skip to main content
survey

Comprehensive Study of Continuous Orthogonal Moments—A Systematic Review

Published:30 August 2019Publication History
Skip Abstract Section

Abstract

Orthogonal moments provide an efficient mathematical framework for computer vision, image analysis, and pattern recognition. They are derived from the polynomials that are relatively perpendicular to each other. Orthogonal moments are more efficient than non-orthogonal moments for image representation with minimum attribute redundancy, robustness to noise, invariance to rotation, translation, and scaling. Orthogonal moments can be both continuous and discrete. Prominent continuous moments are Zernike, Pseudo-Zernike, Legendre, and Gaussian-Hermite. This article provides a comprehensive and comparative review for continuous orthogonal moments along with their applications.

References

  1. Milton Abramowitz and Irene A. Stegun. 1964. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Vol. 55. Courier Corporation. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Milton Abramowitz, Irene A. Stegun, et al. 1966. Handbook of mathematical functions. Appl. Math. Ser. 55, 62 (1966), 39.Google ScholarGoogle Scholar
  3. Ashutosh Aggarwal and Chandan Singh. 2016. Zernike moments-based Gurumukhi character recognition. Appl. Artif. Intell. 30, 5 (2016), 429--444. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Ashutosh Aggarwal and Karamjeet Singh. 2015. Zernike moments-based retrieval of CT and MR images. In Proceedings of the 2015 Annual IEEE India Conference (INDICON’15). IEEE, 1--6.Google ScholarGoogle ScholarCross RefCross Ref
  5. A. Ahmadian, E. Faramarzi, et al. 2003. Image indexing and retrieval using gabor wavelet and legendre moments. In Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 2003, Vol. 1. IEEE, 560--563.Google ScholarGoogle ScholarCross RefCross Ref
  6. Dev Amos. 1986. Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Softw. 12, 3 (1986), 265--273. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Guleng Amu, Surong Hasi, Xingyu Yang, and Ziliang Ping. 2004. Image analysis by pseudo-Jacobi (p= 4, q= 3)--Fourier moments. Appl. Opt. 43, 10 (2004), 2093--2101.Google ScholarGoogle ScholarCross RefCross Ref
  8. S. Annadurai and A. Saradha. 2004. Face recognition using Legendre moments. In Proceedings of the Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP’04). 461--466.Google ScholarGoogle Scholar
  9. Thawar Arif, Zyad Shaaban, Lala Krekor, and Sami Baba. 2009. Object classification via geometrical, Zernike and legendre moments. J. Theor. Appl. Inf. Technol. 7, 1 (2009), 31--37.Google ScholarGoogle Scholar
  10. E. M. Arvacheh and H. R. Tizhoosh. 2005. Pattern analysis using Zernike moments. In Proceedings of the Instrumentation and Measurement Technology Conference 2005 (IMTC’05), Vol. 2. IEEE, 1574--1578.Google ScholarGoogle Scholar
  11. Richard Askey and James Arthur Wilson. 1985. Some Basic Hypergeometric Orthogonal Polynomials That Generalize Jacobi Polynomials. Vol. 319. American Mathematical Soc.Google ScholarGoogle Scholar
  12. Harry Bateman. 1933. Some properties of a certain set of polynomials. Tohoku Math. J. First Ser. 37 (1933), 23--38.Google ScholarGoogle Scholar
  13. Naouar Belghini, Arsalane Zarghili, and Jamal Kharroubi. 2012. 3D face recognition using Gaussian Hermite moments. Spec. Issue Int. J. Comput. Appl. Softw. Eng. Datab. Exp. Syst. 1 (2012), 1--4.Google ScholarGoogle Scholar
  14. S. Benzoubeir, A. Hmamed, and H. Qjidaa. 2009. Hypergeometric Laguerre moment for handwritten digit recognition. In Proceedings of the International Conference on Multimedia Computing and Systems 2009 (ICMCS’09). IEEE, 449--453.Google ScholarGoogle Scholar
  15. V. Subbiah Bharathi and L. Ganesan. 2008. Orthogonal moments based texture analysis of CT liver images. Pattern Recogn. Lett. 29, 13 (2008), 1868--1872. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Li Bo. 2015. Accurate computation of Pseudo-Zernike moments. In Proceedings of the 2015 12th International Computer Conference onWavelet Active Media Technology and Information Processing (ICCWAMTIP’15). IEEE, 220--223.Google ScholarGoogle Scholar
  17. Nasim Borzue and Karim Faez. 2015. Object contour detecting using pseudo zernike moment and multi-layer perceptron. In Proceedings of the 2015 22nd Iranian Conference on Biomedical Engineering (ICBME’15). IEEE, 304--308.Google ScholarGoogle ScholarCross RefCross Ref
  18. C. Camacho-Bello, C. Toxqui-Quitl, A. Padilla-Vivanco, and J. J. Báez-Rojas. 2014. High-precision and fast computation of Jacobi--Fourier moments for image description. J. Opt. Soc. Am. A 31, 1 (2014), 124--134.Google ScholarGoogle ScholarCross RefCross Ref
  19. Lunshao Chai, Honggang Zhang, Zhen Qin, Jie Yu, and Yonggang Qi. 2011. Multi-feature content-based product image retrieval based on region of main object. In Proceedings of the 2011 8th International Conference on Information, Communications and Signal Processing (ICICS’11). IEEE, 1--5.Google ScholarGoogle Scholar
  20. Tang-You Chang, Shen-Chuan Tai, and Guo-Shiang Lin. 2015. A near-duplicate video retrieval method based on Zernike moments. In Proceedings of the 2015 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA’15). IEEE, 860--864.Google ScholarGoogle ScholarCross RefCross Ref
  21. A. Chiang, S. Liao, Q. Lu, and M. Pawlak. 2002. Gegenbauer moment-based applications for Chinese character recognition. In Proceedings of the 2002 IEEE Canadian Conference on Electrical and Computer Engineering (CCECE’02), Vol. 2. IEEE, 908--911.Google ScholarGoogle Scholar
  22. Chee-Way Chong, Paramesran Raveendran, and Ramakrishnan Mukundan. 2004. Translation and scale invariants of Legendre moments. Pattern Recogn. 37, 1 (2004), 119--129.Google ScholarGoogle ScholarCross RefCross Ref
  23. Wang Chun-peng, Wang Xing-yuan, and Xia Zhi-qiu. 2016. Geometrically invariant image watermarking based on fast Radial Harmonic Fourier Moments. Sign. Process.: Image Commun. 45 (2016), 10--23. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Carmine Clemente, Luca Pallotta, Ian Proudler, Antonio De Maio, John J. Soraghan, and Alfonso Farina. 2014. Multi-sensor full-polarimetric SAR automatic target recognition using pseudo-Zernike moments. In Proceedings of the 2014 International Radar Conference (Radar’14). IEEE, 1--5.Google ScholarGoogle ScholarCross RefCross Ref
  25. Claudio Coriano and Çetin Şavkli. 1999. QCD evolution equations: Numerical algorithms from the Laguerre expansion. Comput. Phys. Commun. 118, 2-3 (1999), 236--258.Google ScholarGoogle ScholarCross RefCross Ref
  26. Miklós K. Dede, András Demény, and Judit Kuti Darai. 1996. Evaluation of neutron pulse measurements with Granada-Sibona plexiglass kernel. Nucl. Instrum. Methods A 372, 1--2 (1996), 233--238.Google ScholarGoogle ScholarCross RefCross Ref
  27. C. Lakshmi Deepika, A. Kandaswamy, C. Vimal, and B. Sathish. 2010a. Invariant feature extraction from fingerprint biometric using pseudo Zernike moments. In Proceedings of the International Joint Journal Conference on Engineering and Technology. 104--108.Google ScholarGoogle Scholar
  28. C. Lakshmi Deepika, A. Kandaswamy, C. Vimal, and B. Satish. 2010b. Palmprint authentication using modified legendre moments. Proc. Comput. Sci. 2 (2010), 164--172.Google ScholarGoogle ScholarCross RefCross Ref
  29. Mehdi Dehghan and Karim Faez. 1997. Farsi handwritten character recognition with moment invariants. In Proceedings of the 1997 13th International Conference on Digital Signal Processing (DSP’97), Vol. 2. IEEE, 507--510.Google ScholarGoogle ScholarCross RefCross Ref
  30. Stephane Derrode and Faouzi Ghorbel. 2001. Robust and efficient Fourier--Mellin transform approximations for gray-level image reconstruction and complete invariant description. Comput. Vis. Image Understand. 83, 1 (2001), 57--78. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. Jia-li Dong, Guo-rui Yin, and Zi-liang Ping. 2009. Geometrically robust image watermarking based on Jacobi-Fourier moments. Optoelectr. Lett. 5 (2009), 387--390.Google ScholarGoogle ScholarCross RefCross Ref
  32. Guanyuan Feng, Lin Ma, and Xuezhi Tan. 2014. Ground traffic signs recognition based on Zernike moments and SVM. In Proceedings of the 2014 IEEE 7th Joint International Information Technology and Artificial Intelligence Conference (ITAIC’14). IEEE, 478--481.Google ScholarGoogle ScholarCross RefCross Ref
  33. Alban Foulonneau, Pierre Charbonnier, and Fabrice Heitz. 2003. Geometric shape priors for region-based active contours. In Proceedings of the 2003 International Conference on Image Processing (ICIP’03), Vol. 3. IEEE, III--413.Google ScholarGoogle ScholarCross RefCross Ref
  34. Dariusz Frejlichowski. 2011. Application of zernike moments to the problem of general shape analysis. Contr. Cybernet. 40, 2 (2011), 515--526.Google ScholarGoogle Scholar
  35. Yun Guo, Chunping Liu, and Shengrong Gong. 2015. Improved algorithm for Zernike moments. In Proceedings of the 2015 International Conference on Control, Automation and Information Sciences (ICCAIS’15). IEEE, 307--312.Google ScholarGoogle Scholar
  36. Javad Haddadnia, Majid Ahmadi, and Karim Faez. 2003. An efficient feature extraction method with pseudo-Zernike moment in RBF neural network-based human face recognition system. EURASIP J. Adv. Sign. Process. 2003, 9 (2003), 890--901. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Christopher Hirata and Uroš Seljak. 2003. Shear calibration biases in weak-lensing surveys. Mon. Not. Roy. Astron. Soc. 343, 2 (2003), 459--480.Google ScholarGoogle ScholarCross RefCross Ref
  38. Khalid M. Hosny. 2007. Exact Legendre moment computation for gray level images. Pattern Recogn. 40, 12 (2007), 3597--3605. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Khalid M. Hosny. 2010. Refined translation and scale Legendre moment invariants. Pattern Recogn. Lett. 31, 7 (2010), 533--538. Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. Khalid M. Hosny. 2011a. Fast and low-complexity method for exact computation of 3D Legendre moments. Pattern Recogn. Lett. 32, 9 (2011), 1305--1314. Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. Khalid M. Hosny. 2011b. Image representation using accurate orthogonal Gegenbauer moments. Pattern Recogn. Lett. 32, 6 (2011), 795--804. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. Khalid M. Hosny. 2012. Fast computation of accurate Gaussian--Hermite moments for image processing applications. Digital Sign. Process. 22, 3 (2012), 476--485. Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. Khalid M. Hosny, George A. Papakostas, and Dimitris E. Koulouriotis. 2013. Accurate reconstruction of noisy medical images using orthogonal moments. In Proceedings of the 2013 18th International Conference on Digital Signal Processing (DSP’13). IEEE, 1--6.Google ScholarGoogle Scholar
  44. Haitao Hu and Lingqian Kong. 2012. Image recognition based on Radial Harmonic Fourier moments and SVM. In Proceedings of the 2012 6th International Conference on Internet Computing for Science and Engineering (ICICSE’12). IEEE, 138--142. Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Hai-tao Hu and Lei Qiao. 2011. Radical harmonic Fourier moments. In Proceedings of the 2011 International Conference on Intelligent Computation Technology and Automation (ICICTA’11), Vol. 1. IEEE, 468--471. Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. Ming-Kuei Hu. 1962. Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8, 2 (1962), 179--187.Google ScholarGoogle ScholarCross RefCross Ref
  47. Yongjing Jiang, Ziliang Ping, and Laibin Gao. 2010. A fast algorithm for computing Chebyshev-Fourier moments. In Proceedings of the 2010 International Conference on Future Information Technology and Management Engineering (FITME’10), Vol. 2. IEEE, 425--428.Google ScholarGoogle Scholar
  48. Sahar Kahyaei and Mohamad-Shahram Moin. 2016. Robust matching of fingerprints using pseudo-Zernike moments. In Proceedings of the 2016 4th International Conference on Control, Instrumentation, and Automation (ICCIA’16). IEEE, 116--120.Google ScholarGoogle ScholarCross RefCross Ref
  49. Chao Kan and Mandyam D. Srinath. 2002. Invariant character recognition with Zernike and orthogonal Fourier--Mellin moments. Pattern Recogn. 35, 1 (2002), 143--154.Google ScholarGoogle ScholarCross RefCross Ref
  50. Soon-Yi Kang, Sung-Geun Lim, and Jaebum Sohn. 2005. The continuous symmetric Hahn polynomials found in Ramanujan’s lost notebook. J. Math. Anal. Appl. 307, 1 (2005), 153--166.Google ScholarGoogle ScholarCross RefCross Ref
  51. Harman Preet Kaur and Anmol Sharma. 2015. Offline handwritten signature verification using Zernike moments. In Proceedings of the 2015 5th National Conference on Computer Vision, Pattern Recognition, Image Processing and Graphics (NCVPRIPG’15). IEEE, 1--4.Google ScholarGoogle ScholarCross RefCross Ref
  52. Parminder Kaur and Husanbir Singh Pannu. 2017. Comparative analysis of continuous and discrete orthogonal moments for face recognition. In Proceedings of the 2017 International Conference of Electronics, Communication and Aerospace Technology (ICECA’17), Vol. 1. IEEE, 449--453.Google ScholarGoogle ScholarCross RefCross Ref
  53. Parminder Kaur and Husanbir Singh Pannu. 2018. Comprehensive review of continuous and discrete orthogonal moments in biometrics. Int. J. Comput. Math.: Comput. Syst. Theory 3, 2 (2018), 64--91.Google ScholarGoogle ScholarCross RefCross Ref
  54. Parminder Kaur, Husanbir Singh Pannu, and Avleen Kaur Malhi. 2017. Plant disease recognition using fractional-order Zernike moments and SVM classifier. Neural Computing and Applications (2017), 1--20.Google ScholarGoogle Scholar
  55. Staffs Keele. 2007. Guidelines for performing systematic literature reviews in software engineering. In EBSE Technical Report, Ver. 2.3.Google ScholarGoogle Scholar
  56. Wang Kejia, Zhang Honggang, Ping Ziliang, et al. 2011. Chinese chess character recognition with radial harmonic fourier moments. In Proceedings of the 2011 International Conference on Document Analysis and Recognition (ICDAR’11). IEEE, 1369--1373. Google ScholarGoogle ScholarDigital LibraryDigital Library
  57. Mumtaz Ahmad Khan, Abdul Hakim Khan, and Sayed Mohammad Abbas. 2013. A note on pseudo Jacobi polynomials. Ain Shams Eng. J. 4, 1 (2013), 127--131.Google ScholarGoogle ScholarCross RefCross Ref
  58. Alireza Khotanzad and Yaw Hua Hong. 1990. Invariant image recognition by Zernike moments. IEEE Trans. Pattern Anal. Mach. Intell. 12, 5 (1990), 489--497. Google ScholarGoogle ScholarDigital LibraryDigital Library
  59. Barbara Kitchenham, O. Pearl Brereton, David Budgen, Mark Turner, John Bailey, and Stephen Linkman. 2009. Systematic literature reviews in software engineering—A systematic literature review. Inf. Softw. Technol. 51, 1 (2009), 7--15. Google ScholarGoogle ScholarDigital LibraryDigital Library
  60. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw. 2010. Hypergeometric Orthogonal Polynomials and Their q-analogues. Springer Science 8 Business Media.Google ScholarGoogle Scholar
  61. H. Koelink. 1996. On Jacobi and continuous Hahn polynomials. Proc. Am. Math. Soc. 124, 3 (1996), 887--898.Google ScholarGoogle ScholarCross RefCross Ref
  62. Ban-Hoe Kwan, Kok-Meng Ong, and Raveendran Paramesran. 2006. Noise removal of ECG signals using Legendre moments. In Proceedings of the 27th IEEE Annual International Conference of the Engineering in Medicine and Biology Society (EMBS’05). IEEE, 5627--5630.Google ScholarGoogle Scholar
  63. Vasudevan Lakshminarayanan and Andre Fleck. 2011. Zernike polynomials: A guide. J. Mod. Opt. 58, 7 (2011), 545--561.Google ScholarGoogle ScholarCross RefCross Ref
  64. Bo Li, Guojun Zhang, and Bo Fu. 2010a. Accurate computation and error analysis of Pseudo-Zernike moments. In Proceedings of the 2010 2nd International Conference on Education Technology and Computer (ICETC’10), Vol. 4. IEEE, V4--85.Google ScholarGoogle Scholar
  65. Bo Li, Guojun Zhang, and Bo Fu. 2010b. A novel algorithm for accurate computation of Pseudo-Zernike moments in cartesian coordinates. In Proceedings of the 2010 International Conference on Computer and Communication Technologies in Agriculture Engineering (CCTAE’10), Vol. 2. IEEE, 334--337.Google ScholarGoogle Scholar
  66. Simon Liao and Jing Chen. 2013. Object recognition with lower order gegenbauer moments. Lect. Not. Softw. Eng. 1, 4 (2013), 387.Google ScholarGoogle ScholarCross RefCross Ref
  67. Simon Liao, Amy Chiang, Qin Lu, and Miroslaw Pawlak. 2002. Chinese character recognition via Gegenbauer moments. In Proceedings of the 16th International Conference on Pattern Recognition 2002, Vol. 3. IEEE, 485--488. Google ScholarGoogle ScholarDigital LibraryDigital Library
  68. Wang Lin and Dai Mo. 2006. Localization of singular points in fingerprint images based on the Gaussian-Hermite moments {J}. J. Softw. 17, 2 (2006), 242--249.Google ScholarGoogle ScholarCross RefCross Ref
  69. Rongfang Luo and Tusheng Lin. 2007. Finger crease pattern recognition using Legendre moments and principal component analysis. Chin. Opt. Lett. 5, 3 (2007), 160--163.Google ScholarGoogle Scholar
  70. Li Ma, Tieniu Tan, Yunhong Wang, and Dexin Zhang. 2004. Local intensity variation analysis for iris recognition. Pattern Recogn. 37, 6 (2004), 1287--1298.Google ScholarGoogle ScholarCross RefCross Ref
  71. Xiaojuan Ma, Renlong Pan, and Lin Wang. 2010. License plate character recognition based on Gaussian-Hermite moments. In Proceedings of the 2010 2nd International Workshop on Education Technology and Computer Science (ETCS’10), Vol. 3. IEEE, 11--14.Google ScholarGoogle ScholarCross RefCross Ref
  72. Shahbaz Majeed. 2016. Face recognition using fusion of Local Binary Pattern and Zernike moments. In Proceedings of the IEEE International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES’16). IEEE, 1--5.Google ScholarGoogle ScholarCross RefCross Ref
  73. Mrinal K. Mandal, Tyseer Aboulnasr, and Sethuraman Panchanathan. 1996. Image indexing using moments and wavelets. IEEE Trans. Consum. Electr. 42, 3 (1996), 557--565. Google ScholarGoogle ScholarDigital LibraryDigital Library
  74. Emilio Marengo, Marco Bobba, Maria Cristina Liparota, Elisa Robotti, and Pier Giorgio Righetti. 2005. Use of Legendre moments for the fast comparison of two-dimensional polyacrylamide gel electrophoresis maps images. J. Chromatogr. A 1096, 1 (2005), 86--91.Google ScholarGoogle ScholarCross RefCross Ref
  75. John C. Mason and David C. Handscomb. 2002. Chebyshev polynomials. Chapman and Hall/CRC.Google ScholarGoogle Scholar
  76. R. Mukundan and K. R. Ramakrishnan. 1995. Fast computation of Legendre and Zernike moments. Pattern Recogn. 28, 9 (1995), 1433--1442.Google ScholarGoogle ScholarCross RefCross Ref
  77. A. Nabatchian, E. Abdel-Raheem, and M. Ahmadi. 2008. Human face recognition using different moment invariants: A comparative study. In Proceedings of the Congress on Image and Signal Processing 2008 (CISP’08), Vol. 3. IEEE, Los Alamitos, CA, 661--666. Google ScholarGoogle ScholarDigital LibraryDigital Library
  78. A. J. Nor’aini, P. Raveendran, and N. Selvanathan. 2006. Human face recognition using Zernike moments and nearest neighbor classifier. In Proceedings of the 4th Student Conference on Research and Development 2006 (SCOReD’06). IEEE, Los Alamitos, CA, 120--123.Google ScholarGoogle Scholar
  79. Ying-Han Pang. 2005. Enhanced pseudo Zernike moments in face recognition. IEICE Electr. Expr. 2, 3 (2005), 70--75.Google ScholarGoogle ScholarCross RefCross Ref
  80. Ying-Han Pang, T. B. J. Andrew, N. C. L. David, and F. S. Hiew. 2004. Palmprint verification with moments. Journal of WSCG 12, 1--3 (2004), 325--332.Google ScholarGoogle Scholar
  81. G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios. 2010a. Efficient computation of Zernike and Pseudo-Zernike moments for pattern classification applications. Pattern Recogn. Image Anal. 20, 1 (2010), 56--64.Google ScholarGoogle ScholarCross RefCross Ref
  82. George A. Papakostas, Yiannis S. Boutalis, Dimitris A. Karras, and Basil G. Mertzios. 2007. A new class of Zernike moments for computer vision applications. Inf. Sci. 177, 13 (2007), 2802--2819. Google ScholarGoogle ScholarDigital LibraryDigital Library
  83. George A. Papakostas, Evangelos G. Karakasis, and Dimitris E. Koulouriotis. 2010b. Accurate and speedy computation of image Legendre moments for computer vision applications. Image Vis. Comput. 28, 3 (2010), 414--423. Google ScholarGoogle ScholarDigital LibraryDigital Library
  84. Pravin S. Patil. 2012. Iris recognition based on Gaussian-Hermite moments. Int. J. Comput. Sci. Eng. 4, 11 (2012), 1794.Google ScholarGoogle Scholar
  85. Ziliang Ping, Haiping Ren, Jian Zou, Yunlong Sheng, and Wurigen Bo. 2007. Generic orthogonal moments: Jacobi--Fourier moments for invariant image description. Pattern Recogn. 40, 4 (2007), 1245--1254. Google ScholarGoogle ScholarDigital LibraryDigital Library
  86. ZiLiang Ping, RiGeng Wu, and YunLong Sheng. 2002. Image description with Chebyshev--Fourier moments. J. Opt. Soc. Am. A 19, 9 (2002), 1748--1754.Google ScholarGoogle ScholarCross RefCross Ref
  87. Hasan Abdel Qader, Abdul Rahman Ramli, and Syed Al-Haddad. 2007. Fingerprint recognition using Zernike moments. Int. Arab J. Inf. Technol. 4, 4 (2007), 372--376.Google ScholarGoogle Scholar
  88. Yang Qing-Yue, Gao Fei, and N. I. E. Qing. 2009. A modified L-iterative algorithm for fast computation of Pseudo-Zernike moments. In Proceedings of the 2009 2nd International Congress on Image and Signal Processing.Google ScholarGoogle Scholar
  89. Hassan Qjidaa. 2006. Image reconstruction by Laguerre moments. In Proceedings of the 2nd International Symposium on Communication, Control and Signal Processing (ISCCSP)’06, Vol. 6.Google ScholarGoogle Scholar
  90. Hassan Qjidaa and L. Radouane. 1999. Robust line fitting in a noisy image by the method of moments. IEEE Trans. Pattern Anal. Mach. Intell. 21, 11 (1999), 1216--1223. Google ScholarGoogle ScholarDigital LibraryDigital Library
  91. S. M. Mahbubur Rahman, Shahana Parvin Lata, and Tamanna Howlader. 2015. Bayesian face recognition using 2D Gaussian-Hermite moments. EURASIP J. Image Vid. Process. 2015, 1 (2015), 35.Google ScholarGoogle ScholarCross RefCross Ref
  92. Haiping Ren, Ziliang Ping, Wurigen Bo, Wenkai Wu, and Yunlong Sheng. 2003. Multidistortion-invariant image recognition with radial harmonic Fourier moments. J. Opt. Soc. Am. A 20, 4 (2003), 631--637.Google ScholarGoogle ScholarCross RefCross Ref
  93. Mohammed Saaidia, Narima Zermi, and Messaoud Ramdani. 2014. Facial expression recognition using neural network trained with Zernike moments. In Proceedings of the 2014 4th International Conference on Artificial Intelligence with Applications in Engineering and Technology (ICAIET’14). IEEE, 187--192. Google ScholarGoogle ScholarDigital LibraryDigital Library
  94. Jorge Sánchez-Ruiz. 2001. Linearization and connection formulae involving squares of Gegenbauer polynomials. Appl. Math. Lett. 14, 3 (2001), 261--267.Google ScholarGoogle ScholarCross RefCross Ref
  95. Arup Sarmah and Chandan Jyoti Kumar. 2013. Iris verification using Legendre moments and KNN classifier. International Journal of Engineering Science Invention 2, 8 (2013), 52--59.Google ScholarGoogle Scholar
  96. S. Sreehari Sastry, K. Mallika, B. Gowri Sankara Rao, Sie Tiong Ha, and S. Lakshminarayana. 2012. Novel approach to study liquid crystal phase transitions using Legendre moments. Phase Transit. 85, 8 (2012), 735--749.Google ScholarGoogle ScholarCross RefCross Ref
  97. Zhuhong Shao, Huazhong Shu, Jiasong Wu, Beijing Chen, and Jean Louis Coatrieux. 2014. Quaternion Bessel--Fourier moments and their invariant descriptors for object reconstruction and recognition. Pattern Recogn. 47, 2 (2014), 603--611. Google ScholarGoogle ScholarDigital LibraryDigital Library
  98. Kalpana Sharma, Garima Joshi, and Maitreyee Dutta. 2015. Analysis of shape and orientation recognition capability of Complex Zernike Moments for signed gestures. In Proceedings of the 2015 2nd International Conference on Signal Processing and Integrated Networks (SPIN’15). IEEE, Los Alamitos, CA, 730--735.Google ScholarGoogle ScholarCross RefCross Ref
  99. S. Sharma Ashok, M. Patel Mitul, and P. Chaudhari Jitendra. 2013. Palm print identification using Zernike moments. International Journal of Engineering and Innovative Technology (IJEIT) 4, 11 (2013), 116--118.Google ScholarGoogle Scholar
  100. Jun Shen, Wei Shen, and Danfei Shen. 2000. On geometric and orthogonal moments. Int. J. Pattern Recogn. Artif. Intell. 14, 07 (2000), 875--894.Google ScholarGoogle ScholarCross RefCross Ref
  101. Yunlong Sheng and Lixin Shen. 1994. Orthogonal Fourier--Mellin moments for invariant pattern recognition. J. Opt. Soc. Am. A 11, 6 (1994), 1748--1757.Google ScholarGoogle ScholarCross RefCross Ref
  102. Huazhong Shu, Limin Luo, Xudong Bao, Wenxue Yu, and Guoniu Han. 2000. An efficient method for computation of Legendre moments. Graph. Models 62, 4 (2000), 237--262. Google ScholarGoogle ScholarDigital LibraryDigital Library
  103. Huazhong Shu, Limin Luo, and Jean-louis Caatrieux. 2007. Moment-based approaches in imaging. 1. Basic features {A Look At...}. IEEE Eng. Med. Biol. Mag. 26, 5 (2007), 70--74.Google ScholarGoogle ScholarCross RefCross Ref
  104. Chandan Singh, S. Pooja, and Rahul Upneja. 2011. On image reconstruction, numerical stability, and invariance of orthogonal radial moments and radial harmonic transforms. Pattern Recogn. Image Anal. 21, 4 (2011), 663--676. Google ScholarGoogle ScholarDigital LibraryDigital Library
  105. Chandan Singh and Sukhjeet K. Ranade. 2013. A high capacity image adaptive watermarking scheme with radial harmonic Fourier moments. Digital Sign. Process. 23, 5 (2013), 1470--1482. Google ScholarGoogle ScholarDigital LibraryDigital Library
  106. Chandan Singh and Sukhjeet K. Ranade. 2014. Image adaptive and high-capacity watermarking system using accurate Zernike moments. IET Image Process. 8, 7 (2014), 373--382.Google ScholarGoogle ScholarCross RefCross Ref
  107. Chandan Singh and Rahul Upneja. 2012a. Accurate computation of orthogonal fourier-mellin moments. J. Math. Imag. Vis. 44, 3 (2012), 411--431.Google ScholarGoogle ScholarCross RefCross Ref
  108. C. Singh and R. Upneja. 2012b. A computational model for enhanced accuracy of radial harmonic Fourier moments. In Proceedings of the World Congress of Engineering. 1189--1194.Google ScholarGoogle Scholar
  109. N. V. S. Sree, Rathna Lakshmi, and C. Manoharan. 2011. An automated system for classification of micro calcification in mammogram based on Jacobi moments. Int. J. Comput. Theory Eng. 3, 3 (2011), 431.Google ScholarGoogle Scholar
  110. Gábor Szego. 1939. Orthogonal polynomials, 23. In Proceedings of the American Mathematical Society Colloquium Publi, Vol. 79.Google ScholarGoogle Scholar
  111. Michael Reed Teague. 1980. Image analysis via the general theory of moments. J. Opt. Soc. Am. 70, 8 (Aug. 1980), 920--930.Google ScholarGoogle ScholarCross RefCross Ref
  112. C.-H. Teh and Roland T. Chin. 1988. On image analysis by the methods of moments. IEEE Trans. Pattern Anal. Mach. Intell. 10, 4 (1988), 496--513. Google ScholarGoogle ScholarDigital LibraryDigital Library
  113. Rahul Upneja. 2016. Accurate and fast Jacobi-Fourier moments for invariant image recognition. Int. J. Light Electr. Opt. 127, 19 (2016), 7925--7940.Google ScholarGoogle ScholarCross RefCross Ref
  114. Ekta Walia, Chandan Singh, and Anjali Goyal. 2012. On the fast computation of orthogonal Fourier--Mellin moments with improved numerical stability. J. Real-Time Image Process. 7, 4 (2012), 247--256.Google ScholarGoogle ScholarCross RefCross Ref
  115. Guojiang Wang. 2016. Facial expression recognition method based on Zernike moments and MCE based HMM. In Proceedings of the 2016 9th International Symposium on Computational Intelligence and Design (ISCID’16), Vol. 2. IEEE, 408--411.Google ScholarGoogle ScholarCross RefCross Ref
  116. Kejia Wang, Ziliang Ping, and Yunlong Sheng. 2016. Development of image invariant moments—A short overview. Chin. Opt. Lett. 14, 9 (2016), 091001.Google ScholarGoogle ScholarCross RefCross Ref
  117. Lin Wang and Mo Dai. 2005. Extraction of singular points in fingerprints by the distribution of Gaussian-Hermite moment. In Proceedings of the 1st International Conference on Distributed Frameworks for Multimedia Applications 2005 (DFMA’05). IEEE, 206--209. Google ScholarGoogle ScholarDigital LibraryDigital Library
  118. Lin Wang and Mo Dai. 2007. Application of a new type of singular points in fingerprint classification. Pattern Recogn. Lett. 28, 13 (2007), 1640--1650. Google ScholarGoogle ScholarDigital LibraryDigital Library
  119. Tiansheng Wang and Simon Liao. 2014. Chinese character recognition by Zernike moments. In Proceedings of the 2014 International Conference on Audio, Language and Image Processing (ICALIP’14). IEEE, Los ALamitos, CA, 771--774.Google ScholarGoogle ScholarCross RefCross Ref
  120. Xiang-Yang Wang, E.-No Miao, and Hong-Ying Yang. 2012. A new SVM-based image watermarking using Gaussian--Hermite moments. Appl. Soft Comput. 12, 2 (2012), 887--903. Google ScholarGoogle ScholarDigital LibraryDigital Library
  121. George Neville Watson. 1995. A Treatise on the Theory of Bessel Functions. Cambridge University Press.Google ScholarGoogle Scholar
  122. Chong-Yaw Wee and Raveendran Paramesran. 2007. On the computational aspects of Zernike moments. Image Vis. Comput. 25, 6 (2007), 967--980. Google ScholarGoogle ScholarDigital LibraryDigital Library
  123. Eric W. Weisstein. 2002. Legendre polynomial. (2002).Google ScholarGoogle Scholar
  124. Youfu Wu, Mo Dai, Hongmei Liu, and Gang Zhou. 2008. Discrete Gaussian-Hermite moments and its applications. In Proceedings of the 4th International Conference on Wireless Communications, Networking and Mobile Computing 2008 (WiCOM’08). IEEE, Los Alamitos, CA, 1--4.Google ScholarGoogle ScholarCross RefCross Ref
  125. Youfu Wu and Jun Shen. 2005. Properties of orthogonal Gaussian-Hermite moments and their applications. EURASIP J. Adv. Sign. Process. 2005, 4 (2005), 439420. Google ScholarGoogle ScholarDigital LibraryDigital Library
  126. Youfu Wu and Jing Wu. 2010. Recognizing moving objects based on Gaussian-Hermite moments and ART neural networks. J. Converg. Inf. Technol. 5, 8 (2010), 63--70.Google ScholarGoogle Scholar
  127. Ting Xia, Hongqing Zhu, Huazhong Shu, Pascal Haigron, and Limin Luo. 2007. Image description with generalized pseudo-Zernike moments. J. Opt. Soc. Am. A 24, 1 (2007), 50--59.Google ScholarGoogle ScholarCross RefCross Ref
  128. Bin Xiao, Jian-Feng Ma, and Xuan Wang. 2010. Image analysis by Bessel--Fourier moments. Pattern Recogn. 43, 8 (2010), 2620--2629. Google ScholarGoogle ScholarDigital LibraryDigital Library
  129. Yongqi ng Xin, Simon Liao, and Miroslaw Pawlak. 2004. Geometrically robust image watermarking via pseudo-Zernike moments. In Proceedings of the Canadian Conference on Electrical and Computer Engineering 2004, Vol. 2. IEEE, 939--942.Google ScholarGoogle Scholar
  130. Ma Xing, Mu Chunyang, Wang Yan, Wang Xiaolong, and Chen Xuetao. 2016. Traffic sign detection and recognition using color standardization and Zernike moments. In Proceedings of the 2016 Chinese Control and Decision Conference (CCDC’16). IEEE, 5195--5198.Google ScholarGoogle ScholarCross RefCross Ref
  131. Bo Yang and Mo Dai. 2012. Image reconstruction from continuous Gaussian--Hermite moments implemented by discrete algorithm. Pattern Recogn. 45, 4 (2012), 1602--1616. Google ScholarGoogle ScholarDigital LibraryDigital Library
  132. Bo Yang, Jitka Kostková, Jan Flusser, and Tomáš Suk. 2017. Scale invariants from Gaussian--Hermite moments. Sign. Process. 132 (2017), 77--84.Google ScholarGoogle ScholarCross RefCross Ref
  133. G. Y. Yang, H. Z. Shu, Christine Toumoulin, Guo-Niu Han, and Limin M. Luo. 2006. Efficient Legendre moment computation for grey level images. Pattern Recogn. 39, 1 (2006), 74--80. Google ScholarGoogle ScholarDigital LibraryDigital Library
  134. Xiao-Chen Yuan, Chi-Man Pun, and C.-L. Philip Chen. 2013. Geometric invariant watermarking by local Zernike moments of binary image patches. Sign. Process. 93, 7 (2013), 2087--2095. Google ScholarGoogle ScholarDigital LibraryDigital Library
  135. Hui Zhang, Huazhong Shu, Gouenou Coatrieux, Jie Zhu, Q. M. Jonathan Wu, Yue Zhang, Hongqing Zhu, and Limin Luo. 2011. Affine Legendre moment invariants for image watermarking robust to geometric distortions. IEEE Trans. Image Process. 20, 8 (2011), 2189--2199. Google ScholarGoogle ScholarDigital LibraryDigital Library
  136. Yan Zhang, Stephen J. Kiselewich, and William A. Bauson. 2006. Legendre and Gabor moments for vehicle recognition in forward collision warning. In Proceedings of the Intelligent Transportation Systems Conference 2006 (ITSC’06). IEEE, Los Alamitos, CA, 1185--1190.Google ScholarGoogle Scholar
  137. J. D. Zhou, H. Z. Shu, L. M. Luo, and W. X. Yu. 2002. Two new algorithms for efficient computation of Legendre moments. Pattern Recogn. 35, 5 (2002), 1143--1152.Google ScholarGoogle ScholarCross RefCross Ref
  138. You Zhou, Yan-ying Liu, Chun-min Wang, Jing Chen, Qiu-ping Chen, and Ya-juan Wei. 2009. Contrast research of several human motion detection algorithm {J}. J. Jilin Univ. 6 (2009), 017.Google ScholarGoogle Scholar
  139. Hongqing Zhu, Yan Yang, Zhiguo Gui, Yu Zhu, and Zhihua Chen. 2016. Image analysis by generalized Chebyshev--Fourier and generalized pseudo-Jacobi--Fourier moments. Pattern Recogn. 51 (2016), 1--11. Google ScholarGoogle ScholarDigital LibraryDigital Library
  140. Weixing Zhu, Yan Zhu, Xincheng Li, and Dengting Yuan. 2015. The posture recognition of pigs based on Zernike moments and support vector machines. In Proceedings of the 2015 10th International Conference on Intelligent Systems and Knowledge Engineering (ISKE’15). IEEE, 480--484.Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Comprehensive Study of Continuous Orthogonal Moments—A Systematic Review

          Recommendations

          Comments

          Login options

          Check if you have access through your login credentials or your institution to get full access on this article.

          Sign in

          Full Access

          PDF Format

          View or Download as a PDF file.

          PDF

          eReader

          View online with eReader.

          eReader

          HTML Format

          View this article in HTML Format .

          View HTML Format