fringe pattern demodulation with a two-frame digital phase-locked loop algorithm

Fringe pattern demodulation with a two-framedigital phase-locked loop algorithm

Munther A. Gdeisat, David R. Burton, and Michael J. Lalor

A novel technique called a two-frame digital phase-locked loop for fringe pattern demodulation is pre-sented. In this scheme, two fringe patterns with different spatial carrier frequencies are grabbed for anobject. A digital phase-locked loop algorithm tracks and demodulates the phase difference between bothfringe patterns by employing the wrapped phase components of one of the fringe patterns as a referenceto demodulate the second fringe pattern. The desired phase information can be extracted from thedemodulated phase difference. We tested the algorithm experimentally using real fringe patterns. Thetechnique is shown to be suitable for noncontact measurement of objects with rapid surface variations,and it outperforms the Fourier fringe analysis technique in this aspect. Phase maps produced with thisalgorithm are noisy in comparison with phase maps generated with the Fourier fringe analysis technique.© 2002 Optical Society of America

OCIS codes: 120.2650, 100.5070, 100.0100.

1. Introduction

Fringe pattern demodulation has many industrialand medical applications.1,2 There are many meth-ods that can be used to demodulate fringe patternssuch as Fourier fringe analysis,3 phase stepping,4 anddirect phase detection.5 Probably the most com-monly used methods are the first two. These meth-ods use an arctangent function to calculate the phasecomponents of a fringe pattern. The resultant phasemap may contain 2� discontinuities; consequently,an unwrapping algorithm is required to remove the2� steps. The unwrapping technique can be themost time-consuming and difficult step in the abovealgorithms.

Servin and Rodriguez-Vera have proposed use of afirst-order conventional digital phase-locked loop�DPLL� for fringe pattern demodulation.6 Phasemaps produced by this algorithm do not contain 2�discontinuities; consequently, the need for an un-wrapping algorithm is eliminated. This method is

The authors are with the Coherent and Electro-Optics ResearchGroup, School of Engineering, Liverpool John Moores University,James Parsons Building, Room 114, Byrom Street, Liverpool L33AF, United Kingdom. The e-mail address for M. A. Gdeisat [email protected].

Received 28 November 2001; revised manuscript received 23May 2002.

0003-6935�02�265471-08$15.00�0© 2002 Optical Society of America

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not capable of demodulating fringe patterns withrapid phase variations. Also, phase maps producedby this algorithm are corrupted by high-frequencydisturbances generated by the multiplier phase de-tector �PD� of the conventional DPLL.

A number of attempts have been made to improvethe tracking ability of DPLLs to enable them to de-modulate fringe patterns with rapid phase varia-tions. Other attempts have also been made toproduce phase maps without high-frequency distur-bances added by the DPLL itself.

In an earlier paper, we proposed use of a second-order conventional DPLL for fringe pattern demodu-lation, and we implemented the algorithm in realtime.7 This algorithm has both better tracking abil-ity and noise performance than the first-order con-ventional DPLL, but phase maps produced by thisalgorithm still suffer from high-frequency distur-bances added by the DPLL’s multiplier PD.

Kozlowski and Serra have modified the first-orderconventional DPLL to improve its tracking ability.8The PD of the modified DPLL does not generate high-frequency disturbances; consequently, phase mapsproduced by this algorithm have a better signal-to-noise ratio than phase maps produced with the first-order conventional DPLL. However, the noiseperformance of this technique is worse than the first-order conventional DPLL.9

The linear DPLL has better tracking ability thanthe conventional DPLL or its modified version.10

This algorithm has been used for fringe pattern de-

0 September 2002 � Vol. 41, No. 26 � APPLIED OPTICS 5471

modulation, and it has been implemented in realtime.11 Phase maps produced by this algorithm canbe considered better than phase maps generated withthe above-mentioned algorithms in terms of accuracyand signal-to-noise ratio.

Even after all these attempts to improve the track-ing features of the above-stated basic DPLL algo-rithms �e.g., conventional, modified, and linear�, theyare still unable to demodulate fringe patterns withrapid phase variations. The Fourier fringe analysisalgorithm outperforms the basic DPLL algorithms inthis aspect.12

In this paper we present a novel technique called atwo-frame DPLL. In this algorithm, two fringe pat-terns with different spatial carrier frequencies aregrabbed for an object, and the algorithm demodulatesthe phase difference between the two fringe patterns.The required phase information can be extractedfrom this phase difference. This method is differentfrom the above-mentioned basic DPLL algorithms,but it can be used in conjunction with any one ofthem. The two-frame DPLL algorithm can be usedfor noncontact measurement of an object with rapidsurface variations, and it outperforms the Fourierfringe analysis algorithm in this respect.

Use of two fringe patterns with different spatialcarrier frequencies grabbed for an object also hasbeen used in conjunction with the Fourier fringeanalysis technique.13 This use of two fringe pat-terns with different spatial carrier frequencies canproduce a height distribution over a larger rangethan that obtained by use of a single fringe pattern,and it reduces the number of wraps in the image.14

In Section 2 we present an overview of the opera-tion of a DPLL. In Section 3 we present a twin-fiberinterferometer, which is used to produce fringe pat-terns with different spatial carrier frequencies. Thetheoretical background of the two-frame DPLL algo-rithm and its use for fringe pattern demodulation areexplained in Section 4. In Section 5 our results ofapplying the proposed technique to demodulate realfringe patterns are presented and discussed.

2. Digital Phase-Locked Loop

A block diagram of a DPLL is shown in Fig. 1 andconsists of the following basic elements:

• a phase detector,

• a digital filter �DF�, and• a digital controlled oscillator �DCO�.

The PD is a device whose output level depends on thephase difference between its two inputs. The PDcompares the phase of its input signal against thephase of the DCO output; the output of the PD is ameasure of the phase difference between these twosignals. The DF is an infinite impulse response fil-ter that filters the output of the PD. A proportionalpath DF �zero-order DF� yields a first-order DPLL,whereas a first-order infinite impulse response DF,normally a proportional plus accumulator paths,gives a second-order DPLL.7 The DCO is an oscil-lator whose output frequency depends directly on itsinput level. When the control signal applied to theDCO is zero, its output signal will have a constantfrequency that is called the free-running frequency ofthe DCO � fr�. The output of the DF graduallychanges the frequency of the DCO in a direction thatreduces the phase difference between the input signaland the DCO output.

The DPLL can be used as a phase demodulator.Suppose that a phase-modulated signal is applied tothe input of the DPLL and the DPLL locks and tracksits input. Also, consider that the carrier frequencyof the modulated signal is equal to the free-runningfrequency of the DCO. The instantaneous frequencyof the phase-modulated signal depends on the deriv-ative of the modulating signal. When the loop is inthe tracking state, it is necessary that the frequencyof the DCO be close to the instantaneous frequency ofthe input signal, and the frequency of the DCO isproportional to the control signal. Consequently,the control signal should be a close replica of thederivative of the modulating signal. The modulat-ing signal could be recovered by integration of thecontrol signal.

A. Conventional Digital Phase-Locked Loop

A block diagram of a conventional DPLL is shown inFig. 2. The PD of the conventional DPLL is a mul-tiplier. If a phase-modulated signal is applied to theconventional DPLL, a closed replica of the modulat-ing signal can be extracted from the output of theDCO accumulator, which accumulates the controlsignal as shown in Fig. 2.7

B. Linear Digital Phase-Locked Loop

A block diagram of a linear DPLL is shown in Fig.3.10,11 The operation of the linear DPLL can be sum-marized as follows. The input of the linear DPLLc�x� is split, and one is phase shifted by ��2 by use ofa Hilbert transform.13 The phase components of theinput can be calculated when we substitute the inputc�x� and its ��2 phase-shifted version q�x� into theequation11

�� x� � tan�1�q� x�

c� x�� , (1)

Fig. 1. Block diagram of a DPLL.

5472 APPLIED OPTICS � Vol. 41, No. 26 � 10 September 2002

where x is the sample index. The phase componentsof the input ��x� contain 2� steps because we used thearctangent function. The subtractor PD and themodular circuit of the linear DPLL determine thephase difference between the wrapped phase input��x� and the DCO output and remove the 2� stepsfrom this phase difference. The output of the mod-ular circuit is applied to the DF whose output controlsthe phase of the DCO so as to decrease the phasedifference between the input and the DCO output.The linear DPLL unwraps and demodulates thewrapped phase ��x� simultaneously. If the input tothe linear DPLL is a phase-modulated signal, thedemodulated signal can be extracted from the outputof the DCO accumulator as shown in Fig. 3.11

3. Twin-Fiber Interferometer

A block diagram of a twin-fiber interferometer isshown in Fig. 4. The operation of the twin-fiber in-terferometer can be explained as follows. A laserbeam is focused down into an optical fiber. The laserbeam propagates through the optical fiber until itreaches an optical coupler, which splits the laserbeam into two mutually coherent beams. The twolaser beams propagate through two optical fibers;hence the name of twin-fiber interferometer. Thelaser beams are emitted from the ends of the two

fibers and projected onto an object’s surface. Thetwo laser beams interfere, and their interference pro-duces fringes.1

One of the twin optical fibers can be moved in slightsteps by a computer-controlled mechanism. Atranslation of the movable optical fiber as shown inFig. 5 varies the spacing between fringes �P0�. Anexample of a fringe pattern produced by the twin-fiber interferometer is shown in Fig. 8�a�. One of theoptical fibers is moved far away from the other, so thespacing between fringes is decreased. The resultantfringe pattern is shown in Fig. 8�b�. This methodcan be used to produce fringe patterns with differentspatial carrier frequencies.1

4. Demodulation Algorithm

During the demodulation of a real fringe pattern witha DPLL algorithm, the DCO of the DPLL produces acomputer-generated fringe pattern for a planar sur-face. The computer-generated fringe pattern is pro-duced by the term �2�frx� as shown in Figs. 2 and 3.The real and the computer-generated fringe patternsare applied to the DPLL’s PD. The DPLL tracks thephase difference between both fringe patterns, whichis the required phase information.

Consider that an object with rapid surface varia-tions phase modulates a fringe pattern. The real

Fig. 2. Block diagram of a conventional DPLL.


phase-modulated fringe pattern contains rapid phasevariations. If the real fringe pattern is demodulatedwith a DPLL technique, the DCO of the DPLL gen-erates a computer-generated fringe pattern; theDPLL tracks the phase difference between bothfringe patterns. The phase difference contains rapidphase variations; consequently, the DPLL will not becapable of tracking these rapid phase variations anddemodulating the fringe pattern.

Suppose that two real fringe patterns with differ-ent spatial carrier frequencies are generated for anobject as shown in Figs. 8�a� and 8�b�. Thecomputer-generated fringe pattern can be replacedby one of the real fringe patterns, and the second onecan be applied to the DPLL’s input. The DPLLtracks the phase difference between both real fringepatterns, and the required phase information can beextracted from this difference as we explain below.

Consider that two fringe patterns are modulated byan object with rapid surface variations. Also, con-sider that one of the real fringe patterns is used in-stead of the computer-generated fringe pattern. Ifthe difference between the spatial carrier frequenciesof both real fringe patterns is small, the phase differ-ence between both real fringe patterns will also besmall, and the DPLL can track this phase differenceeasily.

Suppose that an object with a height distribution

Fig. 3. Block diagram of a linear DPLL.

Fig. 4. Twin-fiber interferometer.Fig. 5. Moving one of the optical fibers in the twin-fiber inter-ferometer.


h�x, y� is used to phase modulate two fringe patternswith different spatial carrier frequencies. Thefringe intensity in both phase-modulated fringe pat-terns �derived in Appendix A� can be described as

g1� x, y� � a1� x, y� � b1� x, y�cos�2�f1 x

� 2�f1 h� x, y�sin ��, (2)

g2� x, y� � a2� x, y� � b2� x, y�cos�2�f2 x

� 2�f2 h� x, y�sin ��, (3)

where a1�x, y� and a2�x, y� are the background illu-minations of both fringe patterns, b1�x, y� and b2�x, y�are the amplitude modulation of the fringes, f1 and f2are the spatial carrier frequencies, � is the projectionangle, and x and y are the indices of the x and ycoordinates. The background illuminations of bothfringe patterns must be removed prior to their appli-cation in the demodulation process. Servin andRodriguez-Vera suggested differentiating a fringepattern with respect to the sample index to remove itsbackground illumination.6

Suppose that the background illuminations of bothfringe patterns are filtered out. The intensity of thefirst filtered fringe pattern can be expressed as

c1� x, y� � b1� x, y�cos�2�f1 x � 2�f1 h� x, y�sin ��.(4)

The phase components of the first fringe pattern canbe calculated as follows. The filtered fringe patternis phase shifted by ��2 with a Hilbert transform.13

The Hilbert transform can be implemented with afinite impulse response �FIR� DF �a FIR Hilberttransformer� whose coefficients are given in Table1.13 The fringe pattern image should be applied tothe Hilbert transformer row by row. The phase-shifted fringe pattern can be given by

q1� x, y� � b1� x, y�sin�2�f1 x � 2�f1 h� x, y�sin ��.(5)

The filtered fringe pattern c1�x, y� and its ��2 phase-shifted version q1�x, y� are substituted in Eq. �6� tocalculate the phase components of the first fringepattern:

�� x, y� � tan�1�q1� x, y�

c1� x, y�� tan�1

� �b1� x, y�sin�2�f1 x � 2�f1 h� x, y�sin ��

b1� x, y�cos�2�f1 x � 2�f1 h� x, y�sin �� 2�f1 x � 2�f1 h� x, y�sin ��mod �, (6)

where mod represents the modular function �i.e., rmod � expresses the remainder when r is divided by��. The resultant phase map is wrapped because ofthe arctangent function. The computer-generatedfringe pattern can be replaced by the wrapped phasemap of the first fringe pattern, which can be used asa reference to demodulate the second fringe pattern.The DPLL tracks and demodulates the phase differ-ence between the wrapped phase map and the secondfringe pattern; and this phase difference can be ex-pressed as

�� x, y� � 2�x� f1 � f2� � 2�h� x, y�� f1 � f2�sin �.(7)

The demodulated phase map contains a tilt, whichdepends on the difference between the two spatialcarrier frequencies. This tilt should be removed toextract the required phase information. The rela-tionship between the demodulated phase map �afterremoval of the tilt� and the height distribution of theobject h�x, y� can be described by

� x, y� � 2�h� x, y�� f1 � f2�sin �. (8)

Equation �8� implies that the amplitude of the de-modulated phase map produced with this algorithmis smaller than the amplitude of the demodulatedphase map generated with the technique that uses acomputer-generated fringe pattern. The ratio be-tween both amplitudes can be given by Eq. �9�. Also,the amplitude of the demodulated phase map that isproduced with the two-frame DPLL algorithm in-creases as the frequency difference increases.

�� f1 � f2�

f1, (9)

where is an attenuation factor and � is the absolutevalue operator. This attenuation makes the two-frame DPLL algorithm capable of demodulatingfringe patterns with rapid phase variations.

The signal-to-noise ratio characteristics of phasemaps produced with the two-frame DPLL techniqueare mainly poor in comparison with phase maps gen-erated with the algorithm that utilizes a computer-generated fringe pattern. This can be explained asfollows. The computer-generated fringe patterndoes not contain noise, but the reference real fringepattern is corrupted by noise. Also, phase maps pro-duced by the two-frame DPLL algorithm are attenu-

Table 1. Coefficients of a FIR Hilbert Transformer

Coefficient Value

b0 �0.0529897b1 0.0b2 �0.0882059b3 0.0b4 �0.1868274b5 0.0b6 �0.6278288b7 0.0b8 0.6278288b9 0.0b10 0.1868274b11 0.0b12 0.0882059b13 0.0b14 0.0529897


ated by the factor , whereas the noise level and thehigh-frequency components produced by the DPLL’sPD �in the conventional DPLL case� are not attenu-ated. Both reasons make the two-frame DPLL algo-rithm unsuitable to demodulate fringe patterns withnarrow bandwidths or slow phase variations.

Even though the two-frame DPLL algorithm issuitable for noncontact measurement of objects withrapid surface variations, it is not capable of measur-ing noncontinuous objects that produce abrupt phasechanges in the modulated fringe patterns. Thesephase discontinuities may disturb the tracking pro-cess of a DPLL and make it lose tracking; conse-quently, the demodulated phase maps are distortedby the transient response of the DPLL.

5. Experimental Results

Figures 6�a� and 6�b� show two fringe patterns withdifferent spatial carrier frequencies that were cap-tured for an object. Phase variations in both fringepatterns can be considered large. We differentiatedand smoothed both fringe patterns using 3 � 3 and1 � 5 averaging windows prior to the demodulationprocess. The 1 � 5 averaging window consists of onepixel in the x direction and five pixels in the y direc-tion. The filtered version of the fringe pattern

shown in Fig. 6�a� was phase shifted by ��2 with a15-element FIR Hilbert transformer whose coeffi-cients are given in Table 1. The smoothed and dif-ferentiated fringe pattern and its ��2 phase-shiftedversion were substituted in Eq. �6�. The resultingwrapped phase map is shown in Fig. 6�c�. Thesecond-order conventional DPLL was used in the de-modulation process. The wrapped phase map wasused as a reference to demodulate the fringe patternshown in Fig. 6�b�, and the demodulated phase map isshown in Fig. 6�d�. The demodulated phase mapcontains a tilt that should be removed. The phasemap resulting after removal of the tilt is shown inFig. 6�e�.

The above procedures were repeated, but thesecond-order linear DPLL was used instead of theconventional DPLL. The demodulated phase mapsare shown in Figs. 7�a� and 7�b�.

The Fourier fringe analysis technique12 was usedto demodulate the fringe pattern shown in Fig. 6�b�.The demodulated phase map is shown in Fig. 7�c�. Itcan be concluded from Fig. 7�c� that the Fourier fringeanalysis algorithm that employs Schafer and Oppen-heim’s unwrapper15 failed to analyze this fringe pat-tern in the regions that have rapid phase variations.

As mentioned above, the background illuminationof a fringe pattern must be removed prior to the de-modulation of it by use of a DPLL algorithm. Thefringe pattern can be differentiated to remove itsbackground illumination. The differentiation pro-cess amplifies the high-frequency noise in the fringepattern �e.g., speckle noise�, and it is not suitable forfringe patterns with high sampling rates �e.g., largerthan 20 pixels�fringe� or for noisy fringe patterns.

Lilley et al. have suggested another method to re-move the background illumination of a fringe pat-

Fig. 6. �a� and �b� Two fringe patterns generated for the sameobject with different spatial carrier frequencies. �c� The wrappedphase map of the fringe pattern in �a�. �d� The phase differencebetween both fringe patterns demodulated with the conventionalDPLL and �e� the demodulated phase map after tilt removal.

Fig. 7. �a� Phase difference between the fringe patterns, shown inFigs. 6�a� and 6�b�, detected by use of the second-order linearDPLL. �b� The demodulated phase map after removal of the tilt.�c� The phase map resulting from analysis of the fringe patternshown in Fig. 6�a� with the Fourier fringe analysis technique.


tern.1 In this method, a fringe pattern is capturedby use of the twin-fiber interferometer, and then oneof the optical fibers of the interferometer is vibratedat 500 Hz. A second image is captured during thevibration of one of the fibers. The image can be con-sidered as the background illumination of the fringepattern. During the vibration of one of the fibers,the camera sees a time-averaged image. The back-ground illumination image is subtracted from thefringe pattern, and the mathematical mean of theresultant image is close to zero.

Figures 8�a� and 8�b� show two fringe patterns withdifferent spatial carrier frequencies. Both fringepatterns were captured for the same object. Thephase variations in both fringe patterns can be con-sidered slow. We generated the image shown in Fig.8�c� by vibrating one of the optical fibers of the twin-fiber interferometer. The image can be consideredas the background illumination of both fringe pat-terns. This image was subtracted from the fringepatterns shown in Figs. 8�a� and 8�b� to remove theirbackground illuminations. Figure 8�d� shows theimage that results from the subtraction of the back-ground illumination image from the fringe patternshown in Fig. 8�a�. We then smoothed the resultantfringe patterns �after removal of their background�using a 3 � 3 window once and a 1 � 5 window twice.

The wrapped phase maps of both filtered fringe pat-terns were calculated as explained in the previousexample. One of the wrapped phase maps was ap-plied to a second-order linear DPLL, and the secondone was used instead of the computer-generatedfringe pattern that is produced by the DCO of thelinear DPLL. The linear DPLL demodulates andtracks the phase difference between both wrappedphase maps. The demodulated phase map is shownin Fig. 8�e�. The phase map shown in Fig. 8�f � isproduced after removal of the tilt.

The demodulated phase map shown in Fig. 8�f � isnoisy because the phase variations of the fringe pat-terns shown in Figs. 8�a� and 8�b� are slow. It can beconcluded from this example that the analysis offringe patterns modulated by an object with slow sur-face variations by use of this algorithm will producenoisy phase maps.

6. Conclusions

A novel technique called a two-frame DPLL that issuitable for noncontact measurement of objects withrapid surface variations has been presented. Thetwo-frame DPLL technique outperforms the Fourierfringe analysis algorithm in noncontact measure-ment of objects with rapid surface variations.

Phase maps produced by the two-frame DPLL tech-nique are noisy in comparison with phase maps gen-erated with the Fourier fringe analysis and the basicDPLL schemes.

Appendix A

A fringe pattern’s intensity can be mathematicallyexpressed by the equation3

g� x, y� � a� x, y� � b� x, y�cos�2�f0 x � � x, y��,(A1)

where a�x, y� represents the background illumina-tion; b�x, y� is the amplitude modulation of fringes; f0is the spatial carrier frequency; �x, y� is the phasemodulation of fringes; and x and y are the sampleindices for the x and y axes, respectively.

The relationship between the modulated fringesand the height distribution of an object can be ex-plained by Fig. 9. Consider that fringes with uni-form spacing �P0� are projected down onto an object�where P0 is the spatial carrier period �P0 � 1�f0��.Also, consider that point a is illuminated by point c onthe projected fringes, and a fringe pattern is capturedby a camera. Point a on the fringe pattern appearswith low illumination intensity. Also consider thatpoint a is moved by a distance h to point b, which isilluminated by point d on the fringes, and a secondfringe pattern is captured. In the second fringe pat-tern, point b appears with high illumination inten-sity. The change of the illumination intensity whenpoint a is moved to b results because both points areilluminated by different points on the projectedfringes. The change in illuminating points can beconsidered as a change in the phase of fringes be-cause of the height h of the object. The change in the

Fig. 8. �a� and �b� Two fringe patterns with different spatial car-rier frequencies, �c� the background illumination image for �a� and�c�, �d� image from the subtraction of image �c� from image �a�, �e�the phase difference between both fringe patterns demodulatedwith the linear DPLL, �f � the demodulated phase map after re-moval of the tilt.


phase of fringes or the phase modulation of fringescan be expressed as

�2�h sin �

P0� 2�f0 h sin �. (A2)

The height distribution of the object can be extractedfrom the phase through the equation

h �

2�f0 sin �. (A3)

Substituting Eq. �A2� into Eq. �A1�, a fringe pattern’sintensity can be described as

g� x, y� � a� x, y� � b� x, y�cos�2�f0 x

� 2�f0 h� x, y�sin ��. (A4)

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Fig. 9. Modulation of fringes by the height distribution of anobject.


fringe pattern demodulation with a two-frame digital phase-locked loop algorithm

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