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Journal articles on the topic 'LEGENDRE MOMENTS'

Author: Grafiati

Published: 11 September 2023

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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 (October 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 moments will be established using the recursive property of Legendre polynomials; then the method will be extended to calculate the two-dimensional Legendre moments. This method is completely independent on geometric moment. The complexity analysis shows that the proposed method computes Legendre moments more efficiently than the direct method and the other conventional methods.

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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. Six visual features are then selected to design a partially-decoupled control scheme. Finally simulated and experimental results are presented to illustrate the validity of our proposal.

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SHEN, JUN, WEI SHEN, and DANFEI SHEN. "ON GEOMETRIC AND ORTHOGONAL MOMENTS." International Journal of Pattern Recognition and Artificial Intelligence 14, no. 07 (November 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 image features based on different modes, which is very interesting for pattern analysis and shape classification. Moreover, Gaussian–Hermite moment base functions are much more smoothed, they are thus less sensitive to noise and avoid the artifacts introduced by window function discontinuity. We then analyze the spectral behavior of moments in frequency domain. Theoretical and numerical analyses show that orthogonal Legendre and Gaussian–Hermite moments of different orders separate different frequency bands more effectively. It is also shown that Gaussian–Hermite moments present an approach to construct orthogonal features from the results of wavelet analysis. The orthogonality equivalence theorem is also presented. Our analysis is confirmed by numerical results, which are then reported.

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

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6

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

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7

Dasari, Sridhar, and I. V. Murali Krishna. "Combined Classifier for Face Recognition using Legendre Moments." Computer Engineering and Applications Journal 1, no. 2 (December 29, 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 discrimination of classes in addition to dimensionality reduction. Combination of Legendre moments and Linear Discriminant Analysis is used for improving the capability of Linear Discriminant Analysis when few samples of images are available. Probabilistic Neural network gives fast and accurate classification of face images. Evaluation was performed on two face data bases. First database of 400 face images from Olivetty Research Laboratories (ORL) face database, and the second database of thirteen students are taken. The proposed method gives fast and better recognition rate when compared to other classifiers.DOI:Â 10.18495/comengapp.12.107118

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8

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 by calculating the Euclidean distance of feature vectors. The results of experiments show that Krawtchouk moment is simple and effective to describe image and is independent of rotation, scaling, and translation of the image.

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9

Dhekane, Manasi, Ayan Seal, and Pritee Khanna. "Illumination and Expression Invariant Face Recognition." International Journal of Pattern Recognition and Artificial Intelligence 31, no. 12 (September 17, 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. Legendre moments conserve the spatial structure information of the texture image. The resultant feature vector is classified using k-nearest neighbor classifier with [Formula: see text] norm. To evaluate the proposed method, experiments are performed on IRIS and NVIE databases. The proposed method is tested on both visible and infrared images under different illumination and expression variations and performance is compared with recently published methods in terms of recognition rate, recall, length of feature vector, and computational time. The proposed method gives better recognition rates and outperforms other recent face recognition methods.

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10

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 (February 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 order Chebyshev orthogonal Moments (FCMs). This edge detection filter can be used successfully in the gray level image and color images. The new sets of fractional moments are used to reconstruct the gray level image. The numerical results show FLeMs and FLaMs are promised techniques for image representation. The computational time of the proposed techniques is compared with the computational time of Chebyshev orthogonal Moments techniques and gives better results. Also, the fractional parameters give the flexibility of studying global features of the image at different positions of moments.

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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 (December 21, 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 entropy method.

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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 (January 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 (May 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 (September 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 (August 1, 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 functions are also appraised for their discriminating ability through derivations and experiments. Invariance properties such as scaling, translation and rotational invariance are studied over the new formulation to demonstrate the use of the functions over image processing applications that involve invariance to image transformations.

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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 (February 2000): 341–48. http://dx.doi.org/10.1016/s0031-3203(99)00044-8.

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17

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

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18

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 constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

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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 that the mean of the watermarked image invariants is a predefined number. Watermark detection is as simple as computing the moment invariants of received image. The experiment results demonstrate the proposed method can obtain better visual effect, meanwhile, it is also robust enough to some image degradation process such as adding noise, cropping, filtering and JPEG compression.

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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 (December 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 (June 1, 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 (May 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

Mitchell, Rory, Eibe Frank, and Geoffrey Holmes. "An Empirical Study of Moment Estimators for Quantile Approximation." ACM Transactions on Database Systems 46, no. 1 (April 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 the state-of-the-art KLL quantile estimator on 14,072 real-world datasets drawn from the OpenML database. Our analysis highlights the effectiveness of variants of moment-based quantile approximation for highly space efficient summaries: their average performance using as few as five sample moments can approach the performance of a KLL sketch containing 500 elements. Experiments also illustrate the difficulty of applying the method reliably and showcases which moment-based approximations can be expected to fail or perform poorly.

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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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Hosny, Khalid M., and Mohamed M. Darwish. "Invariant color images representation using accurate quaternion Legendre–Fourier moments." Pattern Analysis and Applications 22, no. 3 (July 30, 2018): 1105–22. http://dx.doi.org/10.1007/s10044-018-0740-1.

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Hosny, Khalid M., and Mohamed M. Darwish. "Robust color image watermarking using invariant quaternion Legendre-Fourier moments." Multimedia Tools and Applications 77, no. 19 (March 5, 2018): 24727–50. http://dx.doi.org/10.1007/s11042-018-5670-9.

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30

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 (October 2020): 655–73. http://dx.doi.org/10.1134/s1054661820040185.

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31

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 (December 2014): 994–1006. http://dx.doi.org/10.1016/j.imavis.2014.09.002.

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32

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 (February 1, 2018): 169–80. http://dx.doi.org/10.1007/s11633-017-1105-8.

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33

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 (August 2012): 735–49. http://dx.doi.org/10.1080/01411594.2012.664275.

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34

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 (July 14, 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 representation of digital images with minimal descriptors. This characteristic is advantageous when analyzing images of living organisms that are not in a static posture. The proposal evaluates the algorithm’s efficiency using different color models, and to enhance its capacity, a subdivision of the VarroaDataset is used. This enhancement allows the algorithm to process additional information about the color and shape of the bee’s legs, wings, eyes, and mouth. To demonstrate the advantages of our approach, we compare it with other deep learning methods, in semantic segmentation techniques, such as DeepLabV3, and object detection techniques, such as YOLOv5. The results suggest that our proposal offers a promising means for the early detection of the Varroa destructor mite, which could be an essential pillar in the preservation of bees and, therefore, in food production.

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SUGI, T., DEJEY, and R. S. RAJESH. "GEOMETRIC ATTACK RESISTANT ROBUST IMAGE WATERMARKING SCHEME." International Journal of Information Acquisition 09, no. 01 (March 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, detection and extraction as they provide synchronization and invariant feature which is necessary for a robust watermarking scheme. The proposed scheme shows resistance to geometric distortion, cropping, filtering, compression, and additive noise than the existing ALMI based scheme [Alghoniemy, M. and Tewfik, A. H. [2004] "Geometric invariance in image watermarking," IEEE Trans. Image Process13(2), 145–153] and affine geometric moment invariant (AGMI) based scheme [Seo, J. S. and Yoo, C. D. [2006] "Image watermarking based on invariant regions of scale-space representation," IEEE Trans. Signal Process. 54(4), 1537–1549].

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36

Harding, Steven N., and Alexander W. N. Riasanovsky. "Moments of the weighted Cantor measures." Demonstratio Mathematica 52, no. 1 (September 3, 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, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xm dμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O((log log(1/ε)) · m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2, 1/2].

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Chuan, Zun Liang, David Chong Teak Wei, Connie Lee Wai Yan, Muhammad Fuad Ahmad Nasser, Nor Azura Md Ghani, Abdul Aziz Jemain, and Choong-Yeun Liong. "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 (August 27, 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 extracting the two-dimensional (2D) statistical moment invariants from the segmented region of interest (ROI) corresponding to the cartridge case and bullets images. To pursue this principal objective, several 2D statistical moment invariants have been compared and tested to determine the most suitable feature set applied in the proposed identification algorithm. The 2D statistical moment invariants employed include Orthogonal Legendre moments (OLM), Hu moments (HM), Tsirikolias-Mertzois moments (TMM), Pan-Keane moments (PKM), and Central Geometric moments (CGM). Moreover, the proposed identification algorithm is also tested in different scenarios, including based on the classification of strength association measurements between the extracted feature sets. The empirical results in this article revealed that the proposed identification algorithm applied with the CGM comprising the weak association classification yielded the best identification accuracy rates, which are >96.5% across all the sample sizes of the training set. These empirical results also conveyed that the superior proposed identification algorithm in this research could be developed as a mobile application for ballistics identification that can significantly reduce the time taken and conveniently perform the ballistics identification tasks.

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38

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 (May 28, 2010): 69–79. http://dx.doi.org/10.5121/ijma.2010.2206.

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39

Vijayalakshmi, B., and V. Subbiah Bharathi. "Classification of CT Liver Images Using Local Binary Pattern with Legendre Moments." Current Science 110, no. 4 (February 1, 2016): 687. http://dx.doi.org/10.18520/cs/v110/i4/687-691.

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40

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, and it can be used in some complex lung field segmentation, especially suitable for the segmentation of lung field with the juxta-pleural pulmonary nodules. Experiments illustrate that the proposed method is able to segment the lung fields in the CT images successfully.

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41

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

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42

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 (June 2, 2014): 146–63. http://dx.doi.org/10.1080/21681163.2014.922437.

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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 (March 23, 2019): 409–27. http://dx.doi.org/10.1007/s00500-019-03922-7.

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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 (November 16, 2014): 433–51. http://dx.doi.org/10.1007/s11554-014-0465-3.

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Camacho-Bello, C. "Exact Legendre–Fourier moments in improved polar pixels configuration for image analysis." IET Image Processing 13, no. 1 (January 10, 2019): 118–24. http://dx.doi.org/10.1049/iet-ipr.2018.5489.

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Huang, Wei, Chuanbo Chen, Mudar Sarem, and Yunping Zheng. "Overlapped rectangle image representation and its application to exact legendre moments computation." Geo-spatial Information Science 11, no. 4 (January 2008): 294–301. http://dx.doi.org/10.1007/s11806-008-0096-6.

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Papakostas, G. A., E. G. Karakasis, and D. E. Koulouriotis. "Accurate and speedy computation of image Legendre moments for computer vision applications." Image and Vision Computing 28, no. 3 (March 2010): 414–23. http://dx.doi.org/10.1016/j.imavis.2009.06.011.

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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 the oscillation differential functions. Considering the independent relationship of the structural random parameters and the parameters of random process, the time-varying moments of the response are evaluated by the point estimate method. Combining with the fourth-moment method theory of reliability analysis, the dynamic reliability response can be evaluated. The dynamic reliability curve is useful for getting the weakness time so as to avoid breakage. Reliability-based sensitivity analysis gives the importance sort of the distribution parameters, which is useful for increasing system reliability. The result obtained by the proposed method is accurate enough compared with that obtained by the Monte Carlo simulation (MCS) method.

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Ege, Ökkeş, and Hakan Öztürk. "The eigenvalues of one-speed neutrons in a slab with forward and backward scattering." Tehnički glasnik 13, no. 2 (June 17, 2019): 81–85. http://dx.doi.org/10.31803/tg-20180116174952.

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The eigenvalue spectrum is studied for one-speed neutrons in a slab with forward and backward scattering. First, the transport equation describing the interaction of neutrons in a system with general geometry is given. Then, the scattering function in transport equation is chosen as the forward-backward-isotropic (FBI) scattering model. The resultant transport equation is solved using the Legendre polynomials expansion (PN method) and the Chebyshev polynomials of second kind expansion (UN method) in neutron angular flux. Then, the PN and UN moments of the equations are obtained using the properties of the Legendre and the Chebyshev polynomials of the second kind. Finally, the eigenvalues for various values of the collision and scattering parameters are calculated using different orders of the presented methods and they are given in the tables for comparison.

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Lin, Zhenyi, Nan Chen, Yongzhen Fan, Wei Li, Knut Stamnes, and Snorre Stamnes. "New Treatment of Strongly Anisotropic Scattering Phase Functions: The Delta-M+ Method." Journal of the Atmospheric Sciences 75, no. 1 (January 2018): 327–36. http://dx.doi.org/10.1175/jas-d-17-0233.1.

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The treatment of strongly anisotropic scattering phase functions is still a challenge for accurate radiance computations. The new delta- M+ method resolves this problem by introducing a reliable, fast, accurate, and easy-to-use Legendre expansion of the scattering phase function with modified moments. Delta- M+ is an upgrade of the widely used delta- M method that truncates the forward scattering peak with a Dirac delta function, where the “+” symbol indicates that it essentially matches moments beyond the first M terms. Compared with the original delta- M method, delta- M+ has the same computational efficiency, but for radiance computations, the accuracy and stability have been increased dramatically.

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