This section introduces our proposed QFr-GLM scheme, derived from quaternion algebra theory, fractional-order orthogonal moments, and GLPs. After developing the basic framework of QFr-GLMs, we analyze the relationship between the quaternion-based method and the single-channel-based 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:

$$ {f}^{rgb}\left(x,y\right)={f}_r\left(x,y\right)i+{f}_g\left(x,y\right)j+{f}_b\left(x,y\right)k. $$

(20)

The remainder of this section is organized as follows. Subsection 3.1 defines and constructs our fractional-order GLPs (Fr-GLPs) and normalized Fr-GLPs (NFr-GLPs), and Subsection 3.2 defines the proposed QFr-GLMs, and relates them to the fractional-order generalized Laguerre moments (Fr-GLMs) of single channels in a traditional RGB color image, and the basic framework is shown in Fig. 1. The QFr-GLMs invariants (QFr-GLMIs) are constructed in subsection 3.3.

### Calculation of Fr-GLPs and NFr-GLPs

Fr-GLPs [37] can be expressed as:

$$ {L}_n^{\left(\alpha, \lambda \right)}(x)={L}_n^{\left(\alpha \right)}\left({x}^{\lambda}\right), $$

(21)

where, *λ* > 0, *x* ∈ [0, +∞], similarly to Eq. (15). The Fr-GLPs satisfy the following orthogonality relation in the interval [0, +∞]:

$$ {\int}_0^{+\infty }{\omega}^{\left(\alpha, \lambda \right)}(x){L}_n^{\left(\alpha, \lambda \right)}(x){L}_m^{\left(\alpha, \lambda \right)}(x) dx={\gamma}_n^{\left(\alpha, \lambda \right)}{\delta}_{nm}, $$

(22)

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 Fr-GLPs can be rewritten as the following binomial expansion [19, 37]:

$$ {L}_n^{\left(\alpha, \lambda \right)}(x)=\sum \limits_{i=0}^n{\psi}_{ni}{x}^{\lambda i}, $$

(23)

where \( {\psi}_{ni}={\left(-1\right)}^i\frac{\Gamma \left(n+\alpha +1\right)}{\Gamma \left(i+\alpha +1\right)\left(n-i\right)!i!} \), similar to Eq. (19), the Fr-GLPs can be implemented by the following recursive algorithm:

$$ {nL}_n^{\left(\alpha, \lambda \right)}(x)=\left[2\left(n-1\right)+\alpha +1-{x}^{\lambda}\right]{L}_{n-1}^{\left(\alpha, \lambda \right)}(x)-\left(n-1+\alpha \right){L}_{n-2}^{\left(\alpha, \lambda \right)}(x), $$

(24)

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 fractional-order GLPs (NFr-GLPs) are defined as:

$$ {\overline{L}}_n^{\left(\alpha, \lambda \right)}(x)={L}_n^{\left(\alpha, \lambda \right)}(x)\sqrt{\frac{\omega^{\left(\alpha, \lambda \right)}(x)}{\gamma_n^{\left(\alpha, \lambda \right)}}}. $$

(25)

**Theorem 1.** The NFr-GLPs\( {\overline{L}}_n^{\left(\alpha, \lambda \right)}(x) \) are orthogonal on the interval [0, +∞]:

$$ {\int}_0^{+\infty }{\overline{L}}_n^{\left(\alpha, \lambda \right)}(x){\overline{L}}_m^{\left(\alpha, \lambda \right)}(x) dx={\delta}_{nm}. $$

(26)

**Proof of Theorem 1.** Given the NFr-GLPs\( {L}_n^{\left(\alpha, \lambda \right)}(x) \) 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

$$ {\displaystyle \begin{array}{c}{\int}_0^{+\infty }{L}_n^{\left(\alpha, \lambda \right)}(x)\sqrt{\frac{\omega^{\left(\alpha, \lambda \right)}(x)}{\gamma_n^{\left(\alpha, \lambda \right)}}}{L}_m^{\left(\alpha, \lambda \right)}(x)\sqrt{\frac{\omega^{\left(\alpha, \lambda \right)}(x)}{\gamma_m^{\left(\alpha, \lambda \right)}}} dx\\ {}=\frac{1}{\sqrt{\gamma_n^{\left(\alpha, \lambda \right)}{\gamma}_m^{\left(\alpha, \lambda \right)}}}{\int}_0^{+\infty }{\omega}^{\left(\alpha, \lambda \right)}(x){L}_n^{\left(\alpha, \lambda \right)}(x){L}_m^{\left(\alpha, \lambda \right)}(x) dx.\end{array}} $$

(27)

Using Eq. (22), we further obtain:

$$ {\int}_0^{+\infty }{\overline{L}}_n^{\left(\alpha, \lambda \right)}(x){\overline{L}}_m^{\left(\alpha, \lambda \right)}(x) dx=\frac{\gamma_n^{\left(\alpha, \lambda \right)}}{\sqrt{\gamma_n^{\left(\alpha, \lambda \right)}{\gamma}_m^{\left(\alpha, \lambda \right)}}}{\delta}_{nm}, $$

(28)

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 NFr-GLPs are recursively calculated as follows:

$$ {\overline{L}}_n^{\left(\alpha, \lambda \right)}(x)=\left({A}_0+{A}_1{x}^{\lambda}\right){\overline{L}}_{n-1}^{\left(\alpha, \lambda \right)}(x)+{A}_2{\overline{L}}_{n-2}^{\left(\alpha, \lambda \right)}(x), $$

(29)

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 \( {A}_2=-\sqrt{\frac{\left(n+\alpha -1\right)\left(n-1\right)}{n\left(n+\alpha \right)}} \). The detailed proof of the recursive operation is given in Appendix A.

Figure 2 shows the distribution curves of the NFr-GLPs under different parameter settings. Note that the parameter *α* mainly affects the amplitudes of the NFr-GLPs of different orders and the distributions of the zero values along the *x*-axis. Thus, if an image is sampled with NFr-GLPs, the local-feature regions (ROI) are easily extracted from the images. In addition, the parameter *λ* can extend the integer-order polynomials to real-order polynomials (*λ* > 0, *λ* ∈ *R*^{+}). Therefore, traditional GLPs are a special case of Fr-GLPs 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 zero-value distributions of the Fr-GLPs along the *x*-axis (Fig. 2c–e), thus affecting the image-sampling result.

### Definition and calculation of QFr-GLMs

Pan et al. [30] proposed the generalized Laguerre moments (GLMs) for grayscale images in Cartesian coordinates. Recalling the introduction, the corresponding Fr-GLMs can be defined as:

$$ {FrS}_{nm}^{\left(\alpha, \lambda \right)}=w\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{f}^{gray}\left(i,j\right){\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_i\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_j\right), $$

(30)

where *f*^{gray}(*i*, *j*) represents a grayscale digital image. For convenience, we map the original two-dimensionaldigital-image matrix to a square area of [0, *L*] × [0, *L*]. Here, *L* > 0, \( w={\left(\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$N$}\right.\right)}^2 \),\( {x}_i=\frac{iL}{N},{y}_j=\frac{jL}{N},i,j=0,1,2,\dots, N-1 \).

Using Eq. (30) with the help of Eq. (20), the right-sideQFr-GLMs of an original RGB color image in Cartesian coordinates are defined as:

$$ {\displaystyle \begin{array}{c}{\boldsymbol{QFrS}}_{nm}^{\left(\alpha, \lambda \right)}=w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right){f}^{rgb}\left(p,q\right)\mu \\ {}=\frac{1}{\sqrt{3}}w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left({\boldsymbol{if}}_r+{jf}_g+{kf}_b\right)\left(i+j+k\right)\\ {}\begin{array}{c}=-\frac{1}{\sqrt{3}}\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left({f}_r+{f}_g+{f}_b\right)\right]\\ {}+\frac{1}{\sqrt{3}}k\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left({f}_r-{f}_g\right)\right],\end{array}\\ {}+\frac{1}{\sqrt{3}}j\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left({f}_b-{f}_r\right)\right]\\ {}+\frac{1}{\sqrt{3}}i\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left({f}_g-{f}_b\right)\right]\end{array}} $$

(31)

where \( \mu =\left(i+j+k\right)/\sqrt{3} \) is the unit pure imaginary quaternion. The QFr-GLMs expressed in quaternion and the Fr-GLMs of single channels in traditional RGB color images are related as follows:

$$ {QFrS}_{nm}^{\left(\alpha, \lambda \right)}=A+ iB+ jC+ kD,, $$

(32)

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 finite-orderQFr-GLMs. The reconstructed image is represented as:

$$ {\displaystyle \begin{array}{c}{\overline{f}}^{rgb}\left(p,q\right)=w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\boldsymbol{QFrS}}_{nm}^{\left(\alpha, \lambda \right)}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\mu \\ {}=\frac{1}{\sqrt{3}}w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}\left(A+ iB+ jC+ kD\right){\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left(i+j+k\right)\\ {}=-\frac{1}{\sqrt{3}}\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left(B+C+D\right)\right]\\ {}+\frac{1}{\sqrt{3}}k\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left(A+B-C\right)\right],\\ {}+\frac{1}{\sqrt{3}}j\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left(A-B+D\right)\right]\\ {}+\frac{1}{\sqrt{3}}i\left[w\sum \limits_{p=0}^{N-1}\sum \limits_{q=0}^{N-1}{\overline{L}}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_p\right){\overline{L}}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_q\right)\left(A+C-D\right)\right]\end{array}} $$

(33)

### Design of QFr-GLMIs

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 geometric-invariant transformations (rotation, scaling, and translation) of the Krawtchouk moments can also be expressed as the linear combination of their corresponding geometric-invariant moments. Inspired by the Krawtchouk moment invariants, this subsection proposes a new set of QFr-GLMIs. After analyzing the relationship between the quaternion fractional-order geometric moment invariants (QFr-GMIs) and the proposed QFr-GLMIs, we provide a realization scheme of the QFr-GLMIs; specifically, we construct the QFr-GLMIs as a linear combination of QFr-GLMs. Finally, we obtain the invariant transformations (rotation, scaling, and translation) of the proposed QFr-GLMIs.

#### Translation invariance of QFr-GMIs

Extending the traditional integer-order geometric moments to real-order (fractional-order) moments, the quaternion fractional-order geometric moments (QFr-GMs) of an *N* × *N* digital color image can be expressed as follows:

$$ {m}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}=\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{x}_i^{\lambda_1p}{y}_j^{\lambda_2q}{f}^{rgb}\left({x}_i,{y}_j\right). $$

(34)

Similarly to the traditional centralized geometric moments of integer-order, the centralized moments of QFr-GMs, can be defined as:

$$ {u}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}=\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{\left({x}_i-{x}_c\right)}^{\lambda_1p}{\left({y}_i-{y}_c\right)}^{\lambda_2q}{f}^{rgb}\left({x}_i,{y}_j\right), $$

(35)

where the centroid of a digital color image (*x*_{c}, *y*_{c}) is defined as:

$$ {\displaystyle \begin{array}{c}{x}_c=\left({m}_{10}^{\left(r;{\lambda}_1;{\lambda}_2\right)}+{m}_{10}^{\left(g;{\lambda}_1;{\lambda}_2\right)}+{m}_{10}^{\left(b;{\lambda}_1;{\lambda}_2\right)}\right)/{m}_{00}^{\left( rgb;{\lambda}_1;{\lambda}_2\right)}\\ {}{y}_c=\left({m}_{01}^{\left(r;{\lambda}_1;{\lambda}_2\right)}+{m}_{01}^{\left(g;{\lambda}_1;{\lambda}_2\right)}+{m}_{01}^{\left(b;{\lambda}_1;{\lambda}_2\right)}\right)/{m}_{00}^{\left( rgb;{\lambda}_1;{\lambda}_2\right)}.\\ {}{m}_{00}^{\left( rgb;{\lambda}_1;{\lambda}_2\right)}={m}_{00}^{\left(r;{\lambda}_1;{\lambda}_2\right)}+{m}_{00}^{\left(g;{\lambda}_1;{\lambda}_2\right)}+{m}_{00}^{\left(b;{\lambda}_1;{\lambda}_2\right)}\end{array}} $$

(36)

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 zeroth-order and first-order 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 zeroth-order and first-order 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 zeroth-order and first-order 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.

#### Rotation, scaling, and translation invariance of QFr-GMIs

Referring to Eq. (17) in [38], the rotation, scaling, and translation invariants of QFr-GMIs can be expressed as follows:

$$ {\displaystyle \begin{array}{l}{v}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}={\tau}^{-\gamma}\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{\left[\left({x}_i-{x}_c\right)\cos \theta +\left({y}_i-{y}_c\right)\sin \theta \right]}^{\lambda_1p}\\ {}\kern1.50em \times {\left[\left({y}_j-{y}_c\right)\cos \theta -\left({x}_i-{x}_c\right)\sin \theta \right]}^{\lambda_2q}{f}^{rgb}\left(x,y\right),\end{array}} $$

(37)

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 QFr-GMIs are detailed in [38].

#### Rotation, scaling, and translation invariance of the proposed QFr-GLMIs

Substituting Eq. (25) into Eq. (31), we first obtain the following result:

$$ {\boldsymbol{QFrS}}_{nm}^{\left(\alpha, \lambda \right)}=w\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{L}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_i\right){L}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_j\right)\sqrt{\frac{\omega^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_i\right)}{\gamma_n^{\left({\alpha}_x,{\lambda}_x\right)}}}\sqrt{\frac{\omega^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_j\right)}{\gamma_m^{\left({\alpha}_y,{\lambda}_y\right)}}}{f}^{rgb}\left(i,j\right)u. $$

(38)

Let \( {\overline{f}}^{rgb}\left(i,j\right) \) be the following weighted color-image representation:

$$ {\overline{f}}^{rgb}\left(i,j\right)=w\sqrt{\omega^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_i\right)}\sqrt{\omega^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_j\right)}{f}^{rgb}\left(i,j\right)u. $$

(39)

Eq. (38) can then be rewritten as follows:

$$ {QFrS}_{nm}^{\left(\alpha, \lambda \right)}={\sigma}_n{\sigma}_m\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{L}_n^{\left({\alpha}_x,{\lambda}_x\right)}\left({x}_i\right){L}_m^{\left({\alpha}_y,{\lambda}_y\right)}\left({y}_j\right){\overline{f}}^{rgb}\left(i,j\right), $$

(40)

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

$$ {QFrS}_{nm}^{\left(\alpha, \lambda \right)}={\sigma}_n{\sigma}_m\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{\psi}_{np}{\psi}_{mq}{m}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}. $$

(41)

Eq. (41) is derived in Appendix B.

The invariant transformations (rotation, scaling, and translation) of the QFr-GLMIs 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):

$$ {QFrS}_{nm}^{\left(\alpha, \lambda \right)}={\sigma}_n{\sigma}_m\sum \limits_{i=0}^{N-1}\sum \limits_{j=0}^{N-1}{\psi}_{np}{\psi}_{mq}{v}_{pq}^{\left( rgb;{\lambda}_1,{\lambda}_2\right)}. $$

(42)