Post on 19-Dec-2015
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/
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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Let us make the definition clear
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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 26-30 2004, Tehran
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More details about our example
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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An evolving curve and its evolving size function
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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 .
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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:
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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Main properties of Size Functions
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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)
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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Resistance to perturbation
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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 26-30 2004, Tehran
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Resistance to noise
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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This way we get resistance to noise:
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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Resistance to occlusions
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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 26-30 2004, Tehran
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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 .
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
=
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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.
=
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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 26-30 2004, Tehran
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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.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April 26-30 2004, Tehran
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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 26-30 2004, Tehran
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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 26-30 2004, Tehran
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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 26-30 2004, Tehran
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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.