[01] R. Rhanim, Instrumentation Measure and Control laboratory, Chouaib Doukkali University, Sciences Faculty, El Jadida, Morocco. [02] A. Gholaifan, Instrumentation Measure and Control laboratory, Chouaib Doukkali University, Sciences Faculty, El Jadida, Morocco. [03] V. Dembele, Instrumentation Measure and Control laboratory, Chouaib Doukkali University, Sciences Faculty, El Jadida, Morocco. [04] K. Assid, Instrumentation Measure and Control laboratory, Chouaib Doukkali University, Sciences Faculty, El Jadida, Morocco. [05] A. Nassim, Instrumentation Measure and Control laboratory, Chouaib Doukkali University, Sciences Faculty, El Jadida, Morocco.

In this paper, we propose a 2D phase extraction algorithm to retrieve optical phase from a single correlation fringe pattern by employing the Bidimensional Empirical Mode Decomposition (BEMD) followed by the Spiral Phase Transform (SPT). The SPT transform extracts the modal phase from every BIMF which is a zero mean 2D AM–FM component obtained by BEMD decomposition, and then the total phase is computed adding all modal phases. The first BIMF of speckle correlation fringe pattern is dominated by residual speckle noise. Hence, the speckle noise can easily be removed by just skipping the first BIMF. The employ of the BEMD decomposition allowed generating an exact quadrature fringe pattern, and then generates a good accuracy in phase extraction by SPT. Numerical simulation study demonstrate the validity of the proposed method and real fringe patterns of carbon fiber deformation gives results in close agreement with those produced by the phase shifting method.

Phase Extraction Methods, Spiral Phase Transform SPT, AM-FM Model, Bidimensional Empirical Mode Decomposition BEMD, Speckle Correlation Fringes

[01] S. Sirohi and S. Chau, ‘Optical Methods of Measurement’, Marcel Dekker, New York, NY, USA (1999) [02] B. V. Dorrio and J. L. Fernandez, ‘Phase-evaluation methods in whole-field optical measurement techniques’, Measurement Science and Technology, vol. 10, no. 3, pp. R33–R55 (1999) [03] MS. Tageda, S. Kobayashi, ‘Fourier transform methods of fringe pattern analysis for computer based topography and Interferometry’, Opt. Soc. Am, Vol. 72, pp.1156-1160 (1982) [04] M. Afifi, FA. Fassi, M. Marjane, K. Nassim, M. Sidki, S. Rachafi, ‘Paul wavelet based algorithm for optical phase distribution evaluation’, Opt Commun, Vol. 211, pp. 47-51 (2002) [05] K. Creath, ‘Phase – measurement interferometry techniques’, In Wolf E (ed.): Progress in optics Vol. XXVI. pp. 349–393 Elsevier Science, Amsterdam (1988) [06] K. Assid, F. Alaoui, V. Dembele, S. Houmairi, M. Sidki, A. Nassim, ‘Normalized Hilbert Huang Transform-NHHT applied to Phase extraction in Wavelet Domain’, The Open Optics Journal.Vol.6, pp 9-13 (2012) [07] K. G. Larkin, D. J. Bone, and M. A. Oldfield, ‘Natural demodulation of two- dimensional fringe patterns. I. General background of the spiral phase quadrature transform’, J. Opt. Soc. Am. A 18, 1862–1870 (2001) [08] K. G. Larkin, ‘Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform’, J. Opt. Soc. Am. A 18, 1871–1881 (2001) [09] J. Villa, I. De la Rosa, G. Miramontes, and J. A. Quiroga, ‘Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator’, J. Opt. Soc. Am. A, 22, 2766–2773 (2005). [10] J. Vargas, R. Restrepo, J. C. Estrada, C. O. S. Sorzano, Yong-Zhao Du and J. M. Carazo, ‘Shack–Hartmann centroid detection using the spiral phase transform’, Applied Optics, Vol. 51, No. 30, 20 October (2012) [11] M. Trusiak, K. Patorski, and M. Wielgus, ‘Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform’, Opt. Express 20, 23463-23479 (2012) [12] Wei Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, S. G. Hanson, ‘Optical vortex metrology for nanometric speckle displacement measurement’, Opt. Express 14, 120–127 (2006). [13] N.E. Huang, Z. Wu, ‘A review on Hilbert-Huang transform: Method and its applications to geophysical studies’, Geophys 2008; RG2006, 23PP [14] N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, ‘The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis’, Proceedings of the Royal Society London, Part A 454 903–995 (1998) [15] J.C. Nunes, Y. Bouaoune, E. Delecheslle, O. Niang, P. Bunel, ‘Image analysis by bidimensional empirical mode decomposition’, Image and Vision Computing 21 (November) 1019–1026 (2003) [16] J. N. Butters and K. A. Leendertz, ‘Speckle pattern and holographic techniques in engineering metrology’, Meas. Control, 4, 349-354 (1971) [17] J. W. Goodman, ‘Laser Speckle and Related Phenomena’, Vol. 9 of Topics in Applied Physics, Springer-Verlag, Berlin (1975) [18] R. Jones and C. Wykes, ‘Holographic and Speckle Interferometry’, 2nd ed. Cambridge U. Press, Cambridge, England (1989) [19] H.T. Yura, S.G. Hanson, J Opt Soc Am A 1987; 4:1931. [20] S.A. Collins, J Opt Soc Am 1970; 60:1168–77. [21] M.A. Schofield and Y. Zhu, ’Fast phase unwrapping algorithm for interferometric applications’, Optics Letters, Vol. 28, No. 14, July 15 (2003) [22] A. Linderhed, ‘Image Empirical Mode Decomposition: a New Tool for Image Processing’, Advances in Adaptive Data Analysis 1(2), pp. 265-294 (2009) [23] A. Linderhed, ‘Variable sampling of the empirical mode decomposition of two-dimensional signals’, Int. J. Wavelets Multiresolution Inf. Process. 03(03), 435–452 (2005) [24] N.E. Huang, ‘Beyond the Fourier transform: coping with nonlinear, nonstationary time series’, NASA Goddard Space Flight Center, USA, 28 pp. http://www.Physiconet.org/events/hrv-2006/huang.pdf (accessed 26 May, 2006) [25] M. B. Bernini, A. Federico and G. H. Kaufmann, ‘Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition’, Appl. Opt. 47(14), 2592–2598 (2008) [26] K. Assid, V. Dembele, F. Alaoui, and A. Nassim, ‘Bidimensional empirical mode decomposition BEMD applied to speckle denoising for wavelet phase evaluation’, Physical Chemical News, vol. 61, pp. 17–23, (2011) [27] Z. Wang and A. C. Bovik, ‘A universal image quality index’, IEEE Signal Process. Lett., vol. 9, no. 3, pp. 81–84, Mar. (2002) [28] Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, ‘Image quality assessment: from error visibility to structural similarity’, IEEE Trans. Image Processing, vol. 13, n. 4, pp. 600–612, apr. (2004) [29] E. M. Barj, M. Afifi, A. Idrissi, S. Rachafi, A. Nassim,’Speckle Correlation Fringes Denoising using Stationary Wavelet Transform: Application in Wavelet phase evaluation technique’, Optics and Laser Technology, Vol. 38, issue 7, pp 506-511 (2006)