Rounding-based Moves for Metric Labeling

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Rounding-based Moves for Metric Labeling. M. Pawan Kumar École Centrale Paris INRIA Saclay , Île-de-France. Metric Labeling. Variables V = { V 1 , V 2 , …, V n }. Metric Labeling. Variables V = { V 1 , V 2 , …, V n }. Metric Labeling. w ab d (f(a),f(b)). θ b (f(b)). w ab ≥ 0.

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### Rounding-based Movesfor Metric Labeling

M. Pawan Kumar

ÉcoleCentrale Paris

INRIA Saclay, Île-de-France

Metric Labeling

Variables V= { V1, V2, …, Vn}

Metric Labeling

Variables V= { V1, V2, …, Vn}

Metric Labeling

wabd(f(a),f(b))

θb(f(b))

wab ≥ 0

θa(f(a))

d is metric

Va

Vb

minf

E(f)

+ Σ(a,b)wabd(f(a),f(b))

= Σaθa(f(a))

Labels L= { l1, l2, …, lh}

Variables V= { V1, V2, …, Vn}

Labeling f: { 1, 2, …, n}  {1, 2, …, h}

Metric Labeling

Va

Vb

minf

E(f)

+ Σ(a,b)wabd(f(a),f(b))

= Σaθa(f(a))

NP hard

Low-level vision applications

Outline
• Approximate Algorithms
• Comparison
• Rounding-based Moves
• Conclusion

Boykov, Veksler and Zabih

Efficiency

Move-Making Algorithms

Kleinberg and Tardos

Accuracy

Convex Relaxations

Kolmogorov and Zabih

Efficiency

Move-Making Algorithms

Chekuri, Khanna,

Naor and Zosin

Accuracy

Convex Relaxations

Outline
• Approximate Algorithms
• Move-Making Algorithms
• Linear Programming Relaxation
• Comparison
• Rounding-based Moves
• Conclusion
Move-Making Algorithms

Space of All Labelings

f

Expansion Algorithm

Variables take label lα or retain current label

Boykov, Veksler and Zabih, 2001

Slide courtesy PushmeetKohli

Expansion Algorithm

Variables take label lα or retain current label

Tree

Ground

House

Status:

Initialize with Tree

Expand Ground

Expand House

Expand Sky

Sky

Boykov, Veksler and Zabih, 2001

Slide courtesy PushmeetKohli

Multiplicative Bounds

f*: Optimal Labeling

f: Estimated Labeling

Σaθa(f(a)) + Σ(a,b)wabd(f(a),f(b))

Σaθa(f*(a)) + Σ(a,b)wabd(f*(a),f*(b))

Multiplicative Bounds

f*: Optimal Labeling

f: Estimated Labeling

Σaθa(f(a)) + Σ(a,b)wabd(f(a),f(b))

B

Σaθa(f*(a)) + Σ(a,b)wabd(f*(a),f*(b))

Outline
• Approximate Algorithms
• Move-Making Algorithms
• Linear Programming Relaxation
• Comparison
• Rounding-based Moves
• Conclusion
Integer Linear Program

Minimize a linear function over a set of feasible solutions

Indicator xa(i)  {0,1} for each variable Va and label li

Indicator xab(i,k)  {0,1} for each neighbor (Va,Vb) and labels li, lk

Number of facets grows exponentially in problem size

Linear Programming Relaxation

Indicator xa(i)  {0,1} for each variable Va and label li

Indicator xab(i,k)  {0,1} for each neighbor (Va,Vb) and labels li, lk

Schlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003

Linear Programming Relaxation

Indicator xa(i)  [0,1] for each variable Va and label li

Indicator xab(i,k)  [0,1] for each neighbor (Va,Vb) and labels li, lk

Schlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003

Approximation Factor

x*: LP Optimal Solution

x: Estimated Integral Solution

ΣaΣiθa(i)xa(i) + Σ(a,b)Σ(i,k) wabd(i,k)xab(i,k)

ΣaΣiθa(i)x*a(i) + Σ(a,b)Σ(i,k) wabd(i,k)x*ab(i,k)

Approximation Factor

x*: LP Optimal Solution

x: Estimated Integral Solution

ΣaΣiθa(i)xa(i) + Σ(a,b)Σ(i,k) wabd(i,k)xab(i,k)

F

ΣaΣiθa(i)x*a(i) + Σ(a,b)Σ(i,k) wabd(i,k)x*ab(i,k)

Outline
• Approximate Algorithms
• Comparison
• Rounding-based Moves
• Conclusion
Theoretical Guarantees

M = ratio of maximum and minimum non-zero distance

Outline
• Approximate Algorithms
• Comparison
• Rounding-based Moves
• Conclusion
Interval Rounding

Treat xa(i)  [0,1] as probability that f(a) = i

Cumulative probability ya(i) = Σj≤ixa(j)

ya(2)

ya(i)

ya(k)

0

ya(1)

ya(h) = 1

Choose an interval of length h’

Interval Rounding

Treat xa(i)  [0,1] as probability that f(a) = i

Cumulative probability ya(i) = Σj≤ixa(j)

r

0

ya(k)-ya(i)

REPEAT

Choose an interval of length h’

Generate a random number r  (0,1]

Assign the label next to r if it is within the interval

Example

0.25

0.5

0.75

1.0

ya(2)

ya(3)

0

ya(1)

ya(4)

0.7

0.8

0.9

1.0

yb(1)

yb(3)

0

yb(4)

yb(2)

0.2

0.3

0.1

1.0

0

yc(3)

yc(2)

yc(4)

yc(1)

Example

0.25

0.5

r

ya(2)

0

ya(1)

0.7

0.8

r

yb(1)

0

yb(2)

0.2

0.1

r

0

yc(2)

yc(1)

Example

0.25

0.5

0.75

1.0

ya(2)

ya(3)

0

ya(1)

ya(4)

0.7

0.8

0.9

1.0

yb(1)

yb(3)

0

yb(4)

yb(2)

0.2

0.3

0.1

1.0

0

yc(3)

yc(2)

yc(4)

yc(1)

Example

0.2

0.3

0.1

1.0

0

yc(3)

yc(2)

yc(4)

yc(1)

Example

0.1

0.2

r

yc(3)

yc(2)

0

-yc(1)

-yc(1)

Example

0.25

0.5

0.75

1.0

ya(2)

ya(3)

0

ya(1)

ya(4)

0.7

0.8

0.9

1.0

yb(1)

yb(3)

0

yb(4)

yb(2)

0.2

0.3

0.1

1.0

0

yc(3)

yc(2)

yc(4)

yc(1)

Key Observation

If d is submodular

d(i,k) + d(i+1,k+1) ≤ d(i,k+1) + d(i+1,k), for all i, k

energy can be minimized via minimum cut

Schlesinger and Flach, 2003

Interval Move

Choose an interval of length h’

Va

Vb

θab(i,k) = wabd(i,k)

Interval Move

Choose an interval of length h’

Va

Vb

θab(i,k) = wabd(i,k)

Interval Move

Choose an interval of length h’

d’(i,k) ≥ d(i,k)

d’ is submodular

Solve to update labels

Va

Vb

Repeat until convergence

θab(i,k) = wabd’(i,k)

Interval Move

Each problem can be solved using minimum cut

Same multiplicative bound as interval rounding

Multiplicative bound is tight

Outline
• Approximate Algorithms
• Comparison
• Rounding-based Moves
• Conclusion
Theoretical Guarantees

M = ratio of maximum and minimum non-zero distance

Boykov, Veksler and Zabih

Length of interval = 1

Move-Making Algorithms

Kleinberg and Tardos

Length of interval = 1

Convex Relaxations

Boykov, Veksler and Zabih

Length of interval = 1

Move-Making Algorithms

Chekuri, Khanna,

Naor and Zosin

Optimal interval length

Convex Relaxations

Theoretical Guarantees

M = ratio of maximum and minimum non-zero distance

### Questions?

http://cvn.ecp.fr/personnel/pawan

[email protected]