C is a cut in the digraph D

Weight of a Cut

n1 n2

n3 n4

10

5

3 2 Sum of length of allarcs in C

D = (N, A)

Weight of a Cut

n1 n2

n3 n4

10

5

3 2 w(C) = Σ(n1,n2) C l(n1,n2)

D = (N, A)

Weight of a Cut

n1 n2

n3 n4

10

5

3 2

What is w(C)?

D = (N, A)

N1

N2

3

Weight of a Cut

n1 n2

n3 n4

10

5

3 2

What is w(C)?

D = (N, A)N1N2

5

Weight of a Cut

n1 n2

n3 n4

10

5

3 2

What is w(C)?

D = (N, A)N2N1

15

st-Cut

n1 n2

n3 n4

10

5

3 2

A source “s”

C is a cut such that• s N1

• t N2

D = (N, A)

C is an st-cut

s

t

A sink “t”

1 2

7 3

Weight of an st-Cut

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

w(C) = Σ(n1,n2) C l(n1,n2)

Weight of an st-Cut

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

What is w(C)?

3

Weight of an st-Cut

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

What is w(C)?

15

Minimum Cut Problem

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Find a cut with theminimum weight !!

C* = argminC w(C)

[Slide credit: Andrew Goldberg]

Augmenting Path and Push-Relabel

n: #nodes

m: #arcs

U: maximumarc length

Solvers for the Minimum-Cut Problem

Outline

• Minimum Cut Problem

• Submodular Energy Functions

Hammer, 1965; Kolmogorov and Zabih, 2004

Overview

Energy Q

DigraphD

One nodes per element

N = N1 U N2

ComputeMinimum

Cut

+ Additional nodes “s” and “t”

Optimalsolution

na N1 implies xa = 0

na N2 implies xa = 1

Outline

• Minimum Cut Problem

• Submodular Energy Functions• Unary Potentials• Pairwise Potentials

Digraph for Unary Potentials

P

Q

xa = 0

xa = 1

Digraph for Unary Potentials

na

P

Q

s

t

xa = 0

xa = 1

Digraph for Unary Potentials

na

P

Q

s

t

Let P ≥ Q

P-Q

0

Q

Q+

ConstantP-Q

xa = 0

xa = 1

Digraph for Unary Potentials

na

P

Q

s

t

Let P ≥ Q

P-Q

0

Q

Q+

ConstantP-Q

xa = 1

w(C) = 0

xa = 0

xa = 1

Digraph for Unary Potentials

na

P

Q

s

t

Let P ≥ Q

P-Q

0

Q

Q+

ConstantP-Q

xa = 0

w(C) = P-Q

xa = 0

xa = 1

Digraph for Unary Potentials

na

P

Q

s

t

Let P < Q

0

Q-P

P

P+

Constant

Q-P

xa = 0

xa = 1

Digraph for Unary Potentials

na

P

Q

s

t

Let P < Q

0

Q-P

P

P+

Constant

xa = 1

w(C) = Q-P

Q-P

xa = 0

xa = 1

Digraph for Unary Potentials

na

P

Q

s

t

Let P < Q

0

Q-P

P

P+

Constant

xa = 0

w(C) = 0

Q-P

xa = 0

xa = 1

Outline

• Minimum Cut Problem

• Submodular Energy Functions• Unary Potentials• Pairwise Potentials

Digraph for Pairwise Potentials

P R

Q S

xa = 0 xa = 1

xb = 0

xb = 1

0 0

Q-P Q-P

0 S-Q

0 S-Q

0 R+Q-S-P

0 0+ + +

P P

P P

Digraph for Pairwise Potentials

na nb

P R

Q S

0 0

Q-P Q-P

0 S-Q

0 S-Q

0 R+Q-S-P

0 0+ + +

P P

P P

s

t

Constant

xa = 0 xa = 1

xb = 0

xb = 1

Digraph for Pairwise Potentials

na nb

P R

Q S

0 0

Q-P Q-P

0 S-Q

0 S-Q

0 R+Q-S-P

0 0+ +

s

tUnary Potential

xb = 1

Q-P

xa = 0 xa = 1

xb = 0

xb = 1

Digraph for Pairwise Potentials

na nb

P R

Q S

0 S-Q

0 S-Q

0 R+Q-S-P

0 0+

s

t

Unary Potentialxa = 1

Q-PS-Q

xa = 0 xa = 1

xb = 0

xb = 1

Digraph for Pairwise Potentials

na nb

P R

Q S

0 R+Q-S-P

0 0

s

t

Pairwise Potentialxa = 1, xb = 0

Q-PS-Q

R+Q-S-P

xa = 0 xa = 1

xb = 0

xb = 1

Digraph for Pairwise Potentials

na nb

P R

Q S s

t

Q-PS-Q

R+Q-S-P

R+Q-S-P ≥ 0

xa = 0 xa = 1

xb = 0

xb = 1