1. J. Flusser, T. Suk, and B. Zitová Moments and Moment Invariants in Pattern Recognition http://zoi.utia.cas.cz/moment_invariants The slides accompanying the book

2. Copyright notice The slides can be used freely for non-profit education provided that the source is appropriately cited. Please report any usage on a regular basis (namely in university courses) to the authors. For commercial usage ask the authors for permission. The slides containing animations are not appropriate to print. © Jan Flusser, Tomas Suk, and Barbara Zitová, 2009

3. Contents 1. Introduction to moments 2. Invariants to translation, rotation and scaling 3. Affine moment invariants 4. Implicit invariants to elastic transformations 5. Invariants to convolution 6. Orthogonal moments 7. Algorithms for moment computation 8. Applications 9. Conclusion

4. Chapter 3

5. Invariants to affine transform What is affine transform ?

6. Invariants to affine transform What is affine transform ?

7. Why is affine transform important? Affine transform is a good approximation of projective transform Projective transform describes a perspective projection of 3-D objects onto 2-D plane by a central camera

8. Projective deformation

9. Why not projective moment invariants? Do not exist when using any finite set of moments Do not exist when using infinite set of (all) moments Exist formally as infinite series of moments of both positive and negative indexes

10. Theory of algebraic invariants (Fundamental theorem) Graph method Image normalization Cayley-Aronhold equation Hybrid approaches Affine moment invariants All methods lead to equivalent invariants … Many ways how to derive them

11. Two simplest AMI’s, frequently cited … such as

12. AMI’s by means of the Fundamental theorem Binary algebraic form Algebraic invariant of weight w

13. AMI’s by means of the Fundamental theorem

14. AMI’s by means of the graph method - arbitrary points r points, n kj – non-negative integers

15. Affine Moment Invariants AMI’s by means of the graph method where

16. Simple examples of the AMI’s 1),

17. Simple examples of the AMI’s 2),

18. Graph representation of the AMI’s

25. Dependence among invariants Trivial invariants (always zero or identical)

26. Dependence among invariants Trivial invariants, identical invariants Reducible invariants (products, linear combinations) Irreducible invariants (polynomials, polynomials of products) Independent invariants

27. Removing dependence For w ≤ 12 : 2 533 942 752 invariants (graphs) altogether 2 532 349 394 zero invariants 1 575 126 identical invariants 14 538 linear combinations 2 105 products ------------------------------------------------------------- 1589 irreducible invariants 80 independent invariants

28. Removing dependence The most difficult step: How to proceed from irreducible to independent invariants? Exhaustive search of all possible polynomial dependences The dependences themselves may be dependent ! (2 nd -order dependencies)

29. Higher-order dependencies The number of independent invariants:

30. Numerical experiments with the AMI’s

31. Robustness of the AMI’s to distortions

33. Affine invariants via normalization Many possibilities how to define normalization constraints Several possible decompositions of the affine transform

34. Decomposition of the affine transform Horizontal and vertical translations Uniform scaling First rotation Stretching Second rotation Mirror reflection

35. Normalization to partial transforms Horizontal and vertical translation -- m 01 = m 10 = 0 Scaling -- c 00 = 1 First rotation -- c 20 real and positive Stretching -- c 20 = 0 (μ 20 =μ 02 ) Second rotation -- c 21 real and positive

36. Moment values after the normalization Translation, uniform scaling and the first rotation Stretching

37. Moment values after the normalization Second rotation

38. Possible volatility of the normalization

39. Affine invariants via half-normalization “Hybrid” approach. The image is normalized to translation, scaling, first rotation and stretching. Then, rotation invariants are used to handle the second rotation. More stable in some cases.

40. Affine invariants from complex moments

41. Affine invariants from Cayley- Aronhold equation Skewing parameter t

42. Digit recognition by the AMI’s

44. Recognition of symmetric patterns

45. Recognition of children’s mosaic

48. Affine invariants of color images Color moments Algebraic invariants of more than one binary forms

49. Affine invariants of color images Common centroid of color channels Additional invariants

50. Affine invariants in 3D 3D affine transform Analogy with the graph method

51. Affine invariants in 3D An example Corresponding hypergraph

52. Affine invariants in 3D Corresponding hypergraph

53. Affine normalization in 3D Theory based on spherical harmonics (analogy to complex moments)

54. Cayley-Aronhold equation in 3D Analogy to 2D