Invariants to translation and scaling

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Invariants to translation and scaling
Normalized central moments
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Invariants to rotation
M.K. Hu, 1962 - 7 invariants of 3rd order
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Hard to find, easy to prove
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Drawbacks of the Hus invariants
Dependence
Incompleteness
Insufficient number ? low discriminability
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Consequence of the incompleteness of the Hus set
The images not distinguishable by the Hus set
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Normalized position to rotation
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Normalized position to rotation
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Invariants to rotation
M.K. Hu, 1962
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General construction of rotation invariants
Complex moment
Complex moment in polar coordinates
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Basic relations between moments
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Rotation property of complex moments
The magnitude is preserved, the phase is shifted
by (p-q)a. Invariants are constructed by phase
cancellation
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Rotation invariants from complex moments
Examples
How to select a complete and independent subset
(basis) of the rotation invariants?
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Construction of the basis
This is the basis of invariants up to the order r
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Inverse problem
Is it possible to resolve this system ?
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Inverse problem - solution
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The basis of the 3rd order
This is basis B3 (contains six real elements)
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Comparing B3 to the Hus set
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Drawbacks of the Hus invariants
Dependence
Incompleteness
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Comparing B3 to the Hus set - Experiment
The images distinguishable by B3 but not by Hus
set
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Difficulties with symmetric objects
Many moments and many invariants are zero
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Examples of N-fold RS
N 1 N 2 N 3
N 4 N 8
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Difficulties with symmetric objects
Many moments and many invariants are zero
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Difficulties with symmetric objects
The greater N, the less nontrivial invariants
Particularly
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Difficulties with symmetric objects
It is very important to use only non-trivial
invariants
The choice of appropriate invariants (basis of
invariants) depends on N
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The basis for N-fold symmetric objects
Generalization of the previous theorem
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Recognition of symmetric objects Experiment 1
5 objects with N 3
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Recognition of symmetric objects Experiment 1
Bad choice p0 2, q0 1
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Recognition of symmetric objects Experiment 1
Optimal choice p0 3, q0 0
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Recognition of symmetric objects Experiment 2
2 objects with N 1 2 objects with N 2 2
objects with N 3 1 object with N 4 2
objects with N 8
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Recognition of symmetric objects Experiment 2
Bad choice p0 2, q0 1
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Recognition of symmetric objects Experiment 2
Better choice p0 4, q0 0
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Recognition of symmetric objects Experiment 2
Theoretically optimal choice p0 12, q0 0
Logarithmic scale
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Recognition of symmetric objects Experiment 2
The best choice mixed orders
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Recognition of circular landmarks
Measurement of scoliosis progress during
pregnancy
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The goal to detect the landmark centers The
method template matching by invariants
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Normalized position to rotation
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Rotation invariants via normalization