1. Graph Cut
Weizhen Jing

2. Image Segmentation methods
Threshold based method
Edge detecting based method
Energy functionals based method
Snake model 、Level Set
region based method
region growing 、split-and-merge
Graph based method
GraphCut 、GrabCut、Random Walk
......

3. Introduction Application stereo vision Image matting
Image segmentation
Minimum cut of Graph

4. Introduction
► goal:Divide an image into two segments:"background" and "object" .
► pixel label problem
Boykov Y, Funka-Lea G. Graph cuts and efficient ND image segmentation[J]. International journal of computer vision, 2006, 70(2):

5. Construct S-T Graph G=<V,E> V=P {S ,T} P : pixel nodes in image
s : terminal node, object
t : terminal node,background
E=n-Links t-Links
n-Links : edges between two pixel nodes
t-Links: edges between s or t and a pixel node t
Wp(t)
WP(s) s
Wp(t): penalty for assigning pixel p to t(background).
Wp(s): penalty for assigning pixel p to s(object).
weights on n-links: penalty for a discontinuity between two adjacent nodes

6. Graph Cut cut a cut C={S,T} s S,t T,S T= , Cost of cut
sum of the costs of the edges from S to T
Min Cut
a cut t s

7. steps of Graph Cut
► Initialize a binary vector A=(A1,A2,...,AP) (a cut)
► Generate a energy function E(A)
► Minimize E(A),find the min cut

8. Initialize a binary vector A
A=(A1,A2,...,AP)
a binary vector which represents certain segmentation
each AP can be either 1(obj)or 0(bkg) 1

9. Initialize a binary vector A
► Automatic Segmentation
● reflect on how the intensity of pixel P fits into a known intensity model(e.g. histogram)of the object and background
► Interactive Segmentation
● Accurately positioned at the desired
boundary (easily generalized to 3D data)
● Indicate segmentation regions rather than boundary , some pixel are marked as internal and some as external for the given object of interest.
Automatic Segmentation
Interactive Segmentation
Automatic Segmentation

10. Energy Founction
Boykov Y Y, Jolly M P. Interactive graph cuts for optimal boundary & region segmentation of objects in ND images[C]//Computer Vision, ICCV etc. Times Cited: 3118.

11. (1) E(A) — Energy function global cost of cut A R(A) — regional term
B(A) — boundary term
— the coefficient specifies a relative important of R(A) versus B(A)
A=(A1,A2,...,AP)
a binary vector which represents certain segmentation
each AP can be either 1(obj)or 0(bkg)

12. Reginal term (2) RP(AP) : penalty for assigning pixel
P to "object"or "background"
Rp(obj) = -ln Pr(Ip|obj)
Rp(bkg) = -ln Pr(Ip|bkg)
RP(.) may reflect on how the intensity
of pixel P fits into a known intensity
model(e.g. histogram)of the object and background.
(2)

13. Boundary term B{p,q} : penalty for a discontinuity between p and q
(AP,Aq) :only consider pixel
pairs on the boundary

14. Weights of edges

15. Max flow/Min Cut s 16 13 10 a b b 4 12 14 9 c 7 d d 4 20 t
● flow : f(u,v) is a flow from u to v
● max flow : the max flow from s to t
● cut : C={S,T} s S,t T,S T= ,
● min cut : sum of the costs of the edges from S to T is minimum
● |f|max=C{S,T} s 16 13 10 a b b 4 12 14 9 c 7 d d 4 20 t

16. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 16 13 10 a b 4 12 14 9 c 7 d 4 20 t
FLOW=0

17. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 16 13 10 a b 4 12 14 9 c 7 d 4 20 t
FLOW=4

18. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s
16-4 13 +4 10 a b 4
12-4 +4
9-4
14-4 +4 +4 c 7 d
4-4 20 +4 t
FLOW=4

19. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 12 13 4 10 a b 4 8 4 5 10 4 4 c 7 d 20 4 t
FLOW=4

20. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 12 13 4 10 a b 4 8 4 5 10 4 4 c 7 d 20 4 t
FLOW=4

21. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 12 13 4 10 a b 4 8 4 5 10 4 4 c 7 d 20 4 t
FLOW=11

22. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s
12-7 13
4+7
10-7 a b
4+7 8 4 5
10-7
4+7 4
7-7 c d +7
20-7 4 +7 t
FLOW=11

23. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 5 13 11 3 a b 11 8 4 5 3 11 4 c d 7 13 4 7 t
FLOW=11

24. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 5 13 11 3 a b 11 8 4 5 3 11 4 c d 7 13 4 7 t
FLOW=11

25. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 5 13 11 3 a b 11 8 4 5 3 4 c d 7 13 4 7 t
FLOW=19

26. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s +8 5
13-8 11
3+8 a b
11-8
8-8
4+8 5 3 11 4 c d 7
13-8 4
7+8 t
FLOW=19

27. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 8 5 5 11 11 a b 3 12 5 3 11 4 c 7 d 5 4 15 t
FLOW=19

28. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 8 5 5 11 11 a b 3 12 5 3 11 4 c 7 d 5 4 15 t
FLOW=19

29. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 8 5 5 11 11 a b 3 12 5 3 11 4 c 7 d 5 4 15 t
FLOW=23

30. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s
8+4 5
5-4 11 11 a b 3 12
5+4 3 11
4-4 c 7 d
5-4 4
15+4 t
FLOW=23

31. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 12 5 1 11 11 a b 3 12 9 3 11 c 7 d 1 4 19 t
FLOW=23

32. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 12 5 1 11 11 a b 3 12 9 3 11 c 7 d 1 4 19 t
FLOW=23

33. Ford & Fulkerson algorithm
Find the path from source to sink
While (path exists)
flow += maximum capacity in the path
Build the residual graph (“subtract” the flow)
Find the augmenting path in the residual graph
End s 12 5 1 11 11 a b 3 12 9 3 11 c 7 d 1 4 19 t
FLOW=23

34. Max flow/Min Cut DFS Gf from S The min cut : S={a,b,d},T={c} s b a 125 1 11
DFS Gf from S
The min cut : S={a,b,d},T={c} 11 a b 3 12 9 3 11 c 7 d 1 4 19 t
FLOW = 23

35. Image Segmentation

36. Conclusion Advantage Disadvantage
●Robust and Efficient Optimization in N-D images
●global optimal ,better Segmentation
●reginal term and boundary term,better Segmentation
●Suitable for color image segmentation
●Simple and effective user interaction
Disadvantage
●gray value of background and object are similar
●background and object model are grey level histogram

37. Thank you!