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**MAP Estimation Algorithms in**

Computer Vision - Part II M. Pawan Kumar, University of Oxford Pushmeet Kohli, Microsoft Research

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**Example: Image Segmentation**

E: {0,1}n → R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑ cij xi(1-xj) i i,j n = number of pixels Move to front Image (D)

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**Example: Image Segmentation**

E: {0,1}n → R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑ cij xi(1-xj) i i,j n = number of pixels Unary Cost (ci) Dark (negative) Bright (positive)

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**Example: Image Segmentation**

E: {0,1}n → R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑ cij xi(1-xj) i i,j n = number of pixels Discontinuity Cost (cij)

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**Example: Image Segmentation**

E: {0,1}n → R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑ cij xi(1-xj) i i,j n = number of pixels x* = arg min E(x) x How to minimize E(x)? Global Minimum (x*)

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**Outline of the Tutorial**

The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

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**Outline of the Tutorial**

The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

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**The st-Mincut Problem Graph (V, E, C) 2 9 1 2 5 4 Source**

Vertices V = {v1, v2 ... vn} Edges E = {(v1, v2) ....} Costs C = {c(1, 2) ....} 2 9 1 v1 v2 2 5 4 Sink

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The st-Mincut Problem What is a st-cut? Source 2 9 1 v1 v2 2 5 4 Sink

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**What is the cost of a st-cut?**

The st-Mincut Problem What is a st-cut? An st-cut (S,T) divides the nodes between source and sink. Source 2 9 What is the cost of a st-cut? 1 Sum of cost of all edges going from S to T v1 v2 2 5 4 Sink = 16

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**What is the cost of a st-cut?**

The st-Mincut Problem What is a st-cut? An st-cut (S,T) divides the nodes between source and sink. Source 2 9 What is the cost of a st-cut? 1 Sum of cost of all edges going from S to T v1 v2 2 5 4 What is the st-mincut? Sink st-cut with the minimum cost = 7

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**How to compute the st-mincut?**

Solve the dual maximum flow problem Compute the maximum flow between Source and Sink Source Constraints Edges: Flow < Capacity Nodes: Flow in = Flow out 2 9 1 v1 v2 2 5 4 Min-cut\Max-flow Theorem Sink In every network, the maximum flow equals the cost of the st-mincut

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 0 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 2 9 1 v1 v2 2 5 4 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 0 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 2 9 1 v1 v2 2 5 4 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 0 + 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 2-2 9 1 v1 v2 2 4 5-2 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 9 1 v1 v2 2 3 4 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 9 1 v1 v2 2 3 4 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 9 1 v1 v2 2 3 4 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 2 + 4 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 5 1 v1 v2 2 3 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 6 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 5 1 v1 v2 2 3 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 6 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 5 1 v1 v2 2 3 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 6 + 1 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 4 1-1 v1 v2 2+1 2 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 7 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 4 v1 v2 3 2 Sink Algorithms assume non-negative capacity

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**Augmenting Path Based Algorithms**

Maxflow Algorithms Flow = 7 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 4 v1 v2 3 2 Sink Algorithms assume non-negative capacity

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**History of Maxflow Algorithms**

n: #nodes m: #edges U: maximum edge weight Augmenting Path and Push-Relabel Algorithms assume non-negative edge weights [Slide credit: Andrew Goldberg]

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**History of Maxflow Algorithms**

n: #nodes m: #edges U: maximum edge weight Augmenting Path and Push-Relabel Algorithms assume non-negative edge weights [Slide credit: Andrew Goldberg]

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**Augmenting Path based Algorithms**

Ford Fulkerson: Choose any augmenting path Source 1000 1000 1 a1 a2 1000 1000 Sink

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**Augmenting Path based Algorithms**

Ford Fulkerson: Choose any augmenting path Source 1000 1000 1 a1 a2 Bad Augmenting Paths 1000 1000 Sink

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**Augmenting Path based Algorithms**

Ford Fulkerson: Choose any augmenting path Source 1000 1000 1 a1 a2 Bad Augmenting Path 1000 1000 Sink

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**Augmenting Path based Algorithms**

Ford Fulkerson: Choose any augmenting path Source 999 1000 a1 a2 1 1000 999 Sink

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**Augmenting Path based Algorithms**

n: #nodes m: #edges Ford Fulkerson: Choose any augmenting path Source 999 1000 a1 a2 1 1000 999 Sink 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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**Augmenting Path based Algorithms**

n: #nodes m: #edges Dinic: Choose shortest augmenting path Source 1000 1000 1 a1 a2 1000 1000 Sink Worst case Complexity: O (m n2)

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**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

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**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 web

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**Outline of the Tutorial**

The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

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**St-mincut and Energy Minimization**

Minimizing a Qudratic Pseudoboolean function E(x) Functions of boolean variables Pseudoboolean? E: {0,1}n → R E(x) = ∑ ci xi + ∑ cij xi(1-xj) cij≥0 i i,j Polynomial time st-mincut algorithms require non-negative edge weights

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**So how does this work? E(x) Construct a graph such that:**

Any st-cut corresponds to an assignment of x The cost of the cut is equal to the energy of x : E(x) st-mincut T S Solution E(x)

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Graph Construction E(a1,a2) Source (0) a1 a2 Sink (1)

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Graph Construction E(a1,a2) = 2a1 Source (0) 2 a1 a2 Sink (1)

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Graph Construction E(a1,a2) = 2a1 + 5ā1 Source (0) 2 a1 a2 5 Sink (1)

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**Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 2 9 5 4 Source (0)**

Sink (1)

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**Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 2 9 2 5 4**

Source (0) 2 9 a1 a2 2 5 4 Sink (1)

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**Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 2 9**

Source (0) 2 9 1 a1 a2 2 5 4 Sink (1)

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**Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 2 9**

Source (0) 2 9 1 a1 a2 2 5 4 Sink (1)

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**Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 2 9**

Source (0) 2 9 Cost of cut = 11 1 a1 a2 a1 = 1 a2 = 1 2 E (1,1) = 11 5 4 Sink (1)

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**Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 2 9**

Source (0) 2 9 st-mincut cost = 8 1 a1 a2 a1 = 1 a2 = 0 2 E (1,0) = 8 5 4 Sink (1)

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**Energy Function Reparameterization**

Two functions E1 and E2 are reparameterizations if E1 (x) = E2 (x) for all x For instance: E1 (a1) = 1+ 2a1 + 3ā1 E2 (a1) = 3 + ā1 a1 ā1 1+ 2a1 + 3ā1 3 + ā1 1 4 3

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**Flow and Reparametrization**

E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 Source (0) 2 9 1 a1 a2 2 5 4 Sink (1)

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**Flow and Reparametrization**

E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 Source (0) 2 9 2a1 + 5ā1 1 a1 a2 = 2(a1+ā1) + 3ā1 2 = 2 + 3ā1 5 4 Sink (1)

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**Flow and Reparametrization**

E(a1,a2) = 2 + 3ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 Source (0) 9 2a1 + 5ā1 1 a1 a2 = 2(a1+ā1) + 3ā1 2 = 2 + 3ā1 3 4 Sink (1)

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**Flow and Reparametrization**

E(a1,a2) = 2 + 3ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 Source (1) 9 9a2 + 4ā2 1 a1 a2 = 4(a2+ā2) + 5ā2 2 = 4 + 5ā2 3 4 Sink (0)

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**Flow and Reparametrization**

E(a1,a2) = 2 + 3ā1+ 5a a1ā2 + ā1a2 Source (1) 5 9a2 + 4ā2 1 a1 a2 = 4(a2+ā2) + 5ā2 2 = 4 + 5ā2 3 Sink (0)

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**Flow and Reparametrization**

E(a1,a2) = 6 + 3ā1+ 5a2 + 2a1ā2 + ā1a2 Source (1) 5 1 a1 a2 2 3 Sink (0)

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**Flow and Reparametrization**

E(a1,a2) = 6 + 3ā1+ 5a2 + 2a1ā2 + ā1a2 Source (1) 5 1 a1 a2 2 3 Sink (0)

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**Flow and Reparametrization**

E(a1,a2) = 6 + 3ā1+ 5a2 + 2a1ā2 + ā1a2 3ā1+ 5a2 + 2a1ā2 = 2(ā1+a2+a1ā2) +ā1+3a2 Source (1) = 2(1+ā1a2) +ā1+3a2 5 F1 = ā1+a2+a1ā2 F2 = 1+ā1a2 1 a1 a2 2 a1 a2 F1 F2 1 2 3 Sink (0)

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**Flow and Reparametrization**

E(a1,a2) = 8 + ā1+ 3a2 + 3ā1a2 3ā1+ 5a2 + 2a1ā2 = 2(ā1+a2+a1ā2) +ā1+3a2 Source (1) = 2(1+ā1a2) +ā1+3a2 3 F1 = ā1+a2+a1ā2 F2 = 1+ā1a2 3 a1 a2 a1 a2 F1 F2 1 2 1 Sink (0)

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**Flow and Reparametrization**

E(a1,a2) = 8 + ā1+ 3a2 + 3ā1a2 Source (1) 3 3 No more augmenting paths possible a1 a2 1 Sink (0)

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**Flow and Reparametrization**

E(a1,a2) = 8 + ā1+ 3a2 + 3ā1a2 Residual Graph (positive coefficients) Source (1) Total Flow 3 bound on the optimal solution 3 a1 a2 1 Sink (0) Inference of the optimal solution becomes trivial because the bound is tight

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**Flow and Reparametrization**

E(a1,a2) = 8 + ā1+ 3a2 + 3ā1a2 Residual Graph (positive coefficients) Source (1) Total Flow 3 st-mincut cost = 8 bound on the optimal solution 3 a1 a2 a1 = 1 a2 = 0 E (1,0) = 8 1 Sink (0) Inference of the optimal solution becomes trivial because the bound is tight

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**Example: Image Segmentation**

E: {0,1}n → R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑ cij xi(1-xj) i i,j x* = arg min E(x) x How to minimize E(x)? Global Minimum (x*)

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**How does the code look like?**

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); g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1) Source (0) Sink (1)

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**How does the code look like?**

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); g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1) Source (0) bgCost(a1) bgCost(a2) a1 a2 fgCost(a1) fgCost(a2) Sink (1)

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**How does the code look like?**

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)); g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1) Source (0) bgCost(a1) bgCost(a2) cost(p,q) a1 a2 fgCost(a1) fgCost(a2) Sink (1)

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**How does the code look like?**

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)); g->compute_maxflow(); label_p = g->is_connected_to_source(nodeID(p)); // is the label of pixel p (0 or 1) Source (0) bgCost(a1) bgCost(a2) cost(p,q) a1 a2 fgCost(a1) fgCost(a2) Sink (1) a1 = bg a2 = fg

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**Image Segmentation in Video**

n-links st-cut = 0 x* E(x) t = 1 correct Image Flow Global Optimum

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**Image Segmentation in Video**

correct Global Optimum Flow

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**Dynamic Energy Minimization**

SA minimize EA Recycling Solutions Can we do better? EB SB computationally expensive operation Boykov & Jolly ICCV’01, Kohli & Torr (ICCV05, PAMI07)

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**Dynamic Energy Minimization**

SA minimize EA Simpler energy EB* differences between A and B similar Reuse flow Reparametrization cheaper operation EB SB 3 – time speedup! computationally expensive operation Boykov & Jolly ICCV’01, Kohli & Torr (ICCV05, PAMI07) Kohli & Torr (ICCV05, PAMI07)

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**Dynamic Energy Minimization**

Original Energy E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 + ā1a2 Reparametrized Energy E(a1,a2) = 8 + ā1+ 3a2 + 3ā1a2 New Energy E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 7a1ā2 + ā1a2 New Reparametrized Energy E(a1,a2) = 8 + ā1+ 3a2 + 3ā1a2 + 5a1ā2 Boykov & Jolly ICCV’01, Kohli & Torr (ICCV05, PAMI07) Kohli & Torr (ICCV05, PAMI07)

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**Outline of the Tutorial**

The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

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**Minimizing Energy Functions**

General Energy Functions NP-hard to minimize Only approximate minimization possible Easy energy functions Solvable in polynomial time Submodular ~ O(n6) MAXCUT NP-Hard Submodular Functions Functions defined on trees Space of Function Minimization Problems

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**Submodular Set Functions**

2|E| = #subsets of E Let E= {a1,a2, .... an} be a set Set function f 2|E| ℝ

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**Submodular Set Functions**

2|E| = #subsets of E Let E= {a1,a2, .... an} be a set Set function f 2|E| ℝ is submodular if f(A) + f(B) f(AB) + f(AB) for all A,B E E A B Important Property Sum of two submodular functions is submodular

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**Minimizing Submodular Functions**

Minimizing general submodular functions O(n5 Q + n6) where Q is function evaluation time [Orlin, IPCO 2007] Symmetric submodular functions E (x) = E (1 - x) O(n3) [Queyranne 1998] Quadratic pseudoboolean Can be transformed to st-mincut One node per variable [ O(n3) complexity] Very low empirical running time

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**Submodular Pseudoboolean Functions**

Function defined over boolean vectors x = {x1,x2, .... xn} Definition: All functions for one boolean variable (f: {0,1} -> R) are submodular A function of two boolean variables (f: {0,1}2 -> R) is submodular if f(0,1) + f(1,0) f(0,0) + f(1,1) A general pseudoboolean function f 2n ℝ is submodular if all its projections fp are submodular i.e. fp(0,1) + fp (1,0) fp (0,0) + fp (1,1)

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**Quadratic Submodular Pseudoboolean Functions**

E(x) = ∑ θi (xi) + ∑ θij (xi,xj) i,j i For all ij θij(0,1) + θij (1,0) θij (0,0) + θij (1,1)

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**Quadratic Submodular Pseudoboolean Functions**

E(x) = ∑ θi (xi) + ∑ θij (xi,xj) i,j i For all ij θij(0,1) + θij (1,0) θij (0,0) + θij (1,1) Equivalent (transformable) E(x) = ∑ ci xi + ∑ cij xi(1-xj) cij≥0 i i,j i.e. All submodular QPBFs are st-mincut solvable

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**How are they equivalent?**

A = θij (0,0) B = θij(0,1) C = θij (1,0) D = θij (1,1) xj A B C D C-A D-C B+C-A-D xi = A + + + 1 1 1 1 if x1=1 add C-A if x2 = 1 add D-C θij (xi,xj) = θij(0,0) + (θij(1,0)-θij(0,0)) xi + (θij(1,0)-θij(0,0)) xj + (θij(1,0) + θij(0,1) - θij(0,0) - θij(1,1)) (1-xi) xj B+C-A-D 0 is true from the submodularity of θij

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**How are they equivalent?**

A = θij (0,0) B = θij(0,1) C = θij (1,0) D = θij (1,1) xj A B C D C-A D-C B+C-A-D xi = A + + + 1 1 1 1 if x1=1 add C-A if x2 = 1 add D-C θij (xi,xj) = θij(0,0) + (θij(1,0)-θij(0,0)) xi + (θij(1,0)-θij(0,0)) xj + (θij(1,0) + θij(0,1) - θij(0,0) - θij(1,1)) (1-xi) xj B+C-A-D 0 is true from the submodularity of θij

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**How are they equivalent?**

A = θij (0,0) B = θij(0,1) C = θij (1,0) D = θij (1,1) xj A B C D C-A D-C B+C-A-D xi = A + + + 1 1 1 1 if x1=1 add C-A if x2 = 1 add D-C θij (xi,xj) = θij(0,0) + (θij(1,0)-θij(0,0)) xi + (θij(1,0)-θij(0,0)) xj + (θij(1,0) + θij(0,1) - θij(0,0) - θij(1,1)) (1-xi) xj B+C-A-D 0 is true from the submodularity of θij

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**How are they equivalent?**

A = θij (0,0) B = θij(0,1) C = θij (1,0) D = θij (1,1) xj A B C D C-A D-C B+C-A-D xi = A + + + 1 1 1 1 if x1=1 add C-A if x2 = 1 add D-C θij (xi,xj) = θij(0,0) + (θij(1,0)-θij(0,0)) xi + (θij(1,0)-θij(0,0)) xj + (θij(1,0) + θij(0,1) - θij(0,0) - θij(1,1)) (1-xi) xj B+C-A-D 0 is true from the submodularity of θij

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**How are they equivalent?**

A = θij (0,0) B = θij(0,1) C = θij (1,0) D = θij (1,1) xj A B C D C-A D-C B+C-A-D xi = A + + + 1 1 1 1 if x1=1 add C-A if x2 = 1 add D-C θij (xi,xj) = θij(0,0) + (θij(1,0)-θij(0,0)) xi + (θij(1,0)-θij(0,0)) xj + (θij(1,0) + θij(0,1) - θij(0,0) - θij(1,1)) (1-xi) xj B+C-A-D 0 is true from the submodularity of θij

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**Quadratic Submodular Pseudoboolean Functions**

x in {0,1}n E(x) = ∑ θi (xi) + ∑ θij (xi,xj) i,j i For all ij θij(0,1) + θij (1,0) θij (0,0) + θij (1,1) Equivalent (transformable) st-mincut T S

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**Minimizing Non-Submodular Functions**

E(x) = ∑ θi (xi) + ∑ θij (xi,xj) i,j i θij(0,1) + θij (1,0) ≤ θij (0,0) + θij (1,1) for some ij Minimizing general non-submodular functions is NP-hard. Commonly used method is to solve a relaxation of the problem [Slide credit: Carsten Rother]

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**Minimization using Roof-dual Relaxation**

unary pairwise submodular pairwise nonsubmodular [Slide credit: Carsten Rother]

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**Minimization using Roof-dual Relaxation**

Double number of variables: [Slide credit: Carsten Rother]

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**Minimization using Roof-dual Relaxation**

Double number of variables: Submodular Non- submodular

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**Minimization using Roof-dual Relaxation**

Double number of variables: Property of the problem: is submodular ! Ignore (solvable using st-mincut)

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**Minimization using Roof-dual Relaxation**

Double number of variables: Property of the solution: is the optimal label

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**Recap Exact minimization of Submodular QBFs using graph cuts.**

Obtaining partially optimal solutions of non-submodular QBFs using graph cuts.

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**E(x) = ∑ θi (xi) + ∑ θij (xi,xj) + ∑ θc (xc)**

But ... Need higher order energy functions to model image structure Field of experts [Roth and Black] Many problems in computer vision involve multiple labels E(x) = ∑ θi (xi) + ∑ θij (xi,xj) + ∑ θc (xc) i,j i c x ϵ Labels L = {l1, l2, … , lk} Clique c V

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**Transforming problems in QBFs**

Higher order Pseudoboolean Functions Quadratic Pseudoboolean Functions Multi-label Functions Pseudoboolean Functions

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**Transforming problems in QBFs**

Higher order Pseudoboolean Functions Quadratic Pseudoboolean Functions Multi-label Functions Pseudoboolean Functions

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**Higher order to Quadratic**

Simple Example using Auxiliary variables { 0 if all xi = 0 C1 otherwise f(x) = x ϵ L = {0,1}n min f(x) = min C1a + C1 ∑ ā xi x x,a ϵ {0,1} Higher Order Submodular Function Quadratic Submodular Function ∑xi = 0 a=0 (ā=1) f(x) = 0 ∑xi ≥ 1 a=1 (ā=0) f(x) = C1

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**Higher order to Quadratic**

min f(x) = min C1a + C1 ∑ ā xi x x,a ϵ {0,1} Higher Order Submodular Function Quadratic Submodular Function C1∑xi C1 1 2 3 ∑xi

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**Higher order to Quadratic**

min f(x) = min C1a + C1 ∑ ā xi x x,a ϵ {0,1} Higher Order Submodular Function Quadratic Submodular Function C1∑xi a=1 a=0 Lower envelop of concave functions is concave C1 1 2 3 ∑xi

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**Higher order to Quadratic**

min f(x) = min f1 (x)a + f2(x)ā x x,a ϵ {0,1} Higher Order Submodular Function Quadratic Submodular Function f2(x) a=1 Lower envelop of concave functions is concave f1(x) 1 2 3 ∑xi

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**Higher order to Quadratic**

min f(x) = min f1 (x)a + f2(x)ā x x,a ϵ {0,1} Higher Order Submodular Function Quadratic Submodular Function f2(x) a=1 a=0 Lower envelop of concave functions is concave f1(x) 1 2 3 ∑xi

99
**Transforming problems in QBFs**

Higher order Pseudoboolean Functions Quadratic Pseudoboolean Functions Multi-label Functions Pseudoboolean Functions

100
**Multi-label to Pseudo-boolean**

So what is the problem? Em (y1,y2, ..., yn) Eb (x1,x2, ..., xm) yi ϵ L = {l1, l2, … , lk} xi ϵ L = {0,1} Multi-label Problem Binary label Problem such that: Let Y and X be the set of feasible solutions, then For each binary solution x ϵ X with finite energy there exists exactly one multi-label solution y ϵ Y -> One-One encoding function T:X->Y 2. arg min Em(y) = T(arg min Eb(x))

101
**Multi-label to Pseudo-boolean**

Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06]

102
**Multi-label to Pseudo-boolean**

Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] Ishikawa’s result: E(y) = ∑ θi (yi) + ∑ θij (yi,yj) i,j i y ϵ Labels L = {l1, l2, … , lk} Convex Function θij (yi,yj) = g(|yi-yj|) g(|yi-yj|) |yi-yj|

103
**Multi-label to Pseudo-boolean**

Popular encoding scheme [Roy and Cox ’98, Ishikawa ’03, Schlesinger & Flach ’06] Schlesinger & Flach ’06: E(y) = ∑ θi (yi) + ∑ θij (yi,yj) i,j i y ϵ Labels L = {l1, l2, … , lk} θij(li+1,lj) + θij (li,lj+1) θij (li,lj) + θij (li+1,lj+1) Covers all Submodular multi-label functions More general than Ishikawa

104
**Multi-label to Pseudo-boolean**

Problems Applicability Only solves restricted class of energy functions Cannot handle Potts model potentials Computational Cost Very high computational cost Problem size = |Variables| x |Labels| Gray level image denoising (1 Mpixel image) (~2.5 x 108 graph nodes)

105
**Outline of the Tutorial**

The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

106
**St-mincut based Move algorithms**

E(x) = ∑ θi (xi) + ∑ θij (xi,xj) i,j i x ϵ Labels L = {l1, l2, … , lk} Commonly used for solving non-submodular multi-label problems Extremely efficient and produce good solutions Not Exact: Produce local optima

107
**Move Making Algorithms**

Energy Solution Space

108
**Move Making Algorithms**

Current Solution Search Neighbourhood Optimal Move Energy Solution Space

109
**Computing the Optimal Move**

Current Solution Search Neighbourhood Optimal Move xc (t) Key Property Move Space Energy Solution Space Better solutions Finding the optimal move hard Bigger move space

110
**Moves using Graph Cuts Expansion and Swap move algorithms**

[Boykov Veksler and Zabih, PAMI 2001] Makes a series of changes to the solution (moves) Each move results in a solution with smaller energy Current Solution Search Neighbourhood N Number of Variables Move Space (t) : 2N L Number of Labels Space of Solutions (x) : LN

111
**Moves using Graph Cuts Expansion and Swap move algorithms**

[Boykov Veksler and Zabih, PAMI 2001] Makes a series of changes to the solution (moves) Each move results in a solution with smaller energy Current Solution Move to new solution How to minimize move functions? Construct a move function Minimize move function to get optimal move

112
**General Binary Moves x = t x1 + (1- t) x2 Em(t) = E(t x1 + (1- t) x2)**

New solution Current Solution Second solution Em(t) = E(t x1 + (1- t) x2) Minimize over move variables t to get the optimal move Move energy is a submodular QPBF (Exact Minimization Possible) Boykov, Veksler and Zabih, PAMI 2001

113
**Swap Move Variables labeled α, β can swap their labels**

Chi square [Boykov, Veksler, Zabih]

114
**Swap Move Variables labeled α, β can swap their labels Swap Sky, House**

Tree Ground Swap Sky, House House Sky [Boykov, Veksler, Zabih]

115
**Swap Move Move energy is submodular if:**

Variables labeled α, β can swap their labels Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Semimetric θij (la,lb) ≥ 0 θij (la,lb) = a = b Chi square Examples: Potts model, Truncated Convex [Boykov, Veksler, Zabih]

116
**Expansion Move Variables take label a or retain current label**

[Boykov, Veksler, Zabih] [Boykov, Veksler, Zabih]

117
**Expansion Move Variables take label a or retain current label**

Tree Ground House Status: Initialize with Tree Expand House Expand Sky Expand Ground Sky [Boykov, Veksler, Zabih] [Boykov, Veksler, Zabih]

118
**θij (la,lb) + θij (lb,lc) ≥ θij (la,lc)**

Expansion Move Variables take label a or retain current label Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Metric Semi metric + Triangle Inequality θij (la,lb) + θij (lb,lc) ≥ θij (la,lc) Examples: Potts model, Truncated linear Cannot solve truncated quadratic [Boykov, Veksler, Zabih] [Boykov, Veksler, Zabih]

119
**Minimize over move variables t**

General Binary Moves x = t x1 + (1-t) x2 New solution First solution Second solution Minimize over move variables t Move Type First Solution Second Solution Guarantee Expansion Old solution All alpha Metric Fusion Any solution Move functions can be non-submodular!!

120
**Solving Continuous Problems using Fusion Move**

x = t x1 + (1-t) x2 x1, x2 can be continuous Optical Flow Example Solution from Method 2 x2 Solution from Method 1 F Final Solution x1 x (Lempitsky et al. CVPR08, Woodford et al. CVPR08)

121
**x = (t ==1) x1 + (t==2) x2 +… +(t==k) xk**

Range Moves Move variables can be multi-label Optimal move found out by using the Ishikawa Useful for minimizing energies with truncated convex pairwise potentials x = (t ==1) x1 + (t==2) x2 +… +(t==k) xk T θij (yi,yj) = min(|yi-yj|,T) θij (yi,yj) |yi-yj| O. Veksler, CVPR 2007

122
**Move Algorithms for Solving Higher Order Energies**

E(x) = ∑ θi (xi) + ∑ θij (xi,xj) + ∑ θc (xc) i,j i c x ϵ Labels L = {l1, l2, … , lk} Clique c V Higher order functions give rise to higher order move energies Move energies for certain classes of higher order energies can be transformed to QPBFs. [Kohli, Kumar and Torr, CVPR07] [Kohli, Ladicky and Torr, CVPR08]

123
**Outline of the Tutorial**

The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

124
**Solving Mixed Programming Problems**

x – binary image segmentation (xi ∊ {0,1}) ω – non-local parameter (lives in some large set Ω) E(x,ω) = C(ω) + ∑ θi (ω, xi) + ∑ θij (ω,xi,xj) i,j i unary potentials pairwise potentials ≥ 0 constant ω θi (ω, xi) Pose Shape Prior Stickman Model Rough Shape Prior

125
**Open Problems Characterization of Problems Solvable using st-mincut**

What functions can be transformed to submodular QBFs? Submodular Functions st-mincut Equivalent

126
**Minimizing General Higher Order Functions**

We saw how simple higher order potentials can be solved How more sophisticated higher order potentials can be solved?

127
**Summary Labelling Problem Submodular Quadratic Pseudoboolean Function**

Exact Transformation (global optimum) Or Relaxed transformation (partially optimal) Labelling Problem Submodular Quadratic Pseudoboolean Function st-mincut T S Sub-problem Move making algorithms

128
Thanks. Questions?

129
**Use of Higher order Potentials**

E(x1,x2,x3) = θ12 (x1,x2) + θ23 (x2,x3) { 0 if xi=xj C otherwise θij (xi,xj) = 8 E(6,6,6) = = 0 7 Disparity Labels 6 5 P1 P2 P3 Pixels Stereo - Woodford et al. CVPR 2008

130
**Use of Higher order Potentials**

E(x1,x2,x3) = θ12 (x1,x2) + θ23 (x2,x3) { 0 if xi=xj C otherwise θij (xi,xj) = 8 E(6,6,6) = = 0 E(6,7,7) = = 1 7 Disparity Labels 6 5 P1 P2 P3 Pixels Stereo - Woodford et al. CVPR 2008

131
**Use of Higher order Potentials**

E(x1,x2,x3) = θ12 (x1,x2) + θ23 (x2,x3) { 0 if xi=xj C otherwise θij (xi,xj) = 8 E(6,6,6) = = 0 E(6,7,7) = = 1 E(6,7,8) = = 2 7 Disparity Labels 6 5 Pairwise potential penalize slanted planar surfaces P1 P2 P3 Pixels Stereo - Woodford et al. CVPR 2008

132
**Computing the Optimal Move**

Current Solution Search Neighbourhood Optimal Move xc T Transformation function (t) E(x) x T(xc, t) = xn = xc + t

133
**Computing the Optimal Move**

Current Solution Search Neighbourhood Optimal Move xc T Transformation function (t) E(x) Em Move Energy x T(xc, t) = xn = xc + t Em(t) = E(T(xc, t))

134
**Computing the Optimal Move**

Current Solution Search Neighbourhood Optimal Move xc T Transformation function (t) E(x) Em Move Energy x T(xc, t) = xn = xc + t minimize t* Em(t) = E(T(xc, t)) Optimal Move

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