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Orthogonal moments

Orthogonal moments. - set of orthogonal polynomials. Motivation for using OG moments Stable calculation by recurrent relations Easier and stable image reconstruction. Numerical stability. How to avoid numerical problems with high dynamic range of geometric moments?. Standard powers.

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Orthogonal moments

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  1. Orthogonal moments - set of orthogonal polynomials • Motivation for using OG moments • Stable calculation by recurrent relations • Easier and stable image reconstruction

  2. Numerical stability How to avoid numerical problems with high dynamic range of geometric moments?

  3. Standard powers

  4. Orthogonal polynomials Calculation using recurrent relations

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

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

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

  8. Legendre polynomials Definition Orthogonality

  9. Legendre polynomials explicitly

  10. Legendre polynomials in 1D

  11. Legendre polynomials in 2D

  12. Legendre polynomials Recurrent relation

  13. Legendre moments

  14. Moments orthogonal on a disk Radial part Angular part

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

  16. Zernike polynomials Definition Orthogonality

  17. Zernike polynomials – radial part in 1D

  18. Zernike polynomials – radial part in 2D

  19. Zernike polynomials

  20. Zernike moments Mapping of Cartesian coordinates x,y to polar coordinates r,φ: • Whole image is mapped inside the unit disk • Translation and scaling invariance

  21. Zernike moments

  22. Rotation property of Zernike moments The magnitude is preserved, the phase is shifted by ℓθ. Invariants are constructed by phase cancellation

  23. Zernike rotation invariants Phase cancellation by multiplication Normalization to rotation

  24. Recognition by Zernike rotation invariants

  25. Insufficient separability

  26. Sufficient separability

  27. Image reconstruction • Direct reconstruction from geometric moments • Solution of a system of equations • Works for very small images only • For larger images the system is ill-conditioned

  28. Image reconstruction by direct computation 12 x 12 13 x 13

  29. Image reconstruction • Reconstruction from geometric moments via FT

  30. Image reconstruction via Fourier transform

  31. Image reconstruction • Image reconstruction from OG moments on a square • Image reconstruction from OG moments on a disk (Zernike)

  32. Image reconstruction from Legendre moments

  33. Image reconstruction from Zernike moments Better for polar raster

  34. Image reconstruction from Zernike moments

  35. Reconstruction of a noise-free image

  36. Reconstruction of a noisy image

  37. Reconstruction of a noisy image

  38. Summary of OG moments • OG moments are used because of their favorable numerical properties, not because of theoretical contribution • OG moments should be never used outside the area of orthogonality • OG moments should be always calculated by recurrent relations, not by expanding into powers • Moments OG on a square do not provide easy rotation invariance • Even if the reconstruction from OG moments is seemingly simple, moments are not a good tool for image compression

  39. Algorithms for moment computation Various definitions of moments in a discrete domain depending on the image model Sum of Dirac δ-functions Nearest neighbor interpolation Bilinear interpolation

  40. Moments in a discrete domain exact formula

  41. Moments in a discrete domain zero-order approximation

  42. Moments in a discrete domain exact formula

  43. Algorithms for binary images • Decomposition methods • Boundary-based methods

  44. Decomposition methods The object is decomposed into K disjoint (usually rectangular) “blocks” such that

  45. Decomposition methods differ from each other by - the decomposition algorithms • the shape of the blocks • the way how the moments of the blocks are • calculated

  46. Delta method (Zakaria et al.) Decomposition into rows

  47. Recursive formulae for the summations where

  48. Delta method (Zakaria et al.) Decomposition into rows Simplification by direct integration

  49. Delta method (Zakaria et al.)

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