Optics and Lasers in Engineering 50 (2012) 1052–1058

Contents lists available at SciVerse ScienceDirect

Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Recent progress in two-dimensional continuous wavelet transformtechnique for fringe pattern analysis

Zhaoyang Wang n, Jun Ma, Minh Vo

Department of Mechanical Engineering, The Catholic University of America, Washington, DC 20064, USA

a r t i c l e i n f o

Available online 17 February 2012

Keywords:

Two-dimensional continuous wavelet

transform

Fringe pattern

Phase extraction

66/$ - see front matter & 2012 Elsevier Ltd. A

016/j.optlaseng.2012.01.029

esponding author.

ail address: wangz@cua.edu (Z. Wang).

a b s t r a c t

The two-dimensional continuous wavelet transform (2D-CWT) technique for fringe pattern analysis has

recently drawn considerable attentions because of its superior characteristics over other fringe analysis

techniques. However, the conventional 2D-CWT technique has a few shortcomings that have restricted

its applications. In this paper, a few important advances to cope with the limitations of the 2D-CWT

technique are presented. With these advances, the 2D-CWT technique is capable of accurately, quickly

and automatically analyzing fringe patterns that contain complex fringes as well as noise and defects.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In the field of fringe pattern analysis, the two-dimensionalcontinuous wavelet transform (2D-CWT) technique has recentlydrawn considerable attentions because of its unique characteristics[1–7], such as the insensitivity to noise and the capability ofcorrupted data recovery. In spite of its superior advantages over theconventional techniques, the conventional 2D-CWT technique has afew shortcomings that have hampered its real-world applications.This paper will present a few important advances that we recentlymade on the 2D-CWT technique for optical fringe pattern analysis,and these advances aim to enhance the applications of the 2D-CWTtechnique to broader scientific research and engineering problems.

Specifically, the following four questions about the 2D-CWTtechnique for optical fringe pattern analysis will be answered anddescribed in this paper. (1) What is the rigorous governingequation? (2) Which mother wavelet should be used? (3) Howto improve the analysis speed? (4) How to identify the fringeorders in complex fringe patterns?

2. Technical description

2.1. Governing equation of the 2D-CWT for fringe pattern analysis

In general, the intensity of an optical fringe pattern can beexpressed as

IðxÞ ¼ IbðxÞþ IaðxÞcos½fðxÞ� ð1Þ

where x indicates the 2D coordinates of each individual pixel inthe fringe pattern, Ib is the background intensity, Ia is themodulation amplitude, and f is the phase distribution of the

ll rights reserved.

fringe pattern. Depending on the measurement technique, there isnormally a direct relation between the physical quantity to beacquired and the fringe phase. For this reason, the fringe patternanalysis often involves phase extraction or determination.

The 2D-CWT of a fringe pattern is defined as [8, 9]:

Wðu,s,yÞ �/I,cu,s,yS¼ s�n

ZR2

IðxÞcnðs�1r�yðx�uÞÞd2x

¼ sn

ZR2

IðxÞcn

ðsr�yðxÞÞeiUxUud2x ð2Þ

where W is the wavelet transform coefficient, u is a translationparameter, s is a scale factor, y is a rotation angle, r�y is theconventional 2�2 rotation matrix corresponding to y, c is the 2Dwavelet function, x is the frequency coordinate, n is the normal-ization parameter, the symbol ^designates the Fourier transform,and the symbol n denotes the complex conjugate operator.

Denoting the local fringe period and orientation at an arbitrarypoint u in a fringe pattern as S (S40) and Y (0rYo2p),respectively, the local fringe pattern at x around u can betheoretically expressed as:

IðxÞ ¼ Ibþ Ia cos½2pS�1nUðx�uÞþfðuÞ� ð3Þ

where Ib and Ia are treated as constants in the local region, andn¼ ðcosY, sinYÞ.

For the purpose of rigorous derivation, the most widely used2D Morlet wavelet is employed here. The wavelet is defined as

cMðxÞ ¼ expðix0UxÞexp �9x92

2s2

!ð4Þ

where s controls the width of the wavelet function, and s¼0.5 isrecommended for general fringe patterns. For fringe patterns withlocal defects where fringes are lacking, a larger s may be used torecover the fringes there. In addition, to make the wavelet scalefactor match with the fringe period, the modulated frequency

Fig. 1. Simulated fringe pattern with noise.

Z. Wang et al. / Optics and Lasers in Engineering 50 (2012) 1052–1058 1053

x0¼(2p,0) is adopted here. Accordingly, the wavelet coefficientcan be derived from Eqs. (2), (3), and (4) as

Wðu,s,yÞ ¼ s2�n 2ps2Ibexpð�2p2s2Þ�

þps2Iaexp �2p2s2 s

S�1

� �2

þ2s

S1�cosðY�yÞ� �� �

exp ifðuÞ� �

þps2Iaexp �2p2s2 s

Sþ1

� �2

�2s

S1�cosðY�yÞ� �� �

�exp½�ifðuÞ��

ð5Þ

The equation indicates that the normalization parameter n

should be set to 2 to normalize the wavelet transform coefficient.It also reveals that the maximum wavelet transform coefficientcan be obtained when s¼S and y¼Y. In this case, point(u,s,y) iscalled a wavelet ridge, and the corresponding wavelet coefficientbecomes

WðuÞridge ¼Wðu,s,yÞridge ¼ 2ps2Ibexpð�2p2s2Þþps2IaexpðifðuÞ�

þps2Iaexpð�8p2s2Þexp½�ifðuÞ� ð6Þ

From Eq. (6), the phase can be calculated as

fðuÞ ¼ tan�1I WðuÞridge

h i1þexpð�8p2s2Þ� �

R WðuÞridge

h i�2ps2Ibexpð�2p2s2Þ

n o1�expð�8p2s2Þ� �

ð7Þ

where I and R denote the imaginary and real parts of a complexvalue, respectively. It is noteworthy that in the relevant literatureof the wavelet transform technique for fringe pattern analysis[10–12], the phase is normally calculated from the followingequation:

fðuÞ ¼ tan�1 I½WðuÞridge�

R½WðuÞridge�ð8Þ

It is clear that Eq. (8) is not a rigorous governing equation but anapproximate one.

Eq. (7) is valid for the 2D Morlet wavelet only; for otherwavelet functions, a similar approach can be used to derive thecorresponding governing equations. In practice, however, usingthe approximate equation, i.e., Eq. (8), typically brings smallerrors that may be neglected in many applications.

2.2. Wavelet selection

With respect to the detecting direction, the wavelets can beclassified as isotropic wavelets and anisotropic or directionalwavelets [8]. The most commonly used isotropic wavelet is the2D Mexican hat wavelet (2D-MHW) defined in the time domainas

cMHðxÞ ¼�r2exp �

1

2s29x9

¼ð2�9x92

Þ

s2exp �

1

2s29x92

ð9Þ

and in the frequency domain as

cMHðxÞ ¼ s29x92exp �

s2

29x92

ð10Þ

The 2D-MHW is real-valued wavelet and is rotationally invar-iant, so it cannot be directly used for fringe pattern analysis whichdemands an analytic wavelet. A solution to this problem is toadapt the 2D-MHW in the frequency domain to an analyticfunction by using the Heaviside function H(x) to enforce thenegative frequency components being zeros. This wavelet, namedthe 2D analytic Mexican hat wavelet (2D-AMHW), is given by

cAMHðxÞ ¼ s29x92expð�s

2

2 9x92ÞHðxÞ ð11Þ

where H(x)¼1 for xxZ0 or xyZ0, and H(x)¼0 otherwise.

Another commonly used isotropic wavelet is the 2D Paulwavelet (2D-PW) [13], and it is defined as

cPðxÞ ¼2ninn!ð1�i9x9Þ�ðnþ1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pð2nÞ!p ð12Þ

and

cPðxÞ ¼2nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

nð2n�1Þ!p 9x9n

expð�9x9ÞHðxÞ ð13Þ

in the time and frequency domains, respectively. Unlike the 2D-MHW, the 2D-PW can be directly employed in fringe patternanalysis [14].

In the directional wavelet category, a typical wavelet is theaforementioned 2D Morlet wavelet (2D-MW), which is expressedin the time domain as Eq. (4) and in the frequency domain as

cMðxÞ ¼ exp �s2

29x�x09

2

ð14Þ

The performance of the 2D-MW is governed by both x0 and s.Since Eq. (14) indicates that x0 and s are coupled together, theselection of either x0 or s must be based on fixing another ofthem. Generally, x0¼(2p,0) is employed to let the scale factors ofthe 2D-CWT correspond to the fringe periods. In this case, theassociated most effective s for typical fringe pattern analysis isexperimentally shown to be around 0.5. A small s helps detectlarge local fringe variations, while a large s tends to smooth outthe local fringe variations and makes the 2D-MW more robust tonoise and defects.

To evaluate the performance of different wavelets in the 2D-CWTfringe pattern analysis, a fringe pattern of 512�512 pixels wherethe phase distribution follows f(x)¼2p(16x)/(xþ150) is generatedby computer simulation. Moreover, the fringe pattern is contami-nated with white noise of signal-to-noise ratio SNR¼�6 dB, asshown in Fig. 1. Fig. 2 shows the errors of the extracted phasedistributions along the horizontal center line of the fringe pattern.The results indicate that the 2D-MW, where x0¼(2p,0) and s¼0.5are used, works well for the simulated fringe pattern; whereas the2D-AMHW and 2D-PW yield notable errors. The reason is that theisotropic wavelets detect the fringe phase information in all thedirections equally at the same time, and the directional phasedistribution of fringes cannot be well distinguished, which leads tonon-negligible errors in phase extraction when the noise level is

100 200 300 400 500−1

−0.5

0

0.5

1

1.5

2

Position (Pixel)

Phas

e er

ror (

rad)

2D−MW2D−AMHW2D−PW

Fig. 2. Errors of phase extracted by using different wavelets.

Fig. 3. Wavelet modulated window in the frequency domain.

Z. Wang et al. / Optics and Lasers in Engineering 50 (2012) 1052–10581054

high. It is noted that because the fringe periods have rapid varia-tions, the analysis gives relatively large errors at the left and rightends. This inherent limitation of the CWT technique can be over-come by discarding the pattern boundaries or using some carriers inactual experiments.

According to the above investigation, the 2D-MW is recom-mended for being used in the 2D-CWT fringe analysis technique.It may also be helpful to point out that another directionalwavelet, named 2D modulated Mexican hat wavelet, is alsosuitable for fringe pattern analysis. Nevertheless, because the2D modulated Mexican hat wavelet is more complex than thepresented three wavelets, it is not included in this paper.

2.3. Processing speed

In the 2D-CWT fringe pattern analysis, it is desired that thescale factors s cover all the possible fringe periods. This requires asmall increment of s, such as 1 pixel or even smaller, for anaccurate analysis. For the rotation angle y, a similar requirementis demanded. This implies that the 2D-CWT dilation and rotationparameters, s and y, should be digitally continuous. It is for thisreason that the 2D-CWT technique needs longer computationtime than many other fringe pattern analysis techniques, eventhough the fast Fourier transform algorithm has been utilized toimplement the convolution calculation of the 2D-CWT [9].

In the spatial frequency domain, the 2D-CWT algorithmintroduces a series of analysis windows in terms of the effectivesupport of wavelets, which are called cover patterns [15]. Fig. 3illustrates a cover map formed by numerous cover patterns in thefrequency domain, where each cover pattern corresponds to awavelet with certain scale factor and rotation angle. To substan-tially reduce the number of dilation and rotation parameters (i.e.,scale factors and rotation angles) or reduce the number of coverpatterns in the cover map, the redundant parameters can beremoved from the continuous parameters and only a minimumnumber of parameters are kept in the 2D-CWT processing with-out losing the analysis accuracies.

For the minimum data set of the dilation parameters, the ratioof any two adjacent scale factors si and siþ1 should satisfy certainrelation. For the 2D-MW, the relationship is described by thefollowing equation [15]:

q¼siþ1

si¼

2pþðffiffiffi2p

=2sÞ2p�ð

ffiffiffi2p

=2sÞð15Þ

Based on Eq. (15), instead of using a large number of scalefactors, the 2D-CWT analysis can use a small number of scalefactors as the dilation parameters. Suppose that the fringe periodsvary from ll to lh, where ll and lh are estimated values with ll nogreater than the actual minimum period and lh no less than theactual maximum period, the ideal smallest scale factor can bedetermined as sl¼ll[1þ(2)1/2/(4ps)]. This ensures that the lowend of the cover pattern is exactly ll. Similarly, the ideal largestscale factor is sh¼lh[1�(2)1/2/(4ps)] to make the high end of thecover pattern be exactly lh. The actual selection of scale factorscan begin from either sl or sh, so the dilation parameters can bechosen as s¼ fsl,qsl,q

2sl, � � �g, and an alternative way is usings¼ f. . .,q�2sh,q�1sh,shg. With the commonly used s¼0.5, thecorresponding q can be calculated as 1.5809.

The increment of the rotation angle y can be determined as [15]:

Dy¼ 2csin�1

ffiffiffi2p

4ps ð16Þ

where c is a non-zero constant to control the overlap between twoadjacent cover patterns. When c¼1, the increments of y can bedetermined as Dy¼0.4540 rad for s¼0.5.

From the presented theory and concept, for a given s, by usingappropriate scale factors and rotation angles, the 2D-CWT analy-sis is able to produce a complete cover map in the spatialfrequency plane. For instance, Fig. 4 shows the basic cover mapfor analyzing a fringe pattern whose fringe periods vary from 2 to72 pixels and fringe orientations cover the entire range, where thescale factor s¼{2.2607, 3.5740, 5.6501, 8.9323, 14.1211, 22.3242,35.2926, 55.7943} and the rotation angle y¼{�0.0184, 0.4357,0.8897, 1.3438} for s¼0.5. In the figure, because the negativefrequency plane does not need to be considered, only half of thecover map is plotted. Compared with the conventional 2D-CWTanalysis that uses digitally-continuous parameters and exhaustivesearching, the scheme based on the above discretized parametershas the ability to dramatically speed up the 2D-CWT analysis.Specifically, the conventional analysis requires 71�33¼2343pairs of parameters (sA[2,72] pixels with an increment of 1 pixeland the rotation angles yA[0,p/2] with an increment of 0.05 rad);with the cover pattern concept, only 8�4¼32 pairs of para-meters are required. This brings a processing speed over 70 timesfaster with the cover pattern approach without losing analysisaccuracy. Detailed demonstration and discussion on the analysis

−100 0 100 200 300 400 500

−50

0

50

100

150

200

250

300

350

400

fx (Hz)

f y (H

z)

Fig. 4. The cover map for a fringe pattern whose fringe periods range from 2 to

72 pixels.

Fig. 5. Traditional 2D-CWT analysis of a fringe pattern with complex fringes: (a)

original fringe pattern, (b) extracted phase distribution, (c) fringe pattern with 60

carrier fringes, and (d) extracted phase distribution with carriers.

Z. Wang et al. / Optics and Lasers in Engineering 50 (2012) 1052–1058 1055

accuracy can be found in Ref. [15], and experimental support willbe presented in the Experiment section of this paper.

2.4. Complex fringe patterns

2.4.1. Single fringe pattern

The 2D-CWT technique is normally employed to analyze asingle fringe pattern. Because a single fringe pattern does notdirectly contain the sign information of the phase gradients, howto determine the full-field fringe orders of a complex fringepattern is usually challenging. This problem can also be seenfrom a careful observation on Eq. (5), which reveals that at point(u,s¼S,y¼pþY), the wavelet coefficient can be simplified as

Wðu,s,yÞ ¼ 2ps2Ibexpð�2p2s2Þþps2Iaexpð�8p2s2Þexp½ifðuÞ�

þps2Iaexpð�ifðuÞ� ð17Þ

It is easy to see from Eqs. (6) and (17) that points (u,S,Y) and(u,S,pþY) yield identical wavelet coefficient modulus, and thephase values extracted by using the arctangent function inEqs. (7) and (8) are opposite in the two cases. Considering thatthe phase values are usually chosen to be positive, either f(u)or2p�f(u) will be obtained with the ridge detection scheme. Thisbrings phase ambiguity to the analysis of complex fringe patterns.Fig. 5 shows an example with the phase distribution governed by

fðxÞ ¼ 24pX5

i ¼ 1

exp½�20:x�pi:2=ðwhÞ� ð18Þ

where w is the image width, h is the image height, and P¼{(w/4,h/4), (w/4,3h/4), (3w/4,h/4), (3w/4,3h/4), (w/2,h/2)}. In this exam-ple, the phase ambiguity issue in Fig. 5(b) is evident for the initialfringe pattern shown in Fig. 5(a). A typical way to cope with thephase ambiguity problem is to add carrier fringes to the fringepattern so that the fringe orders can be monotonically changingalong certain direction, as the fringe pattern and analysis resultshown in Fig. 5(c) and (d).

In order to solve the phase ambiguity issue without addingcarriers (carriers are generally undesired in real applications), aphase determination rule can be employed for the 2D-CWTtechnique to extract phase information from a complex fringepattern [16]. The rule is established on the continuity of fringes,which requires the difference of rotation angles between twoadjacent pixels to be small. The phase determination rule is

described as follows: suppose ydtA[0,p) is the rotation angleargument of the detected maximum ridge at a pixel, andfdtA[0,2p) is the corresponding phase. If ydt is continuousbetween this pixel and its adjacent pixel that has been wellanalyzed, then ydt and fdt are adopted as the rotation angle andphase, respectively for this pixel. Otherwise, ydtþp and 2p�fdt

should be adopted. In practice, the phase determination rule isachieved by the following handling:

½yridgeðuiÞ, fridgeðuiÞ�

¼

½ydtðuiÞ,fdtðuiÞ�, if 9ydiff ðuiÞ�p94ydiff ðuiÞ

½ydtðuiÞ,fdtðuiÞ�, if 9ydiff ðuiÞ�p949ydiff ðuiÞ�2p9½ydtðuiÞþp, 2p�fdtðuiÞ�, else

8><>:

ð19Þ

where

ydiff ðuiÞ ¼ 9ydtðuiÞ�fridgeðui�1Þ9 ð20Þ

and ui and ui�1 represent two adjacent pixels.To apply the phase determination rule, a conventional

2D-CWT processing with rotation angle yA[0,p) is carried outfirst. Then, an arbitrary starting pixel, where the fringe density isrelatively high (i.e., the scale factor s is relatively small) comparedwith most of the fringes in the fringe pattern, is manually selectedand marked as analyzed. After that, the pixel to be analyzed withthe phase determination rule is the one which has an adjacentneighboring pixel that has been analyzed and has the minimumscale factor s. The analysis uses four-adjacency connectivity; ifthere are more than one pixel for the next candidate, all of themwill be analyzed in no particular order. With this approach, theapplication of the phase determination rule to the traditional2D-CWT analysis results follows a path with the minimum scalefactors. To demonstrate the validity of the rule, Fig. 6 shows theextracted phase distribution from Fig. 5(a). It is noteworthy thatalthough the scale-factor-guided searching approach can bedirectly used to obtain the unwrapped phase, it is better to obtainthe wrapped phase first to ensure a correct analysis.

Z. Wang et al. / Optics and Lasers in Engineering 50 (2012) 1052–10581056

2.4.2. Multiple phase-shifted fringe patterns

In addition to analyzing a single fringe pattern, the 2D-CWTtechnique can also be used to process multiple phase-shiftedfringe patterns. The phase-shifting technique is currently themost widely used technique for fringe pattern analysis becauseit possesses a number of advantages such as automatic and fastanalysis; however, the phase-shifting technique is sensitive tothe noise and defects presented in the fringe patterns. Since the2D-CWT technique can cope with the noise and defect issues verywell but lacks the capability of fully-automatic determination offringe orders, a combination of the 2D-CWT technique and thephase-shifting technique looks ideal for the fringe patternanalysis.

Similar to Eq. (1), the intensity of a phase-shifted fringepattern can be mathematically expressed as

IiðxÞ ¼ IbðxÞþ IaðxÞcos½fðxÞþdi� ð21Þ

where i¼ f1,2, � � � ,N�1,Ng,NZ3 is the total number of phase-shifting steps, Ii is the intensity of the ith of the N fringe patterns,and di¼(i�1)2p/N is the phase-shifting amount. From the N

fringe patterns, a complex signal can be constructed as

IðxÞ ¼XN

i ¼ 1

IiðxÞcosðdiÞ�jXN

i ¼ 1

IiðxÞsinðdiÞ ð22Þ

Fig. 6. Wrapped phase extracted from the fringe pattern shown in Fig. 5 by using

the phase determination rule.

Fig. 7. Simulation results: (a) one of the four phase-shifted fringe patterns, (b) wrapped

extracted by the 2D-CWT technique.

where j is the imaginary unit. Substituting Eq. (21) intoEq. (22) can further simplify Eq. (22) as

IðxÞ ¼ N2IaðxÞexp jfðxÞ

� �ð23Þ

phase map extracted by the phase-shifting technique, and (c) wrapped phase map

Fig. 8. Moire fringe pattern of a PBGA package and the analysis result obtained by

the 2D-CWT technique. (a) Fringe pattern, (b) extracted phase map by fast analysis

with 32 pairs of parameters, and (c) extracted phase map by conventional analysis

with 2343 pairs of parameters.

Z. Wang et al. / Optics and Lasers in Engineering 50 (2012) 1052–1058 1057

The new constructed fringe pattern I(x) can then be analyzedby the 2D-CWT technique to automatically determine the full-field phase distributions even though the original fringe patternmay be very complex.

From the point of view of signal processing, the 2D-CWT actsas a fringe-matching filter to detect the local signal informationwhile suppressing the noise and defects in the fringe pattern. Inthe single fringe analysis, the signal described by Eq. (1) involvesa cosine function. Unless the fringe pattern is simple such that thefull-field phase is monotonically distributed along certain direc-tions, there are two possible solutions for the phase (i.e., f or2p�f). On the contrary, the 2D-CWT analysis uses the complexsignal described by Eq. (23) that contains a cosine function in thereal part and a sine function in the imaginary part. The complexsignal provides sufficient information to determine a uniquesolution for the phase. Consequently, with its original advantagesand new feature, the hybrid 2D-CWT technique has the capabilityto analyze phase-shifted fringe patterns containing complexfringes as well as noise and defects. Fig. 7 shows a simulationexample to verify the technique, where the phase distribution ofthe fringes is determined by

fðxÞ ¼ 5p½1�cosð2px=wÞ�½1�cosð2py=hÞ� ð24Þ

In the simulation, four phase-shifted fringe patterns, withhigh-level noise and some defects, are generated by computer.In the phase-shifting analysis, a Gaussian filter has been pre-applied to the fringe patterns to reduce the noise; in the 2D-CWTanalysis, no filtering is pre-applied. From the results, it is easy tosee the effectiveness of the combined analysis of the 2D-CWT andphase-shifting techniques.

Fig. 10. Analysis of a local fringe pattern of a novel electronic packaging

component. (a) fringe patterns, and (b) phase map extracted by the 2D-CWT

technique.

3. Experiment

A few real fringe patterns captured during the deformationmeasurements of electronic packaging components have beenanalyzed by using the presented 2D-CWT technique to demon-strate the validity and practicability of the technique.

Fig. 8(a) is a single fringe pattern acquired during the defor-mation measurement of a plastic ball grid array (PBGA) electronicpackage subjected to thermal loading using the high-sensitivitymoire interferometry [17]. The size of the image is 764�421 pix-els, and the fringe periods range roughly from 2 to 72 pixels.Therefore, the cover map associated with 32 pairs of parameters,previously shown in Fig. 4, is adopted to analyze the moirefringes. The analysis result demonstrates that although theoriginal fringe pattern contains complex fringes, noise, and afew small defects, the 2D-CWT technique can well process thefringe pattern, as the wrapped phase in Fig. 8(b) shows. As acomparison, Fig. 8(c) shows the phase map extracted by using theconventional analysis where 2343 pairs of parameters are used. In

Fig. 9. Analysis of phase-shifted fringe patterns captured in the warpage measurement

extracted by the phase-shifting technique, and (c) phase map extracted by the 2D-CW

this example, the fast analysis based on the cover map conceptreduces the processing time by a factor of 70 without perceptiblyreducing analysis accuracy.

The second experimental demonstration involves four phase-shifted fringe patterns captured during the warpage measure-ment of a tape-automated bonding (TAB) package using high-sensitivity Twyman-Green interferometry. Fig. 9(a) shows one ofthe four fringe patterns, and Fig. 9(b) and (c) show the corre-sponding analysis results obtained by using the phase-shiftingtechnique and the 2D-CWT technique (with phase-shifted fringepatterns). The results clearly show the robustness of the 2D-CWT

of a TAB package. (a) One of the four phase-shifted fringe patterns, (b) phase map

T technique.

Z. Wang et al. / Optics and Lasers in Engineering 50 (2012) 1052–10581058

technique in terms of noise elimination and corrupted datarecovery.

Fig. 10(a) shows the last experimental fringe pattern to beanalyzed, which represents the mechanical deformation in thevertical direction of a local region in a novel electronic packagingcomponent. It is obvious that the fringe pattern contains complexfringes, evident noise, and notable defects. The analysis result isshown in Fig. 10(b), which again demonstrates the superiorperformance of the 2D-CWT technique in the fringe patternanalysis.

4. Conclusion

Compared with many other fringe pattern analysis techniques,the 2D-CWT technique has a number of advantages, such as theinsensitivity to noise and the capability of corrupted data recov-ery. However, the conventional 2D-CWT technique has a fewdrawbacks which have hampered its practical applications. In thispaper, a few important advances that can cope with the limita-tions of the conventional 2D-CWT technique are presented, whichinclude the derivation of rigorous governing equations, theinvestigation on mother wavelet selection, the cover map conceptfor fast processing, and two schemes for automatic determinationof complex fringe orders. These advances allow the 2D-CWTtechnique to be capable of accurately, quickly and automaticallyanalyzing fringe patterns that contain complex fringes as well asnoise and defects, and therefore may help enhance the applica-tions of the 2D-CWT technique to broader scientific research andengineering problems.

Acknowledgment

This work was supported by the National Science Foundation(NSF) under grant 0825806 and the United States Army ResearchOffice (USARO) under grant W911NF-10-1-0502.

References

[1] Niu H, Quan C, Tay C. Phase retrieval of speckle fringe pattern with carriersusing 2D wavelet transform,. Opt. Lasers Eng. 2009;47:1334–9.

[2] Weng J, Zhong J, Hu C. Phase reconstruction of digital holography with thepeak of the two-dimensional Gabor wavelet transform,. Appl. Opt.2009;48:3308–16.

[3] Li S, Su X, Chen W. Spatial carrier fringe pattern phase demodulation by useof a two-dimensional real wavelet,. Appl. Opt. 2009;48:6893–906.

[4] Li S, Su X, Chen W. Wavelet ridge techniques in optical fringe patternanalysis,. J. Opt. Soc. Am. A 2010;27:1245–54.

[5] Huang L, Kemao Q, Pan B, Asundi AK. Comparison of fourier transform,windowed fourier transform, and wavelet transform methods for phaseextraction from a single fringe pattern in fringe projection profilometry,.Opt. Lasers Eng. 2010;48:141–8.

[6] Pokorski K, Patorski K. Visualization of additive type moire and time-averagefringe patterns using the continuous wavelet transform,. Appl. Opt.2010;49:3640–51.

[7] Patorski K, Pokorski K. Examination of singular scalar light fields usingwavelet processing of fork fringes,. Appl. Opt. 2011;50:773–81.

[8] Antoine J, Murenzi R, Vandergheynst P, Ali ST. Two-Dimensional Waveletsand their Relatives. Cambridge University; 2004.

[9] Wang Z, Ma H. Advanced continuous wavelet transform algorithm for digitalinterferogram analysis and processing,. Opt. Eng. 2006;45:045601.

[10] Kadooka K, Kunoo K, Uda N, Ono K, Nagayasu T. Strain analysis for moireinterferometry using the twodimensional continuous wavelet transform,.Exp. Mech. 2003;43:45–51.

[11] Liu H, Cartwright AN, Basaran C. Moire interferogram phase extraction: aridge detection algorithm for continuous wavelet transforms,. Appl. Opt.2004;43:850–7.

[12] Gdeisat MA, Burton DR, Lalor MJ. Spatial carrier fringe pattern demodulationby use of a two-dimensional continuous wavelet transform,. Appl. Opt.2006;45:8722–32.

[13] Afifi M, Fassi-Fihri A, Marjane M, Nassim K, Sidki M, Rachafi S. Paul wavelet-based algorithm for optical phase distribution evaluation,. Opt. Commun.2002;211:47–51.

[14] Gdeisat M, Burton D, Lilley F, Lalor M, and Moore C, Spatial carrier fringepattern demodulation by use of a Two-Dimensional continuous paul wavelettransform, AIP Conference Proceedings 1236, pp. 112–117 (2010).

[15] Ma J, Wang Z, Vo M, Luu L. Parameter discretization in two-dimensionalcontinuous wavelet transform for fast fringe pattern analysis. Appl. Opt.2011;50(34):6399–408.

[16] Ma J, Wang Z, Pan B, Hoang T, Vo M, Luu L. Two-dimensional continuouswavelet transform for phase determination of complex interferograms,. Appl.Opt. 2011;50:2425–30.

[17] Post D, Han B, Ifju P. High Sensitivity Moire: Experimental Analysis forMechanics and Materials. Verlag: Springer; 1994.