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Invariants to translation and scaling

Invariants to translation and scaling. Normalized central moments. Invariants to rotation. M.K. Hu, 1962 - 7 invariants of 3rd order. Hard to find, easy to prove:. Drawbacks of the Hu’s invariants. Dependence. Incompleteness. Insufficient number  low discriminability.

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Invariants to translation and scaling

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  1. Invariants to translation and scaling Normalized central moments

  2. Invariants to rotation M.K. Hu, 1962 - 7 invariants of 3rd order

  3. Hard to find, easy to prove:

  4. Drawbacks of the Hu’s invariants Dependence Incompleteness Insufficient number low discriminability

  5. Consequence of the incompleteness of the Hu’s set The images not distinguishable by the Hu’s set

  6. Normalized position to rotation

  7. Normalized position to rotation

  8. Invariants to rotation M.K. Hu, 1962

  9. General construction of rotation invariants Complex moment Complex moment in polar coordinates

  10. Basic relations between moments

  11. Rotation property of complex moments The magnitude is preserved, the phase is shifted by (p-q)α. Invariants are constructed by phase cancellation

  12. Rotation invariants from complex moments Examples: How to select a complete and independent subset (basis) of the rotation invariants?

  13. Construction of the basis This is the basis of invariants up to the order r

  14. Inverse problem Is it possible to resolve this system ?

  15. Inverse problem - solution

  16. The basis of the 3rd order This is basis B3 (contains six real elements)

  17. Comparing B3to the Hu’s set

  18. Drawbacks of the Hu’s invariants Dependence Incompleteness

  19. Comparing B3to the Hu’s set - Experiment The images distinguishable by B3 but not by Hu’s set

  20. Difficulties with symmetric objects Many moments and many invariants are zero

  21. Examples of N-fold RS N = 1 N = 2 N = 3 N = 4 N = ∞

  22. Difficulties with symmetric objects Many moments and many invariants are zero

  23. Difficulties with symmetric objects The greater N, the less nontrivial invariants Particularly

  24. 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

  25. The basis for N-fold symmetric objects Generalization of the previous theorem

  26. Recognition of symmetric objects – Experiment 1 5 objects with N = 3

  27. Recognition of symmetric objects – Experiment 1 Bad choice: p0 = 2, q0 = 1

  28. Recognition of symmetric objects – Experiment 1 Optimal choice: p0 = 3, q0 = 0

  29. 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 = ∞

  30. Recognition of symmetric objects – Experiment 2 Bad choice: p0 = 2, q0 = 1

  31. Recognition of symmetric objects – Experiment 2 Better choice: p0 = 4, q0 = 0

  32. Recognition of symmetric objects – Experiment 2 Theoretically optimal choice: p0 = 12, q0 = 0 Logarithmic scale

  33. Recognition of symmetric objects – Experiment 2 The best choice: mixed orders

  34. Recognition of circular landmarks Measurement of scoliosis progress during pregnancy

  35. The goal: to detect the landmark centers The method: template matching by invariants

  36. Normalized position to rotation

  37. Rotation invariants via normalization

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