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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

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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

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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

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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

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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

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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

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

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

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

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

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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

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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).

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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

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

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

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

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

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

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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

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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!

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

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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.

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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

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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

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

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

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

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

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

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

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

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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).

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

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

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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

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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

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

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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).

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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.

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

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

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

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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

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

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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

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

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

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

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

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

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

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

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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.

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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

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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

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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.

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

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