From metric homotopies to nD persistence Massimo Ferri Mathematics Department Univ. of Bologna ferri/e.htm [email protected].

From metric homotopies to nD From metric homotopies to nD persistencepersistence

Massimo Ferri

Mathematics Department

Univ. of Bologna

http://www.dm.unibo.it/~ferri/e.htm

[email protected]

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

From metric homotopies to nD persistenceFrom metric homotopies to nD persistence

• The University of Bologna• Metric homotopies• Size Functions and Natural Pseudodistance• Applications• nD persistence• Work in progress and conclusions


The University of BolognaThe 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


The University of BolognaThe University of Bologna

Some “visiting professors”:• Dante Alighieri• Thomas Becket • Erasmus of Rotterdam

A good student:• Nicolaus Copernicus

Thomas Becket


The University of BolognaThe University of BolognaMathematics 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





Metric homotopiesMetric homotopies





Two minimal paths…



…showing the lack of associativity



• No way of obtaining a group• Just a fake one…

… which may be fairly complicated.

A minimal path on a cube



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).





Size Functions and Natural PseudodistanceSize Functions and Natural Pseudodistance









Approximation translates into “blind strips”.



All information carried by a size function can be condensed in the formal series of its cornerpoints

The matching distance



The matching distance between formal series of cornerpoints is stable under perturbation of the measuring function!





It turns out that:

i.e. the matching distance between size functions yields a lower bound (and an optimal one!) to the natural pseudodistance.





ApplicationsApplications

• Classification problems:Bologna

• Leukocytes• Monograms• Sketches• Melanocytic lesions

Genova• Tree leaves• Numerals• Alphabet of the deaf• Cars









Similitudes



Affine transformations



Homographies



melanoma

naevus



An image and one of its splittings.

The curve of the image(meas. fct.: luminance).





• Image retrievalSea faunaSilhouettesTrade marks

• “Keypics”



Query 1 2 3

The challenge of a public database.

CSS

CSS

our system

our system


ApplicationsApplicationsQuery 1 2 3

The challenge of a public database.

CSS

CSS

our system

our system



Trade marks: two queries and the first eight retrieved images



• 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).



A Data Manager might wish to index the image of a saxophone by its geometrical outline, but also (or only) with a musical note.



Some different keypic drawing conceptions.



A retrieval experiment



A retrieval experiment





nD persistencenD persistence

• k-dimensional measuring functions• Higher degree homology



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









Higher homology modules (persistent homology / size functor)





Persistent homology for k-dimensional measuring functions





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.





Work in progress and conclusionsWork 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


Work in progress and conclusionsWork 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.


THANK YOU FOR YOUR ATTENTION !

From metric homotopies to nD persistence Massimo Ferri Mathematics Department Univ. of Bologna ferri/e.htm [email protected].

Documents

natural pseudodistance

nd persistence657

nd persistence1357

nd persistence2357

nd persistence357

nd persistence557

nd persistence457

barbarossa slide