Discrete Optimization Lecture 5 – Part 2 M. Pawan Kumar Slides available online

Interactive Binary Segmentation Foreground histogram of RGB values FG Background histogram of RGB values BG ‘1’ indicates foreground and ‘0’ indicates background

Interactive Binary Segmentation More likely to be foreground than background

Interactive Binary Segmentation More likely to be background than foreground θ a (0) proportional to -log(BG(d a )) θ a (1) proportional to -log(FG(d a ))

Interactive Binary Segmentation More likely to belong to same label

Interactive Binary Segmentation Less likely to belong to same label θ ab (i,k) proportional to exp(-(d a -d b ) 2 ) if i ≠ k θ ab (i,k) = 0 if i = k

Outline Minimum Cut Problem Submodular Energy Functions

Directed Graph n1n1 n2n2 n3n3 n4n Important restriction Positive arc lengths D = (N, A)

Cut n1n1 n2n2 n3n3 n4n Let N 1 and N 2 such that N 1 “union” N 2 = N N 1 “intersection” N 2 = Φ C is a set of arcs such that (n 1,n 2 )  A n 1  N 1 n 2  N 2 D = (N, A) C is a cut in the digraph D

Cut n1n1 n2n2 n3n3 n4n What is C? D = (N, A) N1N1 N2N2 {(n 1,n 2 ),(n 1,n 4 )} ? {(n 1,n 4 ),(n 3,n 2 )} ? {(n 1,n 4 )} ? ✓

Cut n1n1 n2n2 n3n3 n4n What is C? D = (N, A) N1N1 N2N2 {(n 1,n 2 ),(n 1,n 4 ),(n 3,n 2 )} ? {(n 1,n 4 ),(n 3,n 2 )} ? {(n 4,n 3 )} ? ✓

Cut n1n1 n2n2 n3n3 n4n What is C? D = (N, A) N2N2 N1N1 {(n 1,n 2 ),(n 1,n 4 ),(n 3,n 2 )} ? {(n 1,n 4 ),(n 3,n 2 )} ? {(n 3,n 2 )} ? ✓

Cut n1n1 n2n2 n3n3 n4n Let N 1 and N 2 such that N 1 “union” N 2 = N N 1 “intersection” N 2 = Φ C is a set of arcs such that (n 1,n 2 )  A n 1  N 1 n 2  N 2 D = (N, A) C is a cut in the digraph D

Weight of a Cut n1n1 n2n2 n3n3 n4n Sum of length of all arcs in C D = (N, A)

Weight of a Cut n1n1 n2n2 n3n3 n4n w(C) = Σ (n 1,n 2 )  C l(n 1,n 2 ) D = (N, A)

Weight of a Cut n1n1 n2n2 n3n3 n4n What is w(C)? D = (N, A) N1N1 N2N2 3

Weight of a Cut n1n1 n2n2 n3n3 n4n What is w(C)? D = (N, A) N1N1 N2N2 5

Weight of a Cut n1n1 n2n2 n3n3 n4n What is w(C)? D = (N, A) N2N2 N1N1 15

st-Cut n1n1 n2n2 n3n3 n4n A source “s” C is a cut such that s  N 1 t  N 2 D = (N, A) C is an st-cut s t A sink “t” 12 73

Weight of an st-Cut n1n1 n2n2 n3n3 n4n D = (N, A) s t w(C) = Σ (n 1,n 2 )  C l(n 1,n 2 )

Weight of an st-Cut n1n1 n2n2 n3n3 n4n D = (N, A) s t What is w(C)? 3

Weight of an st-Cut n1n1 n2n2 n3n3 n4n D = (N, A) s t What is w(C)? 15

Minimum Cut Problem n1n1 n2n2 n3n3 n4n D = (N, A) s t Find a cut with the minimum weight !! C* = argmin C w(C)

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

Outline Minimum Cut Problem Submodular Energy Functions Hammer, 1965; Kolmogorov and Zabih, 2004

Overview Energy Q Digraph D Digraph D One nodes per element N = N 1 U N 2 N = N 1 U N 2 Compute Minimum Cut + Additional nodes “s” and “t” Optimal solution Optimal solution n a  N 1 implies x a = 0 n a  N 2 implies x a = 1

Outline Minimum Cut Problem Submodular Energy Functions Unary Potentials Pairwise Potentials

Digraph for Unary Potentials P Q x a = 0 x a = 1

Digraph for Unary Potentials nana P Q s t x a = 0 x a = 1

Digraph for Unary Potentials nana P Q s t Let P ≥ Q P-Q 0 Q Q + Constant P-Q x a = 0 x a = 1

Digraph for Unary Potentials nana P Q s t Let P ≥ Q P-Q 0 Q Q + Constant P-Q x a = 1 w(C) = 0 x a = 0 x a = 1

Digraph for Unary Potentials nana P Q s t Let P ≥ Q P-Q 0 Q Q + Constant P-Q x a = 0 w(C) = P-Q x a = 0 x a = 1

Digraph for Unary Potentials nana P Q s t Let P < Q 0 Q-P P P + Constant Q-P x a = 0 x a = 1

Digraph for Unary Potentials nana P Q s t Let P < Q 0 Q-P P P + Constant x a = 1 w(C) = Q-P Q-P x a = 0 x a = 1

Digraph for Unary Potentials nana P Q s t Let P < Q 0 Q-P P P + Constant x a = 0 w(C) = 0 Q-P x a = 0 x a = 1

Outline Minimum Cut Problem Submodular Energy Functions Unary Potentials Pairwise Potentials

Digraph for Pairwise Potentials PR QS x a = 0x a = 1 x b = 0 x b = 1 00 Q-P 0S-Q 0 0R+Q-S-P PP PP

Digraph for Pairwise Potentials nana nbnb PR QS 00 Q-P 0S-Q 0 0R+Q-S-P PP PP s t Constant x a = 0x a = 1 x b = 0 x b = 1

Digraph for Pairwise Potentials nana nbnb PR QS 00 Q-P 0S-Q 0 0R+Q-S-P s t Unary Potential x b = 1 Q-P x a = 0x a = 1 x b = 0 x b = 1

Digraph for Pairwise Potentials nana nbnb PR QS 0S-Q 0 0R+Q-S-P 00 + s t Unary Potential x a = 1 Q-PS-Q x a = 0x a = 1 x b = 0 x b = 1

Digraph for Pairwise Potentials nana nbnb PR QS 0R+Q-S-P 00 s t Pairwise Potential x a = 1, x b = 0 Q-PS-Q R+Q-S-P x a = 0x a = 1 x b = 0 x b = 1

Digraph for Pairwise Potentials nana nbnb PR QS s t Q-PS-Q R+Q-S-P R+Q-S-P ≥ 0 x a = 0x a = 1 x b = 0 x b = 1