Transcript Chapter 6
http://zoi.utia.cas.cz/moment_invariants The slides accompanying the book J. Flusser, T. Suk, and B. Zitová Moments and Moment Invariants in Pattern Recognition 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 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 Chapter 6 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 Orthogonal polynomials Calculation using recurrent relations Two kinds of orthogonality • Moments (polynomials) orthogonal on a unit square • Moments (polynomials) orthogonal on a unit disk Moments orthogonal on a square is a system of 1D orthogonal polynomials Common 1D orthogonal polynomials • • • • • • Legendre Chebyshev Gegenbauer Jacobi (generalized) Laguerre Hermite <-1,1> <-1,1> <-1,1> <-1,1> or <0,1> <0,∞) (-∞,∞) Legendre polynomials Definition Orthogonality Legendre polynomials explicitly Legendre polynomials in 1D Legendre polynomials in 2D Legendre polynomials Explicit formula Recurrence relation Legendre moments Legendre and geometric moments Chebyshev polynomials First kind Second kind Chebyshev polynomials on <-1,1> First kind Second kind Chebyshev polynomials on <-2,2> First kind Second kind Chebyshev polynomials in 2D Chebyshev polynomials – orthogonality Continuous They can be normalized to discrete orthogonality Chebyshev moments Gegenbauer polynomials Generalization of both Legendre and Chebyshev polynomials – parameter λ - special initial values: Gegenbauer polynomials Jacobi polynomials Further generalization, parameters α and β Laguerre and Hermite polynomials • Infinite interval of orthogonality • Suitable for particular applications only Discrete OG polynomials on a square Discrete variable Discrete orthogonality Discrete 1D OG polynomials • • • • • • Discrete Chebyshev Discrete Laguerre Krawtchouk Hahn Dual-Hahn Racah Discrete Chebyshev polynomials Krawtchouk polynomials Krawtchouk polynomials Weighted Krawtchouk polynomials p=0.5 p=0.2 Dual-Hahn and Racah polynomials Nonuniform lattice They were adapted such that s is a traditional coordinate in a discrete image. Moments orthogonal on a disk Radial part Angular part Moments orthogonal on a disk • • • • • • Zernike Pseudo-Zernike Orthogonal Fourier-Mellin Jacobi-Fourier Chebyshev-Fourier Radial harmonic Fourier Zernike polynomials Definition Orthogonality Zernike polynomials – radial part in 1D Zernike polynomials – radial part in 2D Zernike polynomials 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 Rotation property of Zernike moments The magnitude is preserved, the phase is shifted by ℓθ. Invariants are constructed by phase cancellation Zernike rotation invariants Phase cancellation by multiplication Normalization to rotation Pseudo - Zernike polynomials need not be even → redundancy Orthogonal Fourier-Mellin moments Orthogonal FM moments – radial part Orthogonal FM moments – the basis functions Jacobi-Fourier and Chebyshev-Fourier moments Combination: radial part – 1D orthogonal polynomials angular part – traditional harmonic function 1D OG polynomials – shifted version: Orthogonality on <0,1> Radial harmonic Fourier moments Tn(r) is not polynomial → they are not moments in a strict sense Recognition by Zernike rotation invariants Insufficient separability Sufficient separability 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 Image reconstruction by direct computation 12 x 12 13 x 13 Image reconstruction • Reconstruction from geometric moments via FT Image reconstruction via Fourier transform Image reconstruction • Image reconstruction from OG moments on a square • Image reconstruction from OG moments on a disk (Zernike) Image reconstruction from Legendre moments Continuous orthogonality Image reconstruction from Zernike moments Better for polar raster Image reconstruction from discrete Chebyshev moments – precise reconstruction Illustrates discrimination power of the moments Reconstruction of large images Discrete Chebyshev moments – precise up to 1075th order Limitation is imposed by floating-point underflow Orthogonal moments in 3D Ω - Rectangular cuboid Ω - Cylinder Ω - Sphere - spherical harmonics Example: 3D Zernike moments 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 • Preferably discrete OG polynomials should be used • 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