1. Orthogonal moments Tomáš Suk Department of Image Processing

2. Orthogonal moments – transformation of features Geometric moment Orthogonal moment

3. Why to use the orthogonal moments ? • Numerical precision

4. Why to use the orthogonal moments ? • Numerical precision log Fmax R= ─────── log N Fmax – maximum precision in a computer N – image size R – maximum moment order

5. Orthogonal moments - set of orthogonal polynomials

6. Orthogonal polynomials – lower dynamic range

7. Two kinds of orthogonality • Moments (polynomials) orthogonal on a unit square • Moments (polynomials) orthogonal on a unit disk

8. Moments orthogonal on a square is a system of 1D orthogonal polynomials

9. Common 1D orthogonal polynomials • Legendre <-1,1> • Chebyshev <-1,1> • Gegenbauer<-1,1> • Jacobi<-1,1> or <0,1> • (generalized) Laguerre <0,∞) • Hermite (-∞,∞)

10. How to define orthogonal polynomials • Integral representation • Generating function • Differential equation • Hypergeometric series • Rodrigues’ formula • Explicit formula • Recurrence relation

11. Legendre polynomials Adrien-Marie Legendre (1752 – 1833) was a French mathematician. Adrien-Marie Legendreová (1752 – 1833) byla francouzská matematička.

12. Legendre polynomials Integral representation , where the contourC encloses the origin and is traversed in a counterclockwise direction. → Legendre polynomials = Spherical polynomials

13. Legendre polynomials Generating function Pn(x) are coefficients in a Taylor series expansion Differential equation

14. Legendre polynomials Hypergeometric series Pochhammer symbol = rising factorial Legendre polynomials

15. Legendre polynomials Rodrigues’ formula

16. Legendre polynomials Explicit formula Recurrence relation

17. Legendre polynomials Relation of orthogonality Generally

18. Legendre polynomials explicitly

19. Legendre polynomials in 1D

20. Legendre polynomials in 2D

21. Legendre moments

22. Chebyshev polynomials ПафнүтийЛьвовичЧебышёв (1821 – 1894) Russian mathematician spelling Russian: Чебышёв→Чебышев French: Tchebichef German: Tschebyschow English: Chebyshev Czech: Čebyšev

23. Chebyshev polynomials First kind Second kind

24. Chebyshev polynomials on <-1,1> Second kind First kind

25. Chebyshev polynomials in 2D

26. Chebyshev polynomials – orthogonality

27. Gegenbauer polynomials Leopold Gegenbauer (1849–1903) Austrian mathematician

28. Gegenbauer polynomials =ultraspherical, generalization of both Legendre and Chebyshev polynomials – parameter λ= 0, 0.5, 1 - special initial values:

29. Gegenbauer polynomials

30. Gegenbauer polynomials

31. Jacobi polynomials Carl Gustav Jacob Jacobi (1804 – 1851) Prussian mathematician

32. Jacobi polynomialson <-1,1> Further generalization,parameters α and β Relation of orthogonality

33. Jacobi polynomialson <0,1> Parameters p and q Relation of orthogonality

34. Laguerre and Hermite polynomials Edmond Nicolas Laguerre (1834 – 1886) Charles Hermite (1822 – 1901) French mathematicians

35. Laguerre and Hermite polynomials • Infinite interval of orthogonality • Suitable for particular applications only Relations of orthogonality – Laguerre Hermite

36. Literature M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables”, 1964.

37. Discrete OG polynomials on a square Discrete variable Discrete orthogonality

38. Discrete 1D OG polynomials • Discrete Chebyshev • Discrete Laguerre • Krawtchouk • Hahn • Dual-Hahn • Racah

39. Recurrence relation in coordinate Discrete Chebyshev polynomials Recurrence relation in order

40. Krawtchouk polynomials Михайло Пилипович Кравчук (1892 – 1942) Also Kravchuk, Ukrainian mathematician

41. Krawtchouk polynomials

42. Krawtchouk polynomials

43. Weighted Krawtchouk polynomials p=0.5 p=0.2

44. Dual-Hahn and Racah polynomials Wolfgang Hahn (1911–1998) Austrian mathematician Giulio (Yoel) Racah (Hebrew: ג'וליו (יואל) רקח1909 – 1965) Italian–Israeli physicist and mathematician.

45. Dual-Hahn and Racah polynomials Nonuniform lattice They were adapted such that s is a traditional coordinate in a discrete image. Zhu, Shu, Zhou, Luo, Coatrieux 2007 Zhu, Shu, Liang, Luo, Coatrieux 2007

46. Orthogonal polynomials 4F3(4) Wilson Racah Continuous dual Hahn Continuous Hahn Hahn 3F2(3) dual Hahn Meixner - Pollaczek 2F1(2) Krawtchouk Jacobi Meixner 1F1(1)/2F0(1) Laguerre Charlier 2F0(0) Hermite R. Koekoek and R. F. Swarttouw, “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,” Report 98-17, Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics, 1996.

47. q-analogue Hypergeometric series Pochhammer symbol q-Hypergeometric series q-analogue of the Pochhammer symbol

48. Continuous q-Legendre polynomials Polynomials For q and 1/q are identical

49. Moments orthogonal on a disk Radial part Angular part

50. Moments orthogonal on a disk • Zernike • Pseudo-Zernike • Orthogonal Fourier-Mellin • Chebyshev-Fourier • Jacobi-Fourier • Radial harmonic Fourier