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Quasi-linear algorithms for the topological watershed

This work explores the powerful segmentation operator of watershed transform and efficient flooding algorithms for grayscale image segmentation. Contributors: Michel Couprie, Gilles Bertrand, Laurent Najman.

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Quasi-linear algorithms for the topological watershed

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  1. Quasi-linear algorithms for the topological watershed Michel Couprie

  2. Contributors to this work Gilles Bertrand Michel Couprie Laurent Najman

  3. Prelude

  4. Watersheds • Powerful segmentation operator from the field of Mathematical Morphology • Introduced as a tool for segmenting grayscale images by S. Beucher, H. Digabel and C. Lantuejoul in the 70s • Efficient algorithms based on immersion simulation were proposed by L. Vincent, F. Meyer, P. Soille (and others) in the 90s

  5. Watershed transform

  6. Flooding paradigm

  7. Flooding paradigm

  8. Meyer’s flooding algorithm Label the regional minima with different colors Repeat Select a pixel p, not colored, not watershed, adjacent to some colored pixels, and having the lowest possible gray level If p is adjacent to exactly one color then label p with this color If p is adjacent to more than one color then label p as watershed Until no such pixel exists

  9. Flooding algorithm 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 5 1 1 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  10. Flooding algorithm 3 3 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 1 20 3 3 3 3 1 20 3 3 5 5 5 10 10 10 1 10 1 15 20 1 3 3 3 3 1 20 3 3 5 5 30 30 30 10 1 15 15 1 20 3 3 3 3 1 20 3 3 5 30 20 20 20 30 1 15 15 1 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 10 1 1 10 10 5 1 5 5 5 10 40 20 40 10 10 1 1 5 5 5 0 1 1 1 3 5 10 1 15 20 1 15 10 5 1 1 1 0 1 1 1 0 1 1 3 5 10 15 20 15 10 5 1 0 1 1 0 1 1 3 5 10 15 20 15 10 5 1 0 1 1

  11. Flooding algorithm 3 3 3 20 1 3 3 3 3 5 5 5 10 10 10 10 1 15 1 1 20 3 3 3 3 3 1 20 3 3 3 3 5 5 5 10 10 10 1 10 1 15 1 20 3 3 3 3 3 1 20 3 3 3 3 5 5 30 30 30 10 15 1 15 20 1 3 3 3 3 3 1 20 3 3 3 3 5 30 20 20 20 30 1 15 15 1 1 20 3 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 5 1 5 5 5 10 40 20 40 10 10 1 1 5 5 5 0 1 1 3 5 10 1 15 20 1 15 10 1 5 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  12. Flooding algorithm 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 1 1 5 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  13. Flooding algorithm 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 1 1 5 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  14. Flooding algorithm 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 1 1 5 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  15. Flooding algorithm 20 1 3 3 5 5 5 10 10 10 1 10 1 15 1 20 3 20 1 3 3 5 5 5 10 10 10 1 10 1 15 1 20 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 1 20 3 3 5 30 20 20 20 30 15 1 15 1 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 1 10 10 10 10 40 20 20 20 40 1 10 10 1 1 10 10 5 1 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 0 1 3 5 10 1 15 20 1 15 10 5 1 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  16. Flooding algorithm 1 20 3 3 5 5 5 10 10 10 1 10 15 1 1 20 3 20 1 3 3 5 5 5 10 10 10 1 10 15 1 1 20 3 1 20 3 3 5 5 30 30 30 30 30 10 15 1 15 20 1 3 20 1 3 3 5 30 30 20 20 20 30 30 15 1 15 1 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 1 10 10 10 10 40 20 20 20 40 1 10 1 10 10 1 10 5 1 5 5 5 10 40 20 40 10 10 1 1 5 5 5 1 0 1 3 5 10 1 15 20 1 15 10 5 1 1 1 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  17. There is no descending path from the 20s to the minimum 3 The contrast between minima is not preserved Flooding algorithm 20 1 3 3 5 5 5 10 10 10 10 1 1 15 1 20 3 1 20 3 3 5 5 5 10 10 10 10 1 1 15 1 20 3 20 1 3 3 5 5 30 30 30 10 1 15 15 1 20 3 20 1 3 3 5 30 20 20 20 30 15 1 15 1 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 10 1 10 1 1 10 10 5 1 5 5 5 10 40 20 40 10 10 1 1 5 5 5 1 0 1 3 5 10 15 1 20 15 1 10 1 5 1 1 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  18. Flooding algorithm • Efficient implementation thanks to a priority queue (non-monotonic) • Preserves the connectivity of the binary set constituted by the regional minima of the original image • Uses the graylevels as a « heuristics » to guide the propagation of the colors associated to the different minima

  19. Topological watershed

  20. Discrete sets and W-destructible points Let (V,E) be a (undirected) graph andlet X be a subset of V.

  21. Graph

  22. Graph

  23. Discrete sets and W-destructible points Let (V,E) be a (undirected) graph and let X be a subset of V. We say that a point x  X is W-destructible for X if x is adjacent to exactly one connected component of X. M. Couprie and G. Bertrand (1997)

  24. W-destructible point

  25. Non W-destructible point

  26. W-destructible point

  27. W-destructible point

  28. Topological watersheds • Let X and Y be two subsets of V. We say that Y is a W-thinning of X if Y may be obtained from X by iteratively removing W-destructible points. • Let X and Y be subsets of V. We say that Y is a watershed of X if Y is a W-thinning of X and if there is no W-destructible point for Y. M. Couprie and G. Bertrand (1997)

  29. W-thinning

  30. W-thinning

  31. W-thinning

  32. W-thinning

  33. W-thinning

  34. W-thinning

  35. Watershed

  36. Watershed 1

  37. Watershed 2

  38. Discrete maps and W-destructible points • Let (V,E) be a connected (undirected) graph. We denote by F (V) the family composed of all maps from V to Z. • Let F  F (V), we set Fk = {x  V; F(x)  k}, Fk is the cross-sectionof F at level k • Let x  V and let k = F(x). We say that x is W-destructible (for F) if x is adjacent to exactly one connected component of Fk M. Couprie and G. Bertrand (1997)

  39. W-destructible points 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 5 1 1 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  40. W-destructible points 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 5 1 1 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 1 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  41. F1 W-destructible points 20 1 3 3 5 5 5 10 10 10 10 1 1 15 1 20 3 20 1 3 3 5 5 5 10 10 10 1 10 1 15 20 1 3 1 20 3 3 5 5 30 30 30 10 15 1 15 20 1 3 1 20 3 3 5 30 20 20 20 30 1 15 1 15 1 20 3 40 40 40 40 40 20 20 20 40 40 40 40 40 1 10 10 10 10 40 20 20 20 40 10 1 10 1 10 1 10 1 5 5 5 5 10 40 20 40 10 10 1 5 1 5 5 1 0 1 3 5 10 15 1 20 15 1 10 1 5 1 1 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  42. W-destructible points 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 5 1 1 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 0 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  43. Topological watershed • Let F and G be in F (V). • We say that G is a W-thinning of F if G may be obtained from F by iteratively lowering W-destructible points (by 1). • If G is a W-thinning of F which has been obtained by lowering a single point p (by 1 or by more than 1) we say that p has been W-lowered. • We say that G is a (topological) watershed of Fif G is a W-thinning of F and if there is no W-destructible point for G. M. Couprie and G. Bertrand (1997)

  44. Topological watershed 20 1 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 1 20 3 3 5 5 5 10 10 10 10 1 15 1 20 1 3 20 1 3 3 5 5 30 30 30 10 15 1 15 1 20 3 20 1 3 3 5 30 20 20 20 30 1 15 1 15 20 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 10 1 10 10 10 40 20 20 20 40 1 10 1 10 1 10 10 1 5 5 5 5 10 40 20 40 10 10 5 1 1 5 5 0 1 1 3 5 10 15 1 20 15 1 10 1 5 1 0 0 1 1 0 1 3 5 10 15 20 15 10 5 1 0 1 0 1 3 5 10 15 20 15 10 5 1 0 1

  45. Topological watershed 1 3 1 3 3 3 3 3 3 3 3 3 3 1 1 3 3 3 1 1 3 3 3 3 3 3 3 3 3 3 1 3 1 3 1 3 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 1 3 3 3 3 30 20 20 20 30 1 3 1 3 3 40 40 40 40 40 20 20 20 40 40 40 40 40 0 1 1 1 1 40 20 20 20 40 1 0 1 0 0 1 1 1 0 1 1 1 1 40 20 40 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 20 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 20 0 0 0 0 0 1 0 1 1 1 1 1 20 0 0 0 0 0 1

  46. Topological watershed 1 3 1 3 3 3 3 3 3 3 3 3 3 1 1 3 3 3 1 1 3 3 3 3 3 3 3 3 3 3 1 3 1 3 1 3 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 1 3 3 3 3 30 20 20 20 30 1 3 1 3 3 40 40 40 40 40 20 20 20 40 40 40 40 40 0 1 1 1 1 40 20 20 20 40 1 0 1 0 0 1 1 1 0 1 1 1 1 40 20 40 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 20 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 20 0 0 0 0 0 1 0 1 1 1 1 1 20 0 0 0 0 0 1

  47. F40 Topological watershed 1 3 1 3 3 3 3 3 3 3 3 3 3 1 1 3 3 3 1 1 3 3 3 3 3 3 3 3 3 3 1 3 1 3 1 3 3 3 3 3 3 30 30 30 3 1 3 3 3 3 1 1 3 3 3 3 30 20 20 20 30 3 1 3 1 3 40 40 40 40 40 20 20 20 40 40 40 40 40 0 1 1 1 1 40 20 20 20 40 1 0 1 0 1 0 1 0 1 1 1 1 1 40 20 40 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 20 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 20 0 0 0 0 0 1 0 1 1 1 1 1 20 0 0 0 0 0 1

  48. Topological watershed 1 3 1 3 3 3 3 3 3 3 3 3 3 1 1 3 3 3 1 1 3 3 3 3 3 3 3 3 3 3 1 3 1 3 1 3 3 3 3 3 3 30 30 30 3 1 3 3 3 1 3 1 3 3 3 3 30 20 20 20 30 1 3 1 3 3 40 40 40 40 40 20 20 20 40 40 30 40 40 0 1 1 1 1 40 20 20 20 40 1 0 1 0 0 1 1 1 0 1 1 1 1 40 20 40 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 20 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 20 0 0 0 0 0 1 0 1 1 1 1 1 20 0 0 0 0 0 1

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