Comparison Of The Squared Binary, Sinusoidal Pulse Width .

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Comparison of the squared binary, sinusoidal pulsewidth modulation, and optimal pulse widthmodulation methods for three-dimensionalshape measurement withprojector defocusingYajun Wang and Song Zhang*Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50011, USA*Corresponding author: song@iastate.eduReceived 22 August 2011; revised 7 November 2011; accepted 9 November 2011;posted 9 November 2011 (Doc. ID 153221); published 24 February 2012This paper presents a comparative study on three sinusoidal fringe pattern generation techniques withprojector defocusing: the squared binary defocusing method (SBM), the sinusoidal pulse width modulation (SPWM) technique, and the optimal pulse width modulation (OPWM) technique. Because the phaseerror will directly affect the measurement accuracy, the comparisons are all performed in the phase domain. We found that the OPWM almost always performs the best, and SPWM outperforms SBM to agreat extent, while these three methods generate similar results under certain conditions. We will brieflyexplain the principle of each technique, describe the optimization procedures for each technique, andfinally compare their performances through simulations and experiments. 2012 Optical Society ofAmericaOCIS codes: 120.0120, 120.2650, 100.5070.1. IntroductionOver the past decades, digital sinusoidal fringe projection techniques have been very successful in thefield of 3D optical profilometry and applied to numerous application areas [1–3]. However, they still encounter challenges due to the speed limitation aswell as the nonlinear gamma effect. Conventionally,the projector switching speed, which is typically lessthan 120 Hz, limits the measurement speed.However, studies, such as high-frequency vibration,require speed faster than 120 Hz. Furthermore,when a common digital projector is used, nonlineargamma effect will introduce errors. And then gammacalibration is required. Numerous methods [4–9]have been proposed to reduce the measurement1559-128X/12/070861-12 15.00/0 2012 Optical Society of Americaerrors caused by the nonlinear gamma. Thoughsuccessful, the residual errors are usually nonnegligible for high-precision measurement applications.The recently proposed squared binary defocusingtechnique (SBM) [10] has demonstrated its potentialto overcome the aforementioned limitations. However, it poses new challenges: (1) the error induced byhigh-order harmonics and (2) the smaller depth measurement range. Endeavors have been made to conquer these challenges: (1) Ayubi et al. proposed atechnique called sinusoidal pulse width modulation(SPWM) [11], and (2) Wang and Zhang proposed atechnique called optimal pulse width modulation(OPWM) [12]. The two methods have demonstratedtheir superiorities over the SBM under certain conditions, while having their own limitations. Thesquare binary method is sensitive to defocusing effect. It can only give good results when the binarystructures are properly defocused to sinusoidal1 March 2012 / Vol. 51, No. 7 / APPLIED OPTICS861

ones. This means the depth range of measurement ispretty small when SBM is adopted. Both recentlyproposed SPWM and OPWM techniques have theability to improve the SBM method. They can produce high-quality 3D shape measurement even whenthe defocusing degree is small, which means thedepth range of measurement for these two methodsis larger than that for the SBM. Since each methodhas its own merits and shortcomings, it would be ofinterest to present thorough comparisons to thesociety among the three methods: SBM, SPWM,and OPWM.It is important to note that this paper examinesthe method differences from the phase perspectivesince the 3D shape measurement quality is mainlydetermined by the quality of phase data for fringeprojection techniques. It is also important to notethat it is almost impossible to compare these methods exhaustively in one paper due to the fact thatthere are numerous variables affecting the measurement quality. Therefore, we limit our paper to a fewcase studies that will provide sufficient critical characteristics of these methods which are vital for 3Dshape measurement. Specifically, we will focus onanalyzing the phase errors caused by the three methods under different conditions, and we will use athree-step phase-shifting algorithm with equalphase shifts to perform phase analysis for simplicityand speed. The phase error is obtained by taking thedifferences between the phase obtained from the defocusing methods and the phase from the traditionalideal sinusoidal fringe projection method. Since weare studying the phase error caused by defocusing,the amount of defocusing needs to be accountedfor. This research will consider two representativescenarios: the nearly focused case and the significantly defocused case. Different breadths of fringepatterns will also be examined for comparisons. Bothsimulations and experimental results will be presented in this paper to demonstrate the differencesamong SBM, SPWM, and OPWM.Section 2 will explain the principle of each technique. Section 3 will present an optimization strategyfor each technique. Section 4 and Section 5 respectively show simulation and experimental resultsunder optimal conditions, and finally Section 6 summarizes the paper.2. PrincipleA.Three-Step Phase-Shifting AlgorithmPhase-shifting algorithms are widely used in opticalmetrology because of their measurement speed andaccuracy [13]. Numerous phase-shifting algorithmshave been developed including three step, four step,double three step, and five step. In this paper, we usea three-step phase-shifting algorithm with a phaseshift of 2π 3 for simplicity and speed. Three fringeimages can be described asI 1 x; y I 0 x; y I 0 x; y cos ϕ 2π 3 ;862APPLIED OPTICS / Vol. 51, No. 7 / 1 March 2012(1)I 2 x; y I 0 x; y I 0 x; y cos ϕ ;(2)I 3 x; y I 0 x; y I 0 x; y cos ϕ 2π 3 ;(3)where I 0 x; y is the average intensity, I 00 x; y theintensity modulation, and ϕ x; y the phase to besolved. Solving these equations simultaneously leadsto p 3 I 1 I 3 2I 2 I 1 I 3 :ϕ x; y tan 1(4)Equation (4) provides the phase ranging π; π with2π discontinuities.B. Squared Binary MethodOur recent study indicated that it is feasible to generate high-quality sinusoidal fringe patterns byproperly defocusing squared binary structured patterns [10]. Therefore, instead of sending sinusoidalfringe images to a focused projector, sinusoidal fringepatterns can be generated by defocusing binarystructured ones. Figure 1 illustrates projector defocusing at different degrees. In this experiment, theprojector projects squared binary structured patterns onto a uniform white board where the camerafocuses. The projector defocusing is realized by adjusting the focal length of the projector graduallyfrom in focus to out of focus. This experiment showsthat if projector is properly defocused, seemingly sinusoidal fringe patterns can be generated. However,the phase error will be significant if there is no errorcompensation and the projector is not properlydefocused [14].C.Sinusoidal Pulse Width ModulationRecently, Ayubi et al. proposed an interesting technique that can significantly reduce phase errors evenwhen the projector is not defocused properly forSBM, which is known as the SPWM technique [11].The SPWM is a well-studied technique in power electronics to generate sinusoidal signal in time by filtering the binary signals. In brief, to generate lowfrequency (f 0 ) sinusoidal fringe patterns, a higherfrequency f c triangular wave Sm f c is used to modulate the desired ideal sinusoidal signal Si f 0 :Sm f c 2f c x 2Nx N f c ; 2N 1 2f c (5); 2f c x 2N 2 x 2N 1 2f c ; N 1 f c Si f 0 0.5 0.5 cos 2πf 0 x :(6)Here f c f 0 , and N is an integer.The modulated fringe pattern for each point canthen be generated as follows:

)6080100120202040X(pixels)6080100120X(pixels)Fig. 1. (Color online) Influences of different defocusing degrees on the squared binary pattern. (a)–(e) show the patterns when theprojector is nearly in focus to significantly defocused. (f)–(j) show the corresponding cross sections. I b x; y 10Sm f c Si f 0 ;.otherwise(7)Optimal Pulse Width ModulationWe recently proposed OPWM to further improve thedefocusing technique [12]. This technique selectivelyeliminates undesired frequency components by inserting different types of notches in a conventionalbinary square wave. Then, with a slightly defocusedprojector, ideal sinusoidal fringe patterns can begenerated.Figure 3 illustrates an OPWM pattern. The squarewave is chopped n times per half cycle. For a 2π periodic waveform, the Fourier series coefficients are110.80.80.60.4004050ak 1πZ2πf θ dθ 0.5;(8)f θ cos kθ dθ;(9)f θ sin kθ dθ.(10)θ 0Z2πθ 0Z2πθ 00.40.22030X (pixels)12π0.60.210a0 1bk πAmplitudeAmplitudeFigure 2 illustrates the SPWM pattern generationnature by modulating squared binary patterns withtriangular waveform.Defocusing is essential to suppress the high-orderharmonics of the binary patterns while maintainingthe fundamental frequency (f 0 ). The idea of SPWM isto shift the third and higher-order harmonics furtheraway from the fundamental frequency so that theycan be easier to be suppressed. Therefore, the modulated signal can then be converted to ideal sinusoidalsignal by using a smaller size low-pass filter. In otherwords, the projector can be less defocused for sinusoidal fringe generation. By this means, the fringecontrast can be increased, and the depth measurement range can be increased comparing with theSBM technique. This technique has been successfully demonstrated by Ayubi et al. to generate bettersinusoidal fringe patterns even with a smaller degreeof defocusing [11].D.102030X (pixels)4050Fig. 2. (Color online) Modulate sinusoidal waveform with binary structured patterns. (a) The sinusoidal and the modulation waveforms.(b) The resultant binary waveform.1 March 2012 / Vol. 51, No. 7 / APPLIED OPTICS863

b7 1 cos 7α1 cos 7α2 cos 7α3 cos 7α4 0.0;(16)b11 1 cos 11α1 cos 11α2 cos 11α3 cos 11α4 0.0:Fig. 3. Quarter-wave symmetric OPWM waveform.Because of the half-cycle symmetry of the OPWMwave, only odd-order harmonics exist. Furthermore,bk can be simplified asZ4 2πbk f θ sin kθ dθ:(11)π θ 0For the binary OPWM waveform, f θ , we havebk 4πZα104 πZsin kθ dθ π 2αn4πZα3α2sin kθ dθ sin kθ dθ(12)(17)The nonlinear equations require an optimizationprocedure to solve for the angles. Numerous researchhas been conducted on how to solve for this type ofnonlinear equations. For example, some research hasbeen conducted to solve for transcendental equations[16]. Because of the ability to eliminate undesiredhigh-order harmonics, OPWM waveform could become sinusoidal after applying a low-pass filter,which is similar to a small degree of defocusing.Therefore, the depth measurement range can alsobe increased in comparison with the SBM technique.Figure 4 shows the representative binary patternsfor the SBM, SPWM, and OPWM. The fringe periodis 60 pixels, SPWM modulation frequency is 6 pixelsper period, and the OPWM is set to eliminate thefifth- and seventh-order harmonics.E. Phase Error Determination 4 1 cos kα1 cos kα2 cos kα3 cos kαn :kπ(13)The n notches in the waveform create n degrees offreedom. It can eliminate n 1 number of selectedharmonics while keeping the fundamental frequencycomponent within a certain magnitude. To do this,one can set the corresponding coefficients in theabove equation to be desired values (0 for the n 1harmonics to be eliminated and the desired magnitude for the fundamental frequency), and solve forthe angles for all notches [15]. For instance, to eliminate fifth, seventh, and 11th-order harmonics, twonotches could be set, and the following equationscould be formulatedb1 1 cos α1 cos α2 cos α3 cos α4 π 4;(14)b5 1 cos 5α1 cos 5α2 cos 5α3 cos 5α4 0.0;(15)For a 3D shape measurement technique based onfringe analysis, because the 3D information isretrieved from the phase, the measurement erroris typically determined by the phase error underthe same calibration circumstance. Therefore, evaluating one technique can be realized by finding thephase error caused by that technique.It is important to note that it is very difficult for areal measurement system to find the phase error itself from the defocused binary patterns because themeasured surface property may play a vital role, andthe lens distortion complicates the problem. To circumvent this problem, we use three additional fringepatterns that are ideal sinusoidal fringe patternswith the same phase shift and the same number ofpixels per fringe period. Therefore, for each measurement, we have four phase values, ϕs x; y from theideal sinusoidal fringe pattern, ϕb x; y from theSBM patterns, ϕp x; y from the SPWM patterns,and ϕo x; y from the OPWM patterns. The phase errors are defined as the differences between the phasevalues obtained from the different methods and thephase from the ideal sinusoidal fringe patterns:Δϕb x; y ϕb x; y ϕs x; y mod 2π ;Fig. 4. (Color online) Example of three different patterns. (a) SBM pattern; (b) SPWM pattern; (c) OPWM pattern.864APPLIED OPTICS / Vol. 51, No. 7 / 1 March 2012 18

Δϕp x; y ϕm x; y ϕs x; y mod 2π; 19 Δϕo x; y ϕo x; y ϕs x; y mod 2π: 20 Here mod is the modulus operator. It should benoted that, theoretically, it does not need a modulusoperation to obtain phase error from the wrappedphase since the patterns are perfectly aligned. However, due to camera sampling, the modulus operationis practically required to remove the 1-pixel 2π phasejump shift from the binary defocused phases to thesinusoidal phase.F.Since each harmonic will contribute to the profile of abinary structured pattern, it would be desirable tounderstand the influence of each harmonic’s contribution to the phase error. The cross section of asquared binary structured pattern is a square wave;thus understanding the effect of a binary structuredpattern can be simplified to the study of a squarewave. A normalized square wave with a period of2π can be written asy x 1x 2n 1 π; 2nπ x 2nπ; 2n 1 π .(26)(27)From this equation, we can see that what matters isthe difference values of I 1 I 3 and 2I 2 I 1 I 3 . It isclear that when the 3nth-order harmonics exist, thedifferences do not change, and no phase error will beintroduced. Therefore, these harmonics do not needto be accounted to improve the measurement quality.3. Optimization CriteriaA. SPWMInfluence of High-Order Harmonics on Phase(0p ϕ x; y tan 1 3 I h1 I h3 2I h2 I h1 I h3 p tan 1 3 I 1 I 3 2I 2 I 1 I 3 :(21)Here, n is an integer number. The square wave can beexpanded as a Fourier series:The difference between SBM and SPWM is thatSPWM utilizes a higher-frequency (f c ) triangularwave to modulate the ideal sinusoidal wave, shiftingthe higher-order frequency harmonics further awayfrom the fundamental frequency f 0. The only variableis the modulation frequency f c. Therefore, the goal ofSPWM optimization is to find an optimal modulationfrequency f oc so that the measurement quality will bethe best under all circumstances, i.e., differentamounts of defocusing for different fundamental frequencies. In addition, to maintain the fundamentaldifference between SPWM and SBM, the modulationfrequency must be much higher than the fundamental frequency, i.e., f c f 0 . The performance is evaluated based upon the criteria of the introduced phaseerror instead of the sinusoidality appearance.B. OPWMy x 0.5 X2sin 2k 1 π . 2k 1 πk 0(22)For a three-step phase-shifting algorithm with equalphase shift, our previous study [17] found that somehigh-order harmonics will not induce measurementerrors. Specifically, we have demonstrated that the3nth-order harmonics will not influence phase errorat all. If the high-order harmonics exist, the fringepatterns can be described asI h1 x; y I 0 x; y I 00 x; y cos ϕ 2π 3 I k x; y cos 2k 1 ϕ 2π 3 ;(23)I h2 x; y I 0 x; y I 00 x; y cos ϕ I k x; y cos 2k 1 ϕ ;(24)Because the OPWM technique has the ability to selectively eliminate high-order harmonics, it providesmore flexibility to control the pattern structure,while complicating the problem. The solution to thenonlinear equation set is essentially a nonlinear optimization problem, which often leads to multiple solutions. However, as addressed in Subsection 2.F., fora three-step phase-shifting algorithm with equalphase shift, we only need to eliminate the high-orderharmonics except those 3nth-order ones. This simplifies the problem since there are fewer harmonics toconsider.In other words, the most dominant phase error iscaused by fifth- and seventh-order harmonics. Therefore, if we could remove these two frequency components, the measurement error should be very smallsince the next harmonics introducing the phase erroris the 11th order. For the defocusing technique, itis quite easy to suppress the 11th order and aboveharmonics by slightly defocusing the projector.4. SimulationsI h3 x; y 000 I x; y I x; y cos ϕ 2π 3 I k x; y cos 2k 1 ϕ 2π 3 ; (25)where k 1; 2; 3; are integers in Eq. (22). Whenjust the 3nth-order harmonics exist (i.e., third,ninth), Eq. (4) can be rewritten asA. Simulation on Influence of High-Order HarmonicsOur first simulation is to verify the influence of highorder harmonics on phase error. This simulation wascarried out by analyzing the phase error introducedby each frequency harmonics. Equation (22) indicates that the magnitudes of high-order harmonics1 March 2012 / Vol. 51, No. 7 / APPLIED OPTICS865

1000.60.40.15Phase error (rad)120AmplitudeIntensity (grayscale)10.88060400.22000200 400 600 800 1000 120005100 0.05 0.115200 400 600 800 1000 1200Harmonics1200.81000.60.460402000x (pixel)0.15800.2200 400 600 800 1000 1200x (pixel)Phase error (rad)1AmplitudeIntensity (grayscale)x (pixel)0.10.050510150.10.050 0.05 0.1200 400 600 800 1000 1200Harmonicsx (pixel)Fig. 5. (Color online) Influence of high-order harmonics on phase error. (a) Fringe pattern containing 11th-order harmonics components. (b) Cross-section of (a). (c) Frequency spectrum of (b). (d) Phase error for signal in (b); (e)–(h) The corresponding results when thesignal includes the fundamental, third, and ninth-order harmonics.decrease gradually. This means that when the orderis very high (e.g., higher than 11th order), its inducederror could be negligible.To demonstrate this, we firstly determine the theoretical phase error on each harmonics. The phaseerror was determined by combining the fundamentalfrequency with only one higher-order component, i.e.,the signal can be described as2sin f 0 x 3π2sin 2k 1 f 0 x : 2k 1 πI k x; y 0.5 no phase error is introduced by 3nth-order harmonics, and the phase error decreases when the otherharmonic order increases. This indicates that it isnot sufficient to look at the appearance of fringe patterns nor their frequency spectra [18]. Example inFig. 5 clearly shows that the less sinusoidal patternscould have less phase errors (thus better measurement quality) than those seemingly better sinusoidalpatterns.B. Simulation on SPWM Optimization 28 A three-step phase-shifting with equal phase shiftwas used to determine the phase, and the associatedphase error was calculated by comparing against theideal one. Figure 5 shows the influence of each individual harmonic on phase error. This confirms thatThe defocusing effect can be approximated as a Gaussian smoothing filter. A 2D Gaussian filter is usuallydefined asG x; y y 21 x x 2 y 2σ 2e.22πσ(29)Here σ is standard deviation and x and y are meanvalues of the x and y axis, respectively.Intensity(gray scale)25020015010050050100 150 200 250 300X(pixels)Fig. 6. (Color online) Influence of modulation frequency and fringe pitch on phase error with a nearly focused projector. (a) The squarebinary pattern after defocusing. (b) Phase error changes with modulation frequencies for different fringe pitches (P 36 6 n,n 1; 2; 10). (c) Phase error changes with the fringe pitches P (modulation period T 6 pixels).866APPLIED OPTICS / Vol. 51, No. 7 / 1 March 2012

Intensity(gray scale)25020015010050050100 150 200 250 300X(pixels)Fig. 7. (Color online) Influence of modulation frequency and fringe pitch on phase error with a significantly defocused projector. (a) Thesquare binary pattern after defocusing. (b) Phase error changes with modulation frequencies for different fringe pitches (P 36 6 n,n 1; 2; 10). (c) Phase error changes with the fringe pitches P (modulation period T 6 pixels).the simulation shown in Fig. 6(b), we determinedhow the modulation frequency affects the measurement error for different fringe pitches, P, the numberof pixels per fringe period. To minimize the digitaleffect on phase shift and binarization, an incrementof 6 pixels is used for the binary patterns, and an increment of 2 pixels is used for the modulation pulsewidth to ensure the symmetry of a triangular waveform. The fringe pitch P starts with 42 pixels becausewhen it is very small, the error appears random(mainly because of the digital effect). This simulationresult shows that when the modulation pulse widthis 6 pixels, the phase error for almost all the fringepitches at their valley points, meaning that 6 pixelmodulation frequency period could be the optimalone to use.Another simulation was carried out to determinewhether the SPWM technique can improve the quality of 3D shape measurement by reducing measurement error. Figure 6(c) shows the results. ThisFor our case, because the structured stripes areeither vertical or horizontal, only one cross sectionperpendicular to the fringe stripes needs to be considered, which means the problem is reduced to1D. A 1D Gaussian filter is defined as21 x x G x p e 2σ2 .2π σ(30)We first simulate the defocusing effect by applyinga Gaussian filter to the signal. The degree of defocusing can be represented as the breadth of theGaussian filter that is determined by the standarddeviation and the filter size. We applied a smallGaussian filter with size of 11 and standard deviation of 1.83 pixels twice to emulate that the nearlyfocused case. Figure 6(a) shows the square binarypattern after applying the filter. It can be seen thatthe binary structure is very clear, which emulates thepattern generated by a nearly focused projector. In2001501501000.2Phase th5101550100Pixels15050100Pixels150thk order 0100Phase error200100505000050100Pixels1500510kth order harmonics150.10 0.1 0.25000 0.1 0.30k order harmonics2500.1 0.2500Intensity(gray scale)2000.3200AmplitudeIntensity(gray scale)250 0.30510kth order harmonics15Fig. 8. (Color online) OPWM optimization example. The first row shows a bad OPWM pattern, and the second row shows the good OPWMpattern. (a) One of the three phase-shifted OPWM patterns. (b) The frequency spectra before smoothing. (c) The frequency spectra afterapplying a smoothing filter. (d) The phase error. (e) One of the three phase-shifted OPWM patterns. (f) The frequency spectra beforesmoothing. (g) The frequency spectra after applying a smoothing filter. (h) The phase error.1 March 2012 / Vol. 51, No. 7 / APPLIED OPTICS867

2Phase error(rad)Phase error(rad)10.50 0.50100200300X(pixels)400x 10 310 1 20100200300X(pixels)400Fig. 9. (Color online) Validation of the ideal sinusoidal fringe patterns utilized as reference. (a) The unwrapped phase map. (b) Theactual phase error that will be coupled into the real measurement.simulation is to compare the phase error between theoriginal binary patterns and the SPWM patterns under the same condition with different fringe pitches.Again the SPWM period T was chosen to be 6 pixelsto minimize the error. This simulation shows thatwhen the fringe pitch is larger than 56, the modulated patterns give smaller phase error. However,when the fringe pitch is small, the original squaredbinary pattern actually works better. One interestingthing to notice is that when the period of fringe patterns increases, the phase error does not significantly increase for the SPWM technique.We then applied the same Gaussian filter 20 timesto represent the significantly defocused cases. Thecorresponding results are shown in Fig. 7. This simulation shows that when the patterns are significantlydefocused, a modulation period of 6 pixels stillgives close to the minimum errors for different frequency of fringe patterns, while between 6 and 12pixels, the results are decent. However, in comparisonwith the phase error for the traditional binary method, only after the fringe pitch increases to a certainlevel, this SPWM method will perform better.Finding the optimal pulse width to minimize thephase error should be the criteria for evaluatingthe performance of different patterns. This simulation results show that: (1) the SPWM techniqueactually deteriorates the measurement quality ifthe fringe pitch is smaller than a certain number;(2) the optimal modulation period is consistently 6pixels if the projector is nearly focused; and (3) theoptimal modulation period ranges 6–10 pixels ifthe projector is significantly defocused.C.Simulation on OPWM optimizationThe simulation presented in Section 4.A. shows that3nth-order harmonics do not introduce any phaseerror for a three-step phase-shifting algorithm withequal phase shift. Therefore, these frequency components should not be considered during the optimization procedure. In other words, although thethird-order harmonics have more influence on thesquare wave than the fifth and seventh, we do notneed to consider its influence on our measurementaccuracy.Figure 8(a) and 8(e) show two different OPWM patterns with a fringe pitch of 90 pixels. Figures. 8(b)and 8(f) show their corresponding frequency spectra.It clearly shows that Fig. 8(f) has smaller fifth andseventh harmonics magnitudes than those shownin Fig. 8(b); therefore, its performance should bebetter. These patterns were then smoothed by aGaussian filter (size of 11 with a standard deviation0.25Phase error(rad)Intensity(gray scale)150100500.150.10.05005051015Modulation period T(pixels)100 150 200 250 300X(pixels)0.08Phase error(rad)150Intensity(gray scale)0.210050050100 150 200 250 300X(pixels)0.060.040.02051015Modulation period T(pixels)Fig. 10. (Color online) Experimental results on modulation frequency selections. The fringe pitches used here are 42, 60, 72, 90, 102, and150 pixels. (a) The binary square pattern when the projector is nearly focused. (b) The cross section of (a). (c) The results when the projectoris nearly focused; (d)–(f) show the corresponding results when the projector is significantly defocused.868APPLIED OPTICS / Vol. 51, No. 7 / 1 March 2012

000204060Harmonics80100Fig. 11. (Color online) OPWM optimization results. (a) The SBM pattern with 90 pixels per period. (b) OPWM pattern with third-orderharmonics but without fifth- or seventh-order harmonics. (c) OPWM pattern with fifth- and seventh-order harmonics but without thirdorder harmonics. (d) Frequency spectra of the SBM pattern shown in (a). (e) Frequency spectra of the OPWM pattern in (b). (f) Frequencyspectra of the OPWM pattern shown in (c).5. ExperimentsA.Experimental System SetupExperiments were also performed to verify thesimulation results. In this research, we used adigital-light-processing projector (Model: SamsungSP-P310MEMX) and a digital CCD camera (Model:Jai Pulnix TM-6740CL). The camera uses a 16 mm focal length Mega-pixel lens (Model: Computar M1614MP) at F 1.4 to 16C. The camera resolution is 640 480 with a maximum frame rate of 200 frames sec.The camera pixel size is 7.4 7.4 μm2. The projectorhas a resolution of 800 600 with a projection distance of 0.49–2.80 m.B. Validation of Ideal Sinusoidal Fringe PatternGenerationThis experiment is to verify that the adopted conventional sinusoidal method does not bring significantphase error, since we used the phase obtained fromthese patterns as our reference. In the experiments,a uniform white plate was used to quantitatively showthe introduced phase errors by this method. Since theprojector is a nonlinear device, the projection nonlinearity needs to be corrected. In this research, weused the method proposed in [4] to actively changethe patterns before projection. Figure 9(a) showsone cross section of the unwrapped phase map. Thelarger profile was introduced by 3D shape measurement system (e.g., lens distortions, board flatness).The general profile will not introduce additionalphase error since it is systematically caused by thehardware system, rather than the fringe quality.The phase error was obtained by removing the general profile of the unwrapped phase, which is shown inFig. 9(b). It can be seen that the projection nonlinearinfluence was effectively alleviated and the introduced random error is negligible in comparison withthe digitization error.C. Validation of Optimal Modulation Frequency Selectionfor SPWMThe optimal modulation frequency determined fromSection 4 was then validated by experiments.Figure 10(a) shows the square binary pattern whenthe projector is nearly focused and Fig. 10(b) showsthe cross section. Figure 10(c) shows the experimental0.4Phase error(rad)of 5 pixels). Figures 8(c) and 8(g) show the Fourierspectra after applying the smoothing filter. Most ofthe

C. Sinusoidal Pulse Width Modulation Recently, Ayubi et al. proposed an interesting techni-que that can significantly reduce phase errors even when the projector is not defocused properly for SBM, which is known as the SPWM technique [11]. The SPWM isawell-studiedtechniqueinpower elec

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