International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and...

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

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SIZE FUNCTIONS: COMPARING SHAPES BY COUNTING

EQUIVALENCE CLASSES

Patrizio FrosiniVision Mathematics GroupUniversity of Bologna - Italy

http://vis.dm.unibo.it/


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Assume a size pair (M,) is given(i.e. M=topological space, :MIR)

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.


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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.


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We call size function (of the size pair (M,)) the function l (M,): IRxIRIN+ 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.


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Let us make the definition clear


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Example: M is the displayed curve, is the distance from C.

This is the size function l (M, ).


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More details about our example


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An evolving curve and its evolving size function


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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 .


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Theorem. Suppose that M is a closed C2-manifold and the measuring function is C1. 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:


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Main properties of Size Functions


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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)

sizeshow


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Size functions change the problem of comparing shapes into a simple comparison of functions (e.g., by using an Lp- norm)

(where K is a compact subset of the real plane)


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Size functions distribute the information all over the real plane, so that they can be used in presence of noise and occlusions.


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Resistance to perturbation


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Measuring function:(P,Q) = -||P-Q||

(Compare the green structures, revealing the presence of the house)


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Resistance to noise


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What about noise?

NOISE

Size functions count the number of connected components. How can they resist noise?


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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.


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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 =-dD/where =max{D(P):PR}.

Example


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This way we get resistance to noise:


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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.


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Resistance to occlusions


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Choosing the right topological space

Example: topological spaces M=AxA, N=BxB

Measuring functions: ((P,Q)) =((P,Q)) = -||P-Q||

A B


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is the fingerprint of the wrench

Measuring function:(P,Q) = -||P-Q||


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Size functions allow to choose the LEVEL LEVEL OF COMPARISONOF COMPARISON: details are described by small triangles close to the diagonal , while more general aspects of shape are described by large triangles .


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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.


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The size function with respect to P=||P-C|| does not see any deformation which preserves the distance from the center of mass C.

=


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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.

=


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However, size functions are mostly useful when there is NO GROUP OF INVARIANCE


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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.


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The algebraic representation of a size function by a formal series of lines and points.

r+a+b+c


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r’+a’+c’

Another size function with its formal series of lines and points.


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The algebraic representation of size functions allows us to compare them by a matching distance:


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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.

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