Development of image invariant moments—a short overview

Kejia Wang; Ziliang Ping; Yunlong Sheng

doi:10.3788/COL201614.091001

Journals >Chinese Optics Letters >Volume 14 >Issue 9 >Page 091001 > Article

Chinese Optics Letters
Vol. 14, Issue 9, 091001 (2016)

Development of image invariant moments—a short overview

Kejia Wang^1、2, Ziliang Ping^3、4, and Yunlong Sheng^1、*

Author Affiliations

¹Center for Optics, Photonics and Lasers, Department of Physics, Physical Engineering and Optics, Laval University, Quebec G1V0A6, Canada

²Inner Mongolia Electronic Information Vocational Technical College, Huhhot 010070, China

³School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China

⁴Inner Mongolia Normal University, Huhhot 010020, China

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DOI: 10.3788/COL201614.091001 Cite this Article Set citation alerts

Kejia Wang, Ziliang Ping, Yunlong Sheng. Development of image invariant moments—a short overview[J]. Chinese Optics Letters, 2016, 14(9): 091001 Copy Citation Text

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Orthogonal radial polynomials: (a) Zernike polynomials with the degrees n=10–28 for circular harmonic order m=10, (b) orthogonal Fourier-Mellin polynomials with the degrees n=0–9.

Fig. 1. Orthogonal radial polynomials: (a) Zernike polynomials with the degrees n=10–28 for circular harmonic order m=10, (b) orthogonal Fourier-Mellin polynomials with the degrees n=0–9.

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(a) Original image of a letter E in the unit disk; (b) reconstructed from first 64 orthogonal Fourier-Mellin moments Φn,m with n, m=0–7; (c) reconstructed from Zernike moments Rn,m with circular harmonic orders m=0–7 and for each m using the eight lowest degrees, satisfying n≥|m|+2.

Fig. 2. (a) Original image of a letter E in the unit disk; (b) reconstructed from first 64 orthogonal Fourier-Mellin moments Φn,m with n, m=0–7; (c) reconstructed from Zernike moments Rn,m with circular harmonic orders m=0–7 and for each m using the eight lowest degrees, satisfying n≥|m|+2.

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Fig. 3. Shifted Chebyshev polynomial Rn(r) with n=1, 2, 9, 10, [25]

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Fig. 4. Bessel Radial polynomial J1(λnr) with n=0,1,…,9, [28].

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Non-orthogonal moments	Moment invariants (Hu’s) Fourier-Mellin moments complex moments wavelet moments
Orthogonal moments	Continuous orthogonal moments	Cartesian moments	Legendre moments Gaussian-Hermite moments Gegenbauer moments
	Continuous orthogonal moments	Circular moments	Zernike moments pseudo-Zernike moments orthogonal Fourier-Mellin moments Chebyshev-Fourier moments Jacobi-Fourier moments Bessel-Fourier moments radial-harmonic-Fourier transform
	Discrete orthogonal moments	Cartesian moments	Tchebichef moments Krawtchouk moments Hahn moments dual Hahn moments Racah moments
	Discrete orthogonal moments	Circular moments	Radial Tchebichef moments radial Krawtchouk moments