![Page 1: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/1.jpg)
Rounding-based Movesfor Metric Labeling
M. Pawan Kumar
Center for Visual Computing
Ecole Centrale Paris
![Page 2: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/2.jpg)
PostMetric LabelingRandom variables V = {v1, v2, …, vn}
Label set L = {l1, l2, …, lh}
Labelings quantatively distinguished by energy E(y)
Labeling y L∈ n
Unary potential of variable va V∈
∑a θa(ya)
![Page 3: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/3.jpg)
PostMetric LabelingRandom variables V = {v1, v2, …, vn}
Label set L = {l1, l2, …, lh}
Labelings quantatively distinguished by energy E(y)
Labeling y L∈ n
Pairwise potential of variables (va,vb)
∑a θa(ya) + ∑(a,b) wab d(ya,yb)
wab is non-negative d(.,.) is a metric distance function
miny
![Page 4: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/4.jpg)
Post• Existing Work
– Move-Making Algorithms (Efficient)– Linear Programming Relaxation (Accurate)
• Rounding-based Moves– Equivalence– Complete Rounding and Complete Move– Interval Rounding and Interval Move– Hierarchical Rounding and Hierarchical Move
Outline
![Page 5: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/5.jpg)
PostExpansion Algorithm
Sky
House
Tree
GroundInitialize with TreeExpand GroundExpand HouseExpand Sky
Variables take label lα or retain current label
Boykov, Veksler and Zabih, ICCV 1999
![Page 6: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/6.jpg)
PostMove-Making AlgorithmsIteration t
Define St L⊆ n containing current labeling yt
∑a θa(ya) + ∑(a,b) wab d(ya,yb)argminy
s.t. y S∈ t
Sometimes it can even be solved exactly
Above problem is easier than original problem
yt+1 =
Start with an initial labeling y0
![Page 7: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/7.jpg)
Post• Existing Work
– Move-Making Algorithms (Efficient)– Linear Programming Relaxation (Accurate)
• Rounding-based Moves– Equivalence– Complete Rounding and Complete Move– Interval Rounding and Interval Move– Hierarchical Rounding and Hierarchical Move
Outline
![Page 8: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/8.jpg)
PostLinear Programming Relaxation
Chekuri, Khanna, Naor and Zosin, SODA 2001
Binary indicator xa(i) {0,1} ∈
If variable ‘a’ takes the label ‘i’ then xa(i) = 1
∑i xa(i) = 1Each variable ‘a’ takes one label
Similarly, binary indicator xab(i,k) {0,1} ∈
![Page 9: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/9.jpg)
PostLinear Programming Relaxation
Minimize a linear function over feasible x
Indicators xa(i), xab(i,k) {0,1} Relaxed xa(i), xab(i,k) [0,1]
Rounding
Chekuri, Khanna, Naor and Zosin, SODA 2001
![Page 10: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/10.jpg)
Post• Existing Work
– Move-Making Algorithms (Efficient)– Linear Programming Relaxation (Accurate)
• Rounding-based Moves– Equivalence– Complete Rounding and Complete Move– Interval Rounding and Interval Move– Hierarchical Rounding and Hierarchical Move
Outline
![Page 11: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/11.jpg)
PostMove-Making Bound
y*: Optimal Labeling y: Estimated Labeling
Σa θa(ya) + Σ(a,b) wabd(ya,yb)
Σa θa(y*a) + Σ(a,b) wabd(y*a,y*b)
≥
![Page 12: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/12.jpg)
PostMove-Making Bound
y*: Optimal Labeling y: Estimated Labeling
Σa θa(ya) + Σ(a,b) wabd(ya,yb)
Σa θa(y*a) + Σ(a,b) wabd(y*a,y*b)B
≤
For all possible values of θa(i) and wab
![Page 13: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/13.jpg)
PostRounding Approximation
x*: LP Optimal Solution x: Rounded Solution
Σa Σi θa(i)xa(i) + Σ(a,b) Σ(i,k) wabd(i,k)xab(i,k)
≥
Σa Σi θa(i)x*a(i) + Σ(a,b) Σ(i,k) wabd(i,k)x*ab(i,k)
![Page 14: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/14.jpg)
PostRounding Approximation
x*: LP Optimal Solution x: Rounded Solution
Σa Σi θa(i)xa(i) + Σ(a,b) Σ(i,k) wabd(i,k)xab(i,k)
≤
Σa Σi θa(i)x*a(i) + Σ(a,b) Σ(i,k) wabd(i,k)x*ab(i,k)A
For all possible values of θa(i) and wab
![Page 15: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/15.jpg)
PostEquivalence
For any known rounding with approximation A
there exists a move-making algorithm
such that the move-making bound B = A
We know how to design such an algorithm
![Page 16: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/16.jpg)
Post• Existing Work
– Move-Making Algorithms (Efficient)– Linear Programming Relaxation (Accurate)
• Rounding-based Moves– Equivalence– Complete Rounding and Complete Move– Interval Rounding and Interval Move– Hierarchical Rounding and Hierarchical Move
Outline
![Page 17: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/17.jpg)
PostComplete Rounding
Treat x*a(i) [0,1] as probability that ya = li
Cumulative probability za(i) = Σj≤i x*a(j)
0 za(1) za(2) za(h) = 1za(k)za(i)
Generate a random number r (0,1]
Assign the label next to r
r
![Page 18: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/18.jpg)
PostComplete Rounding - Example
0 za(1) za(4)za(3)za(2)
0.25 0.5 0.75 1.0
0 zb(1) zb(4)zb(3)zb(2)
0.7 0.8 0.9 1.0
0 zc(1) zc(4)zc(3)zc(2)
0.1 0.2 0.3 1.0
r
r
r
![Page 19: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/19.jpg)
PostEquivalent Move
Complete Move !!
![Page 20: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/20.jpg)
PostComplete MoveIteration t
Define St ⊆ Ln
∑a θa(ya) + ∑(a,b) wab d(ya,yb)argminy
s.t. y S∈ t
yt+1 =
Start with an initial labeling y0
![Page 21: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/21.jpg)
PostComplete MoveIteration t
Define St = Ln
∑a θa(ya) + ∑(a,b) wab d(ya,yb)argminy
s.t. y S∈ t
How do we solve this problem?
Above problem is the same as the original problem
yt+1 =
Start with an initial labeling y0
![Page 22: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/22.jpg)
PostComplete Move
Define St = Ln
∑a θa(ya) + ∑(a,b) wab d’(ya,yb)argminy
s.t. y S∈ t
How do we solve this problem?
Above problem is the same as the original problem
yt+1 =
![Page 23: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/23.jpg)
PostComplete Move
Define St = Ln
∑a θa(ya) + ∑(a,b) wab d’(ya,yb)argminy
s.t. y S∈ t
Obtained by solving a small LP
Submodular overestimation d’ of d
yt+1 =
![Page 24: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/24.jpg)
PostSubmodular Overestimation
maxi,k d’(li,lk)/d(li,lk)mind’
d’(li,lk) ≥ d(li,lk)s.t.
d’(li,lk+1) + d’(li+1,lk) ≥ d(li,lk) + d(li+1,lk+1)
![Page 25: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/25.jpg)
PostSubmodular Overestimation
bmind’
d’(li,lk) ≥ d(li,lk)s.t.
d’(li,lk+1) + d’(li+1,lk) ≥ d(li,lk) + d(li+1,lk+1)
bd(li,lk) ≥ d’(li,lk)
Dual provides worst-case instance for complete rounding
![Page 26: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/26.jpg)
Post• Existing Work
– Move-Making Algorithms (Efficient)– Linear Programming Relaxation (Accurate)
• Rounding-based Moves– Equivalence– Complete Rounding and Complete Move– Interval Rounding and Interval Move– Hierarchical Rounding and Hierarchical Move
Outline
![Page 27: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/27.jpg)
PostInterval Rounding
Treat x*a(i) [0,1] as probability that ya = li
Cumulative probability za(i) = Σj≤i x*a(j)
0 za(1) za(2) za(h) = 1za(k)za(i)
Choose an interval of length h’
![Page 28: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/28.jpg)
PostInterval Rounding
Treat x*a(i) [0,1] as probability that ya = li
Cumulative probability za(i) = Σj≤i x*a(j)
r
Generate a random number r (0,1]
Assign the label next to r if it is within the interval
za(k)-za(i)0
Choose an interval of length h’ REPEAT
![Page 29: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/29.jpg)
PostInterval Rounding - Example
0 za(1) za(4)za(3)za(2)
0.25 0.5 0.75 1.0
0 zb(1) zb(4)zb(3)zb(2)
0.7 0.8 0.9 1.0
0 zc(1) zc(4)zc(3)zc(2)
0.1 0.2 0.3 1.0
![Page 30: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/30.jpg)
PostInterval Rounding - Example
0 za(1) za(2)
0.25 0.5
0 zb(1) zb(2)
0.7 0.8
0 zc(1) zc(2)
0.1 0.2
r
r
r
![Page 31: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/31.jpg)
PostInterval Rounding - Example
0 za(1) za(4)za(3)za(2)
0.25 0.5 0.75 1.0
0 zb(1) zb(4)zb(3)zb(2)
0.7 0.8 0.9 1.0
0 zc(1) zc(4)zc(3)zc(2)
0.1 0.2 0.3 1.0
![Page 32: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/32.jpg)
PostInterval Rounding - Example
0 zc(1) zc(4)zc(3)zc(2)
0.1 0.2 0.3 1.0
![Page 33: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/33.jpg)
PostInterval Rounding - Example
0 zc(3)zc(2)
0.1 0.2r
-zc(1) -zc(1)
![Page 34: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/34.jpg)
PostInterval Rounding - Example
0 za(1) za(4)za(3)za(2)
0.25 0.5 0.75 1.0
0 zb(1) zb(4)zb(3)zb(2)
0.7 0.8 0.9 1.0
0 zc(1) zc(4)zc(3)zc(2)
0.1 0.2 0.3 1.0
![Page 35: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/35.jpg)
PostEquivalent Move
Interval Move !!
![Page 36: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/36.jpg)
PostInterval MoveIteration t
y S∈ t iff ya = yta or ya interval of labels ∈
∑a θa(ya) + ∑(a,b) wab d(ya,yb)argminy
s.t. y S∈ t
yt+1 =
Start with an initial labeling y0
Choose an interval of labels of length h’
How do we solve this problem?
![Page 37: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/37.jpg)
PostInterval MoveIteration t
y S∈ t iff ya = yta or ya interval of labels ∈
∑a θa(ya) + ∑(a,b) wab d’(ya,yb)argminy
s.t. y S∈ t
yt+1 =
Start with an initial labeling y0
Choose an interval of labels of length h’
Submodular overestimation d’ of d
![Page 38: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/38.jpg)
Post• Existing Work
– Move-Making Algorithms (Efficient)– Linear Programming Relaxation (Accurate)
• Rounding-based Moves– Equivalence– Complete Rounding and Complete Move– Interval Rounding and Interval Move– Hierarchical Rounding and Hierarchical Move
Outline
![Page 39: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/39.jpg)
PostHierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Hierarchical clustering of labels (e.g. r-HST metrics)
![Page 40: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/40.jpg)
PostHierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels L1, L2 or L3
Move down the hierarchy until the leaf level
![Page 41: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/41.jpg)
PostHierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels l1, l2 or l3
![Page 42: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/42.jpg)
PostHierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels l4, l5 or l6
![Page 43: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/43.jpg)
PostHierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels l7, l8 or l9
![Page 44: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/44.jpg)
PostEquivalent Move
Hierarchical Move !!
![Page 45: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/45.jpg)
PostHierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Hierarchical clustering of labels (e.g. r-HST metrics)
![Page 46: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/46.jpg)
PostHierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Obtain labeling y1 restricted to labels {l1,l2,l3}
![Page 47: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/47.jpg)
PostHierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Obtain labeling y2 restricted to labels {l4,l5,l6}
![Page 48: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/48.jpg)
PostHierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Obtain labeling y3 restricted to labels {l7,l8,l9}
![Page 49: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/49.jpg)
PostHierarchical Move
L1 L2 L3
Va Vb
y1(a)
y2(a)
y3(a)
Move up the hierarchy until we reach the root
y1(b)
y2(b)
y3(b)
![Page 51: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/51.jpg)
PostSimple Example - Rounding
θa(1)xa(1) + θa(2)xa(2) + θb(1)xb(1) + θb(2)xb(2)minx≥0
+ d(1,1)xab(1,1) + d(1,2)xab(1,2)+ d(2,1)xab(2,1) + d(2,2)xab(2,2)
xa(1) + xa(2) = 1s.t.
xb(1) + xb(2) = 1
xab(1,1) + xab(1,2) = xa(1)
xab(2,1) + xab(2,2) = xa(2)
xab(1,1) + xab(2,1) = xb(1)
xab(1,2) + xab(2,2) = xb(2)
![Page 52: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/52.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va is assigned label l1? x*a(1)
Probability that Va is assigned label l2? x*a(2)
![Page 53: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/53.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l1 and l1?
min{x*a(1), x*b(1)}
![Page 54: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/54.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l1 and l1?
min{x*ab(1,1)+x*ab(1,2), x*ab(1,1) + x*ab(2,1)}
x*ab(1,1) + min{x*ab(1,2), x*ab(2,1)}
![Page 55: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/55.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l1 and l2?
max{0,x*a(1) - x*b(1)}
![Page 56: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/56.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l1 and l2?
x*ab(1,2) - min{x*ab(1,2), x*ab(2,1)}
max{0,x*ab(1,2) - x*ab(2,1)}
![Page 57: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/57.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l2 and l1?
max{0,x*b(1) - x*a(1)}
![Page 58: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/58.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l2 and l1?
x*ab(2,1) - min{x*ab(1,2), x*ab(2,1)}
max{0,x*ab(2,1) - x*ab(1,2)}
![Page 59: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/59.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l2 and l2?
1-max{x*a(1), x*b(1)}
![Page 60: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/60.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l2 and l2?
min{x*a(2), x*b(2)}
![Page 61: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/61.jpg)
PostSimple Example - Rounding
x*a(1) + x*a(2) = 1x*a(1)0
x*b(1) + x*b(2) = 1x*b(1)0
Generate a uniform random number r (0,1]
Assign the label next to r
r
r
Probability that Va and Vb are assigned l2 and l2?
min{x*ab(2,2)+x*ab(1,2), x*ab(2,2) + x*ab(2,1)}
x*ab(2,2) + min{x*ab(1,2), x*ab(2,1)}
![Page 62: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/62.jpg)
PostSimple Example - Move
θa(ya) + θb(yb)miny + d(ya,yb)
ya ,yb {1,2}∈
If d is submodular, solve using graph cuts
Otherwise
![Page 63: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/63.jpg)
PostSimple Example - Move
θa(ya) + θb(yb)miny + d’(ya,yb)
ya ,yb {0,1}∈
If d is submodular, solve using graph cuts
Otherwise use submodular overestimation d’
Estimate d’ by minimizing distortion
![Page 64: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/64.jpg)
PostSimple Example - Move
bmind'
d’(1,1) ≤ b d(1,1)s.t. d’(1,2) ≤ b d(1,2)
d’(2,1) ≤ b d(2,1) d’(2,2) ≤ b d(2,2)
d(1,1) ≤ d’(1,1) d(1,2) ≤ d’(1,2)
d(2,1) ≤ d’(2,1) d(2,2) ≤ d’(2,2)
d’(1,1) + d’(2,2) ≤ d’(2,1) + d’(2,2)
Dual LP provides worst-case rounding example
LP in the variables d’(i,k)
![Page 65: Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris](https://reader030.vdocuments.site/reader030/viewer/2022032612/56649ec55503460f94bcfd2d/html5/thumbnails/65.jpg)
PostSimple Example - Moved(1,1)β(1,1)+d(1,2)β(1,2)+d(2,1)β(2,1)+d(2,2)β(2,2)minα,β,γ≥0
s.t. d(1,1)α(1,1)+d(1,2)α(1,2)+d(2,1)α(2,1)+d(2,2)α(2,2) = 1
β(1,1) = α(1,1) + γ
β(1,2) = α(1,2) - γ
β(2,1) = α(2,1) - γ
β(2,2) = α(2,2) + γ
Set xab*(i,k) = α(i,k) Set γ = min{xab*(1,2), xab*(2,1)}