Fractional‐order generalized Laguerre moments and moment invariants for grey‐scale image analysis

IET Image ProcessingVolume 15, Issue 2 p. 523-541 ORIGINAL RESEARCH PAPEROpen Access Fractional-order generalized Laguerre moments and moment invariants for grey-scale image analysis Rachid Benouini, Rachid Benouini orcid.org/0000-0001-8586-5289 Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, MoroccoSearch for more papers by this authorImad Batioua, Corresponding Author Imad Batioua rachid.benouini@usmba.ac.ma Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco Correspondence Rachid Benouini, Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco. Email: rachid.benouini@usmba.ac.maSearch for more papers by this authorKhalid Zenkouar, Khalid Zenkouar Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, MoroccoSearch for more papers by this authorSaid Najah, Said Najah Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, MoroccoSearch for more papers by this author Rachid Benouini, Rachid Benouini orcid.org/0000-0001-8586-5289 Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, MoroccoSearch for more papers by this authorImad Batioua, Corresponding Author Imad Batioua rachid.benouini@usmba.ac.ma Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco Correspondence Rachid Benouini, Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco. Email: rachid.benouini@usmba.ac.maSearch for more papers by this authorKhalid Zenkouar, Khalid Zenkouar Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, MoroccoSearch for more papers by this authorSaid Najah, Said Najah Laboratory of Intelligent Systems and Application (LSIA), Faculty of Sciences and Technology, University Sidi Mohamed Ben Abdellah, Fez, MoroccoSearch for more papers by this author First published: 15 December 2020 https://doi.org/10.1049/ipr2.12044Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Here, a new set of fractional-order moments, named fractional-order generalized Laguerre moments (FGLM), is introduced. These proposed moments are defined on the Cartesian coordinate system and their basis functions are represented by the fractional-order generalized Laguerre polynomials. Contrary to the classical Chebyshev, Legendre and Gegenbauer moments, which provide only global feature, our proposed FGLM have the ability to extract both global and local features. Moreover, a new set of rotation, scale and translation invariants of the FGLM, is derived and introduced for image classification and invariant pattern recognition. Just as important, we have presented a systematic parameter selection method for finding the optimal fractional parameter values with respect to pattern recognition applications. Finally, several recursive methods for reducing the computation time of our proposed invariants are also provided in this study. Therefore, to demonstrate the performance of the introduced fractional-order moments and moment invariants, a number of experimental analysis are performed in terms of global and local features extraction, robustness to noise, invariance to geometric deformations, object recognition and computational speed. The presented theoretical and experimental results clearly show that the proposed fractional-order moments and their corresponding invariants could be extremely useful in the field of image analysis. 1 INTRODUCTION Image moments and moment functions have been extensively applied in the domains of image analysis [2, 8], pattern recognition [9-69], image retrieval [14, 17], medical image analysis [18, 21] and image watermarking [22, 25]. This is due to their attractive properties and interpretations, such as the compact representation of the image content, the robustness to noise and the invariance against geometric deformations 1. Historically, the notion of image moments has been first introduced by Hu 9, where the author presented a set of seven moment invariants with respect to rotation, scale and translation (RST) transforms, used particularly for two-dimensional pattern recognition. Unfortunately, Hu's invariants are not independent nor complete 26, since the geometric basis functions are not orthogonal. Actually, the most widely used families of image moments are the continuous and the discrete orthogonal moments. Their utility is founded on the fact that if the basis functions are orthogonal, only a few image moments terms are needed to represent the original image with high accuracy 2. The continuous moments are defined by using continuous orthogonal polynomials as a basis function, implying a suitable transformation of the image coordinates and approximation of the continuous integrals 38. While the discrete moments are based on discrete orthogonal polynomials, which are directly defined in the discrete space, eliminating the need of any numerical approximation 4. Moreover, according to the definition domain of the polynomials basis function, each of the continuous and discrete moments can be divided into two subgroups: moments defined in polar coordinate, called radial moments, and moments defined in the Cartesian coordinate. A detailed taxonomy of the existing families of image moments is illustrated in Figure 1. One common characteristic of all the aforementioned types of moments is that they are limited to being of integer order, since their basis functions are expressed by orthogonal polynomials of integer order. FIGURE 1Open in figure viewerPowerPoint Taxonomy of the existing families of image moments Quite recently, a new category of moments called fractional-order moments (FM) [40, 41] was introduced. The basis functions of this new category of moments are represented by orthogonal polynomials of fractional order, instead of the traditional polynomials of integer order. In fact, the fractional-order orthogonal polynomials are defined by introducing a fractional parameter λ > 0 , in order to generalise the notion of integer-order n with n ∈ N to fractional-order λ n , also called arbitrary order. These sets of polynomials have appeared in the field of mathematic as numerical solutions of some fractional differential equations [42, 45]. However, the fractional-order moments have not been sufficiently studied and their capabilities in the fields of image analysis has not been fully investigated. So far, only few papers concerning the definition of fractional-order moments have been published in the field of image analysis. Zhang et al. 40 introduced the fractional-order orthogonal Fourier–Mellin moments for binary character recognition, based on a new fractional-order radial polynomials. Xiao et al. 41 presented two forms of fractional-order orthogonal moments defined in polar and Cartesian coordinates, based, respectively, on radial shifted and shifted Legendre polynomials of fractional order. The authors investigated the capabilities of fractional-order moments for image reconstruction and rotation invariant image recognition. Even though all these fractional-order moments have shown an improved description capabilities compared to the traditional moments of integer order, the use of fractional moments for image analysis, has not been sufficiently explored and it is a subject of much interest currently. In addition, their applications in the domains of computer vision and pattern recognition constitute a new field of action. In this paper, we introduce a new set of fractional-order moments, named fractional-order generalised Laguerre moments (FGLM), to the field of image analysis. This set of moments is defined on the Cartesian coordinate system based on the generalised Laguerre polynomials of fractional order 44. In addition, the proposed set of moments have the capability of being able to extract both global and local features, where the extraction of local features can be easily achieved by adjusting polynomials parameters associated with the gamma distribution 48. Also, we propose a new set of RST moment invariants, baptised fractional-order generalised Laguerre moment invariants (FGLMI), based on using the algebraic relation between the FGLM and the fractional-order geometric moments. What is more, we developed new recurrence relations for the computation of the polynomials coefficients, which can be used to efficiently reduce the numerical instability and computation cost of the proposed FGLMI. To validate the effectiveness of the proposed FGLM and FGLMI, several numerical experimental results are performed. First, the global feature extraction capability is assessed by measuring the image reconstruction errors under noisy and noise-free conditions. While their ability to extract locale feature is clearly illustrated through several examples for brain tumour isolation. Further, the appropriate values of the locality parameters according to any specific region of interest (ROI) are provided in this work. Subsequently, to validate the effectiveness of the proposed fractional-order invariant descriptors, we are conducted to test their discrimination power, recognition performance and computational speed on several publicly available image databases. The outline is as follows: In Section 2 we highlight the contribution of the paper and we provided the related literature survey. Then, we recall the basic properties of the classical generalised Laguerre polynomials in Section 3. In Section 4, we present the definition of the new FGLM. Section 5 is devoted to presenting the derivation process of the RST invariants of the proposed FGLM. Experimental results and discussion are provided in Section 6. Finally, concluding remarks and future works are presented in Section 7. 2 RELATED WORK AND MOTIVATION One of the earliest classes of orthogonal polynomials is the generalised Laguerre polynomials (as a generalisation of Laguerre polynomials), which have been extensively used in the theory of approximation, quantum mechanics and electromagnetic waves propagation 60. In particular case, the Laguerre polynomials have been used as basis function to define two-dimensional moments for the purpose of digital image representation and analysis [58, 59]. To our best knowledge, very few publications can be found in the literature that discuss the use the Laguerre moments in the field of image analysis. Qjidaa 58 proposed the Laguerre moments for image analysis and showed their potential capabilities in image reconstruction. Pan et al. 59 introduced a type of radial orthogonal moments for gray-level and colour image representation, based on the generalised Laguerre polynomials and the theory of quaternion algebra. A key limitation of this research is that they only address the problem of rotation invariance. Although the Laguerre moments have been successfully applied for image representation and recognition, there are still some interesting and relevant aspects to be addressed. As a recent advance in the field of image analysis by using moment functions, the fractional-order moments is considered as a generalisation of the existing classical moments of integer order, and has provided more flexible image representation capability with much wider applicability. So far, many significant works have been reported during the past decade with the aim to define and apply fractional-order moments for pattern recognition and image representation. In fact, the first concrete definition of fractional-order moments and fractional-order moments invariants can be traced back to Xiao et al. 41, where the authors presented two types of fractional-order moments, defined in radial and Cartesian coordinates, based on fractional-order radial shifted Legendre polynomials and fractional-order shifted Legendre polynomials, respectively. These new types of moments have been used particularly in global and local feature extraction, image reconstruction and pattern recognition. Inspired by the same approach of 41, Zhang et al. 40 presented the fractional-order Fourier–Mellin moments, which has been particularly employed for binary image reconstruction and character recognition. While, Kaur et al. 71 presented a new set of fractional-order Zernike moments for pattern recognition application, where the authors used this rotation invariants of the fractional-order Zernike moments and support vector machine for plant disease classification. However, their practical utilisation was substantially limited on rotation invariants and no discussion concerning the computation of scale and translation invariants has been provided in these studies. More recently, in our previous works [61, 70], we have introduced a new sets of fractional-order moments and moment invariants in Cartesian coordinates, where their basis function are based on Chebyshev and Legendre polynomials of fractional order. These sets has been used for image representation and pattern recognition. However, until now the capabilities of generalised Laguerre polynomials of fractional order as basis for defining fractional-order moments has not yet been explored. In addition, the computational aspects of the fractional-order moments has been slightly discussed in the literature. Noticing that only rotation invariants of generalised Laguerre moments have been derived in 59, by using polar coordinate system, and the invariance with respect to other geometric transformations, such as translation and scaling, need to be established. In fact, the FGLM allow the property to extract both global and local features, unlike the classical Chebyshev, Legendre and Gegenbauer moments, which are only global descriptors. This could be considered as a prime advantage of the proposed FGLM, over the existing fractional-order moments, for instance Legendre, Chebyshev and Zernike, which involve only two fractional parameters. By analysing the works done in the literature it has appeared that the Laguerre moments, show suitable capabilities in image recognition and representation: first, adjusting the gamma distribution parameters of the Laguerre moments can be used for both, local and global image representation. Second, the Laguerre moments has demonstrated prominent results in both coordinate systems, when used as a basis function for Cartesian moments 58 and radial moments 59. Last, due to its representation performance, the Laguerre polynomials have been extensively used for the approximation of one- and two-variable functions and for solving differential equations on the half line 60. The aforementioned points highly motivate us to generalise the Laguerre moments and introduce the fractional-order version of the generalised Laguerre moments, which exhibit wider capabilities in local and global features extraction, and they could be highly useful in image representation and object recognition applications. Motivated by the facts summarised above, the main contributions of this paper include the following points: (1) Introduce a new set of fractional-order moments for global feature extraction, based on the generalised Laguerre polynomials of fractional order. (2) Present and exploit the capability of the proposed fractional-order moments for local feature extraction in the field of image analysis. (3) Present and exploit the capability of the proposed fractional-order moments for local feature extraction in the field of image analysis. (4) Propose a new set of fractional-order moment invariants for object recognition and image classification. (5) Present new recursive methods for efficient computation of the polynomials coefficients. (6) Introduce a systematic parameter selection method for finding the optimal fractional parameter values with respect to pattern recognition applications. (7) Introduce a flexible method for invariant features extraction, which rely on the capacity of the fractional-order moments to capture local features from any position of the image. (8) Carry out the appropriate numerical experiments to demonstrate that the proposed fractional-order moments and moment invariants are computationally efficient, numerically stable and exhibit sufficient recognition performance even in the presence of noisy effects, in comparison with the state-of-the-art methods. 3 MATHEMATICAL OVERVIEW OF THE CLASSICAL GENERALISED LAGUERRE POLYNOMIALS The classical generalised (or associated) Laguerre polynomials (GLP), denoted by L n ( α ) ( x ) with α > − 1 , are defined as the nth eigenfunction of the Sturm–Liouville problem on semi-infinite interval [ 0 , + ∞ ) [46, 47]: x d 2 d x 2 L n ( α ) ( x ) + ( 1 + α − x ) d d x L n ( α ) ( x ) + n L n ( α ) ( x ) = 0 . (1) The GLP are orthogonal over the semi-infinite interval and satisfy the following orthogonality relation: ∫ 0 + ∞ w ( α ) ( x ) L n ( α ) ( x ) L m ( α ) ( x ) d x = h n ( α ) δ n m , (2)for n , m = 0 , 1 , 2 , … , with respect to the weight function w ( α ) ( x ) and the squared norm h n ( α ) that are defined by: w ( α ) ( x ) = x α e − x , h n ( α ) = Γ ( n + α + 1 ) n ! , (3)where Γ ( · ) is the gamma function and δ n m denotes the Kronecker symbol, with δ n m = 1 for n = m and 0 if n ≠ m . Definition 1.The explicit form of the GLP is given by: L n ( α ) ( x ) = ∑ k = 0 n ( − 1 ) k Γ ( n + α + 1 ) Γ ( k + α + 1 ) ( n − k ) ! k ! x k . (4) Further, the GLP can be easily obtained by the three terms recurrence formula, for n ≥ 1 : ( n + 1 ) L n + 1 ( α ) ( x ) = ( 2 n + α + 1 − x ) L n ( α ) ( x ) − ( n + α ) L n − 1 ( α ) ( x ) , (5)where L 0 ( α ) ( x ) = 1 and L 1 ( α ) ( x ) = 1 + α − x . Finally, one can refer to [46, 47] for more details about the generalised Laguerre polynomials. 4 PROPOSED FRACTIONAL-ORDER GENERALISED LAGUERRE MOMENTS In this section we will initially review some general properties of the fractional-order generalised Laguerre polynomials (FGLP). In the following, we will provide the definition of the new FGLM. 4.1 Fractional-order generalised Laguerre polynomials Let λ be a real positive number λ > 0 , by introducing the change of variable x = t λ on the generalised Laguerre polynomials, we can define the FGLP, noted L n ( α , λ ) ( t ) , as follows 44: L n ( α , λ ) ( t ) = L n ( α ) ( t λ ) , λ > 0 and t ∈ 0 , + ∞ . (6) Theorem 1.The family of FGLP forms an orthogonal basis on the semi-infinite interval [ 0 , + ∞ ) and satisfy the following orthogonality relation: ∫ 0 + ∞ w ( α , λ ) ( t ) L n ( α , λ ) ( t ) L m ( α , λ ) ( t ) d t = h n ( α , λ ) δ n m , (7)for n , m = 0 , 1 , 2 , … , with respect to the weight function: w ( α , λ ) ( t ) = λ t ( α + 1 ) λ − 1 e − t λ , (8)and the squared norm: h n ( α , λ ) = h n ( α ) = Γ ( n + α + 1 ) n ! . (9) Proof.The proof of theorem 1 is given in 44. □ Based on Definition 1, the explicit form of L n ( α , λ ) ( t ) for the fractional degree λ n is given by 44: L n ( α , λ ) ( t ) = ∑ k = 0 n a n k t λ k , (10)where a n k = ( − 1 ) k Γ ( n + α + 1 ) Γ ( k + α + 1 ) ( n − k ) ! k ! . (11) Based on Equation (5), the FGLP can be obtained from the recurrence relation, for n ≥ 1 : ( n + 1 ) L n + 1 ( α , λ ) ( t ) = ( 2 n + α + 1 − t λ ) L n ( α , λ ) ( t ) − ( n + α ) L n − 1 ( α , λ ) ( t ) , (12)with the initial cases: L 0 ( α , λ ) ( t ) = 1 and L 1 ( α , λ ) ( t ) = 1 + α − t λ . (13) To ensure the numerical stability for the computation of moments, we define the normalised FGLP as: L ∼ n ( α , λ ) ( t ) = L n ( α , λ ) ( t ) w ( α , λ ) ( t ) h n ( α , λ ) , n = 0 , 1 , 2 , … . (14) Theorem 2.The normalised FGLP are orthogonal over the semi-infinite interval [ 0 , + ∞ ) and the orthogonality relation is given by: ∫ 0 + ∞ L ∼ n ( α , λ ) ( t ) L ∼ m ( α , λ ) ( t ) d t = δ n m , n , m = 0 , 1 , 2 , … . (15) For the proof of theorem 2, see Appendix A.1. Proposition 1.The three terms recurrence relation of the normalised FGLP L ∼ n ( α , λ ) ( t ) can be written as: L ∼ n ( α , λ ) ( t ) = ( A n + B n t λ ) L ∼ n − 1 ( α , λ ) ( t ) + C n L ∼ n − 2 ( α , λ ) ( t ) , n ≥ 2 , (16)where A n = 2 n + α − 1 n ( n + α ) , B n = − 1 n ( n + α ) and C n = − ( n + α − 1 ) ( n − 1 ) n ( n + α ) .With the initial cases: L ∼ 0 ( α , λ ) ( t ) = w ( α , λ ) ( t ) Γ ( α + 1 ) and L ∼ 1 ( α , λ ) ( t ) = ( 1 + α − t λ ) w ( α , λ ) ( t ) Γ ( α + 2 ) . For the proof of proposition 1, see Appendix A.2. Figure 2 presents the graphs of the first six orders of the normalised FGLP with different values of the parameters α and λ. FIGURE 2Open in figure viewerPowerPoint Plots of the normalised fractional-order generalised Laguerre polynomials L ∼ n ( α , λ ) ( t ) , for the first six orders with different parameters values: (a) α = 1 , λ = 1 , (b) α = 1 , λ = 1.1 , (c) α = 1 , λ = 1.2 , (d) α = 10 , λ = 1.6 , (e) α = 70 , λ = 1.6 and (f) α = 120 , λ = 1.6 with t ∈ [ 0 , 30 ] As is well known, the generalised Laguerre polynomials are associated with the standard gamma distribution, which is equipped with the parameter α over the semi-infinite interval [ 0 , + ∞ ) 48. Therefore, the variation of the parameter α mainly affects the polynomials values and changes the distribution of their zeros, as shown in Figure 2d–f. Moreover, one must take into consideration that the determination of e − x 2 / 2 L n ( α ) ( x ) for very large values of α and x gives rise to underflow problems 49. Finally, it is important to note that the FGLP inherit all the properties of the classical GLP, including their relation with the standard gamma distribution. What is more, the presented FGLP includes a new fractional parameter λ > 0 , so as to generalise the notion of integer-order n ∈ N to fractional-order λ n , where we can recover the classical GLP as L n ( α , 1 ) ( x ) = L n ( α ) ( x ) by setting λ to 1. In fact, this parameter can be also used for adjusting the distribution and the contraction of the polynomials zeros. Figure 2a–c clearly depicts the influence of the parameter λ on the graph of the FGLP. For more detailed discussion about the fFGLP we refer the readers to 44. 4.2 Fractional-order generalised Laguerre moments The FGLM of the order ( λ x n , λ y m ) with λ x , λ y > 0 , of a continuous two-dimensional function f ( x , y ) over the domain [ 0 , + ∞ ) × [ 0 , + ∞ ) , can be defined by: FGLM n m = ∫ 0 + ∞ ∫ 0 + ∞ L ∼ n ( α x , λ x ) ( x ) L ∼ m ( α y , λ y ) ( y ) f ( x , y ) d x d y . (17)Therefore, for any digital image functions f ( i , j ) of the size N × M pixels, we may have the privilege of mapping the coordinates of the image function on the square region [ 0 , L ] × [ 0 , L ] with L > 0 . And we can give the following definition. Definition 2.The FGLM of any digital image function is defined by using Zeroth Order Approximation (ZOA) method as : FGLM n m = ∑ i = 0 N − 1 ∑ j = 0 M − 1 L ∼ n ( α x , λ x ) ( x i ) L ∼ m ( α y , λ y ) ( y j ) f ( i , j ) Δ x Δ y , (18)where Δ x = L / N , Δ y = L / M and the image coordinates mapping is given as: x i = i L N + Δ x 2 , y j = j L M + Δ y 2 , for i = 0 , 1 , … , N − 1 and j = 0 , 1 , … , M − 1 . Remark 1.As previously mentioned, the determination of the FGLP for large values of x gives rise to underflow problem 49. Thus, we must carefully choose the value of the interval bound L. Definition 3.The inverse moments transform of the FGLM, which can be used for the reconstruction of the image function from a limited set of computed moments, is given by: f ( i , j ) = ∑ n = 0 n max ∑ m = 0 m max L ∼ n ( α x , λ x ) ( x i ) L ∼ m ( α y , λ y ) ( y j ) FGLM n m . (19) Finally, it worth noting that the proposed fractional-order moments FGLM presents a very important characteristic, which is related to their capability to extract both global and local features. As we shall demonstrate in the numerical experiments in Section 6, the ability to extract local features can be easily achieved by adjusting the polynomials parameters α x , α y , λ x and λ y . 4.3 Accurate computation of fractional-order generalised Laguerre moments It is well known that the computation of moments suffers from numerical errors, which are mainly caused by approximation of continuous integrals, when the double integration is replaced by double summation over fixed sampling intervals, based on ZOA method 31. One of the most used techniques to address this issue is by using numerical integration methods like Gaussian quadrature, Simpson's rule and wavelet, which are employed in the case when it is difficult to evaluate the integrals analytically. Accordingly, in this subsection, we will introduce an accurate method for the computation of the FGLM. This computation method is based on the Gaussian quadrature, that can provide very accurate results that are almost equal to the theoretical ones 31. First, we define discrete version of FGLM as follows: FGLM n m = ∑ i = 0 N − 1 ∑ j = 0 M − 1 f ( i , j ) h n m ( x i , y j ) , (20)with h n m ( x i , y j ) = ∫ x i − Δ x 2 x i + Δ x 2 ∫ y j − Δ y 2 y j + Δ y 2 L ∼ n ( α x , λ x ) ( x ) L ∼ m ( α y , λ y ) ( y ) d x d y . (21) We can use the separability property of the moments basis function, to express the double integration as: h n m ( x i , y j ) = I n ( x i ) I m ( y j ) , where I n ( x i ) = ∫ x i − Δ x 2 x i + Δ x 2 L ∼ n ( α x , λ x ) ( x ) d x , I m ( y j ) = ∫ y j − Δ y 2 y j + Δ y 2 L ∼ m ( α y , λ y ) ( y ) d y . (22) In fact, the one-dimensional integrals I n ( x i ) and I m ( y j ) with respect to x-axis and y-axis, respectively, are difficult to be evaluated analytically. Therefore, we suggest the use of Gaussian numerical integration for accurate computation of I n ( x i ) and I m ( y j ) , as follows: I n ( x i ) = Δ x 2 ∑ l = 0 s − 1 w l L ∼ n ( α x , λ x ) V x i l , I m ( y j ) = Δ y 2 ∑ l = 0 s − 1 w l L ∼ m ( α y , λ y ) V y j l , (23)with V x i l = x i + Δ x 2 t l and V y j l = y j + Δ y 2 t l , (24)where w l , t l with l = 0 , 1 , … , s − 1 , are the weights and the points for the Gaussian integration formula on s sapling points. For s = 3 , the weights w l and the points t l are given by: w 0 = 0.555555556 , w 1 = 0.888888888 and w 2 = 0.555555556 , and t 0 = − 0.774596669 , t 1 = 0 and t 2 = 0.774596669 . Definition 4.Based on the Gaussian numerical integration, the accurate computation method of FGLM for any digital image function, is given by: AFGLM n m = ∑ i = 1 N ∑ j = 1 M f ( i , j ) I n ( x i ) I m ( y j ) . (25)This method will be denoted as AFGLM (accurate fractional-order generalised Laguerre moments) in this paper. One can notice that the terms I n ( x i ) and I m ( y j ) are independent of the image. Therefore, these terms can be pre-computed and stored in advance in order to decrease the computational cost. Also, we refer the readers to 31 for fast computation method of moments. 5 PROPOSED FRACTIONAL-ORDER GENERALISED LAGUERRE MOMENT INVARIANTS The usual methodology for obtaining orthogonal moment invariants is by expressing the orthogonal moment as a linear combination of geometric moments