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From metric homotopies to nD persistence Massimo Ferri Mathematics Department Univ. of Bologna

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Presentation on theme: "From metric homotopies to nD persistence Massimo Ferri Mathematics Department Univ. of Bologna"— Presentation transcript:

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 Ljubljana, 12/4/2012 M. Ferri – From metric homotopies to nD persistence 2/57 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

3 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence3/57 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

4 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence4/57 The University of Bologna Some visiting professors: Dante Alighieri Thomas Becket Erasmus of Rotterdam A good student: Nicolaus Copernicus Thomas Becket

5 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence5/57 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

6 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence6/57 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

7 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence7/57 Metric homotopies

8 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence8/57 Metric homotopies

9 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence9/57 Metric homotopies Two minimal paths…

10 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence10/57 Metric homotopies …showing the lack of associativity

11 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence11/57 Metric homotopies No way of obtaining a group Just a fake one… … which may be fairly complicated. A minimal path on a cube

12 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence12/57 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).

13 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence13/57 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

14 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence14/57 Size Functions and Natural Pseudodistance

15 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence15/57 Size Functions and Natural Pseudodistance

16 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence16/57 Size Functions and Natural Pseudodistance

17 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence17/57 Size Functions and Natural Pseudodistance

18 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence18/57 Size Functions and Natural Pseudodistance Approximation translates into blind strips.

19 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence19/57 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

20 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence20/57 Size Functions and Natural Pseudodistance The matching distance between formal series of cornerpoints is stable under perturbation of the measuring function!

21 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence21/57 Size Functions and Natural Pseudodistance

22 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence22/57 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.

23 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence23/57 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

24 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence24/57 Applications Classification problems: Bologna Leukocytes Monograms Sketches Melanocytic lesions Genova Tree leaves Numerals Alphabet of the deaf Cars

25 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence25/57 Applications

26 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence26/57 Applications

27 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence27/57 Applications

28 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence28/57 Applications Similitudes

29 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence29/57 Applications Affine transformations

30 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence30/57 Applications Homographies

31 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence31/57 Applications melanoma naevus

32 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence32/57 Applications An image and one of its splittings. The curve of the image (meas. fct.: luminance).

33 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence33/57 Applications

34 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence34/57 Applications Image retrieval Sea fauna Silhouettes Trade marks Keypics

35 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence35/57 Applications Query 1 2 3 The challenge of a public database. CSS our system

36 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence36/57 Applications Query 1 2 3 The challenge of a public database. CSS our system

37 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence37/57 Applications Trade marks: two queries and the first eight retrieved images

38 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence38/57 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).

39 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence39/57 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.

40 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence40/57 Applications Some different keypic drawing conceptions.

41 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence41/57 Applications A retrieval experiment

42 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence42/57 Applications A retrieval experiment

43 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence43/57 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

44 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence44/57 nD persistence k-dimensional measuring functions Higher degree homology

45 Ljubljana, 12/4/2012 M. Ferri – From metric homotopies to nD persistence 45/57 nD persistence 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. k-dimensional measuring functions

46 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence46/57 nD persistence

47 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence47/57 nD persistence

48 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence48/57 nD persistence

49 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence49/57 nD persistence Higher homology modules (persistent homology / size functor)

50 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence50/57 nD persistence

51 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence51/57 nD persistence Persistent homology for k-dimensional measuring functions

52 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence52/57 nD persistence

53 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence53/57 nD persistence The reduction theorem from k-dimensional to 1- dimensional measuring functions - through maxima along admissible pairs - works unaltered also for higher homology modules! Cagliari, F., Di Fabio, B., Ferri, M., One-dimensional reduction of multidimensional persistent homology, Proc. Amer. Math. Soc. 138 (2010), 3003-3017.

54 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence54/57 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

55 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence55/57 Work in progress and conclusions Present research: Search for optimal admissible pairs Better bounds for the natural pseudodistance Algorithms Applications to colour images and 3D meshes New applications to melanoma diagnosis

56 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence56/57 Work in progress and conclusions The theory of Size Functions is developing along two directions: k-dimensionality and higher homology modules. Together with concrete applications in Pattern Recognition, we would like to find contact points with other theoretical domains.

57 Ljubljana, 12/4/2012M. Ferri – From metric homotopies to nD persistence57/57 THANK YOU FOR YOUR ATTENTION !


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