# zbMATH — the first resource for mathematics

Fast computation of geometric moments using a symmetric kernel. (English) Zbl 1138.68520
Summary: This paper presents a novel set of geometric moments with symmetric kernel (SGM) obtained using an appropriate transformation of image coordinates. By using this image transformation, the computational complexity of Geometric Moments (GM) is reduced significantly through the embedded symmetry and separability properties. In addition, it minimizes the numerical instability problem that occurs in high order GM computation. The novelty of the method proposed in this paper lies in the transformation of GM kernel from interval $$[0,\infty ]$$ to interval $$[-1,1]$$. The transformed GM monomials are symmetry at the origin of principal Cartesian coordinate axes and hence possess symmetrical property. The computational complexity of SGM is reduced significantly from order $$O(N^{4})$$ using the original form of computation to order $$O(N^{3})$$ for the proposed symmetry-separable approach. Experimental results show that the percentage of reduction in computation time of the proposed SGM over the original GM is very significant at about 75.0% and 50.0% for square and non-square images, respectively. Furthermore, the invariant properties of translation, scaling and rotation in Hu’s moment invariants are maintained. The advantages of applying SGM over GM in Zernike moments computation in terms of efficient representation and computation have been shown through experimental results.

##### MSC:
 68T10 Pattern recognition, speech recognition 68U10 Computing methodologies for image processing
SIMPLIcity
Full Text:
##### References:
 [1] Hu, M.K., Visual pattern recognition by moment invariants, IRE trans. inf. theory IT-8, 179-187, (1962) · Zbl 0102.13304 [2] Teh, C.H.; Chin, R.T., On image analysis by the methods of moments, IEEE trans. pattern anal. Mach. intell., 10, 4, 496-512, (1988) [3] Abu-Mostafa, Y.S.; Psaltis, D., Recognitive aspects of moment invariants, IEEE trans. pattern anal. Mach. intell., PAMI-6, 6, 698-706, (1984) [4] Prokop, R.J.; Reeves, A.P., A survey of moment-based techniques for unoccluded object representation and recognition, CVGIP: graphical models image process., 54, 5, 438-460, (1992) [5] Wong, R.Y.; Hall, E.L., Scene matching with invariant moments, Comput. vision, graphics, image process., 8, 1, 16-24, (1978) [6] A.P. Reeves, M.L. Akey, O.R. Mitchell, A moment based two-dimensional edge detector, in: Proceedings of IEEE Conference on CVPR, 1983. [7] Ho, S.-H.; Don, H.-S., 3-d moment forms: their construction and application to object identification and positioning, IEEE trans. pattern anal. Mach. intell., 11, 10, 1053-1064, (1989) [8] Mukundan, R.; Malik, N.K.; Ramakrishnan, K.R., Attitude estimation using moment invariants, Pattern recognition lett., 14, 3, 199-205, (1993) [9] L. Yang, F. Albregtsen, T. Lønnestad, P. Grøttum, J.-G. Iversen, J.S. Røtnes, J.-A. Røttingen, Measuring shape and motion of white blood cells from a sequence of flourescence microscopy images, in: Proceedings of the 9th Scandinavian Conference on Image Analysis, 1995, pp. 219-227. [10] A.P. Reeves, A. Rostampour, Shape analysis of segmented object using moments, in: Proceedings of the PRIP Conference, 1981, pp. 171-174. [11] Flusser, J.; Suk, T., A moment-based approach to registration of images with affine geometric distortion, IEEE trans. geosci. remote sensing, 32, 2, 382-387, (1994) [12] Alghoniemy, M.; Tewfik, A.H., Geometric invariance in image watermarking, IEEE trans. image process., 13, 2, 145-153, (2004) [13] Y. Zhu, L.D. Silva, C. Ko, Using moment invariants and HMM in facial expression recognition, in: Proceedings of the 4th IEEE Southwest Symposium on Image Analysis and Interpretation, 2000, pp. 305-309. · Zbl 0996.68167 [14] B. Jones, G. Schaefer, S. Zhu, Content-based image retrieval for medical infrared images, in: Proceedings of the 26th Annual International Conference of the Engineering in Medicine and Biology Society (EMBC), vol. 2, 2004, pp. 1186-1187. [15] Yap, P.-T.; Paramesran, R., An efficient method for the computation of Legendre moments, IEEE trans. pattern anal. Mach. intell., 27, 12, 1996-2002, (2005) [16] Mukundan, R.; Ong, S.H.; Lee, P.A., Image analysis by tchebichef moments, IEEE trans. image process., 10, 9, 1357-1364, (2001) · Zbl 1037.68782 [17] Yap, P.-T.; Paramesran, R.; Ong, S.-H., Image analysis by krawtchouk moments, IEEE trans. image process., 12, 11, 1367-1377, (2003) [18] Mukundan, R.; Ramakrishnan, K.R., Moment functions in image analysis, (1998), World Scientific Publishing Singapore · Zbl 0998.94506 [19] Zakaria, M.; Vroomen, L.; Zsombor-Murray, P.; van Kessel, J., Fast algorithm for the computation of moment invariants, Pattern recognition, 20, 639-643, (1987) [20] Li, B.-C., A new computation of geometric moments, Pattern recognition, 26, 1, 109-113, (1993) [21] Singer, M., A new fast algorithm for moment computation, Pattern recognition, 26, 11, 1619-1621, (1993) [22] Yang, L.; Albregtsen, F., Fast and exact computation of cartesian geometric moments using discrete Green’s theorem, Pattern recognition, 29, 7, 1061-1073, (1996) [23] Shen, T.-W.; Lun, D.P.K.; Siu, W.C., On the efficient computation of 2-d image moments using the discrete Radon transform, Pattern recognition, 31, 2, 115-120, (1998) [24] Martinez, J.; Thomas, F., Efficient computation of local geometric moments, IEEE trans. image process., 11, 9, 1102-1111, (2002) [25] Wee, C.-Y.; Paramesran, R., Efficient computation of radial moment functions using symmetrical property, Pattern recognition, 39, 11, 2036-2046, (2006) · Zbl 1103.68791 [26] Hwang, S.-K.; Kim, W.-Y., A novel approach to the fast computation of Zernike moments, Pattern recognition, 39, 11, 2065-2076, (2006) · Zbl 1102.68693 [27] K. Fukunaga, Introduction to Statistical Pattern Recognition, second ed., Computer Science and Scientific Computing, Academic Press, San Diego, 1990. · Zbl 0711.62052 [28] Kan, C.; Srinath, M.D., Invariant character recognition with Zernike and orthogonal fourier – mellin moments, Pattern recognition, 35, 143-154, (2002) · Zbl 0988.68159 [29] Khotanzad, A.; Lu, J.H., Classification of invariant image representations using a neural network, IEEE trans. acoust. speech signal process., 38, 1028-1038, (1990) [30] S. Liao, Image analysis by moments, Ph.D. Thesis, University of Manitoba, Winnipeg, Manitoba, Canada, 1993. [31] Ghosal, S.; Mehrotra, R., Segmentation of range images: an orthogonal moment-based integrated approach, IEEE trans. robotics autom., 9, 4, 385-399, (1993) [32] Iskander, D.R.; Collins, M.J.; Davis, B., Optimal modeling of corneal surfaces with Zernike polynomials, IEEE trans. biomed. eng., 48, 1, 87-95, (2001) [33] Y. Xin, S. Liao, M. Pawlak, Geometrically robust image watermarking via pseudo-Zernike moments, in: Proceedings of the 2004 Canadian Conference on Electrical and Computer Engineering, vol. 2, 2004, pp. 939-942. [34] Haddadnia, J.; Ahmadi, M.; Faez, K., An feature extraction method with pseudo-Zernike moments in RBF neural network-based human face recognition system, EURASIP J. appl. signal process., 890-901, (2003) [35] Y.-S. Kim,W.-Y. Kim, Content-based trademark retrieval system using visually salient features, in: Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1997, pp. 307-312. [36] Wee, C.-Y.; Paramesran, R., On the computational aspects of Zernike moments, Image vision comput., 25, 6, 967-980, (2007) [37] Wang, J.Z.; Li, J.; Wiederhold, G., Simplicity: semantics-sensitive integrated matching for picture libraries, IEEE trans. pattern anal. Mach. intell., 23, 9, 947-963, (2001) [38] Li, J.; Wang, J.Z., Automatic linguistic indexing of pictures by a statistical modeling approach, IEEE trans. pattern anal. Mach. intell., 25, 9, 1075-1088, (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.