Statistical Searching of Deformation Phases on Wavelet Transform Maps of Fringe Patterns
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ARTICLE IN PRESS Optics & Laser Technology 39 (2007) 275–281 www.elsevier.com/locate/optlastec Statistical searching of deformation phases on wavelet transform maps of fringe patterns H.J. Li, H.J. Chen, J. ZhangÃ, C.Y. Xiong, J. Fang Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China Received 18 April 2005; received in revised form 29 July 2005; accepted 10 August 2005 Available online 29 September 2005 Abstract A wavelet transform (WT) analysis is presented to obtain the deformation phases from the fringes with non-uniform carrier frequency, which may appear in the pattern of varied-periodic fringes generated in displacement measurement. Based on the phase maps of the Morlet WT coefficients distributed in a space-scale spectrum, a statistical processing is carried out to search the compacted density of the phase intervals over the scale, and from that the phase modulations related to the object deformation can be determined. Numerical simulation demonstrates the validity of the pattern-processing technique, and the experimental results show the applications to the measurement of the in-plane displacement by the digital speckle pattern interferometry (DSPI) and the measurement of the out-of-plane deflection by the projecting Moire´fringes. r 2005 Elsevier Ltd. All rights reserved. Keywords: Fringes pattern processing; Wavelet transform; Deformation phase retrieve 1. Introduction measurements of surface deformation, however, the initial carrier frequencies are not uniformly distributed in the In optical measurement of surface deformation of an fringe pattern. For example, when a grid pattern with object under various loading, deformation phases need to uniform pitch is projected on a curved surface, a fringe be extracted from the generated fringes through optical carrier with varied space frequencies may be produced due transform or image processing. At present, phase shifting to the curvature variation of the surface, and any further [1] and pattern transformation [2] seem to be the two main object deformation will change the fringe distribution branches associated with the automatic image processing based on that non-uniform carrier. In this case, using of fringe patterns. With the characteristics of multi- Fourier transform to extract the deformation phase resolution, the wavelet transform (WT) presents advan- becomes difficult due to the absence of basic frequency in tages to analyze the pattern images in a space–frequency the carrier pattern, and a reasonable filtering window is not combined domain, which not only is sensitive to the local easy to be determined to perform a frequency shift in the singularities or rapid changes of the frequencies involved in demodulation processing so as to obtain the displacement the patterns, but reveal those localizations in the spatial increments generated by the later loads [3]. positions of the images as well [3,4]. In this paper, a wavelet transform analysis is presented To transform the fringe distributions into a spectrum, a to process those fringe patterns with non-uniform spatial fringe carrier is commonly introduced to the pattern- frequencies, which may have resulted from laser inter- recording to increase the spatial frequency so that the ferometry with distortions of the in-plane strains in deformation phases are involved in the fringe changes as heterogeneous materials under external loading, or pro- carrier frequency modulation [5]. In many practical duced by the projection of an optical grating on a plate or shell surface with initially complex profiles. Because ÃCorresponding author. wavelet analysis is based on local spectrum of the fringe E-mail address: [email protected] (J. Zhang). information, it is possible for any subsequent patterns 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.08.004 ARTICLE IN PRESS 276 H.J. Li et al. / Optics & Laser Technology 39 (2007) 275–281 modulated by the displacement increments to use the The localization property of the wavelet makes the previous fringes as the carrier with local uniformity, to integral range of the transform to be limited onÀÁ a support obtain the phase distributions related to the deformation set ½b À as; b þ as for the analyzing wavelet C x À b=a , components. Meanwhile, a statistical technique is proposed when the support set for the mother wavelet CðxÞ is on in the paper to search the deformation phases on the WT ½s; s. This ensures the validity of developing the phase coefficient phase map represented in the space–frequency j ðxÞ in the fringe intensity I(x) as a Taylor series near the spectrum. The purpose of that is to avoid the procedure to position b, by keeping the linear terms of the first derivative find the maximum trace of the WT coefficient modulus and with respect to b and neglecting the higher orders: then to obtain the corresponding phases, as performed in j ðxÞj ðbÞþj0ðbÞðx À bÞ. Substituting this expression the previous work [3,6–8], which may be sometimes into Eq. (1), the intensity can be rewritten in parts to show difficultly realized in an unconfined searching range or clearly the phase modulation and the terms integrated in involve big errors in the discrete processing of the pattern the wavelet transform: with noises. As analyzed in Section 2, the support window È IðxÞ¼I ðbÞ 1 þ KðbÞ cos ½2pf ðbÞþj0ðbÞx þ½j ðbÞ of the wavelet transform is scale dependent and the way to o É 0 use the maximum WT modulus may result in ambiguity in À b j ðbÞ , ð4Þ the determination of a reasonable scaling region. This where KðbÞ is assumed a constant near the point b and f ðbÞ problem can be eliminated by a direct searching of the represents the local frequency of the fringe carrier with compacted phase location on the WT phase map so as to little deviation in that limited support window. Therefore, obtain a stable result of modulation phases. In Section 3, substituting Eqs. (3) and (4) into Eq. (2), the integration of we show the validity of this statistical method by numerical the Morlet analyzing wavelet and the fringe intensity with simulation and the applications to the deformation phase linear terms leads to an analytical expression of the wavelet analyses of the fringe patterns produced by speckle transform, given by interferometry and Moire´projecting. pffiffiffiffiffiffi o2 WT ða; bÞ¼ 2pI ðbÞ exp À o I o 2 2. WT analysis pffiffiffiffiffiffi 2p ÈÉ þ I ðbÞ exp j½2pf ðbÞ b þ jðbÞ 2.1. WT of fringe carrier 2 1 () a2 o2 2 For a fringe pattern with non-uniform carrier frequency, Â exp À 2pf ðbÞþj0ðbÞ o , ð5Þ its intensity distribution along a line parallel to the axis x 2 a can be expressed as ÈÉwhereI 1ðbÞ¼I 0KðbÞ. IðxÞ¼I o 1 þ KðxÞ cos ½2pf ðxÞx þ jðxÞ , (1) 2.2. Deformation-phase determination where Io is the optical background, K is the fringe contrast, and the function f(x) represents the variedly spatial It is obvious from Eq. (5) that, at any spatial position b, frequency of the fringes. The continuous WT of the as the scale parameter a is fixed, the phase of the WT intensity I(x) can be defined by an integral of the signals coefficient, denoted by F , is directly related the with the translation and dilation of the complex conjuga- WT modulation phase j(b)by tion of a mother wavelet C(x): Z F j ðbÞ¼2pf ðbÞb þ jðbÞ. (6) 1 1 x À b WT a WTða; bÞ¼ IðxÞ C dx, (2) For a fringe carrier without a unique frequency in the a 1 a pattern, the frequency f(x) varies in the whole field. But it where a40 is the scale parameter that is related to the can be kept at a certain value f(b) as the position b is within spatial frequency f by a ¼ 2p/f, and b is the shift parameter the local support window ½b À as; b þ as. Thus the phase representing the positions in the x-line. The mother wavelet j(b) in the fringe intensity is related to its WT coefficient C(x) should be localized and satisfy the admissible phase F (b) by Eq. (6) as a localized frequency condition of WT Z modulation. For many cases of displacement measurement, 1 jCðsÞj2 moreover, we are interested in the deformation phase CC ¼ dso1. represented by the difference Dj ¼ j ðbÞj ðbÞ, where 1 jsj 2 1 the phase changes from j1(b)toj2(b) corresponding to the For the image processing of the fringe patterns, we two states of the object surface deformed by external loads. choose the Morlet wavelet as the mother wavelet, given by As we consider that the local carrier frequency f(b)is x2 maintained on the support set during the load step, the C ðxÞ¼exp À exp ðio xÞ, (3) 2 o linear term in Eq. (6) can be automatically canceled by solving the phase difference of the WT coefficients, or where oo is a positive constant of angle frequency that can be chosen as 2p to meet the criterion of finite admissibility. DFWTðbÞ¼FWT_2ðbÞFWT_1ðbÞ¼DjðbÞ. (7) ARTICLE IN PRESS H.J. Li et al. / Optics & Laser Technology 39 (2007) 275–281 277 That means the deformation phase DjðbÞcan be directly integral number ki with m levels obtained by the phase changes of the WT coefficients Pn transformed from the fringe patterns before and after kiðboÞ¼ f i½FWTðZ; boÞ; i ¼ 1; 2; ...; m . (9) deformation. Z¼1 By varying the scale parameter a and the shift parameter This expression represents the density distribution of the b, the continuous wavelet transform coefficient WTI(a, b) phase values of the WT coefficients scattered over the composes a phase map in the scale-position domain.