Abstract
Exponential moments (EMs) are important radial orthogonal moments, which have good image description ability and have less information redundancy compared with other orthogonal moments. Therefore, it has been used in various fields of image processing in recent years. However, EMs can only take integer order, which limits their reconstruction and antinoising attack performances. The promotion of fractional-order exponential moments (FrEMs) effectively alleviates the numerical instability problem of EMs; however, the numerical integration errors generated by the traditional calculation methods of FrEMs still affect the accuracy of FrEMs. Therefore, the Gaussian numerical integration (GNI) is used in this paper to propose an accurate calculation method of FrEMs, which effectively alleviates the numerical integration error. Extensive experiments are carried out in this paper to prove that the GNI method can significantly improve the performance of FrEMs in many aspects.
Highlights
Exponential moments (EMs) are important radial orthogonal moments, which have good image description ability and have less information redundancy compared with other orthogonal moments. erefore, it has been used in various fields of image processing in recent years
Orthogonal moments are divided into discrete orthogonal moments and continuous orthogonal moments. e continuous orthogonal moments use the continuous function as the basis function, and it has the rotation, scaling, and the translation invariance, which have been greatly developed in recent years, including Legendre moments (LMs) [6], Zernike moments (ZMs) [7], pseudo-Zernike moments (PZMs) [7], Security and Communication Networks orthogonal Fourier–Mellin moments (OFMMs) [8], Chebyshev–Fourier moments (CHFMs) [9], radial harmonic Fourier moments (RHFMs) [10], Bessel–Fourier moments (BFMs) [11], polar harmonic transforms (PHTs) [12], and exponential moments (EMs) [13], Among them, EMs have good antinoise performances and less information redundancy, and their basis functions have the simple form, low computational complexity, and good image description performance [14]
In the study of fractional moments, scholars first define the fractional parameter of t(t > 0) and use rt to replace r in the radial basis function of the orthogonal moment. e radial basis function is further modified to maintain the orthogonality of the moment [19]. e orthogonal moment promoted to the fractional order can adjust the gradient of the radial basis function by assigning different values to the fractional variable of t to further alleviate the problem of numerical instability [20]. e existing fractional moments include fractional-order Legendre–Fourier moments (FrOLFMs) [21], orthogonal fractional-order Fourier–Mellin moments (FrOFMMs) [22], fractional-order Zernike moments (FrZMs) [23], fractionalorder polar harmonic transforms (FrPHTs) [24], fractionalorder orthogonal Chebyshev–Fourier moments (FrCFMs) [25], and fractional-order radial harmonic Fourier moments (FrRHFMs) [26]
Summary
Where the fractional parameter t > 0, and the basis function of FrEMs is defined as follows: FrH(ntm) (r, θ) FrA(nt)(r)exp(jmθ). FrE(ntm) f(r, θ)FrH(ntm) ∗(r, θ)r dr dθ It can be known from formulas (1) and (4) that when t 1, the radial basis functions of FrEMs will be those of EMs; EMs can be deemed as a special form of FrEMs. e radial basis function of EMs is orthogonal within the range of 0 ≤ r ≤ 1:. Different fractional parameters t lead to different calculation emphasis areas. erefore, the specific application of fractional exponential moments should be considered when selecting fractional parameter t
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