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Quaternion fractionalorder color orthogonal momentbased image representation and recognition
EURASIP Journal on Image and Video Processing volume 2021, Article number: 17 (2021)
Abstract
Inspired by quaternion algebra and the idea of fractionalorder transformation, we propose a new set of quaternion fractionalorder generalized Laguerre orthogonal moments (QFrGLMs) based on fractionalorder generalized Laguerre polynomials. Firstly, the proposed QFrGLMs are directly constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and fractionalorder transformations, which extract only the global features from color images, our proposed QFrGLMs can extract both the global and local color features. This paper also derives a new set of invariant colorimage descriptors by QFrGLMs, enabling geometricinvariant pattern recognition in color images. Finally, the performances of our proposed QFrGLMs and moment invariants were evaluated in simulation experiments of correlated color images. Both theoretical analysis and experimental results demonstrate the value of the proposed QFrGLMs and their geometric invariants in the representation and recognition of color images.
1 Introduction
In the last decade, image moments and geometric invariance of moments have emerged as effective methods of feature extraction from images [1, 2]. Both methods have made great progress in imagerelated fields. However, most of the existing algorithms extract the image moments only from grayscale images. Color images contain abundant multicolor information that is missing in grayscale images. Therefore, in recent years, research efforts have gradually shifted to the construction of colorimage moments [3, 4]. Colorimage processing is traditionally performed by one of the three main methods: (1) select a single channel or component from the color space of a color image, such a channel from a red–green–blue (RGB) image, as a grayscale image and calculate its corresponding image moments; (2) directly gray a color image, and then calculate its image moments; and (3) calculate the image moments of each monochromatic channel (R, G and B) in a RGB image, and average them to obtain the final result. Although all the three methods are relatively simple to implement, they discard some of the useful image information and cannot determine the relationship among the different color channels of a RGB image. This common defect reduces the accuracy of colorimage representation in image processing or recognition. Owing to loss of correlations among the different color channels and part of the colorimage information, the advantages of color images over grayscale images are not fully exploited in practical application [5].
Recently, quaternion algebrabased color image representation has provided a new research direction in color model spaces [6, 7] such as RGB, luma–chroma (YUV), and hue–saturation–lightness (HSV) [8]. Quaternion algebra has made several achievements in colorimage processing [9, 10]. The quaternion method represents an image as a threedimensional vector describing the components of the color image, which effectively uses the color information of different channels of the color image. Elouariachi et al. [11] derived a new set of quaternion Krawtchouk moments (QKMs) and explicit quaternion Krawtchouk moment invariants (EQKMIs), which can be applied to fingerspelling sign language recognition. Wang et al. [12, 13] constructed a class of quaternion color orthogonal moments based on quaternion theory. In ref [12], they proposed quaternion polar harmonic Fourier moments (QPHFMs) in polar coordinate space and applied them to colorimage analysis. They also proposed a zerowatermarking method based on quaternion exponent Fourier moments (QEFMs) [13], which is applied to copyright protection of digital images. Xia et al. [14] combined Wang et al.’s method with chaos theory and proposed an accurate quaternion polar harmonic transform for a medical image zerowatermarking algorithm. Guo et al. [15] introduced a new set of quaternion moment descriptors for color image, and they are constructed in the quaternion framework and are an extension of complex moment invariants for grayscale images. The above results on quaternion colorimage moments provide theoretical support for exploring newgeneration colorimage moments. However, imagemoment construction based on quaternion theory is complex and increases the time of the colorimage calculation. Moreover, the performance of the existing quaternion image moments in colorimage analysis is not significantly improved from multichannel colorimage processing [10, 16]. Most importantly, the quaternion colorimage moments constructed by the existing methods are similar to grayscaleimage moments [17] and extract only the global features; therefore, they are powerless for localimage reconstruction and regionofinterest (ROI) detection. In conclusion, the new generation of quaternion colorimage moment algorithms requires further research. The new fractionalorder orthogonal moments effectively improve the performance of orthogonal moments in image analysis and can also improve the quaternion colorimage moments. The basis function of fractionalorder orthogonal moments comprises a set of fractionalorder (or realorder) orthogonal polynomials rather than traditional integerorder polynomials.
Fractionalorder image moments have been realized only in the past 3 years, and their research is incomplete. Accordingly, their applications are limited to image reconstruction and recognition. In addition, the technique of the existing fractionalorder orthogonal moments is only an effective supplement and an extension of integerorder grayscale image moments. Few academic achievements and investigations of fractionalorder orthogonal moments have been reported in image analysis. Inspired by fractionalorder Fourier transforms, Zhang et al. [18] introduced fractionalorder orthogonal polynomials in 2016 and constructed fractionalorder orthogonal Fourier–Mellin moments for character recognition in binary images. Xiao et al. [19] constructed fractionalorder orthogonal moments in Cartesian and polar coordinate spaces. They showed how general fractionalorder orthogonal moments can be constructed from integerorder orthogonal moments in different coordinate systems. Benouini et al. [20] recently introduced a new set of fractionalorder Chebyshev moments and moment invariant methods and applied them to image analysis and pattern recognition. Although the existing fractionalorder image moments provide better image descriptions than traditional integerorder image moments, their application to computer vision and pattern recognition remains in the exploratory stage. An improved fractionalorder polynomial that constructs a superior fractionalorder image moment is an expected hotspot of future research. Combining fractionalorder image moments with quaternion theory, Chen et al. [21] newly developed quaternion fractionalorder Zernike moments (QFrZMs), which are mainly used in robust copy–move forgery detection in color images. Prof. K. M. hosny et al. [22,23,24] have made outstanding achievements in the study of fractionalorder orthogonal moments in recent years. In refs [22, 23], using Legendre and shifted Gegenbauer polynomials, respectively, fractionalorder LegendreFourier moments and shifted Gegenbauer moments are constructed, which are applied in the field of image analysis and pattern recognition. Moreover, a novel set of fractionalorder orthogonal polar harmonic transforms for grayscale and color image analysis are introduced in ref [24], and their performances are verified by corresponding experiments. The fractionalorder generalized Laguerre orthogonal moments and modified generalized Laguerre orthogonal moments proposed by H. karmouni, Mohamed sayyouri, and O. El Ogri [25,26,27] are mainly constructed in Cartesian coordinate system, and they completed the fast and accurate calculation algorithm of the related image moments, and also those moments are applied to the reconstruction or invariant recognition of 2D and 3D images.
This paper combines the quaternion method with fractionalorder Laguerre orthogonal moments [28, 29] and hence develops new class of quaternion fractionalorder generalized Laguerre moments (QFrGLMs) for colorimage reconstruction and geometricinvariant recognition. Compared with circularly orthogonal moments constructed in polar coordinates, the proposed QFrGLMs not only have better image description performance, but also have global and local description capability. However, the orthogonal moments in polar coordinates directly have rotation invariance, while the invariance of image moments in Cartesian coordinates needs secondary construction. Therefore, trying to study the image moments in polar coordinates is the goal and task of our next stage. The main contributions of this paper are summarized below.

1.
In this paper, a new set of quaternion fractionalorder generalized Laguerre moments is proposed (QFrGLMs) based on generalized Laguerre polynomials, which combines quaternion theory with fractionalorder transformation. In contrast to recent work, most of those fractionalorder orthogonal moments are devoted to grayscale images; however, in our article, the grayscale images are extended to color images by quaternion algebraic formula. In addition, compared with circularly orthogonal moments constructed in polar coordinates, the proposed QFrGLMs not only have better image description performance, but also have global and local description capability.

2.
Since the construction of the proposed QFrGLMs involves the selection of multiple parameters, this paper proposes a method for the optimal parameter selection. In addition, based on the QFrGLMs, for geometricinvariant pattern recognition in color images, a new set of invariant colorimage descriptors is derived, named QFrGLM invariants (QFrGLMIs).

3.
The performances of our proposed QFrGLMs and QFrGLMIs were evaluated in the MATLAB simulation experiments of correlated color images.
1.1 Preliminaries
In this section, we first introduce the basic concepts of quaternion theory and fractionalorder image moments. The quaternion is a generalized form of complex numbers, a systematic mathematical theory and method proposed by the British mathematician Hamilton in 1843 [30], also fractionalorder orthogonal moments are defined in Cartesian and polar coordinate spaces, and we present the transformation relationship between fractionalorder orthogonal polynomials in Cartesian coordinate space and those in polar coordinate space. Then, we introduce the related contents of generalized Laguerre polynomials.
1.2 Representation quaternion algebra and fractionalorder image moments
The quaternion is a fourdimensional complex number, also known as a hypercomplex. It is composed of one real component and three imaginary part components and is formally defined in [5]:
where a, b, c and d are real numbers, and i, j, k are unit imaginary numbers satisfying the following properties:
To obtain the fractionalorder image moments (FrIMs), we introduce the parameter 휆 and slightly modify the basis of traditional geometric moments [19] as follows:
where 휆 ∈ R^{+}. As evidenced in Eq. (3), the order of the fractionalorder geometric moments is 휆(n + m); that is, the integerorder is extended to realorder (or fractionalorder).
In Cartesian and polar coordinate spaces, the fractionalorder orthogonal moments are respectively defined as follows:
where \( {\overline{P}}_n\left(\lambda, x\right)=\sqrt{\lambda }{x}^{\left(\lambda 1\right)/2}{P}_n\left({x}^{\lambda}\right)=\sqrt{\lambda}\sum \limits_{i=0}^n{c}_{n,i}{x}^{\lambda i+\left(\left(\lambda 1\right)/2\right)} \) are the fractionalorder orthogonal polynomials, and \( {\overline{P}}_n\left(\lambda, r\right)=\sqrt{\lambda }{r}^{\left(\lambda 2\right)/2}{P}_n\left({r}^{\lambda}\right)=\sqrt{\lambda}\sum \limits_{i=0}^n{c}_{n,i}{r}^{\lambda i+\left(\left(\lambda 2\right)/2\right)} \) are the radial orthogonal polynomials. The traditional integerorder orthogonal polynomials P_{n}(x) are expressed as \( {P}_n(x)=\sum \limits_{i=0}^n{c}_{n,i}{x}^i \), where c_{n, i} are the binomial coefficients of the orthogonal polynomials [31, 32].
Similarly to traditional integerorder image moments [33,34,35,36], a twodimensional image f(x, y) or f(r, θ) can be reconstructed from fractionalorder orthogonal moments of finite order, which can be written as:
We now determine the interchangeable relationship between the fractionalorder orthogonal polynomials in Cartesian coordinate space and those in polar coordinate space. First, if Q_{n}(x) is an integerorder orthogonal polynomial in Cartesian coordinates, the fractionalorder orthogonal polynomial is expressed as \( {Q}_n^{(t)}(x)=\sqrt{t}{x}^{\frac{t1}{2}}{Q}_n\left({x}^t\right) \) (The detailed implementation of the conversion from integerorder to fractionalorder is given in ref [19].), and the corresponding fractionalorder radial orthogonal polynomials in polar coordinates is expressed as \( {Q}_n^{(t)}(r)=\sqrt{t}{r}^{\frac{t2}{2}}{Q}_n\left({r}^t\right) \), t ∈ R^{+}. Second, if Q_{n}(r) is an integerorder orthogonal polynomial in polar coordinate space, the fractionalorder radial orthogonal polynomial is given by \( {Q}_n^{(t)}(r)=\sqrt{t}{r}^{t1}{Q}_n\left({r}^t\right) \) (The detailed process is shown in ref [18].). The corresponding fractionalorder orthogonal polynomial in Cartesian coordinates is then given by \( {Q}_n^{(t)}(x)=\sqrt{t}{x}^{t\frac{1}{2}}{Q}_n\left({x}^t\right) \), t ∈ R^{+}.
The specific conversion process between the fractionalorder orthogonal polynomials in Cartesian coordinate space and those in polar coordinate space is as follows:

(1)
Suppose \( {Q}_n^{(t)}(x)=\sqrt{t}{x}^{\frac{t1}{2}}{Q}_n\left({x}^t\right) \) is a polynomial that is fractionalorder orthonormal between the interval [0,1] in Cartesian coordinates, we have:
and carrying out the weighted transformation on Eq. (8), then we have:
letting r replace x, and \( {Q}_n^{(t)}(r)=\frac{1}{\sqrt{x}}{Q}_n\left({x}^t\right)=\sqrt{t}{r}^{\frac{t2}{2}}{Q}_n\left({r}^t\right) \), we obtain:
The Eq. (10) shows that polynomial \( {Q}_n^{(t)}(r) \) is orthogonal in polar coordinate space.

(2)
Suppose \( {Q}_n^{(t)}(r)=\sqrt{t}{r}^{t1}{Q}_n\left({r}^t\right) \) is a polynomial that is fractionalorder orthonormal between the interval [0,1] in polar coordinates, we have:
then, the Eq. (11) is transformed, we obtain:
similarly, letting x replace r, and \( {Q}_n^{(t)}(x)=\sqrt{r}{Q}_n\left({r}^t\right)=\sqrt{t}{x}^{t\frac{1}{2}}{Q}_n\left({x}^t\right) \), we obtain:
Equation (13) shows that polynomial \( {Q}_n^{(t)}(x) \) is orthogonal in Cartesian coordinate space.
1.3 Generalized Laguerre polynomials
The generalized Laguerre polynomials (GLPs), also known as associated Laguerre polynomials [37], are expressed as \( {L}_n^{\left(\alpha \right)}(x) \). When α > − 1, GLPs satisfy the following orthogonal relationship in the range [0, +∞):
For convenience, we let ω^{(α)}(x) = exp(−x)x^{α} be a weighted function, and \( \frac{\Gamma \left(n+\alpha +1\right)}{n!}={\gamma}_n^{\left(\alpha \right)} \) be the weighted normalization coefficient. Here, Γ(•) is the gamma function, and n, m = 0, 1, 2, 3… Equation (14) is then modified as follows:
where δ_{nm} is the Kronecker delta function. \( {L}_n^{\left(\alpha \right)}(x) \) is then expressed as:
where (α)_{k} = α(a + 1)(a + 2)…(a + k − 1), (α)_{0} = 1 is the Pochhammer expression, and _{1}F_{1}(−n, α + 1; x) is a hypergeometric function given by
Using Eq. (17), \( {L}_n^{\left(\alpha \right)}(x) \) in Eq. (16) is redefined as
To facilitate the calculation, we compute \( {L}_n^{\left(\alpha \right)}(x) \) by the following recursive algorithm:
with \( {L}_0^{\left(\alpha \right)}(x)=1 \) and \( {L}_1^{\left(\alpha \right)}(x)=1+\alpha x \). For details, see [29] and [30].
2 Methods
This section introduces our proposed QFrGLM scheme, derived from quaternion algebra theory, fractionalorder orthogonal moments, and GLPs. After developing the basic framework of QFrGLMs, we analyze the relationship between the quaternionbased method and the singlechannelbased approach. As shown in Fig. 1, the components of an image f^{rgb}(x, y) in RGB color space, f_{r}(x, y), f_{g}(x, y), and f_{b}(x, y), correspond to the three imaginary components of a pure quaternion. Therefore, an image f^{rgb}(x, y) in RGB color space can be expressed by the following quaternion:
The remainder of this section is organized as follows. Subsection 3.1 defines and constructs our fractionalorder GLPs (FrGLPs) and normalized FrGLPs (NFrGLPs), and Subsection 3.2 defines the proposed QFrGLMs, and relates them to the fractionalorder generalized Laguerre moments (FrGLMs) of single channels in a traditional RGB color image, and the basic framework is shown in Fig. 1. The QFrGLMs invariants (QFrGLMIs) are constructed in subsection 3.3.
2.1 Calculation of FrGLPs and NFrGLPs
FrGLPs [37] can be expressed as:
where, λ > 0, x ∈ [0, +∞], similarly to Eq. (15). The FrGLPs satisfy the following orthogonality relation in the interval [0, +∞]:
where ω^{(α, λ)}(x) = λx^{(α + 1)λ − 1} exp(−x^{λ}),\( {\gamma}_n^{\left(\alpha, \lambda \right)}=\frac{\Gamma \left(n+\alpha +1\right)}{n!} \). The FrGLPs can be rewritten as the following binomial expansion [19, 37]:
where \( {\psi}_{ni}={\left(1\right)}^i\frac{\Gamma \left(n+\alpha +1\right)}{\Gamma \left(i+\alpha +1\right)\left(ni\right)!i!} \), similar to Eq. (19), the FrGLPs can be implemented by the following recursive algorithm:
where \( {L}_0^{\left(\alpha, \lambda \right)}(x)=1 \),\( {L}_1^{\left(\alpha, \lambda \right)}(x)=1+\alpha {x}^{\lambda } \).
In order to enhance the stability of polynomials, normalized polynomials are generally used instead of conventional polynomials. Therefore, normalized fractionalorder GLPs (NFrGLPs) are defined as:
Theorem 1. The NFrGLPs\( {\overline{L}}_n^{\left(\alpha, \lambda \right)}(x) \) are orthogonal on the interval [0, +∞]:
Proof of Theorem 1. Given the NFrGLPs\( {L}_n^{\left(\alpha, \lambda \right)}(x) \) and substituting \( {\overline{L}}_n^{\left(\alpha, \lambda \right)}(x)={L}_n^{\left(\alpha, \lambda \right)}(x)\sqrt{\frac{\omega^{\left(\alpha, \lambda \right)}(x)}{\gamma_n^{\left(\alpha, \lambda \right)}}} \) into Eq. (26), one obtains
Using Eq. (22), we further obtain:
when n = m, \( \frac{\gamma_n^{\left(\alpha, \lambda \right)}}{\sqrt{\gamma_n^{\left(\alpha, \lambda \right)}{\gamma}_m^{\left(\alpha, \lambda \right)}}}=1 \), n ≠ m, δ_{nm} = 0. Thus, \( {\int}_0^{+\infty }{\overline{L}}_n^{\left(\alpha, \lambda \right)}(x){\overline{L}}_m^{\left(\alpha, \lambda \right)}(x) dx={\delta}_{nm} \), which completes the proof of Theorem 1. To reduce the computational complexity and ensure numerical stability, the NFrGLPs are recursively calculated as follows:
where \( {\overline{L}}_0^{\left(\alpha, \lambda \right)}(x)=\sqrt{\frac{\omega^{\left(\alpha, \lambda \right)}(x)}{\Gamma \left(\alpha +1\right)}} \), \( {\overline{L}}_1^{\left(\alpha, \lambda \right)}(x)=\left(1+\alpha {x}^{\lambda}\right)\sqrt{\frac{\omega^{\left(\alpha, \lambda \right)}(x)}{\Gamma \left(\alpha +2\right)}} \), \( {A}_0=\frac{2n+\alpha 1}{\sqrt{n\left(n+\alpha \right)}} \), \( {A}_1=\frac{1}{\sqrt{n\left(n+\alpha \right)}} \), and \( {A}_2=\sqrt{\frac{\left(n+\alpha 1\right)\left(n1\right)}{n\left(n+\alpha \right)}} \). The detailed proof of the recursive operation is given in Appendix A.
Figure 2 shows the distribution curves of the NFrGLPs under different parameter settings. Note that the parameter α mainly affects the amplitudes of the NFrGLPs of different orders and the distributions of the zero values along the xaxis. Thus, if an image is sampled with NFrGLPs, the localfeature regions (ROI) are easily extracted from the images. In addition, the parameter λ can extend the integerorder polynomials to realorder polynomials (λ > 0, λ ∈ R^{+}). Therefore, traditional GLPs are a special case of FrGLPs with λ = 1, that is, \( {L}_n^{\left(\alpha, 1\right)}(x)={L}_n^{\left(\alpha \right)}(x) \). Note also that changing λ changes the width of the zerovalue distributions of the FrGLPs along the xaxis (Fig. 2c–e), thus affecting the imagesampling result.
2.2 Definition and calculation of QFrGLMs
Pan et al. [30] proposed the generalized Laguerre moments (GLMs) for grayscale images in Cartesian coordinates. Recalling the introduction, the corresponding FrGLMs can be defined as:
where f^{gray}(i, j) represents a grayscale digital image. For convenience, we map the original twodimensionaldigitalimage matrix to a square area of [0, L] × [0, L]. Here, L > 0, \( w={\left(\raisebox{1ex}{$L$}\!\left/ \!\raisebox{1ex}{$N$}\right.\right)}^2 \),\( {x}_i=\frac{iL}{N},{y}_j=\frac{jL}{N},i,j=0,1,2,\dots, N1 \).
Using Eq. (30) with the help of Eq. (20), the rightsideQFrGLMs of an original RGB color image in Cartesian coordinates are defined as:
where \( \mu =\left(i+j+k\right)/\sqrt{3} \) is the unit pure imaginary quaternion. The QFrGLMs expressed in quaternion and the FrGLMs of single channels in traditional RGB color images are related as follows:
where \( A=\frac{1}{\sqrt{3}}\left[{FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_r\right)+{FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_g\right)+{FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_b\right)\right] \),
\( B=\frac{1}{\sqrt{3}}\left[{FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_g\right){FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_b\right)\right] \), \( C=\frac{1}{\sqrt{3}}\left[{FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_b\right){FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_r\right)\right] \),
\( D=\frac{1}{\sqrt{3}}\left[{FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_r\right){FrS}_{nm}^{\left(\alpha, \lambda \right)}\left({f}_g\right)\right] \).
Accordingly, an original color image f^{rgb}(p, q) can be reconstructed by finiteorderQFrGLMs. The reconstructed image is represented as:
2.3 Design of QFrGLMIs
The authors of [38] proposed a geometric invariance analysis method based on Krawtchouk moments. We considered that the Krawtchouk moments can be calculated as a linear combination of their corresponding geometric moments. Therefore, the geometricinvariant transformations (rotation, scaling, and translation) of the Krawtchouk moments can also be expressed as the linear combination of their corresponding geometricinvariant moments. Inspired by the Krawtchouk moment invariants, this subsection proposes a new set of QFrGLMIs. After analyzing the relationship between the quaternion fractionalorder geometric moment invariants (QFrGMIs) and the proposed QFrGLMIs, we provide a realization scheme of the QFrGLMIs; specifically, we construct the QFrGLMIs as a linear combination of QFrGLMs. Finally, we obtain the invariant transformations (rotation, scaling, and translation) of the proposed QFrGLMIs.
2.3.1 Translation invariance of QFrGMIs
Extending the traditional integerorder geometric moments to realorder (fractionalorder) moments, the quaternion fractionalorder geometric moments (QFrGMs) of an N × N digital color image can be expressed as follows:
Similarly to the traditional centralized geometric moments of integerorder, the centralized moments of QFrGMs, can be defined as:
where the centroid of a digital color image (x_{c}, y_{c}) is defined as:
Above, we mentioned that a quaternion color image can be expressed as a linear combination of the single channels of an original color image. Let λ_{1} and λ_{2} be 1, and let \( {m}_{00}^{\left(r;{\lambda}_1,{\lambda}_2\right)} \) and \( {m}_{01}^{\left(r;{\lambda}_1,{\lambda}_2\right)} \) (or\( {m}_{10}^{\left(r;{\lambda}_1,{\lambda}_2\right)} \)) represent the zerothorder and firstorder moments of the R component of the original color image, respectively. Similarly, let \( {m}_{00}^{\left(g;{\lambda}_1,{\lambda}_2\right)} \) and \( {m}_{01}^{\left(g;{\lambda}_1,{\lambda}_2\right)} \) (or\( {m}_{10}^{\left(g;{\lambda}_1,{\lambda}_2\right)} \)) represent the zerothorder and firstorder moments of the G component of the image, respectively, and let \( {m}_{00}^{\left(b;{\lambda}_1,{\lambda}_2\right)} \) and \( {m}_{01}^{\left(b;{\lambda}_1,{\lambda}_2\right)} \) (or\( {m}_{10}^{\left(b;{\lambda}_1,{\lambda}_2\right)} \)) represent the zerothorder and firstorder moment of the B component of the image, respectively. In this case, Eq. (35) satisfies the translation invariance of the original color image. Figure 3 shows an illustration for the processing of translation invariance, and here, the red “+” mark represents the centroid of the image in Fig. 3, T1 indicates that the original image is translated 60 pixels down and right, T2 means that it is translated 60 pixels up and left, and T3 shows that it is translated 60 pixels up and right, and the final proceed image is the centralized image in Cartesian coordinates.
2.3.2 Rotation, scaling, and translation invariance of QFrGMIs
Referring to Eq. (17) in [38], the rotation, scaling, and translation invariants of QFrGMIs can be expressed as follows:
where \( \tau ={m}_{00}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)} \), \( \gamma =\frac{\lambda_1p+{\lambda}_2q+2}{2} \), \( \theta =\frac{1}{2}\arctan \left(\frac{2{u}_{11}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}}{u_{20}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}{u}_{02}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}}\right) \), and −45^{∘} ≤ θ ≤ 45^{∘}.
The calculation steps of the rotational, scaling, and translation invariants of QFrGMIs are detailed in [38].
2.3.3 Rotation, scaling, and translation invariance of the proposed QFrGLMIs
Substituting Eq. (25) into Eq. (31), we first obtain the following result:
Let \( {\overline{f}}^{rgb}\left(i,j\right) \) be the following weighted colorimage representation:
Eq. (38) can then be rewritten as follows:
where \( {\sigma}_n=\frac{1}{\sqrt{\gamma_n^{\left({\alpha}_x,{\lambda}_x\right)}}} \) and \( {\sigma}_m=\frac{1}{\sqrt{\gamma_m^{\left({\alpha}_x,{\lambda}_x\right)}}} \). Given \( {L}_n^{\left(\alpha, \lambda \right)}(x)=\sum \limits_{i=0}^n{\psi}_{ni}{x}^{\lambda i} \) (see Eq. (23)) and using Eq. (34), the above formula becomes
Eq. (41) is derived in Appendix B.
The invariant transformations (rotation, scaling, and translation) of the QFrGLMIs are obtained by substituting \( {m}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)} \) in Eq. (41) with \( {v}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)} \) in Eq. (37):
3 Results and discussion
In this section, the experimental results and analysis are used to validate the theoretical framework developed in the previous sections. The performances of the proposed QFrGLMs and QFrGLMIs in image processing were evaluated in five sets of typical experiments. In the first group of experiments, the global reconstruction performance of the color images was evaluated under noisefree, noisy, and smoothingfilter conditions. The second group of experiments evaluated the proposed QFrGLMs on localimage reconstruction, ROIfeature extraction, and the influence of different parameter conditions on image reconstruction. To improve the reconstruction and classification performance of the proposed QFrGLMs on color images, the parameters were optimized through image reconstruction in the third group of experiments. The fourth group of experiments tested the image classification of the proposed QFrGLMIs under geometric transformation, noisy, and smoothingfilter conditions. These experiments were mainly performed on different colorimage datasets that are openly accessible on the Internet. In the last group, the computational time consumption of the proposed QFrGLMs was compared with those of the latest QFrZMs and other orthogonal moments. All experimental simulations were completed on a PC terminal with the following hardware configuration: Intel (R) core (IM) i5, 2.5 GHz CPU, 8 GB memory, Windows 7 operating system. The simulation software was MATLAB 2013a.
3.1 Experiments on global reconstruction of color images
This subsection evaluates the global featureextraction performance of the proposed QFrGLMs on color images. The evaluation was divided into two steps: imagereconstruction evaluation of the QFrGLMs and other approaches on original color images (i.e., noisefree and unfiltered images), and imagereconstruction evaluation of color images superposed with salt and pepper noise or preprocessed by a conventional smoothing filter. The QFrGLMs and other image moments are then applied to image feature extraction and are finally subjected to colorimage reconstruction experiments. The test image in this experiment was the colored “cat” image selected from the wellknown Columbia Object Image Library (COIL100). The test image was sized 128 × 128. The colorimage reconstruction performance was evaluated by the mean square error (MSE) and peak signaltonoise ratio (PSNR), which are respectively calculated as follows:
Here, MSE^{(r)}, MSE^{(g)}, and MSE^{(b)} denote the MSE values of the grayscale image corresponding to the independent red, green, and blue components of the color image, respectively, which are defined as
In Eq. (45), f(x, y) and \( \overline{f}\left(x,y\right) \) represent the original twodimensionalN × N grayscale image and its reconstructed image, respectively.
To assess the global reconstruction performance of the proposed QFrGLMs, experiments were performed under three parameter settings: (I)α_{x} = α_{y} = 1, λ_{x} = λ_{y} = 1.1, (II) α_{x} = α_{y} = 1, λ_{x} = λ_{y} = 1.2, and (III) α_{x} = α_{y} = 1, λ_{x} = λ_{y} = 1.3. The performances of the proposed QFrGLMs have been compared with those of QFrZMs and other stateoftheart color image moments. The comparative results are shown in Tables 1 and 2, and Fig. 4. The reconstruction performance of the loworder QFrGLMs (n, m < 12) was poorer under parameter setting (III) than under parameters settings (I) and (II) (Fig. 4). Under parameter setting (III), the loworder QFrGLMs were also outperformed by other color image moments (QGLMs, QFrZMs, and QZMs). Note that QGLMs are a special case of QFrGLMs with α_{x} = α_{y} = 1, λ_{x} = λ_{y} = 1. However, when the order of each colorimage moment was sufficiently high (n, m > 20), the QFrGLMs achieved the best imagereconstruction performance under parameter setting (III). The image reconstruction results of the QFrGLMs clearly differed between the low and highorder moments. In the loworder moments, the zerovalue distributions of the QFrGLMs polynomials were concentrated at the image origin under the parameter settings α_{x} = α_{y} = 1, λ_{x} = λ_{y} = 1.3, so the sampling neglected the edges and details of the image. Conversely, in the highorder moments, the zerovalue distributions of the polynomials approximated a uniform distribution, so the image reconstruction was optimal. To intuitively show the visual effect of image reconstruction, Tables 1 and 2 presents the visualization results of the reconstruction experiments with different colorimage moments the lower and higherorder moments, respectively. It can be seen from Tables 1 and 2 that the proposed image moments in this paper are all optimal in terms of lowerorder moments or higherorder moments, the proposed QFrGLMs provided a better visual effect of the image reconstruction than the other color image moments. Especially in the higherorder, when n, m = 50, the image reconstruction of the QFrZMs has failed, while when the order of the moments is equal to 100, the PSNR value of the proposed QFrGLMs can still maintain above 29 dB, and the visualization effect is nice as usual.
To further verify the robustness of the proposed QFrGLMs in noise resistance and nonconventional signal processing, the features of color images infected with salt and pepper noise or subjected to smooth filtering were extracted by the proposed QFrGLMs and other colorimage moments. New color images were reconstructed using the extracted features, and the performances of the image reconstructions were evaluated by the PSNR. Figure 5 shows the color images subjected to salt and pepper noise (noise density = 2%) and smooth filtering (with a 5 × 5 filter window), Fig. 6 and Tables 3 and 4 compare the color images reconstructed from the different image moments. Regardless of the parameter settings, increasing the order of the image moment (especially the highorder moments) reduced the sensitivity of the proposed QFrGLMs to salt and pepper noise and smoothing. Comparing the PSNR values of the different image moments, we find that the 28orderQFrGLMs outperformed the QFrZMs by 8 dB. In addition, as we all know, the image moments are usually more sensitive to noise in higherorder moments. However, compared with other latest image moments, i.e., QFrRHFMs, QFrPCTs, and QFrPSTs (for the sake of fair comparison, all the different types of image moments are constructed without accurate and fast algorithm), the proposed image moments can still maintain good image reconstruction visualization effect when the order of moments is 100, and its PSNR value is more than 25 dB under the condition of noise density of 2% or smooth filtering (filtering window is 5×5). In summary, the proposed QFrGLMs can properly describe color images under noisefree, noisy, and smoothed conditions and also exhibit high global feature extraction performance. Consequently, the proposed QFrGLMs show promising applicability to color image analysis.
3.2 Experiments on local reconstruction of color images
In recent years, localfeatureextraction or ROI detection have presented new challenges for the existing orthogonal moments. The existing image moments, especially most of the orthogonal moments, extract only the global features, and cannot describe the local features. The detection of arbitrary ROIs in images is especially challenging. Among the existing orthogonal moments, only a few discrete orthogonal moments based on Cartesian coordinate space, such as the Krawtchouk [39] and Hahn [40] moments, can perceive the local features in an image. Thus far, the application of such discrete orthogonal moments has been limited to localfeature detection in binary images. Xiao et al. [19] proposed fractionalorder shifting Legendre orthogonal moments, which extract the local features of a grayscale image by changing the parameter values of the fractional order. However, the local image is not well reconstructed (see Fig. 5 in [19]); especially, the details of the ROI are insufficiently protected in the localimage reconstruction. In addition, localfeatureextraction from color images has been little reported in the literature on image moments. In this subsection, we meet the challenge of applying the proposed QFrGLMs to localfeature extraction from color images. The test images were three typical “block” color images selected from the COIL100 database. The local features in the color images at different positions of the three “block” color images were reconstructed using the features extracted by the QFrGLMs with different parameters. The experimental results are summarized in Table 5. This table shows that under different parameter settings, the proposed QFrGLMs provided good image reconstructions in different regions of the original color image (the target areas of ROI extraction from the original color images are enclosed in the rededged boxes). Under the parameter setting α_{x} = 20, α_{y} = 1, λ_{x} = 1.4, λ_{y} = 1.5, the QFrGLMs extracted the upper part of the original color image. Meanwhile, the QFrGLMs with α_{x} = 1, α_{y} = 100, λ_{x} = 1.28, λ_{y} = 1.38 extracted the bottom part of the original color image, those with α_{x} = 25, α_{y} = 1, λ_{x} = 1.7, λ_{y} = 0.8 obtained the left part of the original color image, and those with α_{x} = 85, α_{y} = 1, λ_{x} = 1.4, λ_{y} = 1.5 extracted the rightupper part of the original color image. We conclude that the translation parameters of the proposed QFrGLMs determine the position information of the local features in the original color images, whereas the fractionalorder parameters mainly affect the quality of the localimage feature extraction and the details of the reconstructed color image. Specifically, the proposed QFrGLMs with smaller and larger values of the translation parameter α along the x and yaxes, respectively, mainly extracted the bottom part of the color image; conversely, the proposed QFrGLMs with smaller and larger α values along the x and yaxes, respectively, extracted the upper part of the color image. If the parameter values of different fractional orders along the x and yaxes are combined, the QFrGLMs obtain the local information at different positions in the original color image. As shown in the localimagereconstruction results (Table 5), the proposed QFrGLMs well described the local features at different positions of the block color images, implying their effectiveness as a localfeatureextraction descriptor.
To further verify their localfeature extraction capability, the proposed QFrGLMs were tested on a medical image (a computed tomography (CT) image of the human ankle, CT image seems to be a grayscale image; however, it is composed of R, G, and B three components—thus, in this experiment, it is regarded as a color image). In this experiment, the QFrGLMs were required to detect the ROI (the lesion area) in the humanankle CT image. As shown in Fig. 7, the proposed QFrGLMs properly detected the lesion in the CT image.
3.3 Optimal parameter selection
As presented in Subsection 4.2, the proposed QFrGLMs with determined translation parameters αrequire the proper selection of the fractionalorder parameter λ, because this parameter mainly affects the quality of the localimage feature extraction and the detailed descriptions of the reconstructed image. Therefore, optimizing the parameter λ is the key requirement of image reconstruction and classification by the proposed QFrGLMs. The optimal λ will guarantee the quality of the image reconstruction and the accuracy of image classification.
To study the influence of the parameters λ_{x} and λ_{y} on the performance of the proposed QFrGLMs, we selected 30 color images (e.g., “cat,” “piggybank,” “tomato,” and “block,”) from the COIL100 database. Referring to the different image reconstructions, an approach for selecting the parameter optimization method is proposed in this subsection. To elucidate how image size affects the parameters λ_{x} and λ_{y}, each of the selected images was scaled to different sizes: 256 × 256, 128 × 128, 64 × 64, and 32 × 32. The results are shown in Fig. 8.
To determine the optimal parameters λ_{x} and λ_{y} in combination, this subsection computes the performance of the proposed QFrGLMs by the average statistical normalized image reconstruction error (ASNIRE), which is defined as follows:
Here, the number of testing images L was 30, f_{c} is an original color image, and \( {\overline{f}}_c \) is the reconstruction of that color image. The SNIRE is the statistical normalized image reconstruction error function proposed in [19], defined as
In this experiment, the orders of the QFrGLMs were set to 10 ≤ n, m ≤ 20. Because λ_{x}, λ_{y} ≥ 0, we limited their values to the interval (0, 2] and calculated the combined results of their optimal values. Figure 8 shows the reference selection range of the optimal parameter values λ_{x} and λ_{y} obtained by this method. The ASNIRE values of the four color images were minimized around λ_{x}, λ_{y} = 1 (the blue regions in Fig. 9). Therefore, when selecting the optimal parameter combination for the proposed QFrGLMs, we suggest seeking within the range [1.0, 1.5], and it is suggested that the optimal parameters should be selected between 1 and 1.5, which can also be obtained from the distribution curves of the NFrGLPs under different parameter settings. It can be seen from the subgraphs f, g, and h of Fig. 2 that when the λ is 1.2 or 1.3, respectively, the distribution of the polynomials is close to uniform distribution. According to the zeropoint theory, the closer the polynomial distribution is to the uniform distribution, the better the effect of using the polynomial to sample the image; at this time, the image moments constructed by the polynomials have the best overall description capability for an image.
3.4 Geometricinvariant recognition in color images
This subsection tests and analyzes the recognition of geometricinvariant transformations (rotation, scaling, and translation) by the proposed QFrGLMs, and their robustness to noise and smoothing filter operations. This experiment was performed on two sets of public colorimage databases: (128 × 128)sized color images selected from COIL100 (Fig. 10) and (128 × 128)sized butterfly color images selected from [5] (Fig. 11). To verify that the proposed QFrGLMs recognize geometric invariants, the QFrGLMs were employed with three parameter settings: (I)λ_{x} = λ_{y} = 1.1, (II)λ_{x} = λ_{y} = 1.2, and (III)λ_{x} = λ_{y} = 1.3. In all three cases, α_{x} = α_{y} = 1. The images sets were categorized by a KNN classifier. The amplitudes of the colorimage moments were arranged into a feature vector for classification as follows:
The classification effects of the different image moments were determined by a measure called the correct classification percent (CCPs), expressed as
where N_{c} and N_{t} represent the number of correctly classified objects and the total number of all testing objects, respectively.
3.4.1 Experiment 1
The color image dataset for this experiment was extracted from COIL100. First, 100 color images from the COIL100 dataset were rotated by 0° and 180°, obtaining 200 images (100 × 2) as the training set. Each image in the training set was then translated by (Δx, Δy) ∈ [−45, 45]. The set of rotation vectors was defined as ϕ_{i} = 5 ∗ i, where i ∈ [0, 35] is an integer, and a scale factor α was defined for the scaling operation. Rotating 200 images byϕ_{i}, and scaling by α = 0.5 + (2.5 ∗ ϕ_{i})/360 ∈ [0.5, 3], we obtained 7200 (36 × 200) color images for testing. Finally, salt and pepper noise (with noise density ranging from 0 to 25% in 5% increments) was added to each image in the existing test set, forming a new noisy test set. In this experiment, the vectors V_{nm} of the different image moments were obtained at k = 12 (loworder moment) and k = 28 (highorder moment). Figure 12 compares the correct classification rates (CCPs) of the proposed QFrGLMs and other orthogonal moments (QZMs, QFrZMs, and QGLMs). As seen in the figure, the proposed QFrGLMs outperformed the other moments in both cases (k = 12 and k = 28).
3.4.2 Experiment 2
The dataset for this experiment was extracted from the Butterfly color image database. As described in Experiment 1, the 20 color images in the extracted dataset were rotated by 0°, 90°, and 270°, obtaining 200 images (20 × 3) as the training set. Next, following the steps described in Experiment 1, we obtained 2160 (36 × 60) color images as the test set. Finally, each image in the test set was passed through a smoothing filter with different window sizes (3, 5, 7, and 9), obtaining 2160 new color images as the filtered test set. Again, the vectors V_{nm} of different image moments were obtained at k = 12 (loworder moment) and k = 28 (highorder moment). Figure 13 shows the classification experiment results after smoothing. The proposed QFrGLMs were strongly robust to rotation, scaling, and translation transformations and achieved higher classification accuracy than the QZMs, QFrZMs, and QGLMs.
3.4.3 Experiment 3
In order to further prove the performance of the proposed image moments in geometric invariant recognition and classification, we compare the proposed geometric moment invariants (QFrGLMs, α_{x} = α_{y} = 1, λ_{x} = λ_{y} = 1.3) with the latest image moments (i.e., QFrRHFMs, QFrPCTs, and QFrPSTs). The experimental study on the geometric invariant image recognition accuracy of the proposed QFrGLMs under both noisy and smoothing filter conditions is presented in this subsection. Based on the training set and test set generated in Experiment 1, salt and pepper noise and smoothing filter destroys each image of the test set, and SNR varies from 25 dB to 0 dB with the reduction 5dB. At each SNR value, we obtain a new processed test set, and knearest neighbor (KNN) classifier is adopted to implement classification. As in every testing set, the correct classification percentages (CCPs) are gained from the proposed QFrGLMs, QFrRHFMs, QFrPCTs, and QFrPSTs, and the experimental results are shown in Table 6. From the classification results in Table 6, it can be seen that the CCPs of the proposed QFrGLMs is the highest in both lower and higherorder moments compared with other latest image moments.
3.5 Computational times
This experiment determined the computational times of the proposed QFrGLMs (for notational simplicity, we express the QFrGLMs with the three groups of parameter settings as QFrGLMs (I), QFrGLMs (II), and QFrGLMs (III)). The results are compared with those of the latest QFrZMs and other quaternion orthogonal moments (such as QZMs, QFrRHFMs, QFrPCTs, and QFrPSTs). The simulations were conducted on a Microsoft Window 7 operating system with a 2.5GHz Intel Core and 8 GB memory, and the program was encoded in Matlab2013a. The images were 25 color images of size 128 × 128 pixels, extracted from the Columbia University Image Library. Figure 14 summarizes the average elapsed CPU times of the 25 color images as the (n + m)th order of each image moment increased from 5 to 25 in 5unit increments.
The computational time of all orthogonal moments increased with order. However, as the polynomial of the moment in our approach is calculated by a recursive algorithm, the proposed QFrGLM color image moments in all parameter settings were computed faster than the QZMs and QFrZMs, and the computational time approached that of QGLM, QFrRHFMs, and QFrPSTs. By the way, the basis functions of QFrRHFMs, QFrPCTs, and QFrPSTs are based on trigonometric functions; therefore, compared with generalized Laguerre polynomials and Zernike polynomials, they do not involve accumulative summation and factorial operations, so the polynomial calculation process is relatively fast. However, because the QZMs, QFrZMs, QFrRHFMs, and QFrPSTs are computed in polar coordinates, the color images must be converted from Cartesian coordinates to polar coordinates, whereas the proposed image moments are directly constructed in the Cartesian coordinate system, which further reduces the computational time.
4 Conclusions
This paper proposed a new set of quaternion fractionalorder generalized Laguerre moments (QFrGLMs) based on GLPs and quaternion algebra. As colorimage feature descriptors, the proposed QFrGLMs can be used for colorimage reconstruction and feature extraction, and the image moments are available for global and local color image representations in the field of image analysis. More importantly, based on the local image representation characteristics of the proposed QFrGLMs, the application of the proposed moments in the field of digital watermarking [41,42,43] can effectively solve the problem of resisting largescale cropping and smearing attacks, which is also one of our future work directions. After establishing the relationship between QFrGLMs and FrGLMs, it was found that QFrGLMs can be represented as linear combinations of FrGLMs. We also presented a new set of rotation, scaling, and translation invariants for object recognition applications. In comparison experiments with other stateoftheart moments, i.e., the performance tests included global and localfeature extraction from color images, and geometricinvariant classification of color images. The proposed QFrGLMs demonstrated higher colorimage reconstruction capability and invariant recognition accuracy under noisefree, noisy, and smooth filtering conditions. Thus, the proposed QFrGLMs are potentially useful for colorimage description and digital watermarking [44,45,46,47]. However, the only deficiency is that the perfect geometric invariance [48, 49] cannot be achieved directly for invariant image recognition since the derivation of these QFrGLMs invariants are not based on generalized Laguerre polynomials themselves. In the future, the focus of our work is to construct a new set of generalized Laguerre moment invariants, namely, deriving an explicit generalized Laguerre moment invariants approach, which can be directly applied to the field of image recognition. In addition, combining with the existing color image representation methods based on quaternion algebra [50, 51] and finding a better performance fractionalorder radial orthogonal polynomials to construct quaternion fractionalorder image moments are our other goals.
Availability of data and materials
(1) The datasets generated in our experiments are available from coil100 image database and Butterfly images Database, URL link: http://www.cs.columbia.edu/CAVE/databases/. http://cs.cqupt.edu.cn/info/1078/4189.htm.
(2) The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 QFrGLMs:

Quaternion fractionalorder generalized Laguerre orthogonal moments
 QPHFMs:

Quaternion polar harmonic Fourier moments
 QFrRHFMs:

Quaternion fractionalorder radial harmonicFourier moments
 QFrPCTs:

Quaternion fractionalorder polar cosine transforms
 QFrPSTs:

Quaternion fractionalorder polar sine transforms
 QEFMs:

Quaternion exponent Fourier moments
 ROI:

Region of interest
 QFrZMs:

Quaternion fractionalorder Zernike moments
 QFrGLMIs:

Quaternion fractionalorder generalized Laguerre moment invariants
 FrIMs:

Fractionalorder image moments
 GLPs:

Generalized Laguerre polynomials
 FrGLPs:

Fractionalorder GLPs
 NFrGLPs:

Normalized FrGLPs
 GLMs:

Generalized Laguerre moments
 FrGLMs:

Fractionalorder GLMs
 QFrGMs:

Quaternion fractionalorder geometric moments
 QFrGMIs:

Quaternion fractionalorder geometric moment invariants
 SNIRE:

Statisticalnormalization imagereconstruction error
 CCPs:

Correct classification percentages
 KNN:

Knearest neighbor
 SNR:

Signal noise ratio
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions.
Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 61702403, 61976168, 61976031), Science Foundation of the Shaanxi Key Laboratory of Network Data Analysis and Intelligent Processing (Grant No. XUPTKLND201901), the Fundamental Research Funds for the Central Universities (JBF180301), the Project funded by China Postdoctoral Science Foundation (Grant No. 2018M633473), Shaanxi Province of Key R & D project (Grant No. 2020GY051), and Shaanxi Provincial Department of Education project (Grant No. 20JS044, 18JK0277), and the project funded by Weinan regional collaborative innovation development research (Grant No. WXQY001001, WXQY002007).
Author information
Authors and Affiliations
Contributions
Jun Liu developed the idea for the study proposed moments and contributed the central idea in our manuscript, Bing He did the analyses for the properties of the proposed image moments and analysed most of the data, and wrote the initial draft of the paper, and the remaining authors contributed to refining the ideas, carrying out additional analyses and finalizing this paper. All authors were involved in writing the manuscript. The author(s) read and approved the final manuscript.
Authors’ information
BING HE was born in 1982. He received the B.S. and M.S. degrees in Communication Engineering and Electrical Engineering from Northwestern Polytechnical University (NPU) and Shaanxi Normal University (SNNU), Xi’an, China, in 2006 and 2009, respectively. Now, he is an associate professor in Weinan Normal University and received his Ph.D. degree in computer science from Xidian University, Xi’An, China, in 2020. His research interests include image processing, object recognition, and digital watermarking.
JUN LIU was born in 1973. He received his M.S. and Ph.D. degrees in Computer Science and Technology from Northwest University, Xi’an, China, in 2009 and 2018, respectively. He is currently a professor at the School of Computer Science and Technology in Weinan Normal University and also a member of China Computer Federation. His research interests include pattern recognition and machine learning.
TENGFEI YANG was born in 1987. He received the M.S. degree with the School of Physics and Information Technology from Shaanxi Normal University, Xi’an, China, in 2016, received his Ph.D. degree in computer science from Xidian University, Xi’an, China. He is currently an associate professor with the School of Cyberspace Security, Xi’an University of Posts and Telecommunications. His research interests include image processing, pattern recognition, multimedia security, and applied cryptography.
BIN XIAO was born in 1982. He received his B.S. and M.S. degrees in Electrical Engineering from Shaanxi Normal University, Xi’an, China, in 2004 and 2007, received his Ph.D. degree in computer science from Xidian University, Xi’An, China. He is now working as a professor at Chongqing University of Posts and Telecommunications, Chongqing, China. His research interests include image processing, pattern recognition, and digital watermarking.
YANGUO PENG was born in 1986. He is currently a lecturer in Computer Architecture at Xidian University, Xi’an, China. He received his Ph. D. degree from Xidian University in 2016. His research interests include secure issues in data management, privacy protection, and cloud security.
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Appendices
Appendix A
1.1 Proof of the recursive operation Eq. (29):
From Eq. (24), when n ≥ 2, we have:
Substituting Eq. (28) in Eq. (27), we obtain:
Substituting \( {\gamma}_n^{\left(\alpha, \lambda \right)}=\frac{\Gamma \left(n+\alpha +1\right)}{n!}=\frac{\left(n+\alpha \right)\Gamma \left(n+\alpha \right)}{n\left(n1\right)!}=\frac{\left(n+\alpha \right)}{n}{\gamma}_{n1}^{\left(\alpha, \lambda \right)} \) into the above formula, we have
Letting \( {A}_0=\frac{2n1+\alpha }{\sqrt{n\left(n+\alpha \right)}} \), \( {A}_1=\frac{1}{\sqrt{n\left(n+\alpha \right)}} \), \( {A}_2=\sqrt{\frac{\left(n+\alpha 1\right)\left(n1\right)}{n\left(n+\alpha \right)}} \), we complete the proof.
Appendix B
1.1 Derivation of Eq. (41)
Substituting Eqs. (23) and (34) into Eq. (40), we have:
\( ={\sigma}_n{\sigma}_m\sum \limits_{p=0}^n\sum \limits_{q=0}^m{\psi}_{np}{\psi}_{mq}{m}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)} \), which completes the derivation.
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He, B., Liu, J., Yang, T. et al. Quaternion fractionalorder color orthogonal momentbased image representation and recognition. J Image Video Proc. 2021, 17 (2021). https://doi.org/10.1186/s13640021005537
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DOI: https://doi.org/10.1186/s13640021005537