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1

HOSNY, KHALID M. "EFFICIENT COMPUTATION OF LEGENDRE MOMENTS FOR GRAY LEVEL IMAGES." International Journal of Image and Graphics 07, no. 04 (2007): 735–47. http://dx.doi.org/10.1142/s021946780700288x.

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Direct computation of Legendre orthogonal moments requires huge arithmetic operations, which is very time consuming. Many works have described methods for reducing the computations involved in evaluating Legendre moments. Nevertheless, reduction computational complexity is still an open problem and needs more investigation. Existing algorithms mainly focused on binary images and compute Legendre moments using a set of geometric moments. We propose a fast and efficient method for computation of Legendre moments for binary and gray level images. A recurrence formula of one-dimensional Legendre m
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2

Tamtsia, Aurelien Yeremou, Youcef Mezouar, Philippe Martinet, Haman Djalo, and Emmanuel Tonye. "2D Legendre Moments-Based Visual Control." Applied Mechanics and Materials 162 (March 2012): 487–96. http://dx.doi.org/10.4028/www.scientific.net/amm.162.487.

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Among region-based descriptors, geometric moments have been widely exploited to design visual servoing schemes. However, they present several disadvantages such as high sensitivity to noise measurement, high dynamic range and information redundancy (since they are not computed onto orthogonal basis). In this paper, we propose to use a class of orthogonal moments (namely Legendre moments) instead of geometric moments to improve the behavior of moment-based control schemes. The descriptive form of the interaction matrix related to the Legendre moments computed from a set of points is rst derived
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3

Dasari, Sridhar, and I. V. Murali Krishna. "Combined Classifier for Face Recognition using Legendre Moments." Computer Engineering and Applications Journal 1, no. 2 (2012): 107–18. http://dx.doi.org/10.18495/comengapp.v1i2.12.

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In this paper, a new combined Face Recognition method based on Legendre moments with Linear Discriminant Analysis and Probabilistic Neural Network is proposed. The Legendre moments are orthogonal and scale invariants hence they are suitable for representing the features of the face images. The proposed face recognition method consists of three steps, i) Feature extraction using Legendre moments ii) Dimensionality reduction using Linear Discrminant Analysis (LDA) and iii) classification using Probabilistic Neural Network (PNN). Linear Discriminant Analysis searches the directions for maximum di
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4

SHEN, JUN, WEI SHEN, and DANFEI SHEN. "ON GEOMETRIC AND ORTHOGONAL MOMENTS." International Journal of Pattern Recognition and Artificial Intelligence 14, no. 07 (2000): 875–94. http://dx.doi.org/10.1142/s0218001400000581.

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Moments are widely used in pattern recognition, image processing, computer vision and multiresolution analysis. To clarify and to guide the use of different types of moments, we present in this paper a study on the different moments and compare their behavior. After an introduction to geometric, Legendre, Hermite and Gaussian–Hermite moments and their calculation, we analyze at first their behavior in spatial domain. Our analysis shows orthogonal moment base functions of different orders having different number of zero-crossings and very different shapes, therefore they can better separate ima
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5

Lakshmi Deepika, C., A. Kandaswamy, C. Vimal, and B. Satish. "Palmprint authentication using modified legendre moments." Procedia Computer Science 2 (2010): 164–72. http://dx.doi.org/10.1016/j.procs.2010.11.021.

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6

Fu, Bo, Jianzhong Zhou, Yuhong Li, Guojun Zhang, and Cheng Wang. "Image analysis by modified Legendre moments." Pattern Recognition 40, no. 2 (2007): 691–704. http://dx.doi.org/10.1016/j.patcog.2006.05.020.

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7

Prévost, Marc. "Legendre modified moments for Euler's constant." Journal of Computational and Applied Mathematics 219, no. 2 (2008): 484–92. http://dx.doi.org/10.1016/j.cam.2007.09.015.

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8

Dhekane, Manasi, Ayan Seal, and Pritee Khanna. "Illumination and Expression Invariant Face Recognition." International Journal of Pattern Recognition and Artificial Intelligence 31, no. 12 (2017): 1756018. http://dx.doi.org/10.1142/s0218001417560183.

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An illumination and expression invariant face recognition method based on uniform local binary patterns (uLBP) and Legendre moments is proposed in this work. The proposed method exploits uLBP texture features and Legendre moments to make a feature representation with enhanced discriminating power. The input images are preprocessed to extract the face region and normalized. From normalized image, uLBP codes are extracted to obtain texture image which overcomes the effect of monotonic temperature changes. Legendre moments are computed from this texture image to get the required feature vector. L
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9

Farouk, R. M., and Qamar A. A. Awad. "Image representation based on fractional order Legendre and Laguerre orthogonal moments." International Journal of ADVANCED AND APPLIED SCIENCES 8, no. 2 (2021): 54–59. http://dx.doi.org/10.21833/ijaas.2021.02.007.

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In this paper, we have introduced new sets of fractional order orthogonal basis moments based on Fractional order Legendre orthogonal Functions (FLeFs) and Fractional order Laguerre orthogonal Functions (FLaFs) for image representation. We have generated a novel set of Fractional order Legendre orthogonal Moments (FLeMs) from fractional order Legendre orthogonal functions and a new set of Fractional order Laguerre orthogonal Moments (FLaMs) from the fractional order Laguerre orthogonal functions. The new presented sets of (FLeMs) and (FLaMs) are tested with the recently introduced Fractional o
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10

Hui, Fan, Hai Feng Wang, and Jin Jiang Li. "Image Registration Based on Feature Points Krawtchouk Moments." Applied Mechanics and Materials 40-41 (November 2010): 584–89. http://dx.doi.org/10.4028/www.scientific.net/amm.40-41.584.

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An image registration based on feature points Krawtchouk moments is proposed. Moments are the shape descriptors based on region. Krawtchouk moments are a set of discrete orthogonal moments and are more suitable for describing two-dimensional images compared to Zemike, Legendre moments. In the image registration based on feature points Krawtchouk moments, Krawtchouk moment invariants of the feature points neighborhood that have been extracted are solved, and then these Krawtchouk moment invariants constitute feature vectors used to describe the feature points, finally feature points are matched
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11

Ding, Jiu, Noah H. Rhee, and Chenhua Zhang. "On Polynomial Maximum Entropy Method for Classical Moment Problem." Advances in Applied Mathematics and Mechanics 8, no. 1 (2015): 117–27. http://dx.doi.org/10.4208/aamm.2014.m504.

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AbstractThe maximum entropy method for the Hausdorff moment problem suffers from ill conditioning as it uses monomial basis {1,x,x2,...,xn}. The maximum entropy method for the Chebyshev moment probelm was studied to overcome this drawback in. In this paper we review and modify the maximum entropy method for the Hausdorff and Chebyshev moment problems studied in and present the maximum entropy method for the Legendre moment problem. We also give the algorithms of converting the Hausdorff moments into the Chebyshev and Lengendre moments, respectively, and utilizing the corresponding maximum entr
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12

Chong, Chee-Way, P. Raveendran, and R. Mukundan. "Translation and scale invariants of Legendre moments." Pattern Recognition 37, no. 1 (2004): 119–29. http://dx.doi.org/10.1016/j.patcog.2003.06.003.

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13

Shu, H. Z., L. M. Luo, W. X. Yu, and J. D. Zhou. "Fast computation of Legendre moments of polyhedra." Pattern Recognition 34, no. 5 (2001): 1119–26. http://dx.doi.org/10.1016/s0031-3203(00)00049-2.

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14

Mukundan, R., and K. R. Ramakrishnan. "Fast computation of Legendre and Zernike moments." Pattern Recognition 28, no. 9 (1995): 1433–42. http://dx.doi.org/10.1016/0031-3203(95)00011-n.

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15

Hameed, Vazeerudeen Abdul. "Orthogonal Moment Invariant Function for Image Processing." Journal of Computational and Theoretical Nanoscience 16, no. 8 (2019): 3400–3403. http://dx.doi.org/10.1166/jctn.2019.8299.

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Orthogonal moments are of great importance in image processing due to their high discriminatory capability. Orthogonal moment invariant functions like Legendre moments and Complex Zernike moments are known for high computational complexity and/or they are complex valued. This paper presents a new orthogonal moment function that is real valued. The formulation is appraised to prove that it is computationally less complex when compared to the existing moment functions. The proposed orthogonal moment functions are appraised over their reversible nature to obtain the original data. The new moment
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16

Pravica, David W., Njinasoa Randriampiry, and Michael J. Spurr. "Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function." Abstract and Applied Analysis 2014 (2014): 1–24. http://dx.doi.org/10.1155/2014/890456.

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The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructe
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17

Shu, H. Z., L. M. Luo, W. X. Yu, and Y. Fu. "A new fast method for computing Legendre moments." Pattern Recognition 33, no. 2 (2000): 341–48. http://dx.doi.org/10.1016/s0031-3203(99)00044-8.

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18

Shu, Huazhong, Limin Luo, Xudong Bao, Wenxue Yu, and Guoniu Han. "An Efficient Method for Computation of Legendre Moments." Graphical Models 62, no. 4 (2000): 237–62. http://dx.doi.org/10.1006/gmod.2000.0523.

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19

Bing, He. "Geometrically Robust Image Watermarking Based on Krawtchouk Invariant Moments." Advanced Materials Research 998-999 (July 2014): 951–56. http://dx.doi.org/10.4028/www.scientific.net/amr.998-999.951.

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In this paper an image watermarking based on krawtchouk moment invariants is proposed. krawtchouk moments are selected for image watermarking because image reconstruction with these moments is better than other orthogonal moments like Legendre, Zernike and Tchebichef. Watermarking is composed of the mean of several function of the first and second krawtchouk moment invariants order designed to be invariant to translation, scaling and rotation. The watermarked image is a linear combination of the original image and a weighted nonlinear transformation of original. The weight is computed such tha
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20

Yap, P. T., and R. Paramesran. "Content-based image retrieval using Legendre chromaticity distribution moments." IEE Proceedings - Vision, Image, and Signal Processing 153, no. 1 (2006): 17. http://dx.doi.org/10.1049/ip-vis:20045064.

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21

Pew-Thian Yap and R. Paramesran. "An efficient method for the computation of Legendre moments." IEEE Transactions on Pattern Analysis and Machine Intelligence 27, no. 12 (2005): 1996–2002. http://dx.doi.org/10.1109/tpami.2005.232.

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22

Wee, C. Y., and R. Paramesran. "Derivation of blur-invariant features using orthogonal Legendre moments." IET Computer Vision 1, no. 2 (2007): 66–77. http://dx.doi.org/10.1049/iet-cvi:20070016.

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23

Zhou, J. D., H. Z. Shu, L. M. Luo, and W. X. Yu. "Two new algorithms for efficient computation of Legendre moments." Pattern Recognition 35, no. 5 (2002): 1143–52. http://dx.doi.org/10.1016/s0031-3203(01)00104-2.

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24

Kawabe, Hidekazu. "Character recognition by the moments using associated legendre functions." Systems and Computers in Japan 26, no. 3 (1995): 53–64. http://dx.doi.org/10.1002/scj.4690260305.

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25

Zheng, Weixiong, and Ryan G. McClarren. "Semi-analytic benchmark for multi-group free-gas Legendre moments and the application of Gauss quadrature in generating thermal scattering Legendre moments." Annals of Nuclear Energy 85 (November 2015): 1131–40. http://dx.doi.org/10.1016/j.anucene.2015.07.031.

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26

Aljohani, Abeer. "A novel spectral transformation technique based on special functions for improved chest X-ray image classification." PLOS One 20, no. 6 (2025): e0325058. https://doi.org/10.1371/journal.pone.0325058.

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Chest X-ray image classification plays an important role in medical diagnostics. Machine learning algorithms enhanced the performance of these classification algorithms by introducing advance techniques. These classification algorithms often requires conversion of a medical data to another space in which the original data is reduced to important values or moments. We developed a mechanism which converts a given medical image to a spectral space which have a base set composed of special functions. In this study, we propose a chest X-ray image classification method based on spectral coefficients
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27

Mitchell, Rory, Eibe Frank, and Geoffrey Holmes. "An Empirical Study of Moment Estimators for Quantile Approximation." ACM Transactions on Database Systems 46, no. 1 (2021): 1–21. http://dx.doi.org/10.1145/3442337.

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We empirically evaluate lightweight moment estimators for the single-pass quantile approximation problem, including maximum entropy methods and orthogonal series with Fourier, Cosine, Legendre, Chebyshev and Hermite basis functions. We show how to apply stable summation formulas to offset numerical precision issues for higher-order moments, leading to reliable single-pass moment estimators up to order 15. Additionally, we provide an algorithm for GPU-accelerated quantile approximation based on parallel tree reduction. Experiments evaluate the accuracy and runtime of moment estimators against t
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28

Hosny, Khalid M., Mohamed M. Darwish, and Tarek Aboelenen. "New fractional-order Legendre-Fourier moments for pattern recognition applications." Pattern Recognition 103 (July 2020): 107324. http://dx.doi.org/10.1016/j.patcog.2020.107324.

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29

Hosny, Khalid M., and Mohamed M. Darwish. "Invariant color images representation using accurate quaternion Legendre–Fourier moments." Pattern Analysis and Applications 22, no. 3 (2018): 1105–22. http://dx.doi.org/10.1007/s10044-018-0740-1.

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30

Hosny, Khalid M., and Mohamed M. Darwish. "Robust color image watermarking using invariant quaternion Legendre-Fourier moments." Multimedia Tools and Applications 77, no. 19 (2018): 24727–50. http://dx.doi.org/10.1007/s11042-018-5670-9.

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31

Huang, Jinlong, Liangzhi Cao, Qingming He, and Hongchun Wu. "Development of sensitivity calculation capability of Legendre moments of scattering angular distributions in NECP-MCX." EPJ Web of Conferences 302 (2024): 10001. http://dx.doi.org/10.1051/epjconf/202430210001.

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The capability of sensitivity coefficients calculation of keff and reaction rate ratios to Legendre moment of scattering angular distributions based on a fully continuous method has been developed in the Monte Caro code NECP-MCX. The sensitivity calculation was based on Iterated Fission Probability (IFP) and Contributon-Linked eigenvalue sensitivity/Uncertainty estimation via Track-length importance Characterization (CLUTCH) method for k-eigenvalue sensitivity analysis and Generalized Adjoint Responses in Monte Carlo (GEAR-MC) method for generalized sensitivity analysis. The Jezebel and Flatto
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32

SUGI, T., DEJEY, and R. S. RAJESH. "GEOMETRIC ATTACK RESISTANT ROBUST IMAGE WATERMARKING SCHEME." International Journal of Information Acquisition 09, no. 01 (2013): 1350008. http://dx.doi.org/10.1142/s0219878913500083.

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A new watermarking approach based on affine Legendre moment invariants (ALMIs) and local characteristic regions (LCRs) which allows watermark detection and extraction under affine transformation attacks is presented in this paper. It is a non-blind watermarking scheme. Original image color image is converted into HSV color space and divided into four parts. LCR is constructed and a set of affine invariants are derived on LCRs based on Legendre moments for each part. These invariants can be used for estimating the affine transform coefficients on the LCRs. ALMIs are used for watermark embedding
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33

Hjouji, Amal, Jaouad EL-Mekkaoui, and Mosatafa Jourhmane. "Image Classification by Mixed Finite Element Method and Orthogonal Legendre Moments." Pattern Recognition and Image Analysis 30, no. 4 (2020): 655–73. http://dx.doi.org/10.1134/s1054661820040185.

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34

Xiao, Bin, Guo-yin Wang, and Wei-sheng Li. "Radial shifted Legendre moments for image analysis and invariant image recognition." Image and Vision Computing 32, no. 12 (2014): 994–1006. http://dx.doi.org/10.1016/j.imavis.2014.09.002.

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35

El Mallahi, Mostafa, Jaouad El Mekkaoui, Amal Zouhri, Hicham Amakdouf, and Hassan Qjidaa. "Rotation Scaling and Translation Invariants of 3D Radial Shifted Legendre Moments." International Journal of Automation and Computing 15, no. 2 (2018): 169–80. http://dx.doi.org/10.1007/s11633-017-1105-8.

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36

Sastry, S. Sreehari, K. Mallika, B. Gowri Sankara Rao, Sie Tiong Ha, and S. Lakshminarayana. "Novel approach to study liquid crystal phase transitions using Legendre moments." Phase Transitions 85, no. 8 (2012): 735–49. http://dx.doi.org/10.1080/01411594.2012.664275.

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37

Noriega-Escamilla, Alicia, César J. Camacho-Bello, Rosa M. Ortega-Mendoza, José H. Arroyo-Núñez, and Lucia Gutiérrez-Lazcano. "Varroa Destructor Classification Using Legendre–Fourier Moments with Different Color Spaces." Journal of Imaging 9, no. 7 (2023): 144. http://dx.doi.org/10.3390/jimaging9070144.

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Bees play a critical role in pollination and food production, so their preservation is essential, particularly highlighting the importance of detecting diseases in bees early. The Varroa destructor mite is the primary factor contributing to increased viral infections that can lead to hive mortality. This study presents an innovative method for identifying Varroa destructors in honey bees using multichannel Legendre–Fourier moments. The descriptors derived from this approach possess distinctive characteristics, such as rotation and scale invariance, and noise resistance, allowing the representa
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38

Chuan, Zun Liang, David Chong Teak Wei, Connie Lee Wai Yan, et al. "A Comparative of Two-Dimensional Statistical Moment Invariants Features in Formulating an Automated Probabilistic Machine Learning Identification Algorithm for Forensic Application." Malaysian Journal of Fundamental and Applied Sciences 19, no. 4 (2023): 525–38. http://dx.doi.org/10.11113/mjfas.v19n4.2917.

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IBIS, ALIS, EVOFINDER, and CONDOR are the massive ballistics computerised technological machines that have typically been utilised in forensic laboratories to automatically locate similarities between images of cartridge cases and bullets. However, it imposed a long execution time and requires physical interpretation to consolidate the analysis results when employing these market-available technologies to accomplish ballistics matching tasks. Therefore, the principal objective of this study is to propose an improvised automated probabilistic machine learning identification algorithm by extract
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39

Harding, Steven N., and Alexander W. N. Riasanovsky. "Moments of the weighted Cantor measures." Demonstratio Mathematica 52, no. 1 (2019): 256–73. http://dx.doi.org/10.1515/dema-2019-0026.

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AbstractBased on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying {\mu ^\alpha }(E) = \sum\nolimits_{n = 0}^{N - 1} {{\alpha _n}{\mu ^\alpha }(\varphi _n^{ - 1}(E))} where ϕn : x ↦ (x + n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L_{{\mu ^\alpha }}^2 [0,
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40

Wang, Qianqian, Changchun Wang, and Hui Wang. "A New Method for Reliability-Based Sensitivity Analysis of Dynamic Random Systems." Mathematical Problems in Engineering 2019 (September 11, 2019): 1–13. http://dx.doi.org/10.1155/2019/5437695.

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A novel numerical method for investigating time-dependent reliability and sensitivity issues of dynamic systems is proposed, which involves random structure parameters and is subjected to stochastic process excitation simultaneously. The Karhunen–Loève (K-L) random process expansion method is used to express the excitation process in the form of a series of deterministic functions of time multiplied by independent zero-mean standard random quantities, and the discrete points are made to be the same as Legendre integration points. Then, the precise Gauss–Legendre integration is used to solve th
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41

Rao, Ch Srinivasa, S. Srinivas Kumar, and B. Chandra Mohan. "CONTENT BASED IMAGE RETRIEVAL USING EXACT LEGENDRE MOMENTS AND SUPPORT VECTOR MACHINE." International journal of Multimedia & Its Applications 2, no. 2 (2010): 69–79. http://dx.doi.org/10.5121/ijma.2010.2206.

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42

Vijayalakshmi, B., and V. Subbiah Bharathi. "Classification of CT Liver Images Using Local Binary Pattern with Legendre Moments." Current Science 110, no. 4 (2016): 687. http://dx.doi.org/10.18520/cs/v110/i4/687-691.

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43

Lai, Jun, and Ke Xu. "Segmenting Lung Fields in CT Image Using Legendre Moments and Active Contour." Advanced Materials Research 433-440 (January 2012): 3564–69. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.3564.

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Conventional methods that perform lung segment -ation in CT slices rely on a large contrast in hounsfield units between the lung and surrounding tissues. However, the lung fields are affected by high density pathologies, and they are discontinuities in the pixel intensities, the traditional segment- ation methods can’t get the good results. Here, we present a new segmentation method of the active contour, which is constraining with respect to a set of fixed reference shapes of lung fields. This approach is based on the shapes descriptors by the legendre moments computed from the shape regions,
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44

Hosny, Khalid M. "Fast and low-complexity method for exact computation of 3D Legendre moments." Pattern Recognition Letters 32, no. 9 (2011): 1305–14. http://dx.doi.org/10.1016/j.patrec.2011.03.011.

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45

Lachiondo, José A., Manuel Ujaldón, Regina Berretta, and Pablo Moscato. "Legendre moments as high performance bone biomarkers: computational methods and GPU acceleration." Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization 4, no. 3-4 (2014): 146–63. http://dx.doi.org/10.1080/21681163.2014.922437.

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46

Hosny, Khalid M., Asmaa M. Khalid, and Ehab R. Mohamed. "Efficient compression of volumetric medical images using Legendre moments and differential evolution." Soft Computing 24, no. 1 (2019): 409–27. http://dx.doi.org/10.1007/s00500-019-03922-7.

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47

Bahaoui, Zaineb, Khalid Zenkouar, Hakim El Fadili, Hassan Qjidaa, and Arsalane Zarghili. "Blocking artifact removal using partial overlapping based on exact Legendre moments computation." Journal of Real-Time Image Processing 14, no. 2 (2014): 433–51. http://dx.doi.org/10.1007/s11554-014-0465-3.

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48

Camacho-Bello, C. "Exact Legendre–Fourier moments in improved polar pixels configuration for image analysis." IET Image Processing 13, no. 1 (2019): 118–24. http://dx.doi.org/10.1049/iet-ipr.2018.5489.

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