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Pushmeet Kohli
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E(X) E: {0,1} n → R 0 → fg 1 → bg Image (D) n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al. ‘04] [Rother, Kolmogorov and Blake `04]
![E(X) E: {0,1} n → R 0 → fg 1 → bg Image (D) n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al.](http://images.slideplayer.com/28/9393539/slides/slide_3.jpg "‘04] [Rother, Kolmogorov and Blake `04].")
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Unary Cost (c i ) Dark (negative) Bright (positive) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al. ‘04] [Rother, Kolmogorov and Blake `04] ∑ c i x i Pixel Colour E(X) =
![Unary Cost (c i ) Dark (negative) Bright (positive) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al.](http://images.slideplayer.com/28/9393539/slides/slide_4.jpg "‘04] [Rother, Kolmogorov and Blake `04] ∑ c i x i Pixel Colour E(X) =.")
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Unary Cost (c i ) Dark (negative) Bright (positive) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al. ‘04] [Rother, Kolmogorov and Blake `04] x * = arg min E(x) E(X) = ∑ c i x i Pixel Colour
![Unary Cost (c i ) Dark (negative) Bright (positive) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al.](http://images.slideplayer.com/28/9393539/slides/slide_5.jpg "‘04] [Rother, Kolmogorov and Blake `04] x * = arg min E(x) E(X) = ∑ c i x i Pixel Colour.")
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Discontinuity Cost (d ij ) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al. ‘04] [Rother, Kolmogorov and Blake `04] E(X) = + ∑ d ij |x i -x j | Smoothness Prior ∑ c i x i Pixel Colour
![Discontinuity Cost (d ij ) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al.](http://images.slideplayer.com/28/9393539/slides/slide_6.jpg "‘04] [Rother, Kolmogorov and Blake `04] E(X) = + ∑ d ij |x i -x j | Smoothness Prior ∑ c i x i Pixel Colour.")
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Discontinuity Cost (d ij ) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al. ‘04] [Rother, Kolmogorov and Blake `04] E(X) = + ∑ d ij x i (1-x j ) + d ij x j (1-x i ) Smoothness Prior ∑ c i x i Pixel Colour
![Discontinuity Cost (d ij ) E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al.](http://images.slideplayer.com/28/9393539/slides/slide_7.jpg "‘04] [Rother, Kolmogorov and Blake `04] E(X) = + ∑ d ij x i (1-x j ) + d ij x j (1-x i ) Smoothness Prior ∑ c i x i Pixel Colour.")
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E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al. ‘04] [Rother, Kolmogorov and Blake `04] E(X) = ∑ c i x i Pixel Colour + ∑ d ij x i (1-x j ) + d ij x j (1-x i ) Smoothness Prior Old Solution x * = arg min E(x)
![E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al.](http://images.slideplayer.com/28/9393539/slides/slide_8.jpg "‘04] [Rother, Kolmogorov and Blake `04] E(X) = ∑ c i x i Pixel Colour + ∑ d ij x i (1-x j ) + d ij x j (1-x i ) Smoothness Prior Old Solution x * = arg min E(x).")
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E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al. ‘04] [Rother, Kolmogorov and Blake `04] E(X) = ∑ c i x i Pixel Colour + ∑ d ij x i (1-x j ) + d ij x j (1-x i ) Smoothness Prior x * = arg min E(x)
![E: {0,1} n → R 0 → fg 1 → bg n = number of pixels [Boykov and Jolly ‘ 01] [Blake et al.](http://images.slideplayer.com/28/9393539/slides/slide_9.jpg "‘04] [Rother, Kolmogorov and Blake `04] E(X) = ∑ c i x i Pixel Colour + ∑ d ij x i (1-x j ) + d ij x j (1-x i ) Smoothness Prior x * = arg min E(x).")
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E(x) = ∑ f i (x i ) + ∑ g ij (x i,x j ) + ∑ h c (x c ) i ij c UnaryPairwiseHigher Order How to minimize E(x)? x takes from a label set L = {l 1, l 2,.., l k }
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Space of Problems n = Number of Variables Segmentation Energy CSP MAXCUT NP-Hard Tractability Properties
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Space of Problems n = Number of Variables Segmentation Energy CSP Tree Structured MAXCUT NP-Hard Tractability Properties Structural Tractability
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Space of Problems n = Number of Variables Segmentation Energy Submodular Functions CSP Tree Structured Pair-wise O(n 3 ) MAXCUT O(n 6 ) NP-Hard Tractability Properties Language or Form Tractability g ij (x i,x j ) Constraints on the terms of your energy functions
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Example: n = 2, A = [1,0], B = [0,1] f([1,0]) + f([0,1]) f([1,1]) + f([0,0]) Property : Sum of submodular functions is submodular E(x) = ∑ c i x i + ∑ d ij |x i -x j | ii,j Binary Image Segmentation Energy is submodular for all A,B {0,1} n f(A) + f(B) f(A ˅ B) + f(A ˄ B) (AND)(OR) Pseudo-boolean function f {0,1} n ℝ is submodular if
![Example: n = 2, A = [1,0], B = [0,1] f([1,0]) + f([0,1]) f([1,1]) + f([0,0]) Property : Sum of submodular functions is submodular E(x) = ∑ c i x i + ∑ d ij |x i -x j | ii,j Binary Image Segmentation Energy is submodular for all A,B {0,1} n f(A) + f(B) f(A ˅ B) + f(A ˄ B) (AND)(OR) Pseudo-boolean function f {0,1} n ℝ is submodular if](http://images.slideplayer.com/28/9393539/slides/slide_16.jpg "Example: n = 2, A = [1,0], B = [0,1] f([1,0]) + f([0,1]) f([1,1]) + f([0,0]) Property : Sum of submodular functions is submodular E(x) = ∑ c i x i + ∑ d ij |x i -x j | ii,j Binary Image Segmentation Energy is submodular for all A,B {0,1} n f(A) + f(B) f(A ˅ B) + f(A ˄ B) (AND)(OR) Pseudo-boolean function f {0,1} n ℝ is submodular if")
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Discrete Analogues of Concave Functions [Lovasz, ’83] Widely applied in Operations Research Applications in Machine Learning MAP Inference in Markov Random Fields Clustering [Narasimhan, Jojic, & Bilmes, NIPS 2005] Structure Learning [Narasimhan & Bilmes, NIPS 2006] Maximizing the spread of influence through a social network [Kempe, Kleinberg & Tardos, KDD 2003]
![ Discrete Analogues of Concave Functions [Lovasz, ’83] Widely applied in Operations Research Applications in Machine Learning MAP Inference in Markov Random Fields Clustering [Narasimhan, Jojic, & Bilmes, NIPS 2005] Structure Learning [Narasimhan & Bilmes, NIPS 2006] Maximizing the spread of influence through a social network [Kempe, Kleinberg & Tardos, KDD 2003]](http://images.slideplayer.com/28/9393539/slides/slide_17.jpg " Discrete Analogues of Concave Functions [Lovasz, ’83] Widely applied in Operations Research Applications in Machine Learning MAP Inference in Markov Random Fields Clustering [Narasimhan, Jojic, & Bilmes, NIPS 2005] Structure Learning [Narasimhan & Bilmes, NIPS 2006] Maximizing the spread of influence through a social network [Kempe, Kleinberg & Tardos, KDD 2003]")
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Polynomial time algorithms Ellipsoid Algorithm: [Grotschel, Lovasz & Schrijver ‘81] First strongly polynomial algorithm: [Iwata et al. ’00] [A. Schrijver ’00] Current Best: O(n 5 Q + n 6 ) [Q is function evaluation time] [Orlin ‘07] Symmetric functions: E(x) = E(1-x) Can be minimized in O(n 3 ) Minimizing Pairwise submodular functions Can be transformed to st-mincut/max-flow [Hammer, 1965] Very low empirical running time ~ O(n) E(X) = ∑ f i (x i ) + ∑ g ij (x i,x j ) iij
![ Polynomial time algorithms Ellipsoid Algorithm: [Grotschel, Lovasz & Schrijver ‘81] First strongly polynomial algorithm: [Iwata et al.](http://images.slideplayer.com/28/9393539/slides/slide_18.jpg "’00] [A. Schrijver ’00] Current Best: O(n 5 Q + n 6 ) [Q is function evaluation time] [Orlin ‘07] Symmetric functions: E(x) = E(1-x) Can be minimized in O(n 3 ) Minimizing Pairwise submodular functions Can be transformed to st-mincut/max-flow [Hammer, 1965] Very low empirical running time ~ O(n) E(X) = ∑ f i (x i ) + ∑ g ij (x i,x j ) iij.")
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Source Sink v1v1 v2v2 2 5 9 4 1 2 Graph (V, E, C) Vertices V = {v 1, v 2... v n } Edges E = {(v 1, v 2 )....} Costs C = {c (1, 2)....}
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Source Sink v1v1 v2v2 2 5 9 4 1 2 What is a st-cut?
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Source Sink v1v1 v2v2 2 5 9 4 1 2 What is a st-cut? An st-cut (S,T) divides the nodes between source and sink. What is the cost of a st-cut? Sum of cost of all edges going from S to T 5 + 1 + 9 = 15
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What is a st-cut? An st-cut (S,T) divides the nodes between source and sink. What is the cost of a st-cut? Sum of cost of all edges going from S to T What is the st-mincut? st-cut with the minimum cost Source Sink v1v1 v2v2 2 5 9 4 1 2 2 + 2 + 4 = 8
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Construct a graph such that: 1.Any st-cut corresponds to an assignment of x 2.The cost of the cut is equal to the energy of x : E(x) Solution T S st-mincut E(x) [Hammer, 1965] [Kolmogorov and Zabih, 2002
![Construct a graph such that: 1.Any st-cut corresponds to an assignment of x 2.The cost of the cut is equal to the energy of x : E(x) Solution T S st-mincut E(x) [Hammer, 1965] [Kolmogorov and Zabih, 2002](http://images.slideplayer.com/28/9393539/slides/slide_23.jpg "Construct a graph such that: 1.Any st-cut corresponds to an assignment of x 2.The cost of the cut is equal to the energy of x : E(x) Solution T S st-mincut E(x) [Hammer, 1965] [Kolmogorov and Zabih, 2002")
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E(x) = ∑ θ i (x i ) + ∑ θ ij (x i,x j ) i,ji θ ij (0,1) + θ ij (1,0) θ ij (0,0) + θ ij (1,1)For all ij E(x) = ∑ c i x i + ∑ c ij x i (1-x j ) c ij ≥0 i,ji Equivalent (transformable)
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Sink (1) Source (0) a1a1 a2a2 E(a 1,a 2 )
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Sink (1) Source (0) a1a1 a2a2 E(a 1,a 2 ) = 2a 1 2
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 2 5 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 2 5 9 4 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 2 5 9 4 2 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 2 5 9 4 2 1 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 2 5 9 4 2 1 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 2 5 9 4 2 1 a 1 = 1 a 2 = 1 E (1,1) = 11 Cost of cut = 11 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 2 5 9 4 2 1 Sink (1) Source (0) a 1 = 1 a 2 = 0 E (1,0) = 8 st-mincut cost = 8
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Source Sink v1v1 v2v2 2 5 9 4 2 1 Solve the dual maximum flow problem Compute the maximum flow between Source and Sink s.t. Edges: Flow < Capacity Nodes: Flow in = Flow out Assuming non-negative capacity In every network, the maximum flow equals the cost of the st-mincut Min-cut\Max-flow Theorem
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Augmenting Path Based Algorithms Source Sink v1v1 v2v2 2 5 9 4 2 1 Flow = 0
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Augmenting Path Based Algorithms 1.Find path from source to sink with positive capacity Source Sink v1v1 v2v2 2 5 9 4 2 1 Flow = 0
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Augmenting Path Based Algorithms 1.Find path from source to sink with positive capacity 2.Push maximum possible flow through this path Source Sink v1v1 v2v2 2-2 5-2 9 4 2 1 Flow = 0 + 2
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Source Sink v1v1 v2v2 0 3 9 4 2 1 Augmenting Path Based Algorithms 1.Find path from source to sink with positive capacity 2.Push maximum possible flow through this path Flow = 2
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Source Sink v1v1 v2v2 0 3 9 4 2 1 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 Flow = 2
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Source Sink v1v1 v2v2 0 3 9 4 2 1 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 Flow = 2
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Source Sink v1v1 v2v2 0 3 5 0 2 1 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 Flow = 2 + 4
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Source Sink v1v1 v2v2 0 3 5 0 2 1 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 Flow = 6
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Source Sink v1v1 v2v2 0 3 5 0 2 1 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 Flow = 6
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Source Sink v1v1 v2v2 0 1 3 0 2-2 1+2 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 Flow = 6 + 2
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Source Sink v1v1 v2v2 0 2 4 0 3 0 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 Flow = 8
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Source Sink v1v1 v2v2 0 2 4 0 3 0 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 Flow = 8
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 2 5 9 4 2 1 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 2 5 9 4 2 1 Sink (1) Source (0) 2a 1 + 5ā 1 = 2(a 1 +ā 1 ) + 3ā 1 = 2 + 3ā 1
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Sink (1) Source (0) a1a1 a2a2 E(a 1,a 2 ) = 2 + 3ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 0 3 9 4 2 1 2a 1 + 5ā 1 = 2(a 1 +ā 1 ) + 3ā 1 = 2 + 3ā 1
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a1a1 a2a2 E(a 1,a 2 ) = 2 + 3ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 0 3 9 4 2 1 9a 2 + 4ā 2 = 4(a 2 +ā 2 ) + 5ā 2 = 4 + 5ā 2 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 2 + 3ā 1 + 5a 2 + 4 + 2a 1 ā 2 + ā 1 a 2 0 3 5 0 2 1 9a 2 + 4ā 2 = 4(a 2 +ā 2 ) + 5ā 2 = 4 + 5ā 2 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 6 + 3ā 1 + 5a 2 + 2a 1 ā 2 + ā 1 a 2 0 3 5 0 2 1 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 6 + 3ā 1 + 5a 2 + 2a 1 ā 2 + ā 1 a 2 0 3 5 0 2 1 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 6 + 3ā 1 + 5a 2 + 2a 1 ā 2 + ā 1 a 2 0 3 5 0 2 1 3ā 1 + 5a 2 + 2a 1 ā 2 = 2(ā 1 +a 2 +a 1 ā 2 ) +ā 1 +3a 2 = 2(1+ā 1 a 2 ) +ā 1 +3a 2 F1 = ā 1 +a 2 +a 1 ā 2 F2 = 1+ā 1 a 2 a1a1 a2a2 F1F2 0011 0122 1011 1111 Sink (1) Source (0)
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a1a1 a2a2 E(a 1,a 2 ) = 8 + ā 1 + 3a 2 + 3ā 1 a 2 0 1 3 0 0 3 3ā 1 + 5a 2 + 2a 1 ā 2 = 2(ā 1 +a 2 +a 1 ā 2 ) +ā 1 +3a 2 = 2(1+ā 1 a 2 ) +ā 1 +3a 2 a1a1 a2a2 F1F2 0011 0122 1011 1111 F1 = ā 1 +a 2 +a 1 ā 2 F2 = 1+ā 1 a 2 Sink (1) Source (0)
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a1a1 a2a2 0 1 3 0 0 3 E(a 1,a 2 ) = 8 + ā 1 + 3a 2 + 3ā 1 a 2 No more augmenting paths possible Sink (1) Source (0)
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a1a1 a2a2 0 1 3 0 0 3 E(a 1,a 2 ) = 8 + ā 1 + 3a 2 + 3ā 1 a 2 Total Flow Residual Graph (positive coefficients) bound on the optimal solution Tight Bound --> Inference of the optimal solution becomes trivial Sink (1) Source (0)
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a1a1 a2a2 0 1 3 0 0 3 E(a 1,a 2 ) = 8 + ā 1 + 3a 2 + 3ā 1 a 2 a 1 = 1 a 2 = 0 E (1,0) = 8 st-mincut cost = 8 Total Flow bound on the energy of the optimal solution Residual Graph (positive coefficients) Tight Bound --> Inference of the optimal solution becomes trivial Sink (1) Source (0)
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[Slide credit: Andrew Goldberg] Augmenting Path and Push-Relabel n: # nodes m: # edges U: maximum edge weight Algorithms assume non- negative edge weights
![[Slide credit: Andrew Goldberg] Augmenting Path and Push-Relabel n: # nodes m: # edges U: maximum edge weight Algorithms assume non- negative edge weights](http://images.slideplayer.com/28/9393539/slides/slide_59.jpg "[Slide credit: Andrew Goldberg] Augmenting Path and Push-Relabel n: # nodes m: # edges U: maximum edge weight Algorithms assume non- negative edge weights")
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[Slide credit: Andrew Goldberg] n: # nodes m: # edges U: maximum edge weight Algorithms assume non- negative edge weights Augmenting Path and Push-Relabel
![[Slide credit: Andrew Goldberg] n: # nodes m: # edges U: maximum edge weight Algorithms assume non- negative edge weights Augmenting Path and Push-Relabel](http://images.slideplayer.com/28/9393539/slides/slide_60.jpg "[Slide credit: Andrew Goldberg] n: # nodes m: # edges U: maximum edge weight Algorithms assume non- negative edge weights Augmenting Path and Push-Relabel")
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a1a1 a2a2 1000 1 Sink Source 1000 0 Ford Fulkerson: Choose any augmenting path
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a1a1 a2a2 1000 1 Sink Source 1000 0 Good Augmenting Paths Ford Fulkerson: Choose any augmenting path
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a1a1 a2a2 1000 1 Sink Source 1000 0 Bad Augmenting Path Ford Fulkerson: Choose any augmenting path
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a1a1 a2a2 999 0 Sink Source 1000 999 1 Ford Fulkerson: Choose any augmenting path
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a1a1 a2a2 999 0 Sink Source 1000 999 1 Ford Fulkerson: Choose any augmenting path n: # nodes m: # edges We will have to perform 2000 augmentations! Worst case complexity: O (m x Total_Flow) (Pseudo-polynomial bound: depends on flow)
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Dinic: Choose shortest augmenting path n: # nodes m: # edges Worst case Complexity: O (m n 2 ) a1a1 a2a2 1000 1 Sink Source 1000 0
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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 web http://www.adastral.ucl.ac.uk/~vladkolm/software.html
![ 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 web](http://images.slideplayer.com/28/9393539/slides/slide_67.jpg " 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 web")
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E(x) = ∑ c i x i + ∑ d ij |x i -x j | ii,j Global Minimum (x * ) x x * = arg min E(x) How to minimize E(x)? E: {0,1} n → R 0 → fg 1 → bg n = number of pixels
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Sink (1) Source (0) Graph *g; For all pixels p /* Add a node to the graph */ nodeID(p) = g->add_node(); /* Set cost of terminal edges */ set_weights(nodeID(p), fgCost(p), bgCost(p)); end for all adjacent pixels p,q add_weights(nodeID(p), nodeID(q), cost(p,q)); end g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1)
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a1a1 a2a2 fgCost(a 1 ) Sink (1) Source (0) fgCost(a 2 ) bgCost(a 1 ) bgCost(a 2 ) Graph *g; For all pixels p /* Add a node to the graph */ nodeID(p) = g->add_node(); /* Set cost of terminal edges */ set_weights(nodeID(p), fgCost(p), bgCost(p)); end for all adjacent pixels p,q add_weights(nodeID(p), nodeID(q), cost(p,q)); end g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1)
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a1a1 a2a2 fgCost(a 1 ) Sink (1) Source (0) fgCost(a 2 ) bgCost(a 1 ) bgCost(a 2 ) cost(p,q) Graph *g; For all pixels p /* Add a node to the graph */ nodeID(p) = g->add_node(); /* Set cost of terminal edges */ set_weights(nodeID(p), fgCost(p), bgCost(p)); end for all adjacent pixels p,q add_weights(nodeID(p), nodeID(q), cost(p,q)); end g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1)
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Graph *g; For all pixels p /* Add a node to the graph */ nodeID(p) = g->add_node(); /* Set cost of terminal edges */ set_weights(nodeID(p), fgCost(p), bgCost(p)); end for all adjacent pixels p,q add_weights(nodeID(p), nodeID(q), cost(p,q)); end g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1) a1a1 a2a2 fgCost(a 1 ) Sink (1) Source (0) fgCost(a 2 ) bgCost(a 1 ) bgCost(a 2 ) cost(p,q) a 1 = bg a 2 = fg
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Mixed (Real-Integer) Problems Multi-label Problems Ordered Labels ▪ Stereo (depth labels) Unordered Labels ▪ Object segmentation ( ‘car’, `road’, `person’) Higher Order Energy Functions
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Mixed (Real-Integer) Problems Multi-label Problems Ordered Labels ▪ Stereo (depth labels) Unordered Labels ▪ Object segmentation ( ‘car’, `road’, `person’) Higher Order Energy Functions
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x – binary image segmentation (x i ∊ {0,1}) ω – non-local parameter (lives in some large set Ω) constant unary potentials pairwise potentials E(x, ω ) = C( ω ) + ∑ θ i ( ω, x i ) + ∑ θ ij ( ω, x i,x j ) i,ji ≥ 0 ω Template Position Scale Orientation We have seen several of them in the intro...
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x – binary image segmentation (x i ∊ {0,1}) ω – non-local parameter (lives in some large set Ω) constant unary potentials pairwise potentials E(x, ω ) = C( ω ) + ∑ θ i ( ω, x i ) + ∑ θ ij ( ω, x i,x j ) i,ji ≥ 0 {x *, ω*} = arg min E(x,ω) Standard “graph cut” energy if ω is fixed x,ω [Kohli et al, 06,08] [Lempitsky et al, 08]
![x – binary image segmentation (x i ∊ {0,1}) ω – non-local parameter (lives in some large set Ω) constant unary potentials pairwise potentials E(x, ω ) = C( ω ) + ∑ θ i ( ω, x i ) + ∑ θ ij ( ω, x i,x j ) i,ji ≥ 0 {x *, ω*} = arg min E(x,ω) Standard graph cut energy if ω is fixed x,ω [Kohli et al, 06,08] [Lempitsky et al, 08]](http://images.slideplayer.com/28/9393539/slides/slide_77.jpg "x – binary image segmentation (x i ∊ {0,1}) ω – non-local parameter (lives in some large set Ω) constant unary potentials pairwise potentials E(x, ω ) = C( ω ) + ∑ θ i ( ω, x i ) + ∑ θ ij ( ω, x i,x j ) i,ji ≥ 0 {x *, ω*} = arg min E(x,ω) Standard graph cut energy if ω is fixed x,ω [Kohli et al, 06,08] [Lempitsky et al, 08]")
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Local Method: Gradient Descent over ω ω* ω * = arg min min E (x, ω ) xω Submodular [Kohli et al, 06,08]
![Local Method: Gradient Descent over ω ω* ω * = arg min min E (x, ω ) xω Submodular [Kohli et al, 06,08]](http://images.slideplayer.com/28/9393539/slides/slide_78.jpg "Local Method: Gradient Descent over ω ω* ω * = arg min min E (x, ω ) xω Submodular [Kohli et al, 06,08]")
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Local Method: Gradient Descent over ω ω * = arg min min E (x, ω ) xω Submodular Dynamic Graph Cuts 15- 20 time speedup! E (x,ω 1 ) E (x,ω 2 ) Similar Energy Functions [Kohli et al, 06,08]
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Global Method: Branch and Mincut [Lempitsky et al, 08] Produces the global optimal solution Exhaustively explores all ω in Ω in the worst case 30,000,000 shapes
![Global Method: Branch and Mincut [Lempitsky et al, 08] Produces the global optimal solution Exhaustively explores all ω in Ω in the worst case 30,000,000 shapes](http://images.slideplayer.com/28/9393539/slides/slide_80.jpg "Global Method: Branch and Mincut [Lempitsky et al, 08] Produces the global optimal solution Exhaustively explores all ω in Ω in the worst case 30,000,000 shapes")
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Mixed (Real-Integer) Problems Multi-label Problems Ordered Labels ▪ Stereo (depth labels) Unordered Labels ▪ Object segmentation ( ‘car’, `road’, `person’) Higher Order Energy Functions
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Exact Transformation to QPBF Move making algorithms E(y) = ∑ f i (y i ) + ∑ g ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } Min y [Roy and Cox ’98] [Ishikawa ’03] [Schlesinger & Flach ’06] [Ramalingam, Alahari, Kohli, and Torr ’08]
![ Exact Transformation to QPBF Move making algorithms E(y) = ∑ f i (y i ) + ∑ g ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } Min y [Roy and Cox ’98] [Ishikawa ’03] [Schlesinger & Flach ’06] [Ramalingam, Alahari, Kohli, and Torr ’08]](http://images.slideplayer.com/28/9393539/slides/slide_82.jpg " Exact Transformation to QPBF Move making algorithms E(y) = ∑ f i (y i ) + ∑ g ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } Min y [Roy and Cox ’98] [Ishikawa ’03] [Schlesinger & Flach ’06] [Ramalingam, Alahari, Kohli, and Torr ’08]")
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So what is the problem? E b (x 1,x 2,..., x m )E m (y 1,y 2,..., y n ) Multi-label ProblemBinary label Problem y i L = {l 1, l 2, …, l k }x i L = {0,1} such that: Let Y and X be the set of feasible solutions, then 1. One-One encoding function T:X->Y 2. arg min E m (y) = T(arg min E b (x))
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Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2
![Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2](http://images.slideplayer.com/28/9393539/slides/slide_84.jpg "Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2")
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Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2 Ishikawa’s result: E(y) = ∑ θ i (y i ) + ∑ θ ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } θ ij (y i,y j ) = g(|y i -y j |) Convex Function g(|y i -y j |) |y i -y j |
![Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2 Ishikawa’s result: E(y) = ∑ θ i (y i ) + ∑ θ ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } θ ij (y i,y j ) = g(|y i -y j |) Convex Function g(|y i -y j |) |y i -y j |](http://images.slideplayer.com/28/9393539/slides/slide_85.jpg "Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2 Ishikawa’s result: E(y) = ∑ θ i (y i ) + ∑ θ ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } θ ij (y i,y j ) = g(|y i -y j |) Convex Function g(|y i -y j |) |y i -y j |")
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Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2 Schlesinger & Flach ’06: E(y) = ∑ θ i (y i ) + ∑ θ ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } θ ij (l i+1,l j ) + θ ij (l i,l j+1 ) θ ij (l i,l j ) + θ ij (l i+1,l j+1 ) l i +1 lili l j +1 ljlj
![Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2 Schlesinger & Flach ’06: E(y) = ∑ θ i (y i ) + ∑ θ ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } θ ij (l i+1,l j ) + θ ij (l i,l j+1 ) θ ij (l i,l j ) + θ ij (l i+1,l j+1 ) l i +1 lili l j +1 ljlj](http://images.slideplayer.com/28/9393539/slides/slide_86.jpg "Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] # Nodes = n * k # Edges = m * k 2 Schlesinger & Flach ’06: E(y) = ∑ θ i (y i ) + ∑ θ ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } θ ij (l i+1,l j ) + θ ij (l i,l j+1 ) θ ij (l i,l j ) + θ ij (l i+1,l j+1 ) l i +1 lili l j +1 ljlj")
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ImageMAP Solution Scanline algorithm [Roy and Cox, 98]
![ImageMAP Solution Scanline algorithm [Roy and Cox, 98]](http://images.slideplayer.com/28/9393539/slides/slide_87.jpg "ImageMAP Solution Scanline algorithm [Roy and Cox, 98]")
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Applicability Cannot handle truncated costs (non-robust) Computational Cost Very high computational cost Problem size = |Variables| x |Labels| Gray level image denoising (1 Mpixel image) (~2.5 x 10 8 graph nodes) θ ij (y i,y j ) = g(|y i -y j |) |y i -y j | discontinuity preserving potentials Blake&Zisserman’83,87
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Unary Potentials Pair-wise Potentials Complexity Ishikawa Transformation [03] ArbitraryConvex and Symmetric T(nk, mk 2 ) Schlesinger Transformation [06] ArbitrarySubmodularT(nk, mk 2 ) Hochbaum [01] LinearConvex and Symmetric T(n, m) + n log k Hochbaum [01] ConvexConvex and Symmetric O(mn log n log nk) Other “less known” algorithms T(a,b) = complexity of maxflow with a nodes and b edges
![Unary Potentials Pair-wise Potentials Complexity Ishikawa Transformation [03] ArbitraryConvex and Symmetric T(nk, mk 2 ) Schlesinger Transformation [06] ArbitrarySubmodularT(nk, mk 2 ) Hochbaum [01] LinearConvex and Symmetric T(n, m) + n log k Hochbaum [01] ConvexConvex and Symmetric O(mn log n log nk) Other less known algorithms T(a,b) = complexity of maxflow with a nodes and b edges](http://images.slideplayer.com/28/9393539/slides/slide_89.jpg "Unary Potentials Pair-wise Potentials Complexity Ishikawa Transformation [03] ArbitraryConvex and Symmetric T(nk, mk 2 ) Schlesinger Transformation [06] ArbitrarySubmodularT(nk, mk 2 ) Hochbaum [01] LinearConvex and Symmetric T(n, m) + n log k Hochbaum [01] ConvexConvex and Symmetric O(mn log n log nk) Other less known algorithms T(a,b) = complexity of maxflow with a nodes and b edges")
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Exact Transformation to QPBF Move making algorithms E(y) = ∑ f i (y i ) + ∑ g ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } Min y [Boykov, Veksler and Zabih 2001] [Woodford, Fitzgibbon, Reid, Torr, 2008] [Lempitsky, Rother, Blake, 2008] [Veksler, 2008] [Kohli, Ladicky, Torr 2008]
![ Exact Transformation to QPBF Move making algorithms E(y) = ∑ f i (y i ) + ∑ g ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } Min y [Boykov, Veksler and Zabih 2001] [Woodford, Fitzgibbon, Reid, Torr, 2008] [Lempitsky, Rother, Blake, 2008] [Veksler, 2008] [Kohli, Ladicky, Torr 2008]](http://images.slideplayer.com/28/9393539/slides/slide_90.jpg " Exact Transformation to QPBF Move making algorithms E(y) = ∑ f i (y i ) + ∑ g ij (y i,y j ) i,ji y Labels L = {l 1, l 2, …, l k } Min y [Boykov, Veksler and Zabih 2001] [Woodford, Fitzgibbon, Reid, Torr, 2008] [Lempitsky, Rother, Blake, 2008] [Veksler, 2008] [Kohli, Ladicky, Torr 2008]")
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Solution Space Energy
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Search Neighbourhood Current Solution Optimal Move Solution Space Energy
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Search Neighbourhood Current Solution Optimal Move xcxc (t)(t) Key Property Move Space Bigger move space Solution Space Energy Better solutions Finding the optimal move hard
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Minimizing Pairwise Functions [Boykov Veksler and Zabih, PAMI 2001] Series of locally optimal moves Each move reduces energy Optimal move by minimizing submodular function Space of Solutions (x) : L n Move Space (t) : 2 n Search Neighbourhood Current Solution n Number of Variables L Number of Labels Kohli et al. ‘07, ‘08, ‘09 Extend to minimize Higher order Functions
![Minimizing Pairwise Functions [Boykov Veksler and Zabih, PAMI 2001] Series of locally optimal moves Each move reduces energy Optimal move by minimizing submodular function Space of Solutions (x) : L n Move Space (t) : 2 n Search Neighbourhood Current Solution n Number of Variables L Number of Labels Kohli et al.](http://images.slideplayer.com/28/9393539/slides/slide_94.jpg "‘07, ‘08, ‘09 Extend to minimize Higher order Functions.")
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Minimize over move variables t x = t x 1 + (1- t ) x 2 New solution Current Solution Second solution E m ( t ) = E( t x 1 + (1- t ) x 2 ) For certain x 1 and x 2, the move energy is sub-modular QPBF [Boykov, Veksler and Zabih 2001]
![Minimize over move variables t x = t x 1 + (1- t ) x 2 New solution Current Solution Second solution E m ( t ) = E( t x 1 + (1- t ) x 2 ) For certain x 1 and x 2, the move energy is sub-modular QPBF [Boykov, Veksler and Zabih 2001]](http://images.slideplayer.com/28/9393539/slides/slide_95.jpg "Minimize over move variables t x = t x 1 + (1- t ) x 2 New solution Current Solution Second solution E m ( t ) = E( t x 1 + (1- t ) x 2 ) For certain x 1 and x 2, the move energy is sub-modular QPBF [Boykov, Veksler and Zabih 2001]")
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Variables labeled α, β can swap their labels [Boykov, Veksler and Zabih 2001]
![Variables labeled α, β can swap their labels [Boykov, Veksler and Zabih 2001]](http://images.slideplayer.com/28/9393539/slides/slide_96.jpg "Variables labeled α, β can swap their labels [Boykov, Veksler and Zabih 2001]")
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Sky House Tree Ground Swap Sky, House Variables labeled α, β can swap their labels [Boykov, Veksler and Zabih 2001]
![Sky House Tree Ground Swap Sky, House Variables labeled α, β can swap their labels [Boykov, Veksler and Zabih 2001]](http://images.slideplayer.com/28/9393539/slides/slide_97.jpg "Sky House Tree Ground Swap Sky, House Variables labeled α, β can swap their labels [Boykov, Veksler and Zabih 2001]")
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Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Semi-metric θ ij (l a,l b ) ≥ 0 θ ij (l a,l b ) = 0 a = b Examples: Potts model, Truncated Convex [Boykov, Veksler and Zabih 2001] Variables labeled α, β can swap their labels
![ Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Semi-metric θ ij (l a,l b ) ≥ 0 θ ij (l a,l b ) = 0 a = b Examples: Potts model, Truncated Convex [Boykov, Veksler and Zabih 2001] Variables labeled α, β can swap their labels](http://images.slideplayer.com/28/9393539/slides/slide_98.jpg " Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Semi-metric θ ij (l a,l b ) ≥ 0 θ ij (l a,l b ) = 0 a = b Examples: Potts model, Truncated Convex [Boykov, Veksler and Zabih 2001] Variables labeled α, β can swap their labels")
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[Boykov, Veksler, Zabih] Variables take label or retain current label [Boykov, Veksler and Zabih 2001]
![[Boykov, Veksler, Zabih] Variables take label or retain current label [Boykov, Veksler and Zabih 2001]](http://images.slideplayer.com/28/9393539/slides/slide_99.jpg "[Boykov, Veksler, Zabih] Variables take label or retain current label [Boykov, Veksler and Zabih 2001]")
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Sky House Tree Ground Initialize with TreeStatus:Expand GroundExpand HouseExpand Sky [Boykov, Veksler, Zabih] [Boykov, Veksler and Zabih 2001] Variables take label or retain current label
![Sky House Tree Ground Initialize with TreeStatus:Expand GroundExpand HouseExpand Sky [Boykov, Veksler, Zabih] [Boykov, Veksler and Zabih 2001] Variables take label or retain current label](http://images.slideplayer.com/28/9393539/slides/slide_100.jpg "Sky House Tree Ground Initialize with TreeStatus:Expand GroundExpand HouseExpand Sky [Boykov, Veksler, Zabih] [Boykov, Veksler and Zabih 2001] Variables take label or retain current label")
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Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Metric [Boykov, Veksler, Zabih] θ ij (l a,l b ) + θ ij (l b,l c ) ≥ θ ij (l a,l c ) Semi metric + Triangle Inequality Examples: Potts model, Truncated linear Cannot solve truncated quadratic Variables take label or retain current label [Boykov, Veksler and Zabih 2001]
![ Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Metric [Boykov, Veksler, Zabih] θ ij (l a,l b ) + θ ij (l b,l c ) ≥ θ ij (l a,l c ) Semi metric + Triangle Inequality Examples: Potts model, Truncated linear Cannot solve truncated quadratic Variables take label or retain current label [Boykov, Veksler and Zabih 2001]](http://images.slideplayer.com/28/9393539/slides/slide_101.jpg " Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Metric [Boykov, Veksler, Zabih] θ ij (l a,l b ) + θ ij (l b,l c ) ≥ θ ij (l a,l c ) Semi metric + Triangle Inequality Examples: Potts model, Truncated linear Cannot solve truncated quadratic Variables take label or retain current label [Boykov, Veksler and Zabih 2001]")
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Expansion and Swap can be derived as a primal dual scheme Get solution of the dual problem which is a lower bound on the energy of solution Weak guarantee on the solution [Komodakis et al 05, 07] E(x) < 2(d max /d min ) E(x*) d max d min θ ij (l i,l j ) = g(|l i -l j |) |y i -y j |
![ Expansion and Swap can be derived as a primal dual scheme Get solution of the dual problem which is a lower bound on the energy of solution Weak guarantee on the solution [Komodakis et al 05, 07] E(x) < 2(d max /d min ) E(x*) d max d min θ ij (l i,l j ) = g(|l i -l j |) |y i -y j |](http://images.slideplayer.com/28/9393539/slides/slide_102.jpg " Expansion and Swap can be derived as a primal dual scheme Get solution of the dual problem which is a lower bound on the energy of solution Weak guarantee on the solution [Komodakis et al 05, 07] E(x) < 2(d max /d min ) E(x*) d max d min θ ij (l i,l j ) = g(|l i -l j |) |y i -y j |")
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Move TypeFirst Solution Second Solution Guarantee ExpansionOld solutionAll alphaMetric FusionAny solution Minimize over move variables t x = t x 1 + (1-t) x 2 New solution First solution Second solution Move functions can be non-submodular!!
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x = t x 1 + (1-t) x 2 x 1, x 2 can be continuous F x1x1 x2x2 x Optical Flow Example Final Solution Solution from Method 1 Solution from Method 2 [Woodford, Fitzgibbon, Reid, Torr, 2008] [Lempitsky, Rother, Blake, 2008]
![x = t x 1 + (1-t) x 2 x 1, x 2 can be continuous F x1x1 x2x2 x Optical Flow Example Final Solution Solution from Method 1 Solution from Method 2 [Woodford, Fitzgibbon, Reid, Torr, 2008] [Lempitsky, Rother, Blake, 2008]](http://images.slideplayer.com/28/9393539/slides/slide_104.jpg "x = t x 1 + (1-t) x 2 x 1, x 2 can be continuous F x1x1 x2x2 x Optical Flow Example Final Solution Solution from Method 1 Solution from Method 2 [Woodford, Fitzgibbon, Reid, Torr, 2008] [Lempitsky, Rother, Blake, 2008]")
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Move variables can be multi-label Optimal move found out by using the Ishikawa Transform Useful for minimizing energies with truncated convex pairwise potentials θ ij (y i,y j ) = min(|y i -y j | 2,T) |y i -y j | θ ij (y i,y j ) T x = (t == 1) x 1 + (t == 2) x 2 +… +(t == k) x k [Kumar and Torr, 2008] [Veksler, 2008]
![ Move variables can be multi-label Optimal move found out by using the Ishikawa Transform Useful for minimizing energies with truncated convex pairwise potentials θ ij (y i,y j ) = min(|y i -y j | 2,T) |y i -y j | θ ij (y i,y j ) T x = (t == 1) x 1 + (t == 2) x 2 +… +(t == k) x k [Kumar and Torr, 2008] [Veksler, 2008]](http://images.slideplayer.com/28/9393539/slides/slide_105.jpg " Move variables can be multi-label Optimal move found out by using the Ishikawa Transform Useful for minimizing energies with truncated convex pairwise potentials θ ij (y i,y j ) = min(|y i -y j | 2,T) |y i -y j | θ ij (y i,y j ) T x = (t == 1) x 1 + (t == 2) x 2 +… +(t == k) x k [Kumar and Torr, 2008] [Veksler, 2008]")
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[Veksler, 2008] Image Noisy Image Range Moves Expansion Move
![[Veksler, 2008] Image Noisy Image Range Moves Expansion Move](http://images.slideplayer.com/28/9393539/slides/slide_106.jpg "[Veksler, 2008] Image Noisy Image Range Moves Expansion Move")
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