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I THE NATURAL PSEUDODISTANCE: A GEOMETRIC-TOPOLOGICAL TOOL FOR COMPARING SHAPES

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/. A trivial example to point out the main idea.

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I THE NATURAL PSEUDODISTANCE: A GEOMETRIC-TOPOLOGICAL TOOL FOR COMPARING SHAPES

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  1. ITHE NATURAL PSEUDODISTANCE: AGEOMETRIC-TOPOLOGICAL TOOL FOR COMPARING SHAPES Patrizio Frosini Vision Mathematics Group University of Bologna - Italy http://vis.dm.unibo.it/ 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 y A x We want that the previous two topological spaces have a small distance. An idea: let us consider the function(P)=y/(max y – min y) In order to have two similar shapes, a homeomorphism f: A Apreserving the “measuring” function must exist. 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 Acan preserve the values taken by the considered “measuring” function? Let us consider the set H of ALL homeomorphisms from AtoA. For everyfHdefine( f )=maxP A(P)-(f (P)). Define International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

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

  5. Another example 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. An idea:let us consider the functions(P)=||P-B||/max||Q-B|| and (P)=||P-B’||/ max||Q-B’||where BandB’are the centres of mass. In order to have two similar shapes, a value-preservinghomeomorphism f: A  Amust exist. 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  Acan preserve the values taken by the considered “measuring” function? Let us consider the set H of ALL homeomorphisms from AtoA. For everyfH define( f )=maxP M (P)-(f (P)). Define International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

  7. An interesting remark: d=0 The function d does not see any deformation which preserves the distance from the center of mass. 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: N M We want that the previous two topological spaces have a small distance. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

  9. 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. N M In order to have two similar shapes, a value-preservinghomeomorphism between the two sets must exist. 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 everyfHdefine( f )=maxP M (P)-( f (P)). Define International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

  11. The general theoretical setting Consider two continuous functions :MIR, :NIR (called measuring functions). M, Ntopological spaces (or manifolds). H = a subset of the set of all homeomorphisms from M to N. Define where ( f )=maxPM(P)-(f (P)). 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: • ( f )0 • ( f )= ( f -1) • (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). 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 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|| A B International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

  15. The concept of optimality M = ellipsoidx2+4y2+9z2=1N = spherex2+y2+z2 =1. M N • = = Gaussian curvature d=35=max -max. Remark:for everyfH,( f )=d. We say that every homeomorphismfHis optimal. International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

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

  17. The concept of optimality N M • (x,y)= (x,y)=y, d=( (C)- (E))/2. Remark: nofHexists with ( f )=d. We say that no homeomorphismfHis optimal. 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 • the distance between two critical values of the measuring functions • the distance between two critical values of the measuring functions • half the distance between two critical values of the measuring functions International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

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

  20. What happens if we don’t know anything about the existence of an optimal homeomorphism? 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, Nclosed smooth manifolds and smooth measuring functions, the following statement holds: 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 International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

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

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

  26. If M, Nare closed smooth curves and , are smooth measuring functions, then a stronger statement holds: 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. (Proof based on a linearization process) International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

  27. If M, Nare closed smooth surfaces and , are smooth measuring functions, then a stronger statement holds: Theorem. The natural pseudodistance equals either D or D/2or 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) International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

  28. An open problem What happens in higher dimension? (n3) We don’t know (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). International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran

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

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