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indowed Fourier transform for fringe pattern analysis

ian Kemao

Fringe patterns in optical metrology systems need to be demodulated to yield the desired parameters.Time–frequency analysis is a useful concept for fringe demodulation, and a windowed Fourier transformis chosen for the determination of phase and phase derivative. Two approaches are developed: the firstis based on the concept of filtering the fringe patterns, and the second is based on the best match betweenthe fringe pattern and computer-generated windowed exponential elements. I focus on the extraction ofphase and phase derivatives from either phase-shifted fringe patterns or a single carrier fringe pattern.Principles as well as examples are given to show the effectiveness of the proposed methods. © 2004Optical Society of America

OCIS codes: 100.2650, 100.5070, 300.6300, 100.7410, 120.3940.

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. Introduction

n optical interferometry the output is usually in theorm of a fringe pattern, from which the phase dis-ribution is to be determined. Once the phase dis-ribution is obtained, it can be converted to theesired parameters, such as in-plane and out-of-lane deformation, object shape, and refractive index.he determination of the phase derivative is alsoonsidered in this paper. This step is necessary be-ause, for example, in moire interferometry the phaseerivative is in proportion to the desired strain.A fringe pattern can be generally expressed as

f � x� � a� x� � b� x�cos�� x�� n� x�, (1)

here f �x�, a�x�, and b�x� are the recorded intensity,ackground intensity, and fringe amplitude, respec-ively; ��x� is the desired phase distribution; and n�x�s the noise, which is assumed to be additive. Equa-ion �1� and the following equations in this paper areritten in one dimension for simplicity. However

ur method is two dimensional. The extension fromne dimensional to two dimensional is straightfor-ard. I give two-dimensional examples for demon-

tration. To demodulate the phase information ��x�rom the fringe intensity f �x�, many effective meth-ds have been developed and widely used, such as the

Qian K. �[email protected]� is with the School of Computerngineering, Nanyang Technological University, Singapore39798.Received 29 October 2003; revised manuscript received 27 Jan-

ary 2004; accepted 3 February 2004.0003-6935�04�132695-08$15.00�0© 2004 Optical Society of America

hase-shifting technique, the Fourier transform witharrier technique, and regularized phase tracking.1–3

he phase-shifting technique is a pointwise methodnd is sensitive to noise. The Fourier transformechnique, on the other hand, is a global transformethod and hence is more tolerant to noise. How-

ver, as the transform is global, the signal in oneosition will affect the signal in other positions inpectral analysis. Regularized phase tracking is aocalized processing method, which constructs differ-nt cosine elements to find the optimal match to theringe pattern in a local area. It can be applied tohase-shifted fringes, a single-carrier fringes, and aingle closed fringe. It is insensitive to noise, andpatially separated signals will not affect each other.owever the phase and the phase derivative shoulde optimized simultaneously, and hence an effectiveptimization strategy is needed for an optimal solu-ion. Regularized phase tracking also needs to elim-nate a�x� and normalize b�x� during preprocessing,hich adds the difficulty of fringe demodulation.4–6

As is shown in Eq. �1�, the main feature of theringe is that it is a cosine-modulated function. Ifhe spatial gradient of the phase is large, the fringeensity is high, and if the phase changes slowly, theringe density is low. Hence measuring fringe den-ity is similar to fringe demodulation. This enabless to look at the fringe processing from another pointf view: time–�spatial-�frequency analysis. I adoptwindowed Fourier transform �WFT�, or short-timeourier transform, to analyze the fringe patterns,nd both the phase and its derivative can be ex-racted with good quality. The techniques can bepplied to both phase-shifted and carrier fringe pat-erns. It should be noted that wavelet analysis is

1 May 2004 � Vol. 43, No. 13 � APPLIED OPTICS 2695

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nother tool for time–frequency analysis, and theethod of wavelet ridges has already been applied in

emodulation of carrier fringe patterns.7–10 How-ver, the wavelet ridges technique needs carrier fre-uencies in both horizontal and vertical directions.f the carrier is presented in one direction, only ane-dimensional wavelet transform is practically ap-licable.9 Consequently it is inferior to the proposedwo-dimensional windowed Fourier transform. Irovide a comparison in Subsection 4.B.The paper is arranged as follows. The basic con-

ept of a WFT is introduced in Section 2. Two ap-roaches for fringe demodulation by use of a WFT areescribed in Sections 3 and 4. They are based onltering and matching, respectively. Discussion isresented in Section 5, where the quantitative evalu-tion of the methods and their roles in fringe demod-lation will be highlighted. A real application forefractive-index measurement will be given in Section, and finally conclusions are presented in Section 7.

. Introduction to Windowed Fourier Transforms

et us start from the well-known Fourier transformnd inverse Fourier transform, which are widely usedn signal and image processing.11,12 These can bexpressed as

Ff ��

f � x�exp��j�x�dx, (2)

f � x� �1

2 ��

Ff ��exp� j�x�d�, (3)

here Ff �� is the Fourier transform of f �x�. TheFT and inverse WFT can be written as11,12

Sf �u, ��

f � x� g� x � u�exp��j�x�dx, (4)

f � x� �1

2 ��

��

Sf �u, �� g� x � u�

� exp� j�x�d�du, (5)

here Sf �u, �� denotes the WFT spectrum; g�x� is aindow, which can be chosen as a Gaussian function,

g� x� � exp��x2�2�2�, (6)

here the parameter � controls the extension of g�x�.The two transform pairs are similar except for aindow function g�x�. In a Fourier transform, when

�x� is transformed to Ff ��, the frequency informa-ion is maintained but time information is buriedeeply and can hardly be recognized. In otherords, from Ff ��, we know what frequencies appear

rom the spectrum, but we do not know where theyccur in the signal. In contrast, using WFT, we ob-ain Sf �u, ��, from which we know not only the spec-rum components but also where a componentppears in the time domain.

696 APPLIED OPTICS � Vol. 43, No. 13 � 1 May 2004

Two advantages of a WFT are obvious but need toe emphasized:

1. As a WFT is performed over a local area, de-ermined by the extension of g�x�, a signal in oneosition will not affect the signal in another positionn spectral analysis, provided their distance is largerhan the effective radius of the Gaussian window;

2. The spectrum of the signal in a local area isxpected to be simpler than the spectrum of the wholeeld signal. More effective operation of the spec-rum is hence possible.

he spectrum of the WFT can be processed in twoays:

1. Threshold the spectrum and set the spectralomponents with low amplitude to zero. This ap-roach assumes that the spectrum for noise is widelyistributed with low coefficients. As the noise isliminated, a high-quality signal can be recon-tructed.2. Find the peak of the spectrum. The frequency

orresponding to the peak is considered the local fre-uency or instantaneous frequency. This approachssumes that the spectrum of the signal in a localrea is extremely simple and that it consists of onlyne frequency.

Based on the WFT, two different approaches forringe demodulation are thus possible and are calledhe windowed Fourier filtering �WFF� method andhe windowed Fourier ridges �WFR� method, respec-ively. The first approach filters the fringe patternsn the windowed Fourier domain, and the second ap-roach searches for the best match between theringe pattern and computer-generated windowed ex-onential elements.

. Windowed Fourier Filtering for Phase-Shifted andarrier Fringe Patterns

. Principle of Windowed Fourier Filtering

n the phase-shifting technique, at least three phase-hifted fringe patterns are recorded as

fi� x� � a� x� � b� x�cos�� x� � �i � 1��,

i � 1, 2, 3, . . . . (7)

et us assume � �2 and i 1, 2, 3, 4 and combinehe phase-shifted fringe patterns into a new complexattern,

fpse� x� �12

� f1� x� � f3� x� � jf4� x� � jf2� x��

� b� x�exp� j�� x��, (8)

here j ��1. The objectives of generating a com-lex field are, first, that the background a�x� be sup-ressed and second, that the exponential function belemental in the Fourier transform and windowedourier transform.11

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For the carrier fringe method, a carrier frequencys introduced into the fringe pattern, which can beritten as

fc� x� � a� x� � b� x�cos��c x � �� x��, (9)

here �c is the carrier frequency. This can be re-ritten as

fc� x� � a� x� �12

b� x�exp� j�c x � j�� x��

�12

b� x�exp��j�c x � j�� x��

� a� x� � fce� x� � fce*� x�, (10)

here * denotes the complex conjugate. The fringeattern is composed of two exponential functions andbackground intensity function. Since the carrier

requency is introduced, the three spectra are sepa-ated in the frequency domain.

The principle of WFF can be derived by combiningqs. �4� and �5� to yield

f � x� �1

2 ��

�� f � x� � h� x, �� h� x, ��d�, (11)

here h�x, �� g�x�exp� j�x� and R denotes a convo-ution, which is implemented with respect to the vari-ble x. Implementation of WFF uses a modified Eq.11� written as

f� � x� �1

2 �a

b

��f � x� � h� x, �� h� x, ��d�, (12)

n which two modifications are made:

1. f �x� R h�x, �� suggests that f �x� R h�x, �� behresholded. If � f �x� R h�x, �� is smaller than aertain threshold, it is treated as noise and is re-oved by being set to zero. The threshold should be

elected interactively.2. The integration limits are set from a to b, in-

tead of from � to . Hence only the desiredpectrum in a limited range is selected. For thehase-shifting technique, typically one chooses a � 0nd b � 0, since f �x� is composed of negative as well
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s positive frequencies. For the carrier technique,ypically one chooses b � a � 0. Hence the spectrumf signal with negative frequencies and the spectrumf background, as well as some noise, are automati-ally rejected. The values of a and b can be readilystimated from the fringe pattern.

After f� �x� is synthesized, the phase can be deter-ined as

�� x� � angle� f� � x��. (13)

oth the phase-shifting and carrier fringe techniquesre dealt with in a similar fashion. In the carrierringe technique, we could also chose a � b � 0, tobtain ��x�.Once the phase is obtained, the phase derivative

an also be computed by numerical differentiation:

��u� � ��u � 1� � ��u�. (14)

. Experimental Verification

simulated fringe pattern13 is used to show the ef-ectiveness of WFF for both phase-shifting techniquend carrier fringe technique. The simulated fringeattern has known ideal phase distribution, henceomparison of the extracted phase and the originalhase is easier. The simulated phase distribution0�x, y� is14

0� x, y� � � 0.002 � �2 �2 � x2 � y2 � 1272

0.002 � 1272 �2 � x2 � y2 � 1272 (15)

nd is shown in Fig. 1�a� for an image size of 256 �56 pixels. Figure 1�b� shows the wrapped phase.he derivative of phase in the horizontal direction ishown in Fig. 1�c�. The phase and phase derivativen Fig. 1 as well as in the following figures are linearlycaled into the range �0, 255�. Phase-shifted fringesre generated by using

fi� x� � 1 � cos��0� x� � �i � 1��2� � ni� x, y�,

i � 1, 2, 3, 4, (16)

nd are shown in Figs. 2�a�–�d�. The added noise isaussian. The phase obtained by directly taking

he angle of Eq. �8� is shown in Fig. 2�e�. This is theonventional phase-shifting technique.15 The result

Fig. 1. Simulated data: �a� phase, �b� wrapped phase, �c� phase derivative in horizontal.


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s noisy. The phase obtained by the proposedethod is shown in Fig. 2�f �. The noise is effectively

emoved, and the improvement is obvious. Figure�g� is the phase derivative in the horizontal directionbtained by numerical differentiation of the phase inig. 2�f �. In this and all of the following examples

he value of � is 4 pixels.Next a carrier fringe pattern, shown in Fig. 3�a�, is

imulated according to the following equation,

fc� x, y� � 1 � cos�x�2 � �0� x, y�� n� x, y�. (17)

ith the traditional Fourier transform method16 thehase can be determined as shown in Fig. 3�b�.ome ringing is noticeable. With the proposedethod improved results are obtained, as shown inig. 3�c�. The corresponding phase derivative in the

ig. 2. WFF with phase-shifting technique: �a�–�d� Phase-shiftef � phase by the proposed WFF, �g� phase derivative from �f � by n

ig. 3. WFF with carrier technique: �a� carrier fringe pattern,roposed WFF, �d� phase derivative from �c� by numerical differen


orizontal is shown in Fig. 3�d�. Both examples ver-fy the effectiveness of WFF.

. Windowed Fourier Ridges for Phase-Shifted andarrier Fringe Patterns

. Principle of Windowed Fourier Ridges

ext the second approach, WFR, is applied to phase-hifted fringe patterns. Using a Fourier transform,ne obtains the spectrum of a signal. The dominat-ng frequency of the signal corresponds to the peak ofhe amplitude spectrum. Similarly, using a win-owed Fourier transform, we can determine the dom-nating frequencies of the fringe pattern in any localrea. These dominating frequencies are instanta-eous frequencies of the fringe pattern, which is veryseful in optical interferometry.

ge patterns, �e� phase by directly using phase-shifting algorithm,ical differentiation.

hase by traditional Fourier transform method, �c� phase by theon.

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Inserting Eq. �8� into Eq. �4�, one has

Sfpse�u, ��

fpse� x� g� x � u�exp��j�x�dx

� ��

b� x� g� x � u�exp� j�� x� � j�x�dx.

(18)

et us assume that in the local area around u �1� theringe modulation is the same, i.e., b�x� b�u� andhat �2� the phase is linear, i.e., ��x� ��u� ��u��x � u�. Thus it can be shown that

Sfpse�u, �� b�u�exp� j��u� � �u��

g�t�exp��j��

� ��u�� t�dt. (19)

wo tricks are used in getting Eq. �19�: �1� theaussian function is symmetrical, i.e., g�x� g��x�;

2� A substitution of t x � u is used. Designating�� as the Fourier transform of g�t�, one has

Sfpse�u, �� b�u�exp� j��u� � �u��G�� u��.(20)

ecause �G�0�� is the peak of �G��, for a specific u,Sfe�u, �� reaches the maximum when � ��u�.hese maxima are termed WFR, implying that reso-ance occurs when a maximum is reached. Hence��u� can be characterized as

��u� � arg max�

�Sfpse�u, ��, (21)

hich means that ��u� takes the value of argumentwhen �Sfpse�u, �� reaches the maximum.It can be shown that

fpse� x� � h� x, �� exp� j�u�Sfpse�u, ��

� b�u�exp� j��u��G�� u��;(22)

ig. 4. WFR with phase-shifting technique: �a� phase derivativeor comparison.

ence, instead of Sfpse�u, ��, we only need to compute�x� R h�x, ��, which is simply a convolution. Equa-ion �21� can be rewritten as

��u� � arg max�

� fpse�u� � h�u, ��. (23)

Applying the WFR to the carrier fringe technique issimple extension. When a resonance of fce�x� oc-

urs for a certain �, the contribution from a�x� andce*�x� can be ignored, as they are distinct from fce�x�n the windowed Fourier domain. In other words, aesonance of fce�x� is same as a resonance of fc�x�.ence the phase derivative can be determined as

��u� � �c � arg max�

�Sfc�u, �� (24)

r

��u� � �c � arg max�

� fc�u� � h�u, ��. (25)

ore details can be found in Ref. 11. Note that inhe carrier technique fce*�x� can also generate a res-nance. If the resonance of fc�x� in Eq. �22� is con-ributed by fce*�x�, then the obtained phaseerivative is simply ��u�.From Eq. �22�, the phase distribution can be di-

ectly determined from the ridge,

��u� � angle� fpse�u� � h�u, ��u��. (26)

his is because G�0� is a real number �the mean valuef the Gaussian window�. I call this the phase fromidges. Another approach to determine the phase dis-ribution is simply integrating the phase derivative,

��u� � ��u0� � �u0

u

��t�dt (27)

r, numerically,

��u� � ��u � 1� � ��u � 1�. (28)

term this the phase by integration.

. Experimental Verification

first apply WFR to phase-shifting fringe patternshown in Figs. 2�a�–�d�. Phase derivatives in both

phase from ridges, �c� phase by integration; the result is wrapped
, �b�

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orizontal and vertical directions are obtained. Theorizontal phase derivative is shown in Fig. 4�a�.he phase from ridges �Eq. �26�� and phase by inte-ration �Eq. �28�� are computed and shown in Figs.�b� and 4�c�, respectively. In phase by integration,he phase of the central row is obtained first, fromhich the phase of all the columns is integrated.he integration path is shown in Fig. 5. The phase

s continuous, and no phase unwrapping is necessary.owever, the wrapped phase is shown in Fig. 4�c� for

omparison only.Next I apply WFR to the carrier fringe pattern

hown in Fig. 3�a�. The phase derivative, phaserom ridges, and phase by integration are shown in

ig. 5. Path for phase integration: �a� phase of central row isrst integrated from the central point; �b� phase of all columns ishen integrated from the central row.

ig. 6. WFR with carrier technique: �a� phase derivative, �b� pomparison. �d� Phase derivative by one-dimensional wavelet rid

Table 1. Accuraci

Fringe Type Demodulation Method

Phase-shifted Phase-shifting algorithmWFFWFR

Carrier Fourier transformWFFWFR

aData in brackets are obtained from wrapped phase.bThe first number is the phase from ridges; the second number


ig. 6�a�, 6�b�, and 6�c�, respectively. The results areimilar to those of Fig. 4. This is not unexpected inrinciple. I also apply the one-dimensional waveletidges to the same carrier fringe pattern. The ob-ained phase derivative is shown in Fig. 6�d�, which istill noisy. It can be seen that WFR is superior tohe method of wavelet ridges. From Figs. 4 and 6,he fringe demodulation by the proposed WFR is suc-essful.

. Accuracies of Windowed Fourier Transforms andidges and Their Role in Fringe Demodulation

n the previous demonstration the effectiveness of theethods is shown by observing the results. Now I

ive an objective evaluation. I compute the mean ofhe absolute phase errors and the mean of the abso-ute phase derivative errors to quantitatively evalu-te and compare different methods. These meansre defined as follows:

�� mean�� x� � �0� x��, (29)

�� mean �� x� � �0�� x�� , (30)

here �0�x� and �0��u� are ideal values of the phasend the phase derivative, respectively, while ��x� and��u� are obtained by various methods introduced inhis paper. The results are shown in Table 1, fromhich it can be seen that both WFF and WFR can

etrieve the phase and phase derivatives accurately.owever, using phase by integration, one avoids thehase-unwrapping process at the cost of lower accu-

from ridges, �c� phase by integration; the result is wrapped for

the WFF and WFR

Phase Error �rad�Phase Difference

Error �rad�

�0.65�a �0.93�a

0.067 0.0170.082,0.30b 0.0190.22 0.0680.11 0.0260.11, 0.20b 0.025

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acy because the phase error is accumulated whenhase is integrated.In this simulation the parameter �, which indi-

ates the size of the Gaussian window, is 4 pixels.hen � was increased to 10 pixels, similar resultsere obtained, except that the phase error obtainedy use of phase by ridges in WFR obviously increasedo 0.25 rad for both phase-shifted and carrier fringes.his is because the assumption of linear phase withinbigger window is less reasonable. For this reason,hase can be approximated to be quadric instead ofinear, i.e., ��x� ��u� � ��u��x � u� � ��u��x ��2�2. Following the derivation in Refs. 17 and 18, Ibtain a phase correction term due to ��u� as ��1��arctan��2��u��, which should be added to the phaseistribution extracted by using the previous linearodel. The value of ��u� can be estimated from��u� that has already been accurately obtained byse of the linear phase model. In the two-

Fig. 7. Roles of WFF and WFR in fringe demodulation.

Fig. 8. Real application: �a� fringe pattern, �b� phase ma

imensional case, if one ignores the influence ofuv�u, v�, the correction term can be derived as

�12

arctan��2�uu�u, v�� 12

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here

�uu�u, v� ��2�

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�vv�u, v� ��2�

�v2 ,

�uv�u, v� ��2�

�u�v.

ith this correction, the phase error for WFR in thisimulation was reduced to 0.12 rad.From the above description, I have demonstrated

ersatile applications of the WFT for fringe demodu-ation. Both the phase and the phase derivative cane obtained. Their roles and relationship are illus-rated in Fig. 7. As can be seen from the abovexamples, both WFF and WFR are highly effective inignal localization and noise suppression, which mayncourage us to use the WFT as a routine tool inodern fringe analysis.

. Real Application for Refractive-Index Measurement

o keep this paper concise, only one demonstration ofreal fringe pattern is provided. Figure 8�a� showsfringe pattern obtained with a Mach–Zehnder in-

erferometer for measuring the refractive index of aolution, into which a potassium chloride �KCl� crys-al is dissolved.19 The measurement was taken tollow the influence of the solution concentration onhe growth of protein crystals to be investigated later.art of the KCl crystal can be seen in the lower partf Fig. 8�a�. This crystal should not enter into thealculation, which would be unavoidable with the tra-itional Fourier transform method. Since the WFFs a local technique, the influence of the crystal is alsoocalized and does not influence the information atther areas. After filtering, the crystal shape iseadily recognizable, as shown in Fig. 8�b�. It can be

ained using the proposed WFF, �c� unwrapped phase map.
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een that WFF not only provides a high-quality phaseap but also separates the crystal from the solution.s a result of these advantages, the phase can benwrapped easily by a simple line-scanning ap-roach1; the results are shown in Fig. 8�c�.

. Conclusions

emodulation of fringe patterns is necessary forany optical metrological systems. I provide sys-

ematic solutions by using time-frequency analysis.windowed Fourier transform �WFF� is chosen for

he determination of the phase and the phase deriv-tive. Two approaches are developed: One isased on the concept of filtering the fringe pattern,nd the second is based on the best match betweenhe fringe pattern and computer-generated exponen-ial elements. Elegant principles are presented, andruitful results are obtained, which will encouragengineers to use the WFF as a routine tool for modernringe pattern analysis.

I thank Seah Hock Soon for his encouragement inhis research, Anand Asundi for many useful discus-ions, and the reviewers for their great insight, com-ents, and suggestions.

eferences and Notes1. D. W. Robinson and G. T. Reid �eds.�, Interferogram Analysis:

Digital Fringe Pattern Measurement Techniques �Institute ofPhysics, Bristol, England, 1993�.

2. M. Servin and M. Kujawinska, “Modern fringe analysis ininterferometry,” in Handbook of Optical Engineering, D. Ma-lacara and B. J. Thompson, eds. �Marcel Dekker, New York,2001�, pp. 373–426.

3. J. M. Huntley, “Automated fringe pattern analysis in experi-mental mechanics: a review,” J. Strain Anal. 33, 105–125�1998�.

4. M. Servin, J. L. Marroguin, and F. J. Cuevas, “Demodulationof a single interferogram by use of a two-dimensional regular-ized phase-tracking technique,” Appl. Opt. 36, 4540–4548�1997�.

5. M. Servin, R. Rodriguez-Vera, J. L. Marraquin, and D. Ma-lacara, “Phase-shifting interferometry using a two-


dimensional regularized phase-tracking technique,” J. Mod.Opt. 45, 1809–1819 �1998�.

6. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R.Rodriguez-Vera, “Phase unwrapping through demodulation byuse of the regularized phase-tracking technique,” Appl. Opt.38, 1934–1941 �1999�.

7. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination ofinterferometer phase distributions by use of wavelets,” Opt.Lett. 24, 905–907 �1999�.

8. K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu,“Strain analysis for moire interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43,45–51 �2003�.

9. J. Fang, C. Y. Xiong, H. J. Li, and Z. L. Zhang, “Digital trans-form processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 �2001�.

0. K. Qian, H. S. Seah, and A. Asundi, “Instantaneous frequencyand its application in strain extraction in moire interferome-try,” Appl. Opt. 42, 6504–6513 �2003�.

1. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. �Aca-demic, San Diego, 1999�.

2. K. Krochenig, Foundations of Time–Frequency Analysis�Birkhauser, Boston, 2001�.

3. MATLAB codes for both WFF and WFR are available upon re-quest.

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