Statistical Searching of Deformation Phases on Wavelet Transform Maps of Fringe Patterns

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 on a support obtain the phase distributions related to the deformation set ½b as; b þ as for the analyzing wavelet C x 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 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 IðxÞ¼I ðbÞ 1 þ KðbÞ cos ½2pf ðbÞþj0ðbÞx þ½j ðbÞ of the wavelet transform is scale dependent and the way to o 0 use the maximum WT modulus may result in ambiguity in 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 o I o 2 2. WT analysis pffiffiffiffiffiffi 2p 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, exp 2pf ðbÞþj0ðbÞ o , ð5Þ its intensity distribution along a line parallel to the axis x 2 a can be expressed as whereI 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 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 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 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. Those intervals. It is believed that the required modulation phase WT coefficients are discrete data normally including the is located in the interval where the biggest amount of kiðboÞ errors resulted from the discrete computation and the appears in statistics, corresponding to the maximum noises involved in the fringe images. Due to the dilation of possibility of the modulation phase compacted in the WT the scale parameter in the wavelets transform, the phase phase map at that position. That seems similar to the way extraction at a specified scale of the WT phase map may be of determining the maximum trace of the WT modulus and influenced by those errors because the support window of then the corresponding phases [3,7,8], but in fact this the wavelet is dependent on the scale level, even in the trace statistical method avoids the use of single datum adopted where the maximum modulus of WT coefficients existing from the WT coefficients at a specified scale level and [3,6,7]. It is thus often difficult to choose a proper range in searches the most compacted phase interval over the whole the scaling coordinate a to find the WT components related scale, which produces a stable result by eliminating the to the deformation as we keep at a position bo. At a big ambiguity to choose a scaling region and reducing the scale Z 2 a, the support set ½b Zs; b þ Zs corresponds to a computational errors related to the scale-dependent sup- large range that may include more errors distributed on port set. Thus, by finding the maximum value of kiðboÞ,or that relatively large interval. When the scale Z is small in k ðb Þ¼maxfk ; k ; k ; ...; k g, (10) the coordinate, on the other hand, the support window jo o 1 2 3 m becomes narrow such that the WT coefficient data involved we can obtain the modulation-related WT coefficient phase may be not enough to compute the required phases. To FWTðboÞ by using the middle value of the phases involved in solve this problem, we use the following statistical that interval starting at jo, given by procedure to search the WT phases in the whole scaling 2p range so as to determine the deformation phases. FWTðboÞ¼p þðjo 0:5Þ . (11) As we know, the phases of the WT coefficients range in m [p,p], which can be divided into m parts with uniform Repeating this process by shifting the parameter b along intervals. For each scale Z in the scaling coordinate a, the the x-line (b-line) and carrying out the computation line by corresponding phase FWT(Z,b) will belong to one of those line over the pattern, we can obtain the modulation phases phase intervals, from which we can construct a function in the whole field of the fringe pattern. The following fi(F) for the WT phases, given by simulation proves the validity of the algorithm, and the 8 processing is applied to the experimental measurements of > 2p 2p > 1; p þ m ði 1ÞpFWTðZ; bÞo p þ m i; surface deformation. > > i ¼ 1; 2; ...m 1; > < 2p 3. Simulation and experiment 1; p pFWTðZ; bÞpp; f ðFÞ¼ m i > i ¼ m; > To show the procedure of extracting deformation phases > > 0; from the fringe pattern with non-uniform spatial frequen- :> others: cies, we simulate a fringe carrier with intensity distribution (8) of IðxÞ¼cos½8p ð3 þ 2 sin xÞ x. (12) The expression includes m unit functions defined on the A nonlinear modulation to the phase term of the carrier equational intervals of the WT phases FWTðZ; bÞ2½p; p. is introduced to produce a phase increment caused by For any unit function fi within i ¼ 1; 2; ...; m 1, it is ð ; bÞ simulated deformation, which makes the intensity become defined on one of the intervals of FWT Z located in ½p þ 2p=mði 1Þ; p þ 2p=mÞ. For the function fm,on I 0ðxÞ¼ cos ½8p ð3 þ 2 sin xÞ xþDjðxÞ the other hand, it is defined on the closed interval ½p ¼ cos ½8p ð3 þ 2 sin xÞ xþ5ð1 cos 2pxÞ . ð13Þ 2p=m; p that is the last interval near p. In this way, each value of the WT phase FðZ; bÞ has been related to one of the By digitizing the intensity signals with sampling number functions fi (i ¼ 1; 2 ...; m) corresponding to the amounts of N ¼ 2048, the pattern intensity I(x) of Eq. (12) is of the phase FWTðZ; bÞ distributed in [p,p], respectively. transformed with the wavelets by starting at the initial scale Therefore, by varying the scaling parameter Z 2 a but ao ¼ 0:0023 and dilating the scale a with a discrete step fixing the position at bo, a statistical searching can be pa ¼ 0:000012, to realize a transform of n ¼ 300 frequency performed along the scaling direction by adding the above levels in the scaling coordinate a. Fig. 1 presents the phase functional numbers (1 or 0) so as to produce a series of distribution of the WT coefficient FWTða; bÞ by gray levels ARTICLE IN PRESS 278 H.J. Li et al. / Optics & Laser Technology 39 (2007) 275–281

0 Phase -1

-2 Fig. 1. Phase map of WT coefﬁcients for the intensity along a line in the simulating fringe carrier. -3

-4 0 500 1000 1500 2000 2500 x (pixel) range in ½p; p in the scale-position (frequency–space) domain. We divide the gray levels of the phases into m ¼ Fig. 2. Initial phases of the fringe carrier resulting from statistical searching in the WT phase map. 200 parts and transfer them to the function series f iðFWTÞ as described in the last section. By adding those numbers from each intervals to form the statistical data kiði ¼ 1; 2; ...; mÞ over the whole scale coordinate, the modulation-related WT phase FWTðbÞ is determined by locating 4 the maximum of ki and using the middle value of that interval as the required phase. Fig. 2 gives the wrapped 3 phases FWTðbÞ along a horizontal line passing through the 2 fringe carrier, whose jumps at the points of 2kp(k is integral) are then connected by the unwrapping process to 1 generate a continuous distribution along the axis b. The 0 same processing is also performed for the intensity I ðxÞ of 0

Eq. (13) to obtain the WT phase modulated by the Phase simulating deformation, in which the starting point of the -1 scale dilation is ao ¼ 0:00023 and the discrete step is pa ¼ 0:0000003728 that is much less than the former one to -2 reflect details of the nonlinear modulation. Fig. 3 shows the 0 0 -3 WT coefficient phase FWTðbÞ that resulted from I ðxÞ, whose unwrapped data, subtracting those in Fig. 2 -4 produces the phase difference corresponding to the 0 500 1000 1500 2000 2500 0 x (pixel) deformation given by DFWTðbÞ¼FWTðbÞFWTðbÞ, are presented in Fig. 4, in which the computed phases DFWTðbÞ Fig. 3. Deformation-modulated phases searched from the WT phase map. are compared with the preset phases DjðbÞ, showing reasonable agreement with a standard error of 0.0893. The wavelet transform analysis is applied to obtain the in-plane displacement field involved in two fringe patterns of the DSPI test. A rectangular specimen of foamed carrier even though spatial distribution is non-uniform in aluminum alloy that is useful in many high-strength/low- the field. weight engineering structures is compressed in horizontal To reduce the speckle noise involved in the DSPI fringe direction when the right boundary is fixed on a rigid edge patterns, a filtering process is carried out, after the gray and the left side loaded by compressive forces. The data adopted from the images have been interpolated and specimen is illuminated by laser beams and the speckle transformed into a Fourier spectrum. The filter function intensity of the rough surface is recorded by a CCD camera Hd is as the specimen is loaded step-by-step. The foamy material ( produces inhomogeneous deformation under external ejMo=2 0 o 0:07p; HdðoÞ¼ (14) load, as shown in Figs. 5 and 6, to give the fringe patterns 00:07poo p; before and after a loading step. The phase solution of the in-plane deformation corresponding to the load increment where j is the imaginary number, o is frequency and M is a needs to use the pattern of Fig. 5 as the basic fringe parameter that can be chosen as 10 for those patterns. The ARTICLE IN PRESS H.J. Li et al. / Optics & Laser Technology 39 (2007) 275–281 279

6 Phase

0 0 500 1000 1500 2000 2500 x (pixel) Fig. 6. DSPI fringe pattern modulated by the in-plane deformation of a Fig. 4. Comparison of the modulation phase DFWT ðbÞ resulting from WT loading step. processing with the phase DjðbÞ preset in the simulating function (dashed line).

cancels also the local frequencies of the fringe carrier. Repeating the same processing for each line included in the two patterns, the whole field of deformation phase Djðx; yÞ that is proportionally related to the in-plane displacements uðx; yÞ by Djðx; yÞ¼ð4p sin y=lÞ uðx; yÞ is acquired, as illustrated in Fig. 9, showing clearly the non-uniform deformation in the foamed aluminum material produced by the load increment. Another application is the out-of-plane displacement measurement from the Moire´fringe patterns of a buckling plate. A grating pattern with uniformly spaced black and white strips is generated by a computer and projected on the plate surface in an inclined direction. Because the thin plate (1 mm in thickness) is initially buckled as often happens in te thin-wall structures, as shown in Fig. 10, the projected lines are no more parallel to each other due to the local curvatures of the plate that intercepts the projecting Fig. 5. DSPI fringe pattern of a foamed aluminum specimen fixed on the pattern at different space locations. As the plate is loaded right edge and compressed on the left edge. laterally from backside, this fringe carrier with non- uniform frequency is changed by the out-of-plane displacements to produce a new fringe pattern under deformation, as presented in Fig. 11. For those two fringe patterns with product of the transformed pattern with this low-pass filter varied spatial frequencies in the field resulting from the function blocks off the random frequency involved in the initial buckling of the plate and the superposition of the speckle images, and its inverse Fourier transform produced surface deformation, the WT analysis utilizes the local the smoothing fringe patterns and can thus be processed by frequencies of the fringes to extract the displacement the above WT algorithms to obtain the phase map of the phases in the space–frequency spectrum. The phase WT coefficients. The required phases are thus determined statistics are computed as the scale parameter a is changed, by the statistical computation of the compacted phase to search the phases related to the initial distortion to locations in the scale range. As examples, Figs. 7 and 8 the fringes on the bulked plate and the phases combined present the wrapped phase distributions along one of the with the plate deformation. The relative phase change horizontal lines in those fringe patterns, corresponding to corresponding to the out-of-place displacement is then the object states before and after loading. Therefore, the obtained by the difference of the unwrapped phases, or 0 phases of the in-plane displacement are obtained by the DjðbÞ¼DFWTðbÞ¼FWTðbÞFWTðbÞ. The processing is subtraction of those WT phases after unwrapping, which performed by the computational program, line-by-line for ARTICLE IN PRESS 280 H.J. Li et al. / Optics & Laser Technology 39 (2007) 275–281

pi ) b ( 0.0 WT Φ

-pi50 100 150 200 250 300 350 400 450 500

Fig. 7. Statistically searched wrapped phases distribution along a line in the initial DSPI pattern as the fringe carrier.

pi ) b ( 0.0 WT Φ

-pi 50 100 150 200 250 300 350 400 450 500

Fig. 8. Wrapped phase distribution of the line in the DSPI fringe pattern modulated by the in-plane deformation.

50 40 30 20 phase 10 0 -10 600 500 400 400 300 (pixel) 200 200 100 (pixel) 0 0

Fig. 9. Deformation phase ﬁeld of the aluminum specimen resulted from the phase differences of the WT coefﬁcients.

the fringe patterns, which produces the out-of-plane Fig. 10. Fringe carrier pattern resulting from the projection of a grating displacement ﬁeld of the loaded plate, as illustrated in on a thin bulked plate with complex surface proﬁle. Fig. 12.

4. Conclusion frequency in the ﬁeld, which may be produced by local distortion of the measuring surface or by deformation We have presented a wavelet transform method to resulted from loading. The technique utilizes the phase process the fringe patterns without a unique carrier maps of the WT coefﬁcients to statistically search the ARTICLE IN PRESS H.J. Li et al. / Optics & Laser Technology 39 (2007) 275–281 281

compacting ranks of phase density distributed over the scaling space so as to obtain the modulation phases. The deformation phases are obtained by the difference of the unwrapped modulation phases of the WT coefﬁcients corresponding to the loading change of the specimen so that the local carrier frequencies can be removed. The numerical simulation shows the stable solution of the related phases and the experimental results show the applications of the transform technique to in-plane and out-of-plane displacement measurements by those optical methods.

Acknowledgments

The support of the NNSFC Fund (10125211) and the 973 Program Integrated MOEMS (G1999033108) is greatly appreciated. We also appreciate the support of the BioMed-X Research Center of PKU.

Fig. 11. Fringe pattern modulated by the out-of-plane displacement of the plate based on the fringe carrier. References

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