1. S I E M E N S C O R P O R A T E R E S E A R C H 1 1 A Seeded Image Segmentation Framework Unifying Graph Cuts and Random Walker Which Yields A New Algorithm Ali Kemal Sinop * Computer Science Department Carnegie Mellon University Leo Grady Department of Imaging and Visualization Siemens Corporate Research * work done while author was at Siemens Corporate Research

2. S I E M E N S C O R P O R A T E R E S E A R C H 2 2 Outline Review of seeded segmentation – Graph Cuts and Random Walker Our generalized framework The q = ∞ case Comparison Conclusion

3. S I E M E N S C O R P O R A T E R E S E A R C H 3 3 Seeded Segmentation Review Interactive segmentation Four parts: 1)Input foreground/background pixels (seeds) from the user 2)Use image content to establish affinity (metric) relationships between pixels 3)Perform energy minimization over the space of functions defined on pixels 4)Assign a foreground/background label to each pixel corresponding to the value of the function at that pixel

4. S I E M E N S C O R P O R A T E R E S E A R C H 4 4 Seeded Segmentation Review – Graph Cuts Graph cuts Abstract image to a weighted graph Compute min-cut/max-flow 4  4 image 4  4 weighted graph abstraction S T 1 5 1 1 4 2 3 2 6

5. S I E M E N S C O R P O R A T E R E S E A R C H 5 5 Seeded Segmentation Review – Random Walker Random Walker Abstract image to a weighted graph 4  4 image 4  4 weighted graph abstraction Compute probability that a random walker arrives at seed Random walk view Steady-state circuit view 1V 1 1 1 6 2 5 2 4 3

6. S I E M E N S C O R P O R A T E R E S E A R C H 6 6 Outline Review of seeded segmentation – Graph Cuts and Random Walker Our generalized framework The q = ∞ case Comparison Conclusion

7. S I E M E N S C O R P O R A T E R E S E A R C H 7 7 Generalized seeded segmentation framework ‘Algorithm A’ Choice of q determines solution properties: q = 1 Graph Cuts q = 2 Random Walker q = ∞ ?

8. S I E M E N S C O R P O R A T E R E S E A R C H 8 8 Generalized seeded segmentation: q = 1 Graph Cuts Unary terms implicit Note: If x is binary, energy represents cut size

9. S I E M E N S C O R P O R A T E R E S E A R C H 9 9 Generalized seeded segmentation: q = 2 Random Walker Solution to random walk problem equivalent to minimization of the Dirichlet integral with appropriate boundary conditions. The solution is given by a harmonic function, i.e., a function satisfying D [ u ] = 1 2 Z ­ ( g r u ( x ; y )) 2 r ¢ g r u = 0

10. S I E M E N S C O R P O R A T E R E S E A R C H 10 Generalized seeded segmentation: q = 2 Random Walker Subject to boundary conditions at seed locations x F = 1 ; x B = 0 Energy functional: r ¢ g r u = 0 L x = 0 Euler-Lagrange: D [ x ] = 1 2 ¡ x T A T ¢ C ( A x ) = 1 2 x T L x

11. S I E M E N S C O R P O R A T E R E S E A R C H 11 Outline Review of seeded segmentation – Graph Cuts and Random Walker Our generalized framework The q = ∞ case Comparison Conclusion

12. S I E M E N S C O R P O R A T E R E S E A R C H 12 The q = ∞ case q = ∞ How to optimize? where is the minimum distance from pixel i to a background seed

13. S I E M E N S C O R P O R A T E R E S E A R C H 13 The q = ∞ case Problem: Uniqueness Multiple solutions minimize functional Solution: Find the solution that additionally minimizes the (q = 2) energy

14. S I E M E N S C O R P O R A T E R E S E A R C H 14 Outline Review of seeded segmentation – Graph Cuts and Random Walker Our generalized framework The q = ∞ case Comparison Conclusion

15. S I E M E N S C O R P O R A T E R E S E A R C H 15 Comparison - Theoretical Metrication q = 1 (Graph Cuts) q = 2 (Random Walker) q = ∞

16. S I E M E N S C O R P O R A T E R E S E A R C H 16 Comparison - Quantitative Stability relationship

17. S I E M E N S C O R P O R A T E R E S E A R C H 17 Comparison - Qualitative q = 1 (Graph Cuts) q = 2 (Random Walker) q = ∞

18. S I E M E N S C O R P O R A T E R E S E A R C H 18 Outline Review of seeded segmentation – Graph Cuts and Random Walker Our generalized framework The q = ∞ case Comparison Conclusion

19. S I E M E N S C O R P O R A T E R E S E A R C H 19 Conclusion 1) Graph Cuts and Random Walker algorithms may be seen as minimizing the same functional with respect to an L 1 or L 2 norm, respectively 2) The L ∞ case was previously unexplored, may be optimized efficiently and produces “tight” segmentations with minimum sensitivity to seed number More information Random walkers paper: http://cns.bu.edu/~lgrady/grady2006random.pdf MATLAB toolbox for graph theoretic image processing at: http://eslab.bu.edu/software/graphanalysis/ Random walkers MATLAB code: http://cns.bu.edu/~lgrady/random_walker_matlab_code.zip L ∞ paper: http://cns.bu.edu/~lgrady/sinop2007linf.pdf