Overview of graph cuts

1. Overview of graph cuts

2. Outline • Introduction • S-t Graph cuts • Extension to multi-label problems • Compare simulated annealing and alpha-expansion algorithm

3. Introduction • Discrete energy minimization methods that can be applied to Markov Random Fields (MRF) with binary labels or multi-labels.

4. Outline • Introduction • S-t Graph cuts • Extensions to multi-label problems • Compare simulated annealing and alpha-expansion algorithm

5. Max flow / Min cut • Flow network • Maximize amount of flows from source to sink • Equal to minimum capacity removed from the network that no flow can pass from the source to the sink s t Max-flow/Min-cut method ：Augmenting paths (Ford Fulkerson Algorithm)

6. A subset of edges such that source and sink become separated G(C)=<V,E-C> the cost of a cut : Minimum cut : a cut whose cost is the least over all cuts S-t Graph Cut

7. How to separate a graph to two class? • Two pixels p1 and p2 corresponds to two class s and t. • Pixels p in the Graph classify by subtracting p with two pixels p1,p2. d1=(p-p1), d2 = (p-p2) • If d1 is closer zero than d2, p is class s. • Absolute of d1 and d2

8. Noise in the boundary of two class • The classified graph may have the noise occurs nearing the pixel (p1+p2)/2 • Adding another constrain (smoothing) to prevent this problem.

9. a cut C n-links t-link s t-link t energy function Regional term Boundary term n-links t-links

10. a cut C hard constraint n-links hard constraint s t S-t Graph cuts for optimal boundary detection Minimum cost cut can be computed in polynomial time

11. Global minimized for binary energy function Regional term Boundary term • Characterization of binary energies that can be globally minimized by s-t graph cuts t-links n-links E(f) can be minimized by s-t graph cuts (regular function)

12. What Energy Functions Can Be Minimized via Graph Cuts? • Regular F2 functions:

13. Outline • Introduction • S-t Graph cuts • Extensions to multi-label problems • Compare simulated annealing and alpha-expansion algorithm

14. Multi way Graph cut algorithm • NP-hard problem(3 or more labels) • two labels can be solved via s-t cuts (Greig et. al 1989) • Two approximation algorithms(Boykovet.al 1998,2001) Basic idea : break multi-way cut computation into a sequence of binary s-t cuts. • Alpha-expansion Each label competes with the other labels for space in the image • Alpha-beta swap Define a move which allows to change pixels from alpha to beta and beta to alpha

15. other labels a Alpha-expansion move Break multi-way cut computation into a sequence of binary s-t cuts

16. (|L|iterations) Alpha-expansion algorithm Stop when no expansion move would decrease energy

17. Alpha-expansion algorithm • Guaranteed approximation ratio by the algorithm: • Produces a labeling f such that ,where f* is the global minimum and Prove in : efficient graph-based energy minimization methods in computer vision

18. initial solution -expansion -expansion -expansion -expansion -expansion -expansion -expansion alpha-expansion moves

19. Alpha-Beta swap algorithm Handles more general energy function

20. Moves Initial labeling α-βswap αexpansion

21. Metric • Semi-metric • If V also satisfies the triangle inequality

22. Alpha-expansion : Metric • Alpha-expansion satisfy the regular function • Alpha-beta swap Prove in: what energy functions can be minimized via graph cuts?

23. V(dL) Potts model “linear” model V(dL) V(dL) dL=Lp-Lq dL=Lp-Lq dL=Lp-Lq Different types of Interaction V “Convex” Interactions V “discontinuity preserving” Interactions V V(dL) dL=Lp-Lq

24. truncated “linear” V “linear” V convex vs. discontinuity-preserving”

25. The use of Alpha-expansion and alpha-beta swap • Three energy function, each with a quadratic Dp. • E1 = Dp + min(K,|fp-fq|2) • E2 uses the Potts model • E3 = Dp + min(K,|fp-fq|) • E1 : semi-metric (use ) • E2,E3 : metric (can use both)

26. Outline • Introduction • S-t Graph cuts • Extensions to multi-label problems • Compare simulated annealing and alpha-expansion algorithm

27. Single “one-pixel” move (Simulated annealing) Single alpha-expansion move Large number of pixels can change their labels simultaneously Only one pixel change its label at a time Computationally intensive O(2^n) (s-t cuts)

28. 參考文獻 • Graph Cuts in Vision and Graphics: Theories and Application • Fast Approximate Energy Minimization via Graph Cuts , 2001 • What energy functions can be minimized via graph cuts?