Invariants to translation and scaling
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Invariants to translation and scaling
Normalized central moments
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Invariants to rotationM.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 Hu’s invariants
Dependence
Incompleteness
Insufficient number low discriminability
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Consequence of the incompleteness of the Hu’s set
The images not distinguishable by the Hu’s set
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Normalized position to rotation
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Normalized position to rotation
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Invariants to rotationM.K. Hu, 1962
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General construction of rotation invariants
Complex moment in polar coordinates
Complex moment
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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)α.
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 Hu’s set
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Drawbacks of the Hu’s invariants
Dependence
Incompleteness
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Comparing B3 to the Hu’s set - Experiment
The images distinguishable by B3 but not by Hu’s 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 = ∞
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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 = ∞
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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