https://orcid.org/0009-0000-4094-1735
Figures
Abstract
Due to the rapid growth of the digital music industry, music copyrights have become valuable intangible assets for businesses, offering exclusivity and profitability. This article takes music copyrights as an example and designs a copyright protection method for music score digital images. The zero-watermarking algorithm offers an effective and lossless means of copyright protection. Owing to their geometric invariance, orthogonal moments exhibit superior robustness, positioning them as one of the mainstream methods in the research of zero-watermarking algorithms. The current zero-watermarking algorithms based on orthogonal moments face a trade-off between robustness and discriminability. In this paper, we propose a mixed low-order moments method based on quaternion-type fractional-order moments (QTFM), which balances the global information and texture details of color image contained in QTFM. Experimental results show that the mixed low-order moments method based on QTFM exhibits superior performance in terms of robustness. In the context of using mixed low-order moment features for image analysis, Franklin moments achieve higher average structural similarity (SSIM) values than other QTFMs.
Citation: Huang Q, Zhu J, Xian Y, Peng J (2025) Music score copyright protection based on mixed low-order quaternion Franklin moments. PLoS One 20(8): e0323447. https://doi.org/10.1371/journal.pone.0323447
Editor: Sadiq H. Abdulhussain, University of Baghdad, IRAQ
Received: November 16, 2024; Accepted: April 8, 2025; Published: August 25, 2025
Copyright: © 2025 Huang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting information files.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The development of the internet has not only fueled economic growth but also introduced challenges in copyright protection in the digital age. While the online environment could have provided greater benefits to copyright holders through the broader distribution of music, any emerging phenomenon comes with dual aspects. As copyright holders enjoy the profits generated by online music, the distribution of their works has increasingly spiraled out of control, leading to more widespread infringements compared to the traditional music environment. Digital watermarking technology has emerged in recent years as an effective solution for copyright protection, addressing issues such as copyright disputes in digital images. This technique embeds copyright information directly into the original image, allowing for the extraction of the embedded information later, thus ensuring the protection of digital image copyrights [1,2].
Zero-watermarking algorithms can be categorized into three main types: spatial domain zero-watermarking algorithms [3], frequency domain zero-watermarking algorithms [4], and moments domain zero-watermarking algorithms [5]. Spatial domain features are directly used to extract image characteristics. However, when geometric and image processing attacks occur, spatial domain features show significant sensitivity, regardless of whether edge or texture information is employed [6]. Features in the frequency domain lack invariance to rotation and scaling, leading to poor performance in scenarios where such transformations occur [7]. Orthogonal moments which are used to extract and represent local and global features can describe images without information redundancy compared to non-orthogonal moments [8,9]. Therefore, it is widely used in image analysis [10], pattern recognition [11], image watermarking processing [12]. Reference [13] proposes an image moment domain steganography algorithm based on orthogonal polynomials. Reference [14] introduces a face recognition method using hybrid orthogonal polynomials. Abdulhussain et al. [15] employs embedded image kernel technique and support vector machine (SVM) classification for multi-font handwritten numeral recognitinon. In addition, orthogonal moments can be extended to three-dimensional moments [16], enabling their application in the representation of 3D objects.
Quaternions have become a widely used approach in color image processing by decomposing the image into three distinct channels [17]. Examining the development of quaternion-based orthogonal moments, quaternion-type fractional moments (QTFM) can be divided into discrete and continuous moments based on the continuity of the basis function. Discrete moments include Krawtchuk moments (KM) [18], fractional Tchebyshev moments (FrTM) [19], Hahn moments (HM) [20], Dual Hahn moments (DHM) [21], Racah moments (RM) [22], and Mountain Fourier moments (MFM) [23]. Continuous orthogonal moments are grouped by coordinate system: those defined in the Cartesian coordinate system include Legendre moments (LM) [24], Gaussian Hermite moments (GHM) [25], and Chebyshev moments (CM) [26]; while those defined in the polar coordinate system include Zernike moments (ZM) [27], pseudo Zernike moments (PZM) [28], Legendre Fourier moments (LFM) [29], exponent Fourier moments (EFM) [30], log-polar Exponent-Fourier moments (LEFM) [31], polar harmonic Fourier moments (PHFM) [32], and ternary radial harmonic Fourier moments (TRHFM) [33]. Discrete moments do not involve any approximation errors, making them more suitable for high-precision image processing tasks, such as image reconstruction, compression, and denoising. In comparison, polar continuous moments demonstrate superior rotation invariance, making them more suitable for applications requiring higher degree of robustness.
The Franklin function series, notable for its unique construction and properties resembling Haar and Schauder function systems [34–37], has attracted considerable attention in mathematics. It has also been applied in signal and image processing [38,39]. The implicit construction of the classical Franklin function imposes specific constraints on its applicability. Therefore, the development of efficient and accurate computational methods for orthogonal polynomials remains a prominent focus of contemporary research. By optimizing the initial functions and partitioning the discrete Racah polynomials plane into asymmetric parts, reference [40] successfully achieves accurate recursive computation of higher-order Racah orthogonal polynomials. Asli et al. introduced a new four-term recurrence relation to compute Krawtchouk polynomials [41]. Reference [42] implements the efficient computation of Tchebichef polynomials using adaptive threshold along with x and n-directions recurrence algorithms. Mahmmod et al. [43] proposed a method for calculating Charlier polynomials using multi-threaded parallel computation. This threading approach involves distributing independent coefficients across different threads to address performance bottlenecks.
Through the analysis of the above content, the following conclusions can be drawn: (1) Zero-watermarking algorithms based on orthogonal moments have certain advantages over other methods in term of robustness. However, when watermark image has large sizes, some existing zero-watermarking algorithms require the computation of higher-order orthogonal moments to increase the capacity of the algorithm. In practical applications, high-order moments typically correspond to local features of the image, which means they exhibit lower robustness compared to low-order moments. (2) Recursive algorithms are often effective for computing orthogonal polynomials with invariant definition intervals. However, the piecewise intervals of the Franklin polynomials change as the order of the function increases. Therefore, while general recursive algorithms have advantages in terms of calculation accuracy, it results in significant time complexity when computing the Franklin function.
The main contributions of this paper can be summarized as follows:
- The paper employs non-recursive matrix operations to implement the orthogonalization process of Franklin polynomials, avoiding the high time complexity and computational errors associated with numerical integration by directly solving for the elements.
- Construction method of mixed low-order moments feature is used to arrange and quantize QTFM, effectively enhancing the robustness and discriminability of the zero-watermarking algorithm.
- The zero-watermarking algorithm constructed using quaternion Franklin moments achieves a balance between global and local image information, enhancing the algorithm’s stability against various types of image attacks.
The rest of the paper is organized as follows: Sect 2 introduces the classical Franklin system and discuss the quaternion type moments. Sect 3 proposes the accurate computation algorithm for quaternion fractional Franklin moments (QFFM). Sect 4 specifies the proposed zero-watermarking algorithm. Experiments and analysis are provided in Sect 5. Finally, Sect 6 concludes the paper.
2 Preliminaries
This section will introduce the definition of the classical Franklin function and presents a method for its fast and accurate computation. In addition, the quaternion type fractional-order moments (QTFM) is defined and explained.
2.1 Classical Franklin polynomials
Consider a sequence if
,
is everywhere dense in
and each point appears in
at most twist, it is called an admissible sequence on
. By means of a non-decreasing permutation,
is obtained from
.
where Sn denote the space of functions defined on , which is left-continuous and linear on
and continuous at
if
. It is clear that
and
, hence there exists a unique function
which is orthogonal to Sn−1 and
.
The classical Franklin system [44] corresponding to Haar collocation points is defined as follows:
For is the nth Franklin function corresponding to the partition
, where
, n = 2t + m,
,
.
2.2 Fast and accurate computation of Franklin polynomials
Over the years, the computational complexity of orthogonalization process for piecewise functions has limited their practical application [45]. The Franklin system, which consists of continuous piecewise linear functions, faces similar difficulties.
Considering the following linear independent groups:
where , k is the maximum integer that satisfies 2k<2n−1.
To propose a rapid computation method for the Franklin system, the non-recursive form of the orthogonalization process is given in Eq (4), where D0 = 1, for , Dn is the Gram determinant. Throughout the paper, the following notation is used:
and
denote two basis functions forming the element in the r-th row and c-th column. ar and ac represent nodes in sequence
corresponding to
and
, respectively.
represents the maximum value between ar and ac.
The element at any position in the matrix
and Dn can be represented as:
where ‘’ and ‘
’ represents ‘Exclusive or’ and ‘Logical conjunction’ which are logical operator. This method circumvents errors from numerical integration, especially the ill-conditioned matrices appearing in LU decomposition. Fig 1 illustrates the computation process for Franklin polynomials.
2.3 Quaternion type fractional-order moments
A quaternion q has one real part and three imaginary parts, which is given by [46]:
where a, b, c, d are real numbers, if a = 0, q corresponds to a pure quaternion and can be given by .
A RGB color image defined in polar coordinates can be represented as:
where are three channels of the color image respectively. To present QTFM, it is imperative to ensure that applying fractional orders to radial basis functions neither compromises their orthogonality nor incurs significant computational complexity. Let
be a non-zero real number,
, then the fractional-order radial basis functions
can be represented as follow:
As the multiplication of quaternion does not comply with the commutative law, there are two types of QTFM with repetition m and order n:
where is the definition domain of color image,
is a pure quaternion satisfying
. Due to the similar properties of
and
, in the following chapters, we will only discuss
.
3 Accurate quaternion Franklin moments
In this section, the computational process of Franklin moments and necessity of precise calculation are presented. Accurate computation methods for Franklin moments are proposed using Gaussian integration and wavelet integration.
3.1 Definitions
For gray-level image, fractional-order Franklin moments (FFM) defined in polar coordinates is represented as:
where represents a channel of
,
is the order of
,
is the repetition. The Fractional-order Franklin polynomials are orthogonal over the range
. Let
, where
denotes the rotation angle of the original image, we have:
Eq (12) demonstrates rotation invariance of FFM. Assume that the image has dimensions
, the discrete form of
are shown as follows:
where ,
, QFFM can be defined as:
For digital images, the double integral in Eq (14) is replaced by a double summation and its discrete form of zero-order approximation is given by the following formula:
In practice, the infinite series must be truncated at the finite number P. Thus, the inverse transform of Eq (14) is given by:
where is a unit pure quaternion, for
, substituting it into Eq (14), we have:
where
Here, ,
, and
respectively represent
corresponding to red, green, and blue channels. Eq (16) can be expressed as:
where
where is theoretically a zero matrix,
,
,
represent the red, green, and blue channels of the reconstructed color image, respectively.
,
,
,
are the gray-level reconstructed image of
,
,
,
.
3.2 Accurate computation
As shown in Fig 2, information loss occurs during the numerical computation of orthogonal moments. To ensure accurate computation of QFFM, we employ Gaussian and wavelet integration methods.
3.2.1 Gaussian integration.
The zeroth order approximation of double integration results in numerical integration errors. This numerical instability occurs when computing QFFM for high values of n and m. If f(x) represents a 1-D function, its gaussian numerical integration over the interval is expressed as:
where wi and ti denote the weight and position of the image sampling points, g denotes the order of Gaussian integration. The 2-D formulation of Gaussian numerical integration of is expressed as:
We achieve accurate computation of QFFM by employing Gaussian numerical integration for the double integration outlined in the following:
where ,
,
and
.
3.2.2 Wavelet integration.
Let f(x) be an 1-D function, the wavelet numerical integration over the interval is defined as [47]:
where W = 2u and , the integer u represents the level of the wavelet. The 2-D formulation of wavelet numerical integration of
is expressed as:
We now accurately compute the QFFM by resorting to wavelet numerical integration of the double integration given in Eq (13):
where ,
and
.
4 Zero-Watermarking algorithm process
In this section, the construction method of mixed low-order moments feature and the processes of zero-watermarking algorithm are described.
4.1 Asymmetric tent map
Chaos theory primarily studies the behavior of dynamical systems that are highly sensitive to initial conditions. In cryptography, the sensitivity of chaotic systems to initial values is often utilized to design secure pseudo-random number generators. Asymmetric tent map [48] is defined as:
where and
are control parameters. As shown in Fig 3, let n = 500 and taking the last 100 items of qn, it can be observed that the values in the blue region do not exhibit a uniform distribution in the interval
. In contrast, the values in the red region with
better meet the requirements.
4.2 Mixed low-order moments feature
Low-order orthogonal moments represent the global features of an image, while high-order moments are more sensitive to image details. Therefore, to enhance the robustness of feature representation, mixed low-order moments method is an effective approach. Mixed low-order moments feature for any images hC can be defined as:
where represents moments feature given in Eq (17) and
is fractional-order parameter. The feature processing workflow for mixed low-order moments is shown in Fig 4.
For different values of the parameter , the features are arranged horizontally and quantized as given in Eq (29), where
represents the number of the elements within the set
;
represents the median of a discrete sequence
. Vector shift operations used in [49] is adopted to generate encryption feature image.
4.3 Zero-watermarking generation and extraction
The procedure of zero-watermarking generation is shown in Fig 5 and described as follow.
Step 1: Applying the QTFMs on the host image. The QTFMs is performed on the host image hC to obtain the corresponding feature . The fractional-order parameter
.
Step 2: Generating the Mixed low-order moments feature.
Assuming the size of watermarking image Wc is , for the QTFM with
, the maximum order
. For the QTFM with
, the maximum order
. Substituting it in Eq (28) and Eq (17), we get quantization feature.
STEP 3: Generating the encryption image.
Assuming that the , where n is iterations,
and q0 are control parameters. Then substituting key1 in the process given in Fig 4 to generate encryption image.
STEP 4: Scrambling the encryption feature image and watermarking image.
The Arnold transform is used for scrambling images which is defined as:
where a and b are scrambling parameters and represents the positions of the pixels
after transforming. We set that
and
, where iter represents the number of iterations.
STEP 5: Constructing the zero-watermarking.
The zero-watermarking image is generated by performing operation which represents ’Exclusive OR’.
STEP 6: Certifying. The copyright owner can certify using zero-watermarking image, the key1, key2, and key3. The procedure of zero-watermarking verification is given in Fig 6. Performing the XOR of the scrambled feature image and the zero-watermarking, retrieved watermark can be obtained by inverse Arnold transform.
where is obtained from key2 and key3.
5 Experimental results and discussions
In this section, a set of numerical experiments are conducted to evaluate the performance of the proposed zero-watermarking algorithm and compare it with that of the existing methods for real RGB images. The dataset used in this paper is composed of both computer-generated dataset [50] and music score images collected from the internet [58] (https://imslp.org/) which is a publicly available music databases.
5.1 Effectiveness of the Franklin polynomial calculation method
As shown in Fig 7, the time complexity of recursive method, non-recursive method, and the proposed method are compared.
The classical Gram-Schmidt process has a time complexity of O(CR2) [51], where R is the number of column vectors and C is the dimensionality of the vectors. The time complexity of recursive orthogonalization process is O(kn5), depending on the number of nodes in sequence . The time complexity of the non-recursive form in Eq (4) is approximately O(Dn3), where D represents the grid density for numerical integration. Since all elements in the matrix correspond to the partition
, the time complexity of proposed method given in Eq (5) is O(3n3).
5.2 Image reconstruction
Using higher-order moments for image reconstruction is an effective method for assessing the accuracy of quaternion moments, where the reconstructed image can be evaluated using the mean squared reconstruction error (MSRE) [52]:
where represents original color image and
represents reconstructed color image.
The proposed QFFM, FLFM, and FPCET are used to reconstruct the “Baboon” color image of size using both direct and accurate computation methods. The computed results of MSRE are displayed in Fig 8. As given in the comparison in Fig 9, when the maximum order is less than 20, the direct computation method does not introduce significant computational errors compared to the exact computation method. However, when the maximum order exceeds 30, the direct computation method are instable. In contrast, the Gaussian integration, wavelet integration, and FFT-based methods are more suitable for tasks requiring high-precision calculation. Fig 10 illustrates the comparison of time complexity between the direct computation method and the exact computation method. Since the Gaussian integration and wavelet integration methods directly increase the grid density of the discrete numerical integration, their time complexity increases significantly. Therefore, the direct computation method is used in the zero-watermarking algorithm to obtain the mixed low-order moments.
5.3 Robustness of proposed QFFM
Since the zero-watermarking algorithm does not affect the original image, imperceptibility is not a concern; only robustness needs to be considered. Structural similarity (SSIM) [53] used to evaluate the robustness of algorithm is defined as:
where , and
are the local means, standard deviations, and cross-covariance for images
. The value closer to 1 indicates better image quality. The similarity between the original watermark image and the extracted watermark image is evaluated using SSIM as the assessment metric. The Peak Signal-to-Noise Ratio (PSNR) [54] can be used to assess the distortion level of the attacked image, which is defined as:
where MSE denotes the mean square error:
where and h*(x,y) represent the original image and attacked image, respectively, with dimensions of
. The higher PSNR value, the lower distortion degree.
In this section, the robustness of the proposed algorithm is evaluated for the common geometric attacks, image processing attacks, and mixed image attacks. In the conducted experiments , the “saxophone” image of size is selected from the collected musical score images as an example. The “flower” image of size
from Fig 11 is selected as the watermark image. Fig 12 shows the parameter settings for each attack, the PSNR values of the attacked images, and the SSIM values of the watermark images retrieved using the proposed algorithm. The obtained results shows that the SSIM values of the proposed algorithm are all greater than 0.99, indicating that the retrieved watermarks are still recognizable despite the original color image being seriously distorted.
a) ‘Dragonice’ (b) ‘Tambourin’ (c) ‘Klavierstucke’ (d) ‘Fest’ (e) ‘Snow child’ (f) ‘Dvorak’ (g) ‘Saxophone’ (h) Watermark image.
The seven color images in Fig 11 are subjected to single-type image attacks with various parameters. Table 1 provides a summary of the robustness of the proposed algorithm against these image attacks. Table 2 summarizes the robustness of the algorithm against single-type image attacks and mixed image attacks, with the specific parameters provided in Table 3. Table 2 calculates the average test results for 50 collected music score images. The results show that the SSIM values for the proposed algorithm are greater than 0.99 for single-type image attacks and exceed 0.98 for mixed image attacks.
5.4 Comparison with other methods
To obtain more reliable experimental results, Table 3 compares the proposed method with the existing similar zero-watermarking algorithms [55–57] and other QTFMs. The average SSIM values are calculated using 30 test color images. The results demonstrate the robustness of the proposed method, which performs well against various types of image attacks and mixed image attacks. When using the mixed low-order moments method, the average SSIM value increases around compared to other types of orthogonal moments. Compared to references [55–57], the improvement is around
. The stability of the experimental results can be attributed to the following factors: (1) The mixed low-order moments method is used to extract low-frequency information from color images, making it is robust to image attacks that alter pixel values, such as noise, blurring, and cutting. (2) References [56,57] embed watermarking image information into the original image by altering the moments amplitudes. This method makes the algorithm sensitive to variations in image pixel values. The method employed in reference [55] requires the computation of high-order moments when the embedded watermark image has a large size, which increases the time complexity and results in relatively poor robustness. (3) The QFFM demonstrates a smaller MSRE in image reconstruction experience, indicating that, at the same maximum order, the QFFM method retains more image texture information. This enhances its robustness against geometric transformations, including scaling, stretching, and compression. (4) Excessive image texture information will lead to increased sensitivity of the algorithm to local pixel changes. Therefore, combining QFFM with the mixed low-order moments method achieves a balance between global and local image information, enhancing the stability of the proposed algorithm against various types of image attacks.
Although the proposed method has several advantages, it also has some limitations. Compared to other QTFM methods that also utilize mixed low-order moments, Franklin moments demonstrate superior average SSIM values in resisting various types of image attacks. However, for specific types of image attacks, it is possible to identify better QTFM methods, indicating that Franklin moments lack specificity in certain scenarios. Additionally, since Franklin functions are composed of piecewise linear functions, they exhibit higher time complexity as shown in Fig 10.
6 Conclusion
A zero-watermarking algorithm based on mixed low-order Franklin moments is proposed for to address issue of copyright protection for digital music score images. The algorithm is constructed based on a set of fractional-order parameters, which are varied within a specific range. Subsequently, a quantization scheme is applied to the extracted features to generate a binary sequence for watermark generation and extraction. The algorithm combines the ability of mixed low-order moments method to effectively extract global image features with the capacity of Franklin moments to capture detailed local information in color images, thereby enhancing its robustness against various types of image attacks. Numerical experiments obviously show that our proposal is much more robust compared to the other considered methods and QTFMs. These results clearly demonstrate that our proposed method can be effectively against common image processing operations and mixed image attacks.
Although the proposed fast computation approach for Franklin functions has achieved certain success, numerical instability issues persist when dealing with high-order (order >200) Franklin polynomials. Therefore, developing a numerically stable construction of Franklin functions is a crucial direction for future work. Furthermore, the derivatives of classical Franklin functions demonstrate discontinuities at the boundaries of the segments. Consequently, the development of a smoother generalization of the Franklin functions represents a significant objective for future research.
Supporting information
S1 Dataset. Music score dataset is used in this paper.
We obtained the music score images from publicly available music databases, specifically from the IMSLP (International Music Score Library Project). This platform offers a vast collection of music scores that are in the public domain, as well as some modern works with explicit permission. The music score images can be accessed through the following link: https://imslp.org/
The majority of the music score images are sourced from the public domain, and therefore do not have any copyright restrictions. Specifically, the scores selected in our study belong to works in the public domain on the IMSLP platform, which are not protected by current copyright laws
https://doi.org/10.1371/journal.pone.0323447.s001
(ZIP)
References
- 1. Zhao J, Zong T, Xiang Y, Gao L, Zhou W, Beliakov G. Desynchronization attacks resilient watermarking method based on frequency singular value coefficient modification. IEEE/ACM Trans Audio Speech Lang Process. 2021;29:2282–95.
- 2. Darwish MM, Farhat AA, El-Gindy TM. Convolutional neural network and 2D logistic-adjusted-Chebyshev-based zero-watermarking of color images. Multimed Tools Appl. 2023;83(10):29969–95.
- 3. Xi X, Hua Y, Chen Y, Zhu Q. Zero-watermarking for vector maps combining spatial and frequency domain based on constrained Delaunay triangulation network and discrete Fourier transform. Entropy. 2023;25.
- 4. Bai Y, Li L, Zhang S, Lu J, Emam M. Fast frequency domain screen-shooting watermarking algorithm based on ORB feature points. Mathematics. 2023.
- 5. Wang X, Wang L, Lin TJ, Pan NP, Yang H. Color image zero-watermarking using accurate quaternion generalized orthogonal fourier–Mellin moments. J Math Imaging Vision. 2021;63:708–34.
- 6. Ali M, Kumar S. A robust zero-watermarking scheme in spatial domain by achieving features similar to frequency domain. Electronics. 2024.
- 7. B D, Rupa Ch, D H, Y V. Privacy preservation of medical health records using symmetric block cipher and frequency domain watermarking techniques. In: 2022 International Conference on Inventive Computation Technologies (ICICT). 2022. p. 96–103.
- 8. Yamni M, Daoui A, Ogri OE, Karmouni H, Sayyouri M, Qjidaa H. Fractional Charlier moments for image reconstruction and image watermarking. Signal Process. 2020;171:107509.
- 9. Wang X, Zhang Y, Niu P, Yang H. IPHFMs: fast and accurate polar harmonic fourier moments. Signal Process. 2023;211:109103.
- 10. He B, Cui J, Peng Y, Yang T. Image analysis by fast improved radial harmonic‐Fourier moments algorithm. Int J Imaging Syst Tech. 2020;30(4):1033–45.
- 11. Wang C, Ma B, Xia Z, Li J, Li Q, Shi Y-Q. Stereoscopic image description with trinion fractional-order continuous orthogonal moments. IEEE Trans Circuits Syst Video Technol. 2022;32(4):1998–2012.
- 12. Sheng T, Zeng W, Yang B, Fu C. Adaptive multiple zero-watermarking based on local fast quaternion radial harmonic Fourier moments for color medical images. Biomed Signal Process Control. 2023;86:105304.
- 13.
Radeaf HS, Mahmmod BM, Abdulhussain SH, Al-Jumaeily D. A steganography based on orthogonal moments. In: Proceedings of the International Conference on Information and Communication Technology. 2019. p. 147–53.
- 14. Abdulhussain SH, Mahmmod BM, AlGhadhban A, Flusser J. Face recognition algorithm based on fast computation of orthogonal moments. Mathematics. 2022;10(15):2721.
- 15. Abdulhussain SH, Mahmmod BM, Naser MA, Alsabah MQ, Ali R, Al-Haddad SAR. A robust handwritten numeral recognition using hybrid orthogonal polynomials and moments. Sensors. 2021;21.
- 16. Mahmmod BM, Abdulhussain SH, Naser MA, Alsabah M, Hussain A, Al-Jumeily D. 3D object recognition using fast overlapped block processing technique. Sensors (Basel). 2022;22(23):9209. pmid:36501912
- 17. Zhou H, Zhang X, Zhang C, Ma Q. Quaternion convolutional neural networks for hyperspectral image classification. Eng Appl Artif Intell. 2023;123:106234.
- 18. Zhang Q, Liu J, Tian Y. Krawtchouk moments and support vector machines based coal and rock interface cutting thermal image recognition. Optik. 2022.
- 19. Duan C, Zhou J, Gong L, Wu JY, Zhou N. New color image encryption scheme based on multi-parameter fractional discrete Tchebyshev moments and nonlinear fractal permutation method. Optics Lasers Eng. 2022.
- 20. Mahmmod BM, Abdulhussain SH, Suk T, Hussain AJ. Fast computation of Hahn polynomials for high order moments. IEEE Access. 2021;PP:1–1.
- 21. Daoui A, Karmouni H, Yamni M, Sayyouri M, Qjidaa H. On computational aspects of high-order dual Hahn moments. Pattern Recogn. 2022;127:108596.
- 22.
Alami AE, Mesbah A, Berrahou N, Berrahou A, Qjidaa H. Quaternion discrete racah moments convolutional neural network for color face recognition. In: 2022 International Conference on Intelligent Systems and Computer Vision (ISCV). 2022. p. 1–5.
- 23. Sahmoudi Y, Ogri OE, Mekkaoui JE, Idrissi BJ, Hjouji A. Improving the machine learning performance for image recognition using a new set of mountain fourier moments. Image Anal Stereol. 2024.
- 24. Bustacara-Medina C, Ruiz-García E. Accuracy of legendre moments for image representation. Comput Sci Res Notes. 2023.
- 25. Yang B, Shi X, Chen X. Image analysis by fractional-order Gaussian-Hermite moments. IEEE Trans Image Process. 2022;31:2488–502. pmid:35263255
- 26. Pallotta L, Cauli M, Clemente C, Fioranelli F, Giunta G, Farina A. Classification of micro-Doppler radar hand-gesture signatures by means of Chebyshev moments. In: 2021 IEEE 8th International Workshop on Metrology for AeroSpace (MetroAeroSpace). 2021. p. 182–7.
- 27. Chen S, Malik A, Zhang X, Feng G, Wu H. A fast method for robust video watermarking based on zernike moments. IEEE Trans Circuits Syst Video Technol. 2023;33(12):7342–53.
- 28. Agilandeeswari L, Prabukumar M, Alenizi FA. A robust semi-fragile watermarking system using Pseudo-Zernike moments and dual tree complex wavelet transform for social media content authentication. Multimed Tools Appl. 2023:1–53. pmid:37362657
- 29. Noriega-Escamilla A, Camacho-Bello CJ, Ortega-Mendoza RM, Arroyo-Núñez JH, Gutiérrez-Lazcano L. Varroa destructor classification using Legendre-Fourier moments with different color spaces. J Imaging. 2023;9(7):144. pmid:37504821
- 30. Qiu XZ, Peng WC, Ma B, Li Q, Zhang H, Wang M. Geometric attacks resistant double zero-watermarking using discrete Fourier transform and fractional-order exponent-Fourier moments. Digit Signal Process. 2023;140:104097.
- 31. Yang H, Qi S, Wang C, Yang S, Wang X. Image analysis by log-polar Exponent-Fourier moments. Pattern Recogn. 2020;101:107177.
- 32. Wang C, Zhang Q, Ma B, Xia Z, Li J, Luo T, et al. Light-field image watermarking based on geranion polar harmonic Fourier moments. Eng Appl Artif Intell. 2022;113:104970.
- 33. Wang C, Wang X, Qiu X, Zhang C. Ternary radial harmonic Fourier moments based robust stereo image zero-watermarking algorithm. Information Sciences. 2019;470:109–20.
- 34. Gevorkyan GG, Keryan KA. On a generalization of general Franklin system on R. J Contemp Mathemat Anal. 2014;49(6):309–20.
- 35. Gevorkyan GG, Navasardyan KA. On uniqueness of series by general Franklin system. J Contemp Mathemat Anal. 2018;53(4):223–31.
- 36. Gevorkyan GG. On the uniqueness of series in the Franklin system. Sbornik: Mathematics. 2016;207:1650–73.
- 37. Gevorkyan GG, Hakobyan LA. Uniqueness theorems for multiple Franklin series converging over rectangles. Math Notes. 2021;109(1–2):208–17.
- 38. He B, Cui J, Xiao B, Wang X. Image analysis by two types of Franklin-Fourier moments. IEEE/CAA J Automat Sinica. 2019;6:1036–51.
- 39. Zhu J, Huang K, Gao G, Yu D. Accurate and fast quaternion fractional-order Franklin moments for color image analysis. Digit Signal Process. 2024.
- 40. Mahmmod BM, Abdulhussain SH, Suk T, Alsabah M, Hussain A. Accelerated and improved stabilization for high order moments of Racah polynomials. IEEE Access. 2023.
- 41. Asli BHS, Rezaei MH. Four-term recurrence for fast Krawtchouk moments using Clenshaw algorithm. Electronics. 2023.
- 42. Abdulhussain SH, Mahmmod BM, Baker T, Al‐Jumeily D. Fast and accurate computation of high‐order Tchebichef polynomials. Concurr Comput. 2022;34(27).
- 43. Mahmmod BM, Flayyih WN, Abdulhussain SH, Sabir FA, Khan B, Alsabah M, et al. Performance enhancement of high degree Charlier polynomials using multithreaded algorithm. Ain Shams Eng J. 2024;15(5):102657.
- 44. Mikayelyan VG. The Gibbs phenomenon for general Franklin systems. Journal of Contemporary Mathematical Analysis. 2017;52:198–210.
- 45. Mourrain B. Polynomial–exponential decomposition from moments. Found Comput Math. 2016;18:1435–92.
- 46. Wang C, Hao Q, Ma B, Wu X, Li J, Qiu X i a Z. Octonion continuous orthogonal moments and their applications in color stereoscopic image reconstruction and zero-watermarking. Eng Appl Artif Intell. 2021;106:104450.
- 47. Camacho-Bello C, Toxqui-Quitl C, Padilla-Vivanco A, Báez-Rojas JJ. High-precision and fast computation of Jacobi-Fourier moments for image description. J Opt Soc Am A Opt Image Sci Vis. 2014;31(1):124–34. pmid:24561947
- 48. Kopets E, Rybin V, Vasilchenko O, Butusov D, Fedoseev P, Karimov A. Fractal tent map with application to surrogate testing. Fractal Fract. 2024;8(6):344.
- 49. Yang H, Qi S, Niu P, Wang X. Color image zero-watermarking based on fast quaternion generic polar complex exponential transform. Signal Process: Image Commun. 2020;82:115747.
- 50. Lin Y, Dai Z, Kong Q. MusicScore: a dataset for music score modeling and generation. arXiv preprint. 2024.
- 51. Siewert J. On orthogonal bases in the Hilbert-Schmidt space of matrices. J Phys Commun. 2022;6(5):055014.
- 52. Abdulhussain SH, Mahmmod BM. Fast and efficient recursive algorithm of Meixner polynomials. J Real-Time Image Proc. 2021;18(6):2225–37.
- 53. Sharma S, Zou JJ, Fang G, Shukla P, Cai W. A review of image watermarking for identity protection and verification. Multimed Tools Appl. 2023;83(11):31829–91.
- 54. Xia Z, Wang C, Ma B, Li Q, Zhang H, Wang M, et al. Geometric attacks resistant double zero-watermarking using discrete Fourier transform and fractional-order Exponent-Fourier moments. Digital Signal Process. 2023;140:104097.
- 55. Hosny KM, Darwish MM, Fouda MM. New color image zero-watermarking using orthogonal multi-channel fractional-order legendre-fourier moments. IEEE Access. 2021;9:91209–19.
- 56. Niu P, Wang P, Liu Y, Yang H, Wang X. Invariant color image watermarking approach using quaternion radial harmonic Fourier moments. Multimed Tools Appl. 2015;75(13):7655–79.
- 57. Hosny KM, Darwish MM. Resilient color image watermarking using accurate quaternion radial substituted Chebyshev moments. ACM Trans Multimed Comput Commun Appl. 2019;15(2):1–25.
- 58. Guo EW. A librarian’s guide to IMSLP. Fontes Artis Musicae. 2014;267–74.