International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 1 II SIZE FUNCTIONS: COMPARING SHAPES BY COUNTING EQUIVALENCE CLASSES 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 a size pair (M,  ) is given (i.e. M =topological space,  :M  IR) We want to take each size pair into a function describing the shape of M with respect to . Instead of comparing manifolds, we shall compare these descriptors.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 3 Definition of size function Let M be a topological space and assume that a continuous function  : M  IR is given. For each real number y we denote by M   y  the set of all points of M at which the measuring function  takes a value not greater than y. For each real number y we say that two points P,Q are   y  -connected if and only if they belong to the same component in M   y .

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 4 We call size function (of the size pair (M,  )) the function l (M,  ) : IR x IR  IN  +  that takes each point (x,y) of the real plane to the number of equivalence classes of M   x  with respect to   y  -connectedness. EQUIVALENT DEFINITION 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 5 Let us make the definition clear

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 6 Example: M is the displayed curve,  is the distance from C. This is the size function l (M,  ).

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 7 More details about our example

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 8

9

10

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 11 An evolving curve and its evolving size function

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 12 We observe that 1) l (M,  ) (x,y) is non-decreasing in x and non- increasing in y; 2) Discontinuities in x propagate vertically towards the diagonal  ={(x,y):x=y}, while discontinuities in y propagate horizontally towards the diagonal .

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 13 Theorem. Suppose that M is a closed C 2 - manifold and the measuring function  is C 1. If (x,y) is a discontinuity point for the size function and x<y, then either x or y or both are critical values for . The following result locates the discontinuity points of a size function:

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 14 Main properties of Size Functions

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 15 Size functions are easily computable (from the discrete point of view, we only have to count the connected components of a graph) (We shall tell about this in the last lecture)

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 16 Size functions change the problem of comparing shapes into a simple comparison of functions (e.g., by using an L p - norm) (where K is a compact subset of the real plane)

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 17 Size functions distribute the information all over the real plane, so that they can be used in presence of noise and occlusions.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 18 Resistance to perturbation

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 19 Measuring function:  (P,Q) = -||P-Q|| (Compare the green structures, revealing the presence of the house)

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 20 Resistance to noise

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 21 What about noise? NOISE Size functions count the number of connected components. How can they resist noise?

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 22 Changing the topological space We have to remember that the topological space doesn’t need to be the original object. The topological space may be, e.g., the rectangle cointaining the image The measuring function  may be, e.g., depending on the local density of black pixels. This way we get resistance to noise.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 23 Call D(P) the local density of black points at a point P in the rectangle R. Call d(P) the distance of P from the center of mass of the set of black points.We can set  =-d  D/  where  =max{D(P):P  R}. Example

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 24 This way we get resistance to noise:

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 25 Getting resistance to noise by a different kind of measuring function: the  –technique. In this approach, we take the rectangle of the image as a topological space and change the measuring function by adding a function . The function  increases when we leave the shape we are studying. This way we can bridge the gap between two components, provided that the cost  is paid.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 26 Resistance to occlusions

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 27 Choosing the right topological space Example: topological spaces M =AxA, N =BxB Measuring functions:  ((P,Q)) =  ((P,Q)) = - ||P-Q|| AB

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 28 is the fingerprint of the wrench Measuring function:  (P,Q) = -||P-Q||

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 29 LEVEL OF COMPARISON Size functions allow to choose the LEVEL OF COMPARISON: details are described by small triangles close to the diagonal , while more general aspects of shape are described by large triangles.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 30 Size functions inherit invariance from measuring functions. Hence we can get the invariance we want. = The size function with respect to  =y does not see any horizontal deformations.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 31 The size function with respect to  P  =||P-C|| does not see any deformation which preserves the distance from the center of mass C. =

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 32 Measuring functions invariant under affine or projective transformations can easily be obtained. Therefore, we can easily get size functions invariant under affine or projective transformations. =

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 33 However, size functions are mostly useful when there is NO GROUP OF INVARIANCE

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 34 A curvature driven plane curve evolution and the corresponding size function (w.r.t. the distance from the center of mass) Thanks to Frederic Cao for the curvature evolution code and to Michele d’Amico for making this animation.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 35 The algebraic representation of a size function by a formal series of lines and points. r+a+b+c

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 36 r’+a’+c’ Another size function with its formal series of lines and points.

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 37 The algebraic representation of size functions allows us to compare them by a matching distance:

International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 38 We have introduced the size function as another flexible geometrical-topological tool for comparing shapes. Size functions are Summary 1)Suitable for comparing shapes; 2)Easy to compute; 3)Easy to compare; 4)Resistant to noise and occlusions (if suitable topological spaces and measuring functions are chosen); 5)Suitable for multilevel description of shape; 6)Suitable for getting the invariance we want; 7)Describable in a compact way by using formal series.