Size Function Jianwei Hu

Author Patrizio Frosini Ricercatore presso la Facolt à di Ingegneria dell'Universit à di Bologna Ricercatore presso la Facolt à di Ingegneria dell'Universit à di Bologna Dipartimento di Matematica, Piazza di Porta San Donato, 5, BOLOGNA Dipartimento di Matematica, Piazza di Porta San Donato, 5, BOLOGNA

References 1. Frosini, P., A distance for similarity classes of submanifolds of a Euclidean space, Bull. Austral. Math. Soc. 42, 3 (1990), Verri, A., Uras, C., Frosini, P., Ferri, M., On the use of size functions for shape analysis, Biol. Cybern. 70, (1993), Frosini, P., Landi, C., Size Theory as a Topological Tool for Computer Vision, Pattern Recognition and Image Analysis, Vol. 9, No. 4, , Frosini, P., Pittore, M., New methods for reducing size graphs, Intern. J. Computer Math. 70, , Frosini, P., Landi, C., Size functions and formal series, Applicable Algebra in Engin. Communic. Comput., 12(4) (2001), Cerri, A., Ferri, M., Giorgi, D., Retrieval of trademark images by means of size functions, Graph. Models, 68 (2006), 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 – Donatini, P., Frosini, P., Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, Vol. 9 (2007), No. 2, 231 – d'Amico, M., Frosini, P., and Landi, C., Natural pseudo-distance and optimal matching between reduced size functions (submitted).

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

What are Size Functions Size Functions are a new kind of mathematical transform Size Functions are a new kind of mathematical transform Size Functions are a mathematical tool for describing and comparing shapes of topological spaces Size Functions are a mathematical tool for describing and comparing shapes of topological spaces Shape Size graph Natural number Shape Size graph Natural number measuring function size function

Definitions Definition 1: Size Pair Definition 1: Size Pair is a compact topological space. is a compact topological space. is a continuous function from to the set (called measuring function). is a continuous function from to the set (called measuring function). Definition 2: homotopy 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

Definitions (Contd.) Remark 3: Remark 3: For every we shall denote by the set. Definition 4: Size Function 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

Example

Invariant Properties Euclidean Invariance Euclidean Invariance Biological Cybernetics 1993

Invariant Properties “ Ad hoc ” Invariance“ Ad hoc ” Invariance Biological Cybernetics 1993

Resistant to Noise Biological Cybernetics 1993

Resistant to Occlusions Biological Cybernetics 1993

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

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

Reduction of Size Graphs A global method:  -reduction A global method:  -reduction A local method: ⊿ -reduction A local method: ⊿ -reduction International Journal of Computer Mathematics 1999

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

 -reduction International Journal of Computer Mathematics 1999

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

⊿ -reduction International Journal of Computer Mathematics 1999

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

 -reduction vs ⊿ -reduction  KO ⊿ ⊿  KO  International Journal of Computer Mathematics 1999

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

Measuring Functions Distance from points Distance from points Projections Projections Jumps Jumps Graphical Models 2006

Images Retrieval Graphical Models 2006

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

3D Shape Matching Size graph reduction 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.