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# Rounding-based Moves for Metric Labeling - PowerPoint PPT Presentation

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

Variables V= { V1, V2, …, Vn}

Variables V= { V1, V2, …, Vn}

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}

Va

Vb

minf

E(f)

+ Σ(a,b)wabd(f(a),f(b))

= Σaθa(f(a))

NP hard

Low-level vision applications

• 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

• Approximate Algorithms

• Move-Making Algorithms

• Linear Programming Relaxation

• Comparison

• Rounding-based Moves

• Conclusion

Space of All Labelings

f

Variables take label lα or retain current label

Boykov, Veksler and Zabih, 2001

Slide courtesy PushmeetKohli

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

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))

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))

• Approximate Algorithms

• Move-Making Algorithms

• Linear Programming Relaxation

• Comparison

• Rounding-based Moves

• Conclusion

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

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

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

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)

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)

• Approximate Algorithms

• Comparison

• Rounding-based Moves

• Conclusion

M = ratio of maximum and minimum non-zero distance

• Approximate Algorithms

• Comparison

• Rounding-based Moves

• Conclusion

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’

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

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)

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)

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)

0.2

0.3

0.1

1.0

0

yc(3)

yc(2)

yc(4)

yc(1)

0.1

0.2

r

yc(3)

yc(2)

0

-yc(1)

-yc(1)

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)

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

Choose an interval of length h’

Va

Vb

θab(i,k) = wabd(i,k)

Choose an interval of length h’

Add the current labels

Va

Vb

θab(i,k) = wabd(i,k)

Choose an interval of length h’

Add the current labels

d’(i,k) ≥ d(i,k)

d’ is submodular

Solve to update labels

Va

Vb

Repeat until convergence

θab(i,k) = wabd’(i,k)

Each problem can be solved using minimum cut

Same multiplicative bound as interval rounding

Multiplicative bound is tight

• Approximate Algorithms

• Comparison

• Rounding-based Moves

• Conclusion

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

M = ratio of maximum and minimum non-zero distance

### Questions?

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

[email protected]