Probabilistic Inference Lecture 3

Probabilistic InferenceLecture 3

M. Pawan [email protected]

Slides available online http://cvc.centrale-ponts.fr/personnel/pawan/

Recap of lecture 1

Exponential FamilyP(v) = exp{-Σα θαΦα(v) - A(θ)}

SufficientStatistics

Parameters Log-PartitionFunction

Random Variables V = {V1,V2,…,Vn}

Labeling V = vva L = {l1,l2,…,lh}

Random Variable Va takes a value or label va

Overcomplete RepresentationP(v) = exp{-Σα θαΦα(v) - A(θ)}

SufficientStatistics

Parameters Log-PartitionFunction

There exists a non-zero c such that Σα cαΦα(v) = Constant

Pairwise MRF

Random Variable V = {V1, V2, …,Vn}

Neighborhood over variables specified by edges E

Label set L = {l1, l2, …, lh}

P(v) = exp{-Σα θαΦα(v) - A(θ)}

Sufficient Statistics Parameters

Ia;i(va) θa;i for all Va V, li L

θab;ik for all (Va,Vb) E, li, lk L

Iab;ik(va,vb)

Pairwise MRF




P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}

A(θ) : log Z

Probability P(v) =Πa ψa(va) Π(a,b) ψab(va,vb)

Z

ψa(li) : exp(-θa;i) ψa(li,lk) : exp(-θab;ik)

Parameters θ are sometimes also referred to as potentials

Pairwise MRF




P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}

Labeling as a function f : {1, 2, … , n} {1, 2, …, h}

Variable Va takes a label lf(a)

Pairwise MRF




P(f) = exp{-Σa θa;f(a) -Σa,b θab;f(a)f(b) - A(θ)}



Energy Q(f) = Σa θa;f(a) + Σa,b θab;f(a)f(b)

Pairwise MRF




P(f) = exp{-Q(f) - A(θ)}



Energy Q(f) = Σa θa;f(a) + Σa,b θab;f(a)f(b)

Inference

maxv ( P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)} )

Maximum a Posteriori (MAP) Estimation

minf ( Q(f) = Σa θa;f(a) + Σa,b θab;f(a)f(b) )

Energy Minimization

P(va = li) = Σv P(v)δ(va = li)

Computing Marginals

P(va = li, vb = lk) = Σv P(v)δ(va = li)δ(vb = lk)

Recap of lecture 2

DefinitionsEnergy Minimization

f* = arg min Q(f; )Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

Min-marginals

qa;i = min Q(f; ) s.t. f(a) = i

Q(f; ’) = Q(f; ), for all f ’ Reparameterization

Belief PropagationPearl, 1988

General form of Reparameterization

’a;i = a;i

’ab;ik = ab;ik

+ Mab;k

- Mab;k

+ Mba;i

- Mba;i

’b;k = b;k

Reparameterization of (a,b) in Belief Propagation

Mab;k = mini { a;i + ab;ik } Mba;i = 0

Belief Propagation on Trees

Vb

Va

Forward Pass: Leaf Root

All min-marginals are computed

Backward Pass: Root Leaf

Vc

Vd Ve Vg Vh

Computational Complexity• Each constant takes O(|L|)

• Number of constants - O(|E||L|)

O(|E||L|2)• Memory required ?

O(|E||L|)

Belief Propagation on Cycles

Va Vb

Vd Vc

a;0

a;1

b;0

b;1

d;0

d;1

c;0

c;1

Remember my suggestion?Fix the label of Va


Va Vb

Vd Vc

a;0 b;0

b;1

d;0

d;1

c;0

c;1

Equivalent to a tree-structured problem


Va Vb

Vd Vc

a;1

b;0

b;1

d;0

d;1

c;0

c;1

Equivalent to a tree-structured problem


Choose the minimum energy solution

Va Vb

Vd Vc

a;0

a;1

b;0

b;1

d;0

d;1

c;0

c;1

This approach quickly becomes infeasible

Vincent Algayres Algorithm

Va Vb

Vd Vc

a;0 b;0

d;0

d;1

c;0

c;1

Compute zero cost paths from all labels of Va to all labels of Vd. Requires fixing Va.

Speed-Ups for Special Cases

ab;ik = 0, if i = k

= C, otherwise.

Mab;k = mini { a;i + ab;ik }

Felzenszwalb and Huttenlocher, 2004


ab;ik = wab|i-k|




ab;ik = min{wab|i-k|, C}




ab;ik = min{wab(i-k)2, C}



Lecture 3

Ising ModelP(v) = exp{-Σα θαΦα(v) - A(θ)}

Random Variable V = {V1, V2, …,Vn} Label set L = {0, 1}





Iab;ik(va,vb)

Ia;i(va): indicator for va = li Iab;ik(va,vb): indicator for va = li, vb = lk

Ising ModelP(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}

Random Variable V = {V1, V2, …,Vn} Label set L = {0, 1}





Iab;ik(va,vb)

Ia;i(va): indicator for va = li Iab;ik(va,vb): indicator for va = li, vb = lk

Interactive Binary Segmentation

Foreground histogram of RGB values FG

Background histogram of RGB values BG

‘1’ indicates foreground and ‘0’ indicates background


More likely to be foreground than background


More likely to be background than foreground

θa;0 proportional to -log(BG(da))

θa;1 proportional to -log(FG(da))


More likely to belong to same label


Less likely to belong to same label

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

θab;ik = 0 if i = k

Outline

• Minimum Cut Problem

• Two-Label Submodular Energy Functions

• Move-Making Algorithms

Directed Graph

n1 n2

n3 n4

10

5

3 2

Two important restrictions

(1) Rational arc lengths

(2) 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



• n2 N2

D = (N, A)


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

Remember …

Two important restrictions

(1) Rational arc lengths(2) Positive arc lengths

Cut

n1 n2

n3 n4

10

5

3 2



• n2 N2

D = (N, A)


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

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)

w(C) = Σ(n1,n2) C l(n1,n2)

Outline




Hammer, 1965; Kolmogorov and Zabih, 2004

Overview

Energy Q

DigraphD

One nodes per random variable

N = N1 U N2

ComputeMinimum

Cut

+ Additional nodes “s” and “t”

Labelingf*

na N1 implies f(a) = 0

na N2 implies f(a) = 1

Outline


• Two-Label Submodular Energy Functions• Unary Potentials• Pairwise Potentials• Energy Minimization


Digraph for Unary Potentials

Va

θa;0

θa;1P

Q

f(a) = 0

f(a) = 1


na

P

Q

s

t

f(a) = 0

f(a) = 1


na

P

Q

s

t

Let P ≥ Q

P-Q

0

Q

Q+

ConstantP-Q

f(a) = 0

f(a) = 1


na

P

Q

s

t

Let P ≥ Q

P-Q

0

Q

Q+

ConstantP-Q

f(a) = 1

w(C) = 0

f(a) = 0

f(a) = 1


na

P

Q

s

t

Let P ≥ Q

P-Q

0

Q

Q+

ConstantP-Q

f(a) = 0

w(C) = P-Q

f(a) = 0

f(a) = 1


na

P

Q

s

t

Let P < Q

0

Q-P

P

P+

Constant

Q-P

f(a) = 0

f(a) = 1


na

P

Q

s

t

Let P < Q

0

Q-P

P

P+

Constant

f(a) = 1

w(C) = Q-P

Q-P

f(a) = 0

f(a) = 1


na

P

Q

s

t

Let P < Q

0

Q-P

P

P+

Constant

f(a) = 0

w(C) = 0

Q-P

f(a) = 0

f(a) = 1

Outline




Digraph for Pairwise Potentials

Va

θab;11

Vb

θab;00

θab;01 θab;10

P R

Q S

f(a) = 0 f(a) = 1

f(b) = 0

f(b) = 1

0 0

Q-P Q-P

0 S-Q

0 S-Q

0 R+Q-S-P

0 0+ + +

P P

P P


na nb

P R

Q S

f(a) = 0 f(a) = 1

f(b) = 0

f(b) = 1

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


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 Potentialf(b) = 1

Q-P

f(a) = 0 f(a) = 1

f(b) = 0

f(b) = 1


na nb

P R

Q S

0 S-Q

0 S-Q

0 R+Q-S-P

0 0+

s

t

Unary Potentialf(a) = 1

Q-PS-Q

f(a) = 0 f(a) = 1

f(b) = 0

f(b) = 1


na nb

P R

Q S

0 R+Q-S-P

0 0

s

t

Pairwise Potentialf(a) = 1, f(b) = 0

Q-PS-Q

f(a) = 0 f(a) = 1

f(b) = 0

f(b) = 1

R+Q-S-P


na nb

P R

Q S s

t

Q-PS-Q

f(a) = 0 f(a) = 1

f(b) = 0

f(b) = 1

R+Q-S-P

R+Q-S-P ≥ 0

General 2-label MAP estimation is NP-hard

Outline




st-Flow

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Flow is less than length

Flow is non-negative

For all nodes expect s,t

Incoming flow

= Outgoing flow

st-Flow

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Flow is non-negative


Incoming flow

= Outgoing flow

flow(n1,n2) ≤ l(n1,n2)

st-Flow

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R


Incoming flow

= Outgoing flow

flow(n1,n2) ≥ 0


st-Flow

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Incoming flow

= Outgoing flow

For all a N \ {s,t}

flow(n1,n2) ≥ 0


st-Flow

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

= Outgoing flow

For all a N \ {s,t}

Σ(n,a)A flow(n,a)

flow(n1,n2) ≥ 0


st-Flow

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

For all a N \ {s,t}

Σ(n,a)A flow(n,a)

= Σ(a,n)A flow(a,n)

flow(n1,n2) ≥ 0


Weight of an st-Flow

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Outgoing flow of s

- Incoming flow of s


n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Σ(s,n)A flow(s,n)

- Σ(n,s)A flow(n,s)

= 0


n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Σ(s,n)A flow(s,n)


n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Σ(s,n)A flow(s,n)

= Incoming flow of t


n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Σ(s,n)A flow(s,n)

= Σ(n,t)A flow(n,t)

Max-Flow Problem

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Find the maximum flow!!

Min-Cut Max-Flow Theorem

n1 n2

n3 n4

10

5

3 2

D = (N, A)s

t

1 2

7 3

Function flow: A R

Weight of minimum-cut=

Weight of maximum-flow

Max-Flow via Reparameterization !!

Following slides courtesyPushmeet Kohli

Maxflow Algorithms

Augmenting Path Based Algorithms

1. Find path from source to sink with positive capacity

2. Push maximum possible flow through this path

3. Repeat until no path can be found

Source

Sink

n1 n2

2

5

9

42

1

Algorithms assume non-negative capacity

Flow = 0

Maxflow Algorithms





Source

Sink

2-2

5-2

9

42

1


Flow = 0 + 2

n1 n2

Maxflow Algorithms

Source

Sink

0

3

9

42

1






Flow = 2

n1 n2

Maxflow Algorithms

Source

Sink

0

3

5

02

1






Flow = 2 + 4

n1 n2

Maxflow Algorithms

Source

Sink

0

3

5

02

1






Flow = 6

n1 n2

Maxflow Algorithms

Source

Sink

0

2

4

02+1

1-1






Flow = 6 + 1

n1 n2

Maxflow Algorithms

Source

Sink

0

2

4

03

0






Flow = 7

n1 n2

History of Maxflow Algorithms



n: #nodesm: #arcsU: maximumarc length

Algorithms assume non-negative arc

lengths

History of Maxflow Algorithms



n: #nodesm: #arcsU: maximum arc length

Algorithms assume non-negative arc

lengths

Augmenting Path based Algorithms

a1 a2

1000 1

Sink

Source

1000

1000

1000

0

Ford Fulkerson: Choose any augmenting path

a1 a2

1000 1

Sink

Source

1000

1000

1000

0


Bad Augmenting

Paths


a1 a2

1000 1

Sink

Source

1000

1000

1000

0


Bad Augmenting

Path


a1 a2

9990

Sink

Source

1000

1000

9991



a1 a2

9990

Sink

Source

1000

1000

9991


n: #nodesm: #arcs

We will have to perform 2000 augmentations!

Worst case complexity: O (m x Total_Flow)

(Pseudo-polynomial bound: depends on flow)


Dinic: Choose shortest augmenting path

n: #nodesm: #arcs

Worst case Complexity: O (m n2)


a1 a2

1000 1

Sink

Source

1000

1000

1000

0

Maxflow in Computer Vision

• Specialized algorithms for vision problems– Grid graphs – Low connectivity (m ~ O(n))

• Dual search tree augmenting path algorithm[Boykov and Kolmogorov PAMI 2004]• Finds approximate shortest

augmenting paths efficiently• High worst-case time complexity• Empirically outperforms other

algorithms on vision problems

Maxflow in Computer Vision

• Specialized algorithms for vision problems– Grid graphs – Low connectivity (m ~ O(n))

• Dual search tree augmenting path algorithm[Boykov and Kolmogorov PAMI 2004]• Finds approximate shortest

augmenting paths efficiently• High worst-case time complexity• Empirically outperforms other

algorithms on vision problems• Efficient code available on the

webhttp://pub.ist.ac.at/~vnk/software.html

Outline




Metric LabelingP(v) = exp{-Σα θαΦα(v) - A(θ)}






Iab;ik(va,vb)

θab;ik is a metric distance function over labels

Label set L = {0, …, h-1}

Metric LabelingP(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}






Iab;ik(va,vb)

θab;ik is a metric distance function over labels

Label set L = {0, …, h-1}

Stereo Correspondence

Disparity Map


L = {disparities}

Pixel (xa,ya) in leftcorresponds to

pixel (xa+va,ya) in right


L = {disparities}

θa;i is proportional tothe difference in RGB values


L = {disparities}

θab;ik = wab d(i,k)

wab proportional to exp(-(da-db)2)

Move-Making AlgorithmsSpace of All Labelings

f

Expansion Algorithm

Initialize labeling f = f0 (say f0(a) = 0, for all Va)

For α = 0, 2, … , h-1

End

fα = argminf’ Q(f’)

s.t. f’(a) {f(a)} U {lα}

Update f = fα

Boykov, Veksler and Zabih, 2001

Repeat until

convergence

Expansion AlgorithmVariables take label lα or retain current label

Slide courtesy Pushmeet Kohli

Expansion Algorithm

SkyHouse

TreeGround

Initialize with TreeStatus: Expand GroundExpand HouseExpand Sky

Slide courtesy Pushmeet Kohli

Variables take label lα or retain current label

Expansion Algorithm

Restriction on pairwise potentials?

θab;ik + θab;αα ≤ θab;iα + θab;αk

Metric Labeling

Probabilistic Inference Lecture 3

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