Orthogonal moments Tom Suk Department of Image Processing

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

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