International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 1 III SIZE FUNCTIONS AS A TOOL FOR EVALUATING THE NATURAL PSEUDODISTANCE Patrizio Frosini Vision Mathematics Group University of Bologna - Italy

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 2 Assume that two size pairs (M,  ), (N,  ) are given and H is the set of all homeomorphisms from M to N. We recall the definition of natural pseudodistance between (M,  ) and (N,  ): where  ( f )=max P  M  (P)-  (f (P)) .

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 3 For every size pair (M,  ) we define the size function l (M,  ) : IR x IR  IN  +  : For x<y, l (M,  ) (x,y) is the number of connected components of M   y  that contain at least one point of M   x .

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 4 The lower bound theorem

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 5 Theorem. Assume that l (M,  ) (x 1,y 1 ) > l (N   x 2,y 2 ). Then d((M,  ),(N,  ))  min{x 2 -x 1, y 1 -y 2 }. The previous result allows to get information about the natural pseudodistance by using only the values of two size functions computed at two points. An important link between size functions and the natural pseudistance exists: N.B.: the previous result is useful if (x 1,y 1 ) is on the left of (x 2,y 2 ) and higher than it.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 6 Sketch of proof: d((M,  ),(N,  )) < min{x 2 -x 1, y 1 -y 2 } We prove the contrapositive statement: by assuming that we show that l (M,  ) (x 1,y 1 )  l (N   x 2,y 2 ).

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 7 The key idea is to prove that if two points P,Q belong to the same component C  M   y  and d((M,  ),(N,  )) < h then we can consider a homeomorphism f:M  N with  ( f )<h and say that the points f(P),f(Q) belong to the connected set f(C)  M   y+h .

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 8 M N  distance from C How to use the lower bound theorem  distance from C’

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 9 Difference function l (M,  ) l (N,  ) superimposition

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 10 Difference function

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 11 d((M,  ),(N,  ))  min{x 2 -x 1, y 1 -y 2 } We get a lower bound

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 12 It’s easy to see that when we try to get better lower bounds, we move the points towards the discontinuity set:

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 13 Since each discontinuity point has at least a coordinate that is a critical value for the measuring function, our best lower bound is either the distance between two critical values or half the distance between two critical values of the measuring function.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 14 A method for trying to compute the natural pseudodistance: 1)Compute the size functions of the size pairs (M,  ),(N,  ) ; 2)Compute the best lower bound B for the natural pseudodistance by the lower bound theorem; 3)If we are able to find a sequence of homeomorphisms ( f i ) with lim  ( f i )= B then B is the natural pseudodistance. Is it always possible to find such a sequence? NO

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 15 These two size pairs have the same size function (and hence the corresponding lower bound B vanishes), but their natural pseudodistance does not vanish! (Indeed d=1 ) M N  y  y Therefore we CANNOT find a sequence of homeomorphisms ( f i ) with lim  ( f i ) equal to our lower bound.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 16 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. We recall that f or M, N closed smooth manifolds and smooth measuring functions, the following statement holds: Since our lower bound can only equal D or D/2, the computational method that we have seen cannot prove the possible existence of examples with k  1,2.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 17 Theorem. Call  the matching distance between the size functions l (M,  ) and l (N . Then d((M,  ),(N,  ))  . The previous result allows to get information about the natural pseudodistance by computing the matching distance between the size functions. A greater lower bound exists, involving the matching distance between size functions:

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 18 How to get a lower bound by computing the matching distance between the size functions: d((M,  ),(N,  ))  . 

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 19 The inequality d   assures that a small change of the measuring function induces a small change of the size function. IMPORTANT REMARK Stability of size functions w.r.t. small changes of measuring functions. ( d((M,  ),(M,  )) is small, hence  is small, too!)

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 20 Two lower bound theorems for the natural pseudodistance were given. These theorems give us a method to try to compute the natural pseudodistance by using the size functions, and guarantee the stability of size functions with respect to small changes of the measuring functions. Summary