1. Invariants to translation and scaling Normalized central moments

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

4. Hard to find, easy to prove:

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

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

7. Normalized position to rotation

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

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

14. Basic relations between moments

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

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

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

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

20. Inverse problem - solution

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

22. Comparing B 3 to the Hu’s set

23. Drawbacks of the Hu’s invariants Dependence Incompleteness

24. Comparing B 3 to the Hu’s set - Experiment The images distinguishable by B 3 but not by Hu’s set

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

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

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

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

29. 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

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

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

33. Recognition of symmetric objects – Experiment 1 Bad choice: p 0 = 2, q 0 = 1

34. Recognition of symmetric objects – Experiment 1 Optimal choice: p 0 = 3, q 0 = 0

35. 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 = ∞

36. Recognition of symmetric objects – Experiment 2 Bad choice: p 0 = 2, q 0 = 1

37. Recognition of symmetric objects – Experiment 2 Better choice: p 0 = 4, q 0 = 0

38. Recognition of symmetric objects – Experiment 2 Theoretically optimal choice: p 0 = 12, q 0 = 0 Logarithmic scale

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

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

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

42. Normalized position to rotation

43. Rotation invariants via normalization