From metric homotopies to nD persistence

1. From metric homotopies to nD persistence Massimo Ferri Mathematics Department Univ. of Bologna http://www.dm.unibo.it/~ferri/e.htm ferri@dm.unibo.it

2. From metric homotopies to nD persistence • The University of Bologna • Metric homotopies • Size Functions and Natural Pseudodistance • Applications • nD persistence • Work in progress and conclusions M. Ferri – From metric homotopies to nD persistence

3. The University of Bologna • Oldest in Europe? (Competitor: La Sorbonne) • Active at the end of the 11th century as a Law school • 1158: formal recognition by Emperor Frederick I Barbarossa M. Ferri – From metric homotopies to nD persistence

4. The University of Bologna Some “visiting professors”: • Dante Alighieri • Thomas Becket • Erasmus of Rotterdam A good student: • Nicolaus Copernicus Thomas Becket M. Ferri – From metric homotopies to nD persistence

5. The University of Bologna Mathematics in Bologna: • Luca Pacioli (14th c.): arithmetic and geometry • Rafael Bombelli (16th c.): invention of complex #’s • Scipione Dal Ferro, Gerolamo Cardano, Ludovico Ferrari (16th c.): 3rd and 4th degree formulas • Bonaventura Cavalieri, Pietro Mengoli (17th c.): early integral calculus • Maria Gaetana Agnesi (18th c.): analytical geometry • Luigi Cremona, Eugenio Beltrami, Beniamino Segre (19th-20th c.): algebraic geometry • Cesare Arzela`, Leonida Tonelli (20th c.): analysis M. Ferri – From metric homotopies to nD persistence

6. From metric homotopies to nD persistence • The University of Bologna • Metric homotopies • Size Functions and Natural Pseudodistance • Applications • nD persistence • Work in progress and conclusions M. Ferri – From metric homotopies to nD persistence

7. Metric homotopies M. Ferri – From metric homotopies to nD persistence

8. Metric homotopies M. Ferri – From metric homotopies to nD persistence

9. Metric homotopies Two minimal paths… M. Ferri – From metric homotopies to nD persistence

10. Metric homotopies …showing the lack of associativity M. Ferri – From metric homotopies to nD persistence

11. Metric homotopies • No way of obtaining a group • Just a fake one… A minimal path on a cube … which may be fairly complicated. M. Ferri – From metric homotopies to nD persistence

12. Metric homotopies My student Patrizio Frosini decided for a totally different approach: Size Functions Frosini, P., Measuring shapes by size functions, Proc. of SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA 1607 (1991). M. Ferri – From metric homotopies to nD persistence

13. The University of Bologna Metric homotopies Size Functions and Natural Pseudodistance Applications nD persistence Work in progress and conclusions From metric homotopies to nD persistence M. Ferri – From metric homotopies to nD persistence

14. Size Functions and Natural Pseudodistance M. Ferri – From metric homotopies to nD persistence

15. Size Functions and Natural Pseudodistance M. Ferri – From metric homotopies to nD persistence

16. Size Functions and Natural Pseudodistance M. Ferri – From metric homotopies to nD persistence

17. Size Functions and Natural Pseudodistance M. Ferri – From metric homotopies to nD persistence

18. Size Functions and Natural Pseudodistance Approximation translates into “blind strips”. M. Ferri – From metric homotopies to nD persistence

19. Size Functions and Natural Pseudodistance All information carried by a size function can be condensed in the formal series of its cornerpoints The matching distance M. Ferri – From metric homotopies to nD persistence

20. Size Functions and Natural Pseudodistance The matching distance between formal series of cornerpoints is stable under perturbation of the measuring function! M. Ferri – From metric homotopies to nD persistence

21. Size Functions and Natural Pseudodistance M. Ferri – From metric homotopies to nD persistence

22. Size Functions and Natural Pseudodistance It turns out that: i.e. the matching distance between size functions yields a lower bound (and an optimal one!) to the natural pseudodistance. M. Ferri – From metric homotopies to nD persistence

23. From metric homotopies to nD persistence • The University of Bologna • Metric homotopies • Size Functions and Natural Pseudodistance • Applications • nD persistence • Work in progress and conclusions M. Ferri – From metric homotopies to nD persistence

24. Applications • Classification problems: • Bologna • Leukocytes • Monograms • Sketches • Melanocytic lesions • Genova • Tree leaves • Numerals • Alphabet of the deaf • Cars M. Ferri – From metric homotopies to nD persistence

25. Applications M. Ferri – From metric homotopies to nD persistence

26. Applications M. Ferri – From metric homotopies to nD persistence

27. Applications M. Ferri – From metric homotopies to nD persistence

28. Applications Similitudes M. Ferri – From metric homotopies to nD persistence

29. Applications Affine transformations M. Ferri – From metric homotopies to nD persistence

30. Applications Homographies M. Ferri – From metric homotopies to nD persistence

31. Applications naevus melanoma M. Ferri – From metric homotopies to nD persistence

32. An image and one of its splittings. The curve of the image (meas. fct.: luminance). Applications M. Ferri – From metric homotopies to nD persistence

33. Applications M. Ferri – From metric homotopies to nD persistence

34. Applications • Image retrieval • Sea fauna • Silhouettes • Trade marks • “Keypics” M. Ferri – From metric homotopies to nD persistence

35. CSS our system CSS our system Applications Query 1 2 3 The challenge of a public database. M. Ferri – From metric homotopies to nD persistence

36. CSS our system CSS our system Applications Query 1 2 3 The challenge of a public database. M. Ferri – From metric homotopies to nD persistence

37. Applications Trade marks: two queries and the first eight retrieved images M. Ferri – From metric homotopies to nD persistence

38. Applications • We suggest that images on the Internet should be equipped with simplified sketches representing the essentials of the images themselves: keypics. • Keypics should be provided by the image owner or manager. • This graphical indexing might be extended to whole Web pages. • Encoding of keypics should be standard (e.g. in SVG). M. Ferri – From metric homotopies to nD persistence

39. Applications A Data Manager might wish to index the image of a saxophone by its geometrical outline, but also (or only) with a musical note. M. Ferri – From metric homotopies to nD persistence

40. Applications Some different keypic drawing conceptions. M. Ferri – From metric homotopies to nD persistence

41. Applications A retrieval experiment M. Ferri – From metric homotopies to nD persistence

42. Applications A retrieval experiment M. Ferri – From metric homotopies to nD persistence

43. From metric homotopies to nD persistence • The University of Bologna • Metric homotopies • Size Functions and Natural Pseudodistance • Applications • nD persistence • Work in progress and conclusions M. Ferri – From metric homotopies to nD persistence

44. nD persistence • k-dimensional measuring functions • Higher degree homology M. Ferri – From metric homotopies to nD persistence

45. nD persistence k-dimensional measuring functions Frosini, P., Mulazzani, M., Size homotopy groups for computation of natural size distances, Bull. of the Belgian Math. Soc. - Simon Stevin, 6 (1999), 455-464. M. Ferri – From metric homotopies to nD persistence

46. nD persistence M. Ferri – From metric homotopies to nD persistence

47. nD persistence M. Ferri – From metric homotopies to nD persistence

48. nD persistence M. Ferri – From metric homotopies to nD persistence

49. nD persistence Higher homology modules (persistent homology / size functor) M. Ferri – From metric homotopies to nD persistence

50. nD persistence M. Ferri – From metric homotopies to nD persistence