Size Function

1. Size Function Jianwei Hu 2007-05-23

2. Author Patrizio Frosini • Ricercatore presso la Facoltà di Ingegneria dell'Università di Bologna • Dipartimento di Matematica, Piazza di Porta San Donato, 5, BOLOGNA http://www.dm.unibo.it/~frosini/

3. References • Frosini, P., A distance for similarity classes of submanifolds of a Euclidean space, Bull. Austral. Math. Soc. 42, 3 (1990), 407-416. • Verri, A., Uras, C., Frosini, P., Ferri, M., On the use of size functions for shape analysis, Biol. Cybern. 70, (1993), 99-107. • Frosini, P., Landi, C., Size Theory as a Topological Tool for Computer Vision, Pattern Recognition and Image Analysis, Vol. 9, No. 4, 596-603, 1999. • Frosini, P., Pittore, M., New methods for reducing size graphs, Intern. J. Computer Math. 70, 505-517, 1999. • Frosini, P., Landi, C., Size functions and formal series, Applicable Algebra in Engin. Communic. Comput., 12(4) (2001), 327-349. • Cerri, A., Ferri, M., Giorgi, D., Retrieval of trademark images by means of size functions, Graph. Models, 68 (2006), 451-471. • d'Amico, M., Frosini, P., and Landi, C., Using matching distance in Size Theory: a survey, International Journal of Imaging Systems and Technology, Vol. 16 (2006) , No. 5, 154–161. • Donatini, P., Frosini, P., Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, Vol. 9 (2007), No. 2, 231–253. • d'Amico, M., Frosini, P., and Landi, C., Natural pseudo-distance and optimal matching between reduced size functions (submitted).

4. Outline • General Concepts of Size Function • Definition • Invariant Properties • Comparing Size Function • Corner Points & Formal Series • Reducing Size Graphs • L-reduction • ⊿-reduction • Measuring Functions • Applications • Images Retrieval • 3D Shape Matching

5. What are Size Functions • Size Functions are a new kind of mathematical transform • Size Functions are a mathematical tool for describing and comparing shapes of topological spaces • Shape Size graph Natural number size function measuring function http://vis.dm.unibo.it/sizefcts/FAQ/faq.htm

6. Definitions • Definition 1: Size Pair • is a compact topological space. • is a continuous function from to the set (called measuring function). • Definition 2: homotopy For every we define a relation in by setting if and only if either or there exists a continuous path such that and for every . In this second case we shall say that and are homotopic and call a homotopy from to . The BULLETIN of the Australian Mathematical Society 1990

7. Definitions (Contd.) • Remark 3: For every we shall denote by the set . • Definition 4: Size Function Consider the function defined by setting equal to the (finite or infinite) number of equivalence classes in which is divided by the equivalence relation . Such a function will be called the size function associated with the size pair . The BULLETIN of the Australian Mathematical Society 1990

8. Example http://vis.dm.unibo.it/sizefcts/FAQ/faq.htm

9. Invariant Properties • Euclidean Invariance Biological Cybernetics 1993

10. Invariant Properties • “Ad hoc” Invariance Biological Cybernetics 1993

11. Resistant to Noise Biological Cybernetics 1993

12. Resistant to Occlusions Biological Cybernetics 1993

13. Concepts for Comparison • Cornerpoint • Formal Series • 3A+B+4C+5D+E Applicable Algebra in Engineering, Communication and Computing 2001

14. How to Compare Compare formal series and • Hausdorff distance • Two sets and • Matching distance • Two sets and • is the set of all bijections from to Applicable Algebra in Engineering, Communication and Computing 2001

15. Reduction of Size Graphs • A global method: L-reduction • A local method: ⊿-reduction International Journal of Computer Mathematics 1999

16. L-reduction • is the set of one ring neighbor of • is the set for which takes the largest value • is the single step descent flow function • is the descent flow operator • Minimum vertex • Main saddle International Journal of Computer Mathematics 1999

17. L-reduction International Journal of Computer Mathematics 1999

18. ⊿-reduction • Three simple ⊿-moves International Journal of Computer Mathematics 1999

19. ⊿-reduction International Journal of Computer Mathematics 1999

20. ⊿-reduction • Does a total ⊿-reduction exist? • Two different ways to obtain the same total ⊿-reduction International Journal of Computer Mathematics 1999

21. L-reduction vs ⊿-reduction LKO ⊿ ⊿KO L International Journal of Computer Mathematics 1999

22. L-reduction vs ⊿-reduction • Sometimes L-reduction makes the size graph worse • The procedure of applying simple ⊿-moves cannot proceed indefinitely International Journal of Computer Mathematics 1999

23. Measuring Functions • Distance from points • Projections • Jumps Graphical Models 2006

24. Images Retrieval Graphical Models 2006

25. 3D Shape Matching • Measuring Functions • Distance from the center of mass to each vertex • Transformations invariance • Distance from some fixed planes • Distance from the point user specified • Deformed model retrieval • Curvature of each point (patch) • Feature sensitive

26. 3D Shape Matching • Size graph reduction Salient Geometric Features for Partial Shape Matching and Similarity, Ran Gal and Daniel Cohen-or, ACM Transactions on Graphics, Vol. 25, No. 1, January 2006, Pages 130–150.

27. Thank you