Generic orthogonal moments: Jacobi–Fourier moments for invariant image description

A multi-distorted invariant orthogonal moments, Jacobi–Fourier Moments (JFM), were proposed. The integral kernel of the moments was composed of radial Jacobi polynomial and angular Fourier complex componential factor. The variation of two parameters in Jacobi polynomial, αα and ββ, can form various types of orthogonal moments: Legendre–Fourier Moments (α=1,β=1)(α=1,β=1); Chebyshev–Fourier Moments (α=2,β=32); Orthogonal Fourier–Mellin Moments (α=2,β=2)(α=2,β=2); Zernike Moments and pseudo-Zernike Moments, and so on. Therefore, Jacobi–Fourier Moments are generic expressions of orthogonal moments formed by a radial orthogonal polynomial and angular Fourier complex component factor, providing a common mathematical tool for performance analysis of the orthogonal moments. In the paper, Jacobi–Fourier Moments were calculated for a deterministic image, and the original image was reconstructed with the moments. The relationship between Jacobi–Fourier Moments and other orthogonal moments was studied. Theoretical analysis and experimental investigation were conducted in terms of the description performance and noise sensibility of the JFM.