Discrete Optimization Lecture 5. M. Pawan Kumar

Transcription

1 Discrete Optimization Lecture 5 M. Pawan Kumar pawan.kumar@ecp.fr

2 Exam Question Type 1 v 1 s v v 4 Q. Find the distance of the shortest path from s=v 0 to all vertices in the graph using Dijkstra s method. Show the estimate of the distance at each iteration Iteration = 1 5 v 2 v v 5 v 6 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 0 Iteration = v 7 Iteration =

3 Exam Question Type 2 Q. Provide the code for the following function that implements the Floyd-Warshall algorithm in O(n 3 ) time. void floyd_warshall(int n, double **A, double **D); n = number of vertices in the graph A = nxn array to store the input, A[i][j] = length of arc (i,j), 0 i < n, 0 j < n D = nxn array to store the output Use the following assumptions. (1) The graph has no negative length directed circuit. (2) If (i,j) is not an arc, A[i][j] = infinity (3) double **allocate_memory(int n, int m) returns an nxm array (4) free_memory(double **D, int n, int m) frees an nxm array (5)

4 void floyd_warshall(int n, double **A, double **D) { double **D_prev = allocate_memory(n,n); int i, j, k; for(i=0;i<n;i++) { for(j=0;j<n;j++) { D[i][j] = A[i][j]; } } for(k=0;k<n;k++) { for(i=0;i<n;i++) { for(j=0;j<n;j++) { D_prev[i][j] = D[i][j]; } } } for(i=0;i<n;i++) { for(j=0;j<n;j++) { if(d_prev[i][k] + D_prev[k][j] < D[i][j]) { D[i][j] = D_prev[i][k] + D_prev[k][j]; } } } } free_memory(d_prev,n,n);

5 Exam Question Type 3 Q. Prove that Dijkstra s algorithm is correct when the length of the arcs is non-negative. Let l(a,b) be the length of the arc from vertex a to vertex b. Let d(a) be the current estimate of the distance from the source vertex s to the vertex a. Let dist(a) be the distance of the shortest path from the source vertex s to the vertex a. By definition, d(a) dist(a). We initialize U = V, where V is the set of all vertices. At each iteration, we choose u = argmin a U d(a). We set U = U {u}, and d(v) = min{d(v),d(u)+l(u,v)} We will show that d(u) = dist(u). At the first iteration, u = s and d(u) = 0. Since all arc lengths are non-negative, it follows that d(s) = dist(s) = 0. At iteration t, let us assume that d(u) > dist(u). There exists a shortest path s, v 1,v 2, v k, u from s to u. Let i be the smallest index such that i U. d(v i ) d(v i-1 ) + l(v i-1,v i ) = dist(v i-1 )+l(v i-1,v i ) = dist(v i ) Therefore, d(v i ) = dist(v i ), which implies that d(v i ) < d(u). This implies that u cannot be chosen at the current iteration. Hence, by contradiction d(u) = dist(u).

6 Interactive Image Segmentation Minimum-Cut Outline From MAP Estimation to Minimum-Cut Solving Minimum-Cut

7 Image Segmentation How? Pose the problem as MAP estimation.

8 Random Variables v 1 v 2 v 3 v n Each random variable v a corresponds to a pixel Set of all variables = V = {v 1,v 2,,v n }

9 Edges v 1 v 2 v 3 v n Edge (v a,v b ) connects two neighboring pixels. Neighbors: top, down, left and right Set of all edges = E

10 Labels v 1 v 2 v 3 v n Label set L = {l 0,l 1 } Label l 0 corresponds to object Label l 1 corresponds to background

11 Labeling v 1 v 2 v 3 v n Labeling f: V è L We say v a is assigned a label l f(a) Number of possible labelings? 2 n

12 Unary Potentials θ a;1 θ a;0 v a Create a histogram of RGB values H 0 (R,G,B) = number of pixels with color = (R,G,B) number of pixels

13 Unary Potentials θ a;1 θ a;0 v a θ a;0 = -log H 0 (R(a),G(a),B(a)) H 0 (R,G,B) = number of pixels with color = (R,G,B) number of pixels

14 Unary Potentials θ a;1 θ a;0 v a θ a;0 will be low/high?

15 Unary Potentials θ a;1 θ a;0 v a θ a;0 will be low/high?

16 Unary Potentials θ a;1 θ a;0 v a Create a histogram of RGB values H 1 (R,G,B) = number of pixels with color = (R,G,B) number of pixels

17 Unary Potentials θ a;1 θ a;0 v a θ a;1 = -log H 1 (R(a),G(a),B(a)) H 0 (R,G,B) = number of pixels with color = (R,G,B) number of pixels

18 Unary Potentials θ a;1 θ a;0 v a θ a;1 will be low/high?

19 Unary Potentials θ a;1 θ a;0 v a θ a;1 will be low/high?

20 Pairwise Potentials θ ab;11 θ ab;01 θ ab;10 θ ab;00 v a v b Neighboring pixels tend to have the same label.

21 Pairwise Potentials 0 θ ab;01 θ ab;10 0 v a v b Neighboring pixels tend to have the same label.

22 Pairwise Potentials 0 θ ab;01 θ ab;10 0 v a v b Similar color tends to imply same label. D(a,b) = difference in RGB values of a and b θ ab;01 = θ ab;10 = exp(-d(a,b)).

23 Pairwise Potentials 0 θ ab;01 θ ab;10 0 v a v b θ ab;01 and θ ab;10 will be low/high?

24 Pairwise Potentials 0 θ ab;01 θ ab;10 0 v a v b θ ab;01 and θ ab;10 will be low/high?

25 Energy Function v 1 v 2 v 3 v n Labeling f: V è L We say v a is assigned a label l f(a) Q(f) = Σ a V θ a;f(a) + Σ (a,b) E θ ab;f(a)f(b)

26 MAP Estimation v 1 v 2 v 3 v n Labeling f: V è L We say v a is assigned a label l f(a) min f Q(f) = Σ a V θ a;f(a) + Σ (a,b) E θ ab;f(a)f(b)

27 Outline Interactive Image Segmentation Minimum-Cut (A Special Case) From MAP Estimation to Minimum-Cut Solving Minimum-Cut

28 Directed Graph D = (V, A) 10 v 1 v v 3 v 4 5 Two important restrictions (1) Rational arc lengths (2) Positive arc lengths

29 Cut D = (V, A) 10 v 1 v v 3 v 4 5 Let V 1 and V 2 such that V 1 union V 2 = V V 1 intersection V 2 = Φ C is a set of arcs such that (u,v) A u V 1 v V 2 C is a cut in the digraph D

30 Cut D = (V, A) V 1 10 v 1 v v 3 v 4 5 What is C? {(v 1,v 2 ),(v 1,v 4 )}? {(v 1,v 4 ),(v 3,v 2 )}? {(v 1,v 4 )}? V 2

31 Cut V 2 V 1 10 v 1 v v 3 v 4 5 D = (V, A) What is C? {(v 1,v 2 ),(v 1,v 4 ),(v 3,v 2 )}? {(v 4,v 3 )}? {(v 1,v 4 ),(v 3,v 2 )}?

32 Cut V 1 V 2 10 v 1 v v 3 v 4 5 D = (V, A) What is C? {(v 1,v 2 ),(v 1,v 4 ),(v 3,v 2 )}? {(v 3,v 2 )}? {(v 1,v 4 ),(v 3,v 2 )}?

33 Cut D = (V, A) 10 v 1 v v 3 v 4 5 Let V 1 and V 2 such that V 1 union V 2 = V V 1 intersection V 2 = Φ C is a set of arcs such that (u,v) A u V 1 v V 2 C is a cut in the digraph D

34 Weight of a Cut D = (V, A) 10 v 1 v Sum of length of all arcs in C v 3 v 4 5

35 Weight of a Cut D = (V, A) 10 v 1 v w(c) = Σ (u,v) C l(u,v) v 3 v 4 5

36 Weight of a Cut D = (V, A) V 1 10 v 1 v What is w(c)? 3 v 3 v 4 5 V 2

37 Weight of a Cut V 2 V 1 10 v 1 v D = (V, A) What is w(c)? 5 v 3 v 4 5

38 Weight of a Cut V 1 V 2 10 v 1 v D = (V, A) What is w(c)? 15 v 3 v 4 5

39 st-cut s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) A source vertex s A sink vertex t C is a cut such that s V 1 t V 2 C is an st-cut

40 Weight of an st-cut s 1 2 D = (V, A) 10 v 1 v w(c) = Σ (u,v) C l(u,v) v 3 5 v 4 7 t 3

41 Weight of an st-cut s 1 2 v 1 10 v D = (V, A) What is w(c)? 3 v 3 5 v 4 7 t 3

42 Weight of an st-cut s 1 2 v 1 10 v D = (V, A) What is w(c)? 15 v 3 5 v 4 7 t 3

43 Minimum Cut Problem s 1 2 v 1 10 v D = (V, A) Find a cut with the minimum weight!! C* = argmin C w(c) v 3 5 v 4 7 t 3

44 Solvers for the Minimum-Cut Problem Augmenting Path and Push-Relabel n: #nodes m: #edges U: maximum arc length [Slide credit: Andrew Goldberg]

45 Remember Two important restrictions (1) Rational arc lengths (2) Positive arc lengths

46 Cut D = (V, A) 10 v 1 v v 3 v 4 5 Let V 1 and V 2 such that V 1 union V 2 = V V 1 intersection V 2 = Φ C is a set of arcs such that (u,v) A u V 1 v V 2 C is a cut in the digraph D

47 st-cut s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) A source vertex s A sink vertex t C is a cut such that s V 1 t V 2 C is an st-cut

48 Minimum Cut Problem s 1 2 v 1 10 v D = (V, A) Find a cut with the minimum weight!! C* = argmin C w(c) v 3 v 4 5 w(c) = Σ (u,v) C l(u,v) 7 t 3

49 Interactive Image Segmentation Minimum-Cut Outline From MAP Estimation to Minimum-Cut Solving Minimum-Cut

50 Overview Energy Q One vertex per random variable + Additional vertices s and t Digraph D Compute Minimum Cut Labeling f* v a V 1 implies f(a) = 0 v a V 2 implies f(a) = 1 V = V 1 U V 2

51 Outline Interactive Image Segmentation Minimum-Cut From MAP Estimation to Minimum-Cut Unary Potentials Pairwise Potentials Solving Minimum-Cut

52 Digraph for Unary Potentials P f(a) = 0 θ a;1 Q f(a) = 1 θ a;0 v a

53 Digraph for Unary Potentials P Q f(a) = 0 f(a) = 1 s v a t

54 Digraph for Unary Potentials P f(a) = 0 s Let P Q Q f(a) = 1 Constant v a P-Q P-Q 0 + Q Q t

55 Digraph for Unary Potentials P f(a) = 0 s Let P Q Q f(a) = 1 f(a) = 1 Constant v a P-Q w(c) = 0 P-Q 0 + Q Q t

56 Digraph for Unary Potentials P f(a) = 0 s Let P Q Q f(a) = 1 f(a) = 0 Constant v a P-Q w(c) = P-Q P-Q 0 + Q Q t

57 Digraph for Unary Potentials P f(a) = 0 s Let P < Q Q f(a) = 1 Q-P Constant v a 0 Q-P + P P t

58 Digraph for Unary Potentials P f(a) = 0 s Let P < Q Q f(a) = 1 Q-P f(a) = 1 Constant v a w(c) = Q-P 0 Q-P + P P t

59 Digraph for Unary Potentials P f(a) = 0 s Let P < Q Q f(a) = 1 Q-P f(a) = 0 Constant v a w(c) = 0 0 Q-P + P P t

60 Outline Interactive Image Segmentation Minimum-Cut From MAP Estimation to Minimum-Cut Unary Potentials Pairwise Potentials Solving Minimum-Cut

61 Digraph for Pairwise Potentials f(b) = 0 f(a) = 0 f(a) = 1 P R θ ab;11 f(b) = 1 Q S θ ab;01 θ ab;10 θ ab;00 v a v b P P P P S-Q Q-P Q-P 0 S-Q 0 R+Q-S-P 0 0

62 Digraph for Pairwise Potentials f(b) = 0 f(b) = 1 f(a) = 0 f(a) = 1 P R Q S s Constant v a v b P P P P t S-Q Q-P Q-P 0 S-Q 0 R+Q-S-P 0 0

63 Digraph for Pairwise Potentials f(b) = 0 f(b) = 1 f(a) = 0 f(a) = 1 P R Q S s Q-P v a v b Unary Potential f(b) = 1 t 0 0 Q-P Q-P 0 S-Q S-Q 0 R+Q-S-P 0 0

64 Digraph for Pairwise Potentials f(b) = 0 f(b) = 1 f(a) = 0 f(a) = 1 P R Q S S-Q s Q-P v a v b t Unary Potential f(a) = 1 0 S-Q 0 S-Q + 0 R+Q-S-P 0 0

65 Digraph for Pairwise Potentials f(b) = 0 f(b) = 1 f(a) = 0 f(a) = 1 P R Q S S-Q s Q-P v a R+Q-S-P v b t Pairwise Potential f(a) = 1, f(b) = 0 0 R+Q-S-P 0 0

66 Digraph for Pairwise Potentials f(b) = 0 f(b) = 1 f(a) = 0 f(a) = 1 P R Q S S-Q s Q-P v a R+Q-S-P v b t R+Q-S-P 0 General 2-label MAP estimation is NP-hard

67 Outline Interactive Image Segmentation Minimum-Cut From MAP Estimation to Minimum-Cut Solving Minimum-Cut st-flow, Maximum-Flow Computing the maximum-flow

68 st-flow s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) Function flow: A è R Flow is less than length Flow is non-negative For all vertex expect s,t Incoming flow = Outgoing flow

69 st-flow s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) Function flow: A è R flow(u,v) l(u,v) Flow is non-negative For all vertex expect s,t Incoming flow = Outgoing flow

70 st-flow s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) Function flow: A è R flow(u,v) l(u,v) flow(u,v) 0 For all vertex expect s,t Incoming flow = Outgoing flow

71 st-flow s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) Function flow: A è R flow(u,v) l(u,v) flow(u,v) 0 For all a V \ {s,t} Incoming flow = Outgoing flow

72 st-flow s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) Function flow: A è R flow(u,v) l(u,v) flow(u,v) 0 For all a V \ {s,t} Σ (u,a) A flow(u,a) = Outgoing flow

73 st-flow s 1 2 v 1 10 v v 3 5 v 4 7 t 3 D = (V, A) Function flow: A è R flow(u,v) l(u,v) flow(u,v) 0 For all a V \ {s,t} Σ (u,a) A flow(u,a) = Σ (a,v) A flow(a,v)

74 Weight of an st-flow s 1 2 v 1 10 v D = (V, A) Function flow: A è R Outgoing flow of s - Incoming flow of s v 3 5 v 4 7 t 3

75 Weight of an st-flow s 1 2 v 1 10 v D = (V, A) Function flow: A è R Σ (s,v) A flow(s,v) - Σ (u,s) A flow(u,s) v 3 v 4 5 = 0 7 t 3

76 Weight of an st-flow s 1 2 v 1 10 v 2 D = (V, A) Function flow: A è R Σ (s,v) A flow(s,v) 3 2 v 3 5 v 4 7 t 3

77 Weight of an st-flow s 1 2 v 1 10 v D = (V, A) Function flow: A è R Σ (s,v) A flow(s,v) = Incoming flow of t v 3 5 v 4 7 t 3

78 Weight of an st-flow s 1 2 v 1 10 v D = (V, A) Function flow: A è R Σ (s,v) A flow(s,v) = Σ (u,t) A flow(u,t) v 3 5 v 4 7 t 3

79 Max-Flow Problem s 1 2 v 1 10 v 2 D = (V, A) Function flow: A è R Find the maximum flow!! 3 2 v 3 5 v 4 7 t 3

80 Min-Cut Max-Flow Theorem s 1 2 v 1 10 v v 3 5 v 4 D = (V, A) Function flow: A è R Weight of minimum-cut = Weight of maximum-flow 7 t 3

81 Outline Interactive Image Segmentation Minimum-Cut From MAP Estimation to Minimum-Cut Solving Minimum-Cut st-flow, Maximum-Flow Computing the maximum-flow Slides by Pushmeet Kohli

82 Maxflow Algorithms Source v 1 v 2 5 Flow = Sink 4 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 Algorithms assume non-negative capacity

83 Maxflow Algorithms Source v 1 v 2 5 Flow = Sink 4 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 Algorithms assume non-negative capacity

84 Maxflow Algorithms Source v 1 v Flow = Sink 4 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 Algorithms assume non-negative capacity

85 Maxflow Algorithms Source v 1 v 2 3 Flow = 2 2 Sink 4 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 Algorithms assume non-negative capacity

86 Maxflow Algorithms Source v 1 v 2 3 Flow = 2 2 Sink 4 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 Algorithms assume non-negative capacity

87 Maxflow Algorithms Source v 1 v 2 3 Flow = 2 2 Sink 4 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 Algorithms assume non-negative capacity

88 Maxflow Algorithms Source v 1 v 2 3 Flow = Sink 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 Algorithms assume non-negative capacity

89 Maxflow Algorithms Source v 1 v 2 3 Flow = 6 2 Sink 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 Algorithms assume non-negative capacity

90 Maxflow Algorithms Source v 1 v 2 3 Flow = 6 2 Sink 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 Algorithms assume non-negative capacity

91 Maxflow Algorithms Source v 1 v 2 2 Flow = Sink 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 Algorithms assume non-negative capacity

92 Maxflow Algorithms Source v 1 v 2 2 Flow = 7 3 Sink 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 Algorithms assume non-negative capacity

93 Maxflow Algorithms Source v 1 v 2 2 Flow = 7 3 Sink 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 Algorithms assume non-negative capacity

94 History of Maxflow Algorithms Augmenting Path and Push-Relabel n: #nodes m: #edges U: maximum arc length Algorithms assume nonnegative arc lengths [Slide credit: Andrew Goldberg]

95 History of Maxflow Algorithms Augmenting Path and Push-Relabel n: #nodes m: #edges U: maximum arc length Algorithms assume nonnegative arc lengths [Slide credit: Andrew Goldberg]

96 Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source a 1 a Sink

97 Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source a 1 a Sink Bad Augmenting Paths

98 Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source a 1 a Sink Bad Augmenting Path

99 Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source a 1 a Sink

100 Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path n: #nodes m: #edges Source a 1 a Sink We will have to perform 2000 augmentations! Worst case complexity: O (m x Total_Flow) (Pseudo-polynomial bound: depends on flow)

101 Augmenting Path based Algorithms Dinic: Choose shortest augmenting path n: #nodes m: #edges Source a 1 a Sink Worst case Complexity: O (m n 2 )

102 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

103 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