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 1

5. General motivation How can we recognize objects on non-ideal images?

6. Traffic surveillance - can we recognize the license plates?

7. Recognition (classification) = assigning a pattern/object to one of pre-defined classes The object is described by its features Features – measurable quantities, usually form an n-D vector in a metric space Object recognition

8. Non-ideal imaging conditions  degradation of the image g = D(f) D - unknown degradation operator Problem formulation

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

10. What are invariants? Invariants are functionals defined on the image space such that I(f) = I(D(f)) for all admissible D

11. Example: TRS

12. What are invariants? Invariants are functionals defined on the image space such that I(f) = I(D(f)) for all admissible D I(f 1 ), I(f 2 ) “different enough“ for different f 1, f 2

13. Discrimination power

14. Major categories of invariants Simple shape descriptors - compactness, convexity, elongation,... Transform coefficient invariants - Fourier descriptors, wavelet features,... Point set invariants - positions of dominant points Differential invariants - derivatives of the boundary Moment invariants

15. What are moment invariants? Functions of image moments, invariant to certain class of image degradations, such as Rotation, translation, scaling Affine transform Elastic deformations Convolution/blurring Various combinations

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

17. The most common moments Geometric moments (p + q) - the order of the moment

18. Geometric moments – the meaning 0 th order - area 1 st order - center of gravity 2 nd order - moments of inertia 3 rd order - skewness

19. Uniqueness theorem Geometric moments

20. Complex moments

21. Basic relations between the moments