Orthogonal moments Tom Suk Department of Image Processing
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Orthogonal moments Tomáš Suk Department of Image Processing
Orthogonal moments – transformation of features Geometric moment Orthogonal moment
Why to use the orthogonal moments ? • Numerical precision
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
Orthogonal moments - set of orthogonal polynomials
Orthogonal polynomials – lower dynamic range
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 1 D orthogonal polynomials
Common 1 D orthogonal polynomials • • • Legendre <-1, 1> Chebyshev <-1, 1> Gegenbauer <-1, 1> Jacobi <-1, 1> or <0, 1> (generalized) Laguerre <0, ∞) Hermite (-∞, ∞)
How to define orthogonal polynomials • • Integral representation Generating function Differential equation Hypergeometric series Rodrigues’ formula Explicit formula Recurrence relation
Legendre polynomials Adrien-Marie Legendre (1752 – 1833) was a French mathematician. Adrien-Marie Legendreová (1752 – 1833) byla francouzská matematička.
Legendre polynomials Integral representation , where the contour C encloses the origin and is traversed in a counterclockwise direction. → Legendre polynomials = Spherical polynomials
Legendre polynomials Generating function Pn(x) are coefficients in a Taylor series expansion Differential equation
Legendre polynomials Hypergeometric series Pochhammer symbol = rising factorial Legendre polynomials
Legendre polynomials Rodrigues’ formula
Legendre polynomials Explicit formula Recurrence relation
Legendre polynomials Relation of orthogonality Generally
Legendre polynomials explicitly
Legendre polynomials in 1 D
Legendre polynomials in 2 D
Legendre moments
Chebyshev polynomials Пафнүтий Львович Чебышёв (1821 – 1894) Russian mathematician spelling Russian: Чебышёв → Чебышев French: Tchebichef German: Tschebyschow English: Chebyshev Czech: Čebyšev
Chebyshev polynomials First kind Second kind
Chebyshev polynomials on <-1, 1> First kind Second kind
Chebyshev polynomials in 2 D
Chebyshev polynomials – orthogonality
Gegenbauer polynomials Leopold Gegenbauer (1849– 1903) Austrian mathematician
Gegenbauer polynomials =ultraspherical, generalization of both Legendre and Chebyshev polynomials – parameter λ = 0, 0. 5, 1 - special initial values:
Gegenbauer polynomials
Gegenbauer polynomials
Jacobi polynomials Carl Gustav Jacobi (1804 – 1851) Prussian mathematician
Jacobi polynomials on <-1, 1> Further generalization, parameters α and β Relation of orthogonality
Jacobi polynomials on <0, 1> Parameters p and q Relation of orthogonality
Laguerre and Hermite polynomials Edmond Nicolas Laguerre (1834 – 1886) Charles Hermite (1822 – 1901) French mathematicians
Laguerre and Hermite polynomials • Infinite interval of orthogonality • Suitable for particular applications only Relations of orthogonality – Laguerre Hermite
Literature M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables”, 1964.
Discrete OG polynomials on a square Discrete variable Discrete orthogonality
Discrete 1 D OG polynomials • • • Discrete Chebyshev Discrete Laguerre Krawtchouk Hahn Dual-Hahn Racah
Discrete Chebyshev polynomials Recurrence relation in order Recurrence relation in coordinate
Krawtchouk polynomials Михайло Пилипович Кравчук (1892 – 1942) Also Kravchuk, Ukrainian mathematician
Krawtchouk polynomials
Krawtchouk polynomials
Weighted Krawtchouk polynomials p=0. 5 p=0. 2
Dual-Hahn and Racah polynomials Wolfgang Hahn (1911– 1998) Austrian mathematician Giulio (Yoel) Racah (Hebrew: ( ג'וליו )יואל רקח 1909 – 1965) Italian–Israeli physicist and mathematician.
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
Orthogonal polynomials 4 F 3(4) Wilson Continuous 3 F 2(3) dual Hahn 2 F 1(2) Meixner Pollaczek Continuous Hahn Jacobi 1 F 1(1)/2 F 0(1) Laguerre 2 F 0(0) Hermite Racah dual Hahn Meixner Krawtchouk Charlier 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.
q-analogue Hypergeometric series Pochhammer symbol q-Hypergeometric series q-analogue of the Pochhammer symbol
Continuous q-Legendre polynomials Polynomials For q and 1/q are identical
Moments orthogonal on a disk Radial part Angular part
Moments orthogonal on a disk • • • Zernike Pseudo-Zernike Orthogonal Fourier-Mellin Chebyshev-Fourier Jacobi-Fourier Radial harmonic Fourier
Zernike polynomials Frits Zernike (1888 – 1966) Dutch physicist Groningen
Zernike polynomials Definition Orthogonality
Zernike polynomials – radial part in 1 D
Zernike polynomials – radial part in 2 D
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
Zernike moments - scaling Zernike coefficients an, m are transformed to bn, m in scaling by λ where
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 Teh, Chin 1988
Orthogonal Fourier-Mellin moments (non-orthogonal) Fourier-Mellin moments Yunlong Sheng, Lixin Shen 1994
Orthogonal FM moments – radial part
Orthogonal FM moments – the basis functions
Jacobi-Fourier and Chebyshev-Fourier moments Combination: radial part – 1 D orthogonal polynomials angular part – traditional harmonic function 1 D OG polynomials – shifted version: Orthogonality on <0, 1> Ping, Wu, Sheng 2002 Ping, Ren, Zou, Sheng, Bo 2004
Radial harmonic Fourier moments Tn(r) is not polynomial → they are not moments in a strict sense Ren, Ping, Bo, Wu, Sheng 2003
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 1075 th order Limitation is imposed by floating-point underflow
Orthogonal moments in 3 D Ω - Cube Ω - Cylinder Ω - Sphere - spherical harmonics
Example: 3 D 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 • Main application – pattern recognition • Reconstruction from moments - theoretical meaning - watermarking (limited robustness to attacks) - compression (other methods are better)
Current research - orthogonal invariants They should preserve: - invariance - orthogonality - numerical precision - discriminability - completeness - independence Some demands are contrary Partial problems • Precise scaling invariance • Rotation invariance on a square • Affine invariance not based on normalization • Discrete orthogonal polynomials on a polar raster • Convolution invariance
Thank you for your attention
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