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Segmentation by discrete watersheds Part 1: Watershed cuts Jean Cousty Four-Day Course on Mathematical Morphology in image analysis Bangalore 19-22 October 2010 J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 1/36 An

  1. Segmentation by discrete watersheds Part 1: Watershed cuts Jean Cousty Four-Day Course on Mathematical Morphology in image analysis Bangalore 19-22 October 2010 J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 1/36

  2. An applicative introduction to segmentation in medicine Magnetic Resonance Imagery (MRI) is more and more used for cardiac diagnosis J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 2/36

  3. An applicative introduction to segmentation in medicine A cardiac MRI examination includes three steps: Spatio-temporal acquisition (cin´ e MRI) im J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 2/36

  4. An applicative introduction to segmentation in medicine A cardiac MRI examination includes three steps: Spatio-temporal acquisition (cin´ e MRI) Spatio-temporal acquisition during contrast agent injection (Perfusion) im J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 2/36

  5. An applicative introduction to segmentation in medicine A cardiac MRI examination includes three steps: Spatio-temporal acquisition (cin´ e MRI) Spatio-temporal acquisition during contrast agent injection (Perfusion) Volumic acquisition after the evacuation of the contrast agent (delayed enhanced MRI) im J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 2/36

  6. Medical problem #1 Problem Visualizing objects of interests in 3D or 4D images rendu2 J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 3/36

  7. Medical problem #2 Problem Determining measures useful for cardiac diagnosis Infarcted volumes, ventricular volumes, ejection fraction, myocardial mass, movement . . . J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 4/36

  8. Technical problem Problem Segmentation of object of interest J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 5/36

  9. Technical problem Problem Segmentation of object of interest A morphological solution Watershed J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 5/36

  10. Watershed: introduction J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  11. Watershed: introduction For topographic purposes, the watershed has been studied since the 19th century (Maxwell, Jordan, . . . ) J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  12. Watershed: introduction One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  13. Watershed: introduction One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  14. Watershed: introduction One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  15. Watershed: introduction One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  16. Watershed: introduction One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  17. Watershed: introduction One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 6/36

  18. Watershed: problem #1 Problem How to define the watershed of digital image? J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 7/36

  19. Watershed: problem #1 Problem How to define the watershed of digital image? Which mathematical framework(s)? J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 7/36

  20. Watershed: problem #1 Problem How to define the watershed of digital image? Which mathematical framework(s)? Which properties? J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 7/36

  21. Watershed: problem #1 Problem How to define the watershed of digital image? Which mathematical framework(s)? Which properties? Which algorithms ? J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 7/36

  22. Watershed: problem #2 Problem In practice: over-segmentation J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 8/36

  23. Over-segmentation and region merging Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 9/36

  24. Over-segmentation and region merging Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions Example : delayed enhanced cardiac MRI [DOUBLIER03] J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 9/36

  25. Over-segmentation and region merging Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions Example : delayed enhanced cardiac MRI [DOUBLIER03] J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 9/36

  26. Over-segmentation and region merging Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions Example : delayed enhanced cardiac MRI [DOUBLIER03] J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 9/36

  27. Over-segmentation and region merging Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions cavité sanguine infarctus myocarde Example : delayed enhanced cardiac MRI [DOUBLIER03] J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 9/36

  28. Over-segmentation Solution 2 Seeded watershed (or marker based watershed) J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 10/36

  29. Over-segmentation Solution 2 Seeded watershed (or marker based watershed) Methodology proposed by Beucher and Meyer (1993) 1 Recognition 2 Delineation (generally done by watershed) 3 Smoothing J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 10/36

  30. Over-segmentation Solution 2 Seeded watershed (or marker based watershed) Methodology proposed by Beucher and Meyer (1993) 1 Recognition 2 Delineation (generally done by watershed) 3 Smoothing Semantic information taken into account at steps 1 and 3 To kow more about this framework, wait for the second lecture of today J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 10/36

  31. Outline 1 Defining discrete watersheds is difficult Grayscale image as vertex weighted graphs Region merging problems 2 Watershed in edge-weighted graphs Watershed cuts: definition and consistency Minimum spanning forests: watershed optimality J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 11/36

  32. Defining discrete watersheds is difficult Can we draw a watershed of this image? 2 2 2 2 2 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 1 5 20 5 1 Image equipped with the 4-adjacency J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 12/36

  33. Defining discrete watersheds is difficult Can we draw a watershed of this image? A A A A A 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 B 5 20 5 C Image equipped with the 4-adjacency Label the pixels according to catchment basins letters A , B and C J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 12/36

  34. Defining discrete watersheds is difficult Possible drawings A A A A A A A A A A 40 30 30 30 40 A A A A A 40 20 20 20 40 40 20 20 20 40 40 40 20 40 40 B B 20 C C B 5 20 5 C B B 20 C C Topographical watershed J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 13/36

  35. Defining discrete watersheds is difficult Possible drawings A A A A A A A A A A 40 30 30 30 40 A A A A A 40 20 20 20 40 C C C C C 40 40 20 40 40 B B C C C B 5 20 5 C B B C C C Flooding from the minima J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 13/36

  36. Defining discrete watersheds is difficult Possible drawings A A A A A A A A A A 40 30 30 30 40 A A A A A 40 20 20 20 40 40 A A A 40 40 40 20 40 40 B 40 A 40 C B 5 20 5 C B B 20 C C Flooding with divide J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 13/36

  37. Defining discrete watersheds is difficult Possible drawings A A A A A A A A A A 40 30 30 30 40 40 30 30 30 40 40 20 20 20 40 B B 20 C C 40 40 20 40 40 B B 20 C C B 5 20 5 C B B 20 C C Topological watershed J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology 13/36

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