1. Institute of Information Theory and Automation Prague, Czech Republic Flinders University of South Australia Adelaide, Australia Object Recognition by Implicit Invariants Jan Flusser Jaroslav Kautsky Filip Šroubek

2. General motivation How can we recognize deformed objects?

5. Curved surface  deformation of the image g = D(f) D - unknown deformation operator Problem formulation

6. What are explicit invariants? Functionals defined on the image space L such that E(f) = E(D(f)) for all admissible D Fourier descriptors, moment invariants,...

7. What are explicit invariants? Functionals defined on the image space L such that E(f) = E(D(f)) for all admissible D For many deformations explicit invariants do not exist.

8. What are implicit invariants? Functionals defined on L x L such that I(f,D(f)) = 0 for all admissible D Implicit invariants exist for much bigger set of deformations

9. Our assumption about D Image deformation is a polynomial transform r(x) of order > 1 of the spatial coordinates f’(r(x)) = f(x)

10. What are moments? Moments are “projections” of the image function into a polynomial basis

11. How are the moments transformed? A depends on r and on the polynomial basis A is not a square matrix Transform r does not preserve the order of the moments Explicit moment invariants cannot exist. If they existed, they would contain all moments. m’ = A.m

12. Construction of implicit moment invariants Eliminate the parameters of r from the system Each equation of the reduced system is an implicit invariant m’ = A.m

13. Artificial example

14. Invariance property

15. Robustness to noise

16. Object recognition Amsterdam Library of Object Images http://staff.science.uva.nl/˜aloi/

17. ALOI database 99% recognition rate

18. The bottle

21. The bottle again

27. 100% recognition rate

28. Implementation How to avoid numerical problems with high dynamic range of standard moments?

29. Implementation How to avoid numerical problems with high dynamic range of standard moments? We used orthogonal Czebyshev polynomials

30. Summary We proposed a new concept of implicit invariants We introduced implicit moment invariants to polynomial deformations of images

31. Thank you ! Any questions?

32. Odtud dal uz to nebylo !

33. Common types of moments Geometric moments

34. Special case If an explicit invariant exist, then I(f,g) = |E(f) – E(g)|

35. An example in 1D

36. Orthogonal moments Legendre Zernike Fourier-Mellin Czebyshev Krawtchuk, Hahn

39. Outlook for the future and open problems Discriminability? Robustness? Other transforms?

40. How is it connected with image fusion?

41. Základní přístupy Brute force Normalized position  inverse problem Description of the objects by invariants Basic approaches

42. An example in 2D

43. Our assumption about D Image degradation is a polynomial transform r(x) of the spatial coordinates of order > 1

44. Construction of implicit moment invariants Eliminate the parameters of r from the system Each equation of the reduced system is an implicit invariant

45. How are the moments transformed? A depends on r and on the moment basis A is not a square matrix Transform r does not preserve the moment orders Explicit moment invariants cannot exist. If they existed, they would contain all moments.