Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform

Visualization of additive-type moiré and time-averagefringe patterns using the continuous

wavelet transform

Krzysztof Pokorski* and Krzysztof PatorskiInstitute of Micromechanics and Photonics, Warsaw University of Technology,

8 Sw. A. Boboli Street, 02-525 Warsaw, Poland

*Corresponding author: [email protected]

Received 25 February 2010; accepted 4 June 2010;posted 7 June 2010 (Doc. ID 124597); published 22 June 2010

An application of the continuous wavelet transform to modulation extraction of additive moiré fringesand time-average patterns is proposed. We present numerical studies of the influence of various param-eters of the wavelet transformation itself and a fringe pattern under study on the demodulation results.To facilitate the task of demodulating a signal with zero crossing values, a two-frame approach for wave-let ridge extraction is proposed. Experimental studies of vibration mode patterns by time-average inter-ferometry provide excellent verification of numerical findings. They compare very well with the results ofour previous investigations using the temporal phase-shifting method widely considered as the mostaccurate one. No need of performing phase shifting represents significant simplification of the experi-mental procedure. © 2010 Optical Society of AmericaOCIS codes: 120.2650, 120.3180, 120.4120, 120.7280, 100.7410.

1. Introduction

The moiré fringe technique is a well-established toolin optical metrology [1,2]. It consists of comparing aquasi-periodic structure with deformed lines (carry-ing an information on the phenomenon or elementunder test) with the reference structure. Dependingon the illumination used, one can distinguish twomodes of superimposition of structures. In the firstmode, overlapping structures are incoherently illu-minated or else superimposition is by photographicor video techniques. In the second mode, coherentmoiré fringes are formed by illuminating the struc-tures coherently and filtering out double diffractionorders or by overlapping real-time interferencefringes with a recorded pattern.

In this study, we deal with the incoherently super-posed structures. Three types of incoherent superpo-sition can be distinguished: additive, subtractive,and multiplicative [2,3]. The optical realization

and appearance of the resulting moiré pattern differfor the three types of superposition. In themajority ofapplications, two structures are usually superim-posed. Multiplicative and subtractive superpositionsyield directly visible low spatial frequency beat pat-terns. In the latter case, the operation of subtractionof two composite structures, now most frequentlyperformed by numerical means [4], is followed by rec-tification. Direct appearance of moiré fringes in themultiplicative and subtractive cases can be readilyexplained by their simple mathematical descrip-tion—the lowest spatial frequency beat term (moiré)appears independently in the formula describing theresulting intensity distribution. On the other hand,linear additive superimposition results in moiréfringes in the form of the amplitude modulation ofa carrier. The beat pattern (the fringe pattern macro-structure) does not occur in isolation and can beobserved only when the microstructure (carrier pat-tern) is resolved [2,3].

Most common implementations of the additivetype moiré are as follows:

0003-6935/10/193640-12$15.00/0© 2010 Optical Society of America

3640 APPLIED OPTICS / Vol. 49, No. 19 / 1 July 2010

1. The two-field (two-structure) exposure tech-nique [1–3]. It includes two cases: a) the double ex-posure method with sequential-in-time exposures ofa recording medium (a non-real-time technique), andb) simultaneously exposing a recording medium withtwo mutually incoherent light fields (a real-timetechnique)—see, for example, double projectionmoiré topography [2,5,6].

A simple mathematical description of the additivesuperposition is given by [2,3]:

IðxÞ ¼ I1ðxÞ þ I2ðxÞ¼ 1

þ cosfπ½ð1=d1Þ − ð1=d2Þ�xg cosfπ½ð1=d1Þþ ð1=d2Þ�xg

¼ 1þ cosfπ½ðν1 − ν2Þ�xg cosfπ½ðν1 þ ν2Þ�xg; ð1Þ

where, for simplicity, one-dimensional (1D) struc-tures I1ðxÞ ¼ ð1=2Þ½1þ cosð2πx=d1Þ� and I2ðxÞ ¼ð1=2Þ½1þ cosð2πx=d2Þ�, with slightly different spatialperiods d1 and d2 (spatial frequencies ν1 ¼ 1=d1 andν2 ¼ 1=d2) and mutually parallel lines, have been as-sumed. The first cosine in Eq. (1) corresponds to themoiré pattern (the difference beat frequency ðν1 − ν2Þcarries the required information on the difference be-tween the two superimposed structures), and the sec-ond cosine term describes the sum beat frequencycarrier.

2. The time-average recording of a vibrating per-iodic or quasi-periodic pattern that is transferred onto the tested specimen surface (studies of in-planevibrations [7]) or projected onto it (studies of out-of-plane vibrations [6,8,9]). The latter case, for exam-ple, includes interferometric studies of vibrations ofMEMS silicon elements, such as cantilevers andmembranes [10–15].

In the case of a single frequency sinusoidal vibra-tion, the intensity distribution of a two-beam time-averaged interferogram can be expressed as [6–16]

Ivibðx; yÞ ¼ Kðx; yÞ�1þ Cstatðx; yÞJ0

×�4πλ a0ðx; yÞ

�cosφvibðx; yÞ

�; ð2Þ

where Kðx; yÞ is the interferogram bias, Cstatðx; yÞ isthe contrast of interference fringes (nonvibrating ob-ject), φvibðx; yÞ ¼ ð2π=λÞOPDðx; yÞ, where OPDðx; yÞ isthe optical path difference between the beams, and λis the light wavelength. J0 is the zero-order Besselfunction and a0ðx; yÞ is the vibration amplitudedistribution. Calculating the modulation distri-bution for the dynamic object, one can obtainjKCstatJ0ð4πa0=λÞj. By normalizing it by the modula-tion of the static object jKCstatj (J0 ¼ 1, since a0 ¼ 0)the zero-order Bessel function with the vibration am-plitude encoded in its argument can be obtained.

In both Eqs. (1) and (2), the common feature is theinformation encoded amplitude modulation (the firstcosine term in Eq. (1) and the J0 function in Eq. (2)) ofthe carrier fringes.

The requirement to resolve the carrier to monitorthe modulation function is rather arduous; moreover,the contrast of fringes is very low. The simplest andmost common approach to increase the visibility ofadditive-type moiré is to apply nonlinear detection[2,17–20]. This results in generating an isolated termin the intensity distribution, similarly to the case ofmultiplicative moiré. At the same time, however,many high frequency terms (especially in the caseof superimposing binary amplitude structures) arepresent and constitute a noise to the information-carrying moiré fringes. Various spatial-averaging-based methods, such as defocusing and average ormedian filtering, can be used to smooth out higher-frequency noise terms in the multiplicative moiré.Note that these methods are not directly applicableto the additive-type moiré—the carrier removalcauses disappearance of modulation (moiré) bands.

An elegant solution to visualize and enable quanti-tative analysis of additivemoiré fringes is to calculatethe modulation term of the intensity distributions inEqs. (1) and (2). For example, for the purpose of demo-dulating two-beam time-average interferograms,phase-type techniques of automatic fringe patternanalysis have been proposed, i.e., the Fourier trans-form [10], phase shifting [11–14], and spatial carrierphase shifting [15]. In this paper, for the first time, toour knowledge, the applicability of another powerfulfringe pattern analysis method—the continuouswavelet transform to determine the modulation func-tion of the time-average interferograms and additive-type moirégrams—is investigated. The wavelettransform resulted from works that aimed at the im-provement of effectiveness of the Fourier transformwhen applied to nonstationary signals (with the fre-quency spectrum changing in time). The goal was toenable local analysis of frequency content, in contrastto global analysis performed by the Fourier transform[21,22]. Because of its local analysis capability, thecontinuous wavelet transform (CWT) has attractedconsiderable attention in extracting the phase offringe patterns (see, for example, [23–27] and refer-ences therein).

In the literature, the application of 1D temporal[28,29], as well as 1D and two-dimensional (2D) spa-tial [30,31] wavelet analyses of moiré fringes can befound. In all of them, however, the extraction of thephase of the multiplicative-type moiré was consid-ered. The same remark concerns moiré interferome-try studies [25,32–34], which deal, in fact, with phaseextraction of two-beam interferograms.

We propose the use of 1D and 2D versions of theCWT to calculate the modulation of additive-typesuperposition moiré and time-average interferencepatterns. First, the fundamentals of 1D and 2Dwavelet transformations are recalled. Next, a reviewof the wavelet transform ridge extraction algorithms,

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together with their performance comparison whenapplied to modulation distribution determination,is presented. Computer simulations include studiesof the noise resistance, analyses of spatial carrier fre-quency, orientation and gradient influences, and theinfluence of the modulation function spatial varia-tions. Next, the complex Morlet wavelet, chosen forfurther investigations because of its noise resistance,is studied, in particular, its modified version with anarrower Gaussian windowwidth. The CWT normal-ization issue crucial for modulation determination isconsidered, as well.

Theoretical and numerical findings are corrobo-rated by the wavelet transformation processing oftime-averaged interferograms of vibrating micro-membranes at resonant frequencies [12]. The CWTresults obtained compare very well with previouslypresented results of the phase-shifting method, thelatter one widely considered as the most accurateone. Advantages of the CWT approach are the demo-dulated image quality and that there is no needto meet phase-shifting method requirements for anartifact-free result.

2. Continuous Wavelet Transform

A. One-Dimensional Continuous Wavelet Transform

The 1D-CWT of a signal f ðxÞ is defined as

S1Dðb; sÞ ¼ s−ηZRψ�ðs−1ðx − bÞÞf ðxÞdx; ð3Þ

where S1Dðb; sÞ denotes the wavelet coefficient, η isthe normalization parameter, usually set to η ¼ 0:5[22], and ψ� denotes the complex conjugate of themother wavelet function ψ , as dilated by scaleparameter s ∈ Rþ and translated on the x axis bytranslation parameter b ∈ R. As an output of thetransform, one receives a 2D map of local similaritybetween signal and wavelet.

To meet mathematical requirements, the motherwavelet function has to be square integrable andsatisfy the so-called admissibility condition, whichensures invertibility of the transform [22]:

2πZR

j ψ̂ðkÞj2jkj dk < ∞; ð4Þ

where ψ̂ denotes the Fourier transform of the waveletfunction ψ and k is a spectral coordinate. This condi-tion is not crucial in terms of modulation extraction(transform invertibility is not needed). Therefore,any oscillating, compactly supported function canbe considered as a wavelet.

The Morlet wavelet is often used in interferogramanalysis because of good localization in the spatialand spectral domains [35]. It is defined as a sinewave modulated by a Gaussian window:

ψMorletðxÞ ¼ eik0xe−mx2 ; ð5Þ

where k0 is sine wave frequency andm is a Gaussianwindow width parameter. The wavelet satisfies theadmissibility condition for k0 ≈ 6 and m ¼ 0:5. Fordifferent values of parameterm, the function is calledthe modified Morlet wavelet.

The Paul wavelet is defined as follows [36]:

ψPaulðxÞ ¼2nn!ð1 − ixÞ−ðnþ1Þ

2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2nÞ!=2p ; ð6Þ

where n ∈ N denotes the wavelet order. The Paulwavelet satisfies the admissibility condition for everyn value.

The derivative of Gaussian (DoG) wavelet is thenth derivative of a Gaussian function [37]:

ψDoGðxÞ ¼ −dn

djxjn ðe−jxj2=2Þ: ð7Þ

For n ¼ 2, the DoG wavelet is real valued and iscalled the Mexican Hat wavelet. The complex Mexi-can Hat wavelet can be generated by multiplying thewavelet spectrum by the Heaviside step function.

B. Two-Dimensional Continuous Wavelet Transform

The 2D-CWT of a signal f ðxÞ is defined as [22]

S2Dðs; b; θÞ ¼ s−ηZ Z

R2ψ�ðs−1r−θðx − bÞÞf ðxÞd2x; ð8Þ

where S2Dðs; b; θÞ denote the wavelet coefficients, ψdenotes the 2D wavelet function, x ¼ ðx; yÞ is the spa-tial coordinate, b ¼ ðbx; byÞ (b ∈ R2) is the translationparameter, and s ∈ Rþ is the scale parameter. Thenormalization parameter η for 2D-CWT is usuallyset to η ¼ 1 and rθ is a standard rotation matrix:

rθ ¼�cos θ − sin θsin θ cos θ

�: ð9Þ

The 2D-CWT coefficients reflect the local similaritybetween a transformed signal and the wavelet func-tion rotated, scaled, and translated along the x andy axes.

Two-dimensional wavelets are square integrablefunctions that satisfy the 2D admissibility condition(analogous with the 1D admissibility condition,ensuring invertibility of the transform [22]):

cψ ¼ ð2πÞ2Z Z

R2

j ψ̂ðkÞj2jkj2 d2k < ∞: ð10Þ

For signal modulation extraction purposes, this con-dition can be ignored and any oscillating, compactlysupported function may be used as a wavelet.

The 2D Morlet wavelet [24,37,38] is a plane wavemodulated by a Gaussian function:

ψMoreltðxÞ ¼ eik0xe−mjxj2 ; ð11Þ


where k0 ¼ ðu0; v0Þ sets the plane wave frequencyand direction and m is the Gaussian window widthparameter. This wavelet satisfies the admissibilitycondition for jk0j ≈ 6 and m ¼ 0:5. As for the 1D case,for differentm values, this function is called the mod-ified Morlet wavelet.

The Fan wavelet is created as a sum ofNθ modifiedMorlet wavelets with different kj parameters, settingdifferent wavelet directions θ [24,37]:

ψFanðxÞ ¼XNθ−1

j¼0

eikjxe−mjxj2 ; ð12Þ

where kj ¼ ðk0 cosðjθÞ; k0 sinðjθÞÞ.C. Continuous Wavelet Transform Modulation Extraction

Signal modulation extraction using the CWT differsin the 1D and 2D cases. When using 1D-CWT, the im-age is analyzed on a row-by-row basis. Every imagerow is transformed using a complex wavelet and itsridge, given by maxsjS1Dðs; bÞj, is extracted. The mod-ulus of the received ridge wavelet coefficients givessignal modulation M:

M ¼ jSj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðRe2½S� þ Im2½S�Þ

q: ð13Þ

In the 2D case, the whole image is transformed atonce (using a complex wavelet) and its ridge(maxs;θjS2Dðs; b; θÞj) is extracted. Then modulation isreceived as the modulus of ridge wavelet coefficients.

3. Ridge Extraction

A. Implemented Algorithms

The wavelet transform ridge can be extracted usingdifferent methods. In this work, a few algorithmswere implemented and compared.

The simplest way to find the wavelet transformridge bases directly on its definition. In the 1Ddirect maximum algorithm [24,39,40], the ridgeis extracted as the maximum value of the CWT mod-ulus for every translation parameter. This algorithmis simple and fast, and it works accurately on signalswithout noise and with high fringe contrast. Noisegenerates parasite maxima in the map of waveletcoefficients. These points correspond to noise modu-lation instead of signal modulation, resulting in inac-curacies in ridge extraction. Furthermore, low signalcontrast levels create gaps in the wavelet ridge.These are difficult to detect as ridge parts, especiallyin the presence of noise.

The two-frame1Ddirectmaximum algorithm isproposed to avoid this problem. It utilizes two signals:onewith constantmodulation (e.g., the interferogramof a static, nonvibrating object) and one with modula-tion variations (e.g., the interferogram of a vibratingobject). The ridge of the wavelet transformed signalwith constant modulation is found with the directmaximum algorithm. Because it is identical to the

ridge of the CWT of a modulated signal, its coordi-nates are taken to extract themodulated image ridge.

The 2D direct maximum algorithm [24] workslike its 1D analogue. The ridge is extracted as themaximum value of wavelet coefficients for differentscale parameters s and wavelet rotation angles θ. Itshares the same disadvantages as the 1D algorithm,except for the noise resistance. The 2D-CWT filtersnoise much more efficiently than the 1D-CWT.Therefore, the 2D direct maximum algorithm ex-tracts ridge precisely, even in cases of images withsignificantly low signal-to-noise ratio.

The two-frame 2D direct maximum algorithmutilizes one image with a quasi-constant modulationand one with modulation variations. For every pointof the modulation-free pattern, the values of the ro-tation angle θmax and the scale parameter smax thatcorrespond to the maximum wavelet coefficients arefound. The modulated image ridge is created of coef-ficients that correspond to θmax and smax.

Furthermore, the 1D and 2D cost algorithms pro-posed by Liu et al. [34] and Abid et al. [41] were im-plemented and analyzed. Some modifications weremade because of the modulated image contrastlosses: there was no preliminary candidate ridgepoints search; the algorithms checked all waveletcoefficients as possible ridge points.

B. Algorithm Comparison

The precision of the image modulation determi-nation using different implemented ridge extractionalgorithms was compared. Fringe patterns pðx; yÞwith modulation described by function Mðx; yÞ weregenerated. The image modulation map Oðx; yÞ wascomputed using different ridge extraction algorithms.It was then normalized to [0, 1] range. The root-mean-square (RMS) error, Eq. (14), was computed for everyalgorithm and different input image parameters:

RMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

K−5x¼5

PL−5y¼5½Mðx; yÞ −Oðx; yÞ�2

ðK − 10ÞðL − 10Þ

s; ð14Þ

where K × L are the image dimensions (image edgeeffects were not considered).

1. Algorithm Noise Resistance

The fringe pattern pðx; yÞ ¼ cosðωxþ βϕðx; yÞÞ wasgenerated (ω ¼ 0:5, β ¼ 0:5) and modulated withthe functionMðx; yÞ ¼ expf−½ðx − 256Þ2 þ ðy − 256Þ2�=30; 000g. ϕðx; yÞ is a function generated by a MA-TLAB command peaks:

ϕðx; yÞ ¼ 3ð1 − xÞ2e−x2−ðyþ1Þ2

− 10

�x5− x3 − y5

�e−x

2−y2

−13e−ðxþ1Þ2−y2 : ð15Þ

Gaussian noise (GN) was added. In the output im-age given by Iðx; yÞ ¼ pðx; yÞMðx; yÞ þ 1þGN, thestandard deviation of noise σ was changed from


σ ¼ 0:03 to σ ¼ 0:6, with an increment of δσ ¼ 0:03.Results of modulation extraction are presented inFig. 1.

Increasing the noise level causes high RMS errorvalues for modulations determined with 1D algo-rithms. The introduction of noise has much smallerinfluence on modulations extracted with 2D algo-rithms. The 1D algorithm analyzes images on a row-by-row basis; as a result, the modulation is obtainedwithout information from pixels that are out of theanalyzed row.

Cost algorithms, as designed for noisy images, givebetter results than the direct maximum algorithms,in the 1D case and the 2D case. The best results areobtained using two-frame algorithms because theyextract ridges from images without considerablecontrast modulations (the modulation value neverdecreases below the noise level).

2. Carrier Fringe Frequency Influence

The fringe pattern pðx; yÞ ¼ cosðωxÞ was generatedand modulated with the function Mðx; yÞ ¼expf−½ðx − 256Þ2 þ ðy − 256Þ2�=10; 000g. In the outputimage given by Iðx; yÞ ¼ pðx; yÞMðx; yÞ þ 1, the fre-quency of carrier fringes ω was changed from ω ¼0:05 to ω ¼ 1with an increment of δω ¼ 0:05. Resultsof modulation extraction are presented in Fig. 2.

All implemented algorithms give better results forhigh-frequency fringes. Low values of the CWT scaleparameter provide better modulation extraction inthe areas of fast modulation changes because a smal-ler area is averaged by CWT. Therefore, the overallmodulation extraction error is much smaller.

Note that, without noise presence, all 1D algo-rithms give equal RMS values and all 1D algorithmsalso give equal RMS values.

3. Carrier Fringe Inclination Angle Influence

The fringe pattern pðx; yÞ ¼ cos½0:5ðx cos θ þ y sin θÞ�was generated and modulated with the functionMðx; yÞ ¼ expf−½ðx − 256Þ2 þ ðy − 256Þ2�=10; 000g. Inthe output image given by Iðx; yÞ ¼ pðx; yÞMðx; yÞþ1, the carrier fringe orientation angle θ was changedfrom θ ¼ π

40 to θ ¼ π2 with an increment of δθ ¼ π

40. Re-

sults of modulation extraction are presented inFig. 3.

For 1D algorithms, changing the fringe orientationangle causes an effect analogous to that of frequencychanging. Therefore, when demodulating imageswith fringes that change their direction (e.g., circularfringes) it is better to use 2D algorithms because theyinclude an angle parameter, which allows adjust-ment of the wavelet to a local fringe direction.

4. Modulation Function Frequency Influence

The fringe pattern pðx; yÞ ¼ cosð0:5xÞ was generatedand modulated with the function Mðx; yÞ ¼ 0:5cosf½ðx − 256Þ2 þ ðy − 256Þ2�1=2=μg þ 0:5. In the out-put image given by Iðx; yÞ ¼ pðx; yÞMðx; yÞ þ 1, the μparameter that changes modulation frequency wasvaried from μ ¼ 5 to μ ¼ 100 with an increment ofδμ ¼ 5. Results of modulation extraction are pre-sented in Fig. 4.

The RMS modulation extraction error increaseswith increasing frequency of the modulation func-tion. It assumes maximum values when the carrierfrequency approaches the modulation frequency.The analyzing wavelet frequency is similar to thecarrier frequency. The wavelet width covers the areaof modulation changes and averages the modulationvalue. Therefore, computed modulation is incorrect.

Fig. 1. (Color online) Noise resistance algorithm comparison. Fig. 2. (Color online) Carrier frequency influence comparison.

Fig. 3. (Color online) Carrier inclination angle influencecomparison.


5. Carrier Fringe Frequency Gradient Influence

The fringe pattern pðx; yÞ ¼ cos½0:5xþ βϕðx; yÞ�[ϕðx; yÞ according to Eq. (15)] was generated andmodulated with the function Mðx; yÞ ¼ expf−½ðx−256Þ2 þ ðy − 256Þ2�=10; 000g. In the output imagegiven by Iðx; yÞ ¼ pðx; yÞMðx; yÞ þ 1, the phase modu-lation parameter β, responsible for carrier frequencygradient, was changed from β ¼ 0 to β ¼ 2:5 with anincrement of δβ ¼ 0:25. Results of modulation extrac-tion are presented in Fig. 5.

Increasing the parameter β dramatically increasesthe RMS error value for 2D algorithms. This isbecause the range of parameters applied to the2D-CWT was selected to provide a modulation com-puting time comparable with 1D algorithm computa-tions. Broadening the range of parameters (2D DMHQ in Fig. 5) causes longer computation times, butresults in an RMS error comparable with the onein case of 1D algorithms (Fig. 5).

4. Modified Morlet Wavelet Optimization

Modification of the Morlet wavelet consisted of vary-ing the m parameter, which changes the waveletGaussian window width [23,38,42]. The frequencyof the harmonic wave stays unchanged. Modifyingthe m parameter affects modulation determination.Increasing m results in worse noise filtration (the

wavelet is narrower and a shorter part of a signalis averaged). It also causes the appearance of para-sitic carrier fringes (the wavelet imaginary part ismore affected by narrowing the Gaussian windowbecause its maximum and minimum are located onthe sides of the wavelet center). On the other hand,higher values ofm cause better resistance to a carrierfringe deformation (the wavelet is narrower) [23,38].Therefore, the optimization of the m parameter isessential.

Variant carrier frequency pattern was generated[Fig. 6(a)] according to the formula

pðx; yÞ ¼ 0:5 sin�2π15

x1þ x=500

�þ 0:5: ð16Þ

The first line was selected (pðx; 1Þ) and the 1D wave-let was transformed using the 1D modified Morletwavelet with different m parameters. The ridgewas extracted (the 1D direct maximum algorithm)and the signal modulation was determined. Form > 1, parasitic modulations appear. Therefore, the1D modified Morlet parameter m was set to 1 to pro-vide the most accurate fringe filtration.

Analogously, the pattern given by Eq. (16) was2D wavelet transformed using the 2D modifiedMorlet wavelet with different m parameter values.

Fig. 4. (Color online) Modulation function frequency influencecomparison.

Fig. 5. (Color online) Carrier fringe frequency gradient influencecomparison.

Fig. 6. One-dimensional modified Morlet m optimization: (a) 1Dsignal, (b) extracted modulation of the signal for m ¼ 1, and (c) ex-tracted modulation of the signal for m ¼ 2.


Modulation was extracted (the 2D direct maximumridge extraction algorithm). Parasitic fringes wereapparent for m > 1. The Gaussian window widthparameter for the 2D modified Morlet wavelet wasalso set to 1.

Setting the value of parameterm equal to 1 causesviolation of the admissibility condition. Fringe pat-tern modulation extraction does not require CWT in-version; therefore, the violation can be accepted.

5. Wavelet Selection

A. One-Dimensional Wavelets

The 1D Morlet, Paul, and Mexican Hat waveletswere compared. Two criteria were taken into con-sideration: noise resistance and the carrier fringefiltration.

The modulation of the fringe pattern shown inFig. 7(a) was extracted using the 1D-CWT and thedirect maximum ridge extraction algorithm. Differ-ent wavelets and parameters were used. Extractedmodulations are presented in Fig. 7.

The modified Morlet wavelet (m ¼ 1), the Paulwavelet, and the Mexican Hat wavelet filter out thecarrier fringes quite well. Selecting proper Paulwavelet parameter n that minimizes the numberof oscillations in the wavelet (n ¼ 4) results in a goodresistance to fringe deformation—similar to that ob-tained with the Mexican Hat wavelet. The modifiedMorlet wavelet gives slightly inferior results.

Noise resistance was compared using generatedmodulated fringe patterns with increasing noiselevels. The pattern described by

pðx; yÞ ¼ cos½0:5xþ 0:5ϕðx; yÞ� ð17Þwas modulated according to

Mðx; yÞ ¼ ef−½ðx−256Þ2þðy−256Þ2�=30;000g: ð18ÞGaussian noise with standard deviation σ was added.The resulting fringe pattern to be demodulated isdescribed as

Iðx; yÞ ¼ pðx; yÞMðx; yÞ þ 1þGN: ð19ÞNoise standard deviation was increased from σ ¼0:03 to σ ¼ 0:3 with an increment of δσ ¼ 0:03. Forevery noise level, the image modulation was ex-tracted. The RMS error was calculated accordingto Eq. (14). Results are presented in Fig. 8. The bestnoise resistance was obtained using the modifiedMorlet wavelet.

For further investigations, the modified Morletwavelet (m ¼ 1) was chosen as the one providing bestoverall results in modulation extraction.

B. Two-Dimensional Wavelets

The 2Dmodified Morlet wavelet and the Fan waveletwere analyzed. The main practical difference be-tween these two approaches is that the Fan waveletis quasi-isotropic, which simplifies modulationextraction algorithms (no need to change the θ angle

Fig. 7. Interferograms of (a) modulation extraction using different 1D wavelets: (b) Morlet m ¼ 2, (c) Paul n ¼ 4, (d) Mexican Hat, (e)Morlet m ¼ 0:5, and (f) Paul n ¼ 20.


when computing the 2D-CWT) and makes themfaster. Calculated modulation maps obtained usingthe Fan wavelet and the modified Morlet wavelet(m ¼ 1) were compared. Modulation of a sampletwo-beam time-average interferogram was extractedusing the 2D direct maximum algorithm (Fig. 9).Modulation maps obtained using the modified Mor-let wavelet are much smoother and do not containparasitic artifacts. Therefore, the modified Morletwavelet was selected as a tool for 2D-CWT modula-tion extraction.

6. Continuous Wavelet Transform Normalization

The influence of the CWT normalization param-eter η [Eqs. (3) and (8)] on the image modulation ex-traction was analyzed. The most common selection ofη ¼ 0:5 (1D-CWT) and η ¼ 1 (2D-CWT) results in un-equal frequency representation in the CWT coeffi-cient map (higher wavelet coefficients for low signalfrequencies; analyzed signal of constant amplitude).Telfer and Szu [43] and Wang and Ma [23,42] pro-posed applying η ¼ 1 for the 1D-CWT and η ¼ 2 forthe 2D-CWT. This results in equalizing the CWT fre-quency representation, which is crucial in modula-tion determination because the modulus of CWTcoefficients is taken directly as image modulation.Further simulations were carried out to determinethe normalization parameter value that gives thebest results for the modified Morlet wavelet, m ¼ 1.

A. One-Dimensional CWT Normalization

The normalization parameter value for the 1D-CWTwas determined by simulation. The pattern withvariant frequency and constant modulation was gen-

erated according to Eq. (16). The first row of the pat-tern was wavelet transformed using different ηvalues (0.88 to 1 with an increment of 0.01). TheCWT ridge was found using the direct maximum al-gorithm and signal modulation was extracted. Thebest results were obtained for η ¼ 0:98 (Fig. 10), giv-ing almost equal normalization for all frequencies(modulation drops at the signal edges are not con-nected with normalization and are not treated in thispaper).

B. Two-Dimensional CWT Normalization

Analogous simulations were carried for the 2D-CWT.The 2D signal with variant frequency and constantmodulation was generated according to Eq. (16)and wavelet transformed using different η values(1.9 to 2 with an increment of 0.01). The CWT ridgewas found with the 2D direct maximum algorithmand signal modulation was extracted. The first rowof every signal modulation map was selected andanalyzed. The most homogeneous modulation wasobtained for η ¼ 1:98.

7. Experimental Results

A. Time-Averaged Interferograms

The CWT method’s performance was verified usingexperimental data. Two-beam time-averaged inter-ferograms of vibrating and static micromembraneswere used [12]. The interferograms contain circularcarrier fringes of variant frequency and, because ofthat, are difficult to analyze. So far, only the temporalphase-shifting (TPS) method, which requires at leastthree phase-shifted images, has been used to deter-mine their modulation.

In Fig. 11, modulations obtained using differentridge extraction algorithms are compared. One canrealize that the 2D algorithms outperform the 1D al-gorithms in terms of noise filtration. Furthermore,1D algorithms fail to extract modulation in areaswhere the fringe direction is parallel to the selectedrow. In the case of circular carrier fringes, the hori-zontal analyzing profile selection results in verticaldim stripe in the image’s central part [Fig. 11(a)].

In Fig. 12, membrane modulations obtained usingthe 2D two-frame algorithm and modified Morletwavelets with different values of parameter m arepresented. On the one hand, selecting m close tothe value of 2 results in smaller image blur. Onthe other hand, low values ofm provide better carrier

Fig. 8. (Color online) One-dimensional wavelet noise resistancecomparison.

Fig. 9. Interferograms of (a) a nonvibrating and (b) a vibrating circular silicon micromembrane, and modulation extraction for case (b)using different 2D wavelets: (c) Fan wavelet, (d) Morlet wavelet.


fringe filtration. Considering these two effects, thebest results are obtained for m ¼ 1.

The proposed method was compared with the TPSmethod [12–14,44]. Temporal phase-shifting modula-tion maps were calculated using five interferogramswith the phase step π=2 (algorithm 5L [13]). Contin-uous wavelet transform modulation maps were ex-tracted using the 2D two-frame ridge extractionalgorithm, a modified Morlet wavelet (m ¼ 1), andnormalization parameter η ¼ 1:98 (Fig. 13).

Inspection of Fig. 13 reveals that the CWTand TPSmodulation maps are comparable. The main differ-ences are apparent in the membrane center andon its edges. Dim edges are caused by a sudden car-rier disappearance. In spite of Gaussian window nar-rowing, low carrier frequency near the edges enforcesusing a large scale parameter. Therefore, membranemodulation “leaks” to the area around membraneedges. Central errors are caused by very low and zerocarrier frequency value.

Normalized cross sections of the modulation mapsobtained with the TPS and CWT methods are shownin Fig. 14. The CWT method is similar in its perfor-mance to the TPS method. Furthermore, CWT filtersnoise and carrier fringes much better. The CWTmethod behaves slightly worse in areas where inter-ferogram modulation assumes a zero value. In theseareas, the CWT maps often show nonzero modula-tion. This effect is caused by the presence of bothnoise and low modulation value. Wavelet algorithmstake the noise as a signal and generate its modula-tion instead of signals. This effect is difficult toovercome.

Fig. 10. One-dimensional CWT normalization: (a) 1D signal, (b)extracted modulation for η ¼ 0:98, and (c) extracted modulation forη ¼ 0:88.

Fig. 11. Vibrating membrane modulation maps, f rez ¼ 178 kHz, obtained using different ridge extraction algorithms: (a) 1D direct max-imum, (b) 2D direct maximum, and (c) 2D two-frame algorithms.

Fig. 12. Membrane modulationmaps obtained with different values of parameterm, f rez ¼ 833 kHz: (a)m ¼ 0:5, (b)m ¼ 1, and (c)m ¼ 2.


Fig. 13. Time-average interferogram modulation maps for different resonant modes obtained using the (a), (b), (c) CWT and (d), (e), (f)TPS methods.

Fig. 14. Modulation profiles for a membrane shown in Figs. 13(a) and 13(d) obtained using the (a) CWT and (b) TPS methods (crosssection taken along the central membrane diameter).

Fig. 15. Real-time patterns of (a) ΔU=Δx and (b) ΔU=Δy for a loaded cantilever beam. x and y are horizontal and vertical coordinates,respectively; U is the x component of specimen displacements. Lateral shear values: (a) Δx ¼ 1:2 mm and (b) Δy ¼ 0:9 mm. Note thedouble image of a cross marked on the specimen, which defines the shear magnitude and direction [45].


As mentioned earlier, modulation “leakage” nearthe membrane edges causes modulation drops withinthe membrane; see the four main modulation peaksin Fig. 14. Because they correspond to vibrationnodes, they should assume the same, maximummodulation value. The outer peaks assume, however,lower values.

B. Additive Moiré Fringes

Figure 15 shows examples of additive-type moirépatterns obtained by real-time optical shearing ofin-plane displacement fringes in the moiré interfero-metry technique [45] (implementation of so-calledmechanical differentiation or the moiré of the moirémethod). Moiré fringes obtained using our CWT ap-proach are presented in Fig. 16. Note that very satis-factory demodulation results irrespective of mutualorientation between the carrier fringes (in the caseunder consideration, the two-beam interference, hy-perbolic-shape-like fringes) and the moiré bands,except for the area with very wide two-beam interfer-ence fringes, where the carrier indispensable forwavelet transformation almost vanishes.

Note that, because of a nonuniform slight overex-posure of the original patterns (Fig. 15) the profile ofdemodulated moiré fringes (Fig. 16) departs from atheoretical sinusoidal one.

8. Conclusions

A wavelet-transform-based approach to fringe pat-tern modulation determination was proposed. Theperformance of different ridge extraction algorithmsand mother wavelets was compared. The 2D two-frame algorithm with a modified Morlet waveletturned out to give the most accurate results in thecase of noisy and highly modulated patterns. In the1D and 2D cases, modified Morlet Gaussian windowwidth and CWT normalization parameters were op-timized for the purpose of optimum modulation de-termination. The comparison with the TPS methodrevealed that the CWT method can extract imagemodulation with almost the same fidelity and accu-racy. The TPS (multiframe) technique requirementsare avoided in the CTW (single-frame) approach. Alimitation of the latter one is that the spatial fre-quency of modulation bands (moiré fringes) cannotapproach the carrier spatial frequency. Hence, it co-

incides with the common condition for discerningmoiré fringes from the component structures [2].

To the best of our knowledge, we are the first groupto demodulate additive-type moiré and time-averagefringes without resorting to nonlinear recording. Inthis way, the resulting CWT images aremore suitablefor successive application of single-frame automaticfringe pattern processing algorithms for measuranddetermination.

Potential further applications of the CWT additivemoiré demodulation method can be envisaged inexperimental mechanics in the reflection moiré tech-nique (in particular, various Ligtenberg and Rieder–Ritter configurations) for studying static anddynamicflexures of plates and shells [2] and in conventionalphotoelastic interferometry for separating interact-ing isopachic and isochromatic fringes [46–48].

All the computations and simulations presented inthis paper were conducted in a MATLAB environ-ment aided by Yet AnotherWavelet Toolbox (YAWTb)[49].

The authors thank Adam Styk for calculating mod-ulation maps using the TPS method (Figs. 13 and 14)and for valuable advice. The work described in [12]was co-supported by EU Project OCMMM (the partof the experiments under this project was performedat Laboratorie d’Optique P. M. Duffieux, Universitéde Franche—Comté, Besancon, France). This workwas supported, in part, by the Ministry of Scienceand Higher Education, grant N N505 464238, andstatutory activity funds.

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http://rhea.tele.ucl.ac.be/yawtb/





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Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform