1. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 1 I THE NATURAL PSEUDODISTANCE: A GEOMETRIC-TOPOLOGICAL TOOL FOR COMPARING SHAPES Patrizio Frosini Vision Mathematics Group University of Bologna - Italy http://vis.dm.unibo.it/

2. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 2 A trivial example to point out the main idea Let us suppose to have to compare two planar shapes with respect to translations and dilations: A A We want that the previous two topological spaces have a small distance. In order to have two similar shapes, a homeomorphism f: A  A preserving the “measuring” function must exist. An idea: let us consider the function  (P)=y/(max y – min y) y x

3. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 3 How can we measure how well a homeomorphism f: A  A can preserve the values taken by the considered “measuring” function ? Let us consider the set H of ALL homeomorphisms from A to A. For every f  H define  ( f )=max P  A  (P)-  (f (P)) . Define

4. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 4 d=0d=0 An interesting remark: The function d does not see any horizontal deformations.

5. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 5 Another example In order to have two similar shapes, a value- preserving homeomorphism f: A  A must exist. An idea:let us consider the functions  (P)=||P-B||/max||Q-B|| and  (P)=||P-B’||/ max||Q-B’|| where B and B’ are the centres of mass. Let us suppose to have to compare these two planar shapes with respect to translations, rotations and dilations: We want that the previous two topological spaces have a small distance.

6. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 6 How can we measure how well a homeomorphism f: A  A can preserve the values taken by the considered “measuring” function? Let us consider the set H of ALL homeomorphisms from A to A. For every f  H define  ( f )=max P  M  (P)-  (f (P)) . Define

7. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 7 d=0d=0 An interesting remark: The function d does not see any deformation which preserves the distance from the center of mass.

8. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 8 One more example Let us suppose to have to compare these two shapes with respect to affine transformations: We want that the previous two topological spaces have a small distance. M N

9. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 9 In order to have two similar shapes, a value- preserving homeomorphism between the two sets must exist. An idea: let us consider the functions  and  which take each point P to the ratio between the area of the largest “internal” triangle touching P and the area of the shape. M N

10. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 10 How can we measure how well a homeomorphism f between the two sets can preserve the values taken by the considered “measuring” function? Let us consider the set H of ALL homeomorphisms between the two sets. For every f  H define  ( f )=max P  M  (P)-  ( f (P)) . Define

11. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 11 Consider two continuous functions  :M  IR,  :N  IR ( called measuring functions ). The general theoretical setting M, N topological spaces (or manifolds). H = a subset of the set of all homeomorphisms from M to N. where  ( f )=max P  M  (P)-  (f (P)) . Define

12. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 12 The function d is a pseudodistance between the pairs (M,  ), (N,  ) (called size pairs) In fact: 1)  ( f )  0 2)  ( f )=  ( f -1 ) 3)  (g  f )   (f ) +  (g ) N.B.: we are assuming that H(M,N) is obtained by composing the homeomorphisms in the sets H(M, L) and H(L, N).

13. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 13 Choosing the right topological space The right topological space doesn’t need to be “the object”. It may be “the rectangle of the image”, or something else. Example 1: topological space=rectangle Measuring function= normalized grey-level

14. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 14 Choosing the right topological space Example 2: topological spaces M =AxA, N =BxB Measuring functions:  ((P,Q)) =  ((P,Q)) = - ||P-Q|| AB

15. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 15 M = ellipsoid x 2 +4y 2 +9z 2 =1 N = sphere x 2 +y 2 +z 2 =1. Remark: for every f  H,  ( f )=d. We say that every homeomorphism f  H is optimal. MN The concept of optimality  =  = Gaussian curvature d=35=  max  -max .

16. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 16 Remark: no f  H exists with  ( f )=d. We say that no homeomorphism f  H is optimal. The concept of optimality  (x,y)=  (x,y)=y, d=  (A)-  (E) M N

17. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 17 Remark: no f  H exists with  ( f )=d. We say that no homeomorphism f  H is optimal.  (x,y)=  (x,y)=y, d=(  (C)-  (E))/2. M N The concept of optimality

18. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 18 We observe that in the last three examples the natural pseudodistance is 1) the distance between two critical values of the measuring functions 2) the distance between two critical values of the measuring functions 3) half the distance between two critical values of the measuring functions

19. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 19 Theorem. Suppose an optimal homeomorphism exists between M and N. Then the natural pseudodistance equals the Euclidean distance between two suitable critical values of the measuring functions. Why is the concept of optimality important?

20. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 20 Theorem. The natural pseudodistance equals D/k, where k is a positive integer and D is the Euclidean distance between two suitable critical values of the measuring functions. For M, N closed smooth manifolds and smooth measuring functions, the following statement holds: What happens if we don’t know anything about the existence of an optimal homeomorphism?

21. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 21 Sketch of proof: 1) We define the concept of waggon (P,Q) : where

22. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 22 An example:

23. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 23 2) We prove that we can change every d -approximating sequence into another d -approximating sequence (without increasing the number of waggons), whose maximal trains begin and end at critical points of the measuring functions.

24. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 24 The key move

25. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 25 3) The theorem follows by taking a maximal train Critical value

26. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 26 Theorem. The natural pseudodistance equals either D or D/2, where D is the Euclidean distance between two suitable critical values of the measuring functions. If M, N are closed smooth curves and ,  are smooth measuring functions, then a stronger statement holds: (Proof based on a linearization process)

27. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 27 Theorem. The natural pseudodistance equals either D or D/2 or D/3, where D is the Euclidean distance between two suitable critical values of the measuring functions. (Proof based on the theory of harmonic maps) If M, N are closed smooth surfaces and ,  are smooth measuring functions, then a stronger statement holds:

28. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 28 We don’t know What happens in higher dimension? (n  3) (The higher dimensional cases are important for getting invariance under affine and projective transformations) We don’t know examples for which the minimum value of k is strictly greater than 2. (Remember that we have proved min k  2 just for curves). An open problem

29. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran 29 1) Powerful for comparing shapes; 2) difficult to compute (we usually have to study all the homeomorphisms between two manifolds). We need a tool for studying the natural pseudodistance: the size functions. The natural pseudodistance is We have introduced the natural pseudodistance as a flexible variational tool for comparing shapes. Summary