Comparison of the squared binary, sinusoidal pulse width ... · X(pixels) Intensity 20 40 60 80 100 120 20 40 60 80 100 120 140 X(pixels) Intensity Fig. 1. (Color online) Influences

Comparison of the squared binary, sinusoidal pulsewidth modulation, and optimal pulse widthmodulation methods for three-dimensional

shape measurement withprojector defocusing

Yajun Wang and Song Zhang*Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50011, USA

*Corresponding author: [email protected]

Received 22 August 2011; revised 7 November 2011; accepted 9 November 2011;posted 9 November 2011 (Doc. ID 153221); published 24 February 2012

This paper presents a comparative study on three sinusoidal fringe pattern generation techniques withprojector defocusing: the squared binary defocusing method (SBM), the sinusoidal pulse width modula-tion (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 do-main. 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. Introduction

Over the past decades, digital sinusoidal fringe pro-jection techniques have been very successful in thefield of 3D optical profilometry and applied to numer-ous application areas [1–3]. However, they still en-counter 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 measurement

errors caused by the nonlinear gamma. Thoughsuccessful, the residual errors are usually nonnegli-gible for high-precision measurement applications.

The recently proposed squared binary defocusingtechnique (SBM) [10] has demonstrated its potentialto overcome the aforementioned limitations. How-ever, it poses new challenges: (1) the error induced byhigh-order harmonics and (2) the smaller depth mea-surement range. Endeavors have been made to con-quer 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 con-ditions, while having their own limitations. Thesquare binary method is sensitive to defocusing ef-fect. It can only give good results when the binarystructures are properly defocused to sinusoidal

1559-128X/12/070861-12$15.00/0© 2012 Optical Society of America

1 March 2012 / Vol. 51, No. 7 / APPLIED OPTICS 861

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 pro-duce 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 meth-ods exhaustively in one paper due to the fact thatthere are numerous variables affecting the measure-ment quality. Therefore, we limit our paper to a fewcase studies that will provide sufficient critical char-acteristics of these methods which are vital for 3Dshape measurement. Specifically, we will focus onanalyzing the phase errors caused by the three meth-ods 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 de-focusing 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 signifi-cantly defocused case. Different breadths of fringepatterns will also be examined for comparisons. Bothsimulations and experimental results will be pre-sented in this paper to demonstrate the differencesamong SBM, SPWM, and OPWM.

Section 2 will explain the principle of each techni-que. Section 3 will present an optimization strategyfor each technique. Section 4 and Section 5 respec-tively show simulation and experimental resultsunder optimal conditions, and finally Section 6 sum-marizes the paper.

2. Principle

A. Three-Step Phase-Shifting Algorithm

Phase-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 as

I1�x; y� � I0�x; y� � I0�x; y� cos�ϕ − 2π∕3�; (1)

I2�x; y� � I0�x; y� � I0�x; y� cos�ϕ�; (2)

I3�x; y� � I0�x; y� � I0�x; y� cos�ϕ� 2π∕3�; (3)

where I0�x; y� is the average intensity, I00�x; y� theintensity modulation, and ϕ�x; y� the phase to besolved. Solving these equations simultaneously leadsto

ϕ�x; y� � tan−1

� ��3

p�I1 − I3�∕�2I2 − I1 − I3�

�: (4)

Equation (4) provides the phase ranging �−π; π� with2π discontinuities.

B. Squared Binary Method

Our recent study indicated that it is feasible to gen-erate high-quality sinusoidal fringe patterns byproperly defocusing squared binary structured pat-terns [10]. Therefore, instead of sending sinusoidalfringe images to a focused projector, sinusoidal fringepatterns can be generated by defocusing binarystructured ones. Figure 1 illustrates projector defo-cusing at different degrees. In this experiment, theprojector projects squared binary structured pat-terns onto a uniform white board where the camerafocuses. The projector defocusing is realized by ad-justing the focal length of the projector graduallyfrom in focus to out of focus. This experiment showsthat if projector is properly defocused, seemingly si-nusoidal 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 Modulation

Recently, Ayubi et al. proposed an interesting techni-que 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 elec-tronics to generate sinusoidal signal in time by filter-ing the binary signals. In brief, to generate lowfrequency (f 0) sinusoidal fringe patterns, a higher-frequency f c triangular wave Sm�f c� is used to mod-ulate the desired ideal sinusoidal signal Si�f 0�:

Sm�f c�

��2f cx− 2N x∈ �N∕f c; �2N� 1�∕�2f c��−2f cx� 2N� 2 x∈ ��2N� 1�∕�2f c�; �N� 1�∕f c�

;(5)

Si�f 0� � 0.5� 0.5 cos�2πf 0x�: (6)

Here f c > f 0, and N is an integer.The modulated fringe pattern for each point can

then be generated as follows:

862 APPLIED OPTICS / Vol. 51, No. 7 / 1 March 2012

Ib�x; y� ��1 Sm�f c� > Si�f 0�;0 otherwise

. (7)

Figure 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 modu-lated 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 sinusoi-dal fringe generation. By this means, the fringecontrast can be increased, and the depth measure-ment range can be increased comparing with theSBM technique. This technique has been success-fully demonstrated by Ayubi et al. to generate bettersinusoidal fringe patterns even with a smaller degreeof defocusing [11].

D. Optimal Pulse Width Modulation

We recently proposed OPWM to further improve thedefocusing technique [12]. This technique selectivelyeliminates undesired frequency components by in-serting 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 per-iodic waveform, the Fourier series coefficients are

a0 � 12π

Z2π

θ�0f �θ�dθ � 0.5; (8)

ak � 1π

Z2π

θ�0f �θ� cos�kθ�dθ; (9)

bk � 1π

Z2π

θ�0f �θ� sin�kθ�dθ. (10)

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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.

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Fig. 2. (Color online) Modulate sinusoidal waveform with binary structured patterns. (a) The sinusoidal and the modulation waveforms.(b) The resultant binary waveform.


Because of the half-cycle symmetry of the OPWMwave, only odd-order harmonics exist. Furthermore,bk can be simplified as

bk � 4π

Z2π

θ�0f �θ� sin�kθ�dθ: (11)

For the binary OPWM waveform, f �θ�, we have

bk � 4π

Z α1

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π

Z α3

α2sin�kθ�dθ�…

� 4π

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αnsin�kθ�dθ (12)

� 4kπ �1 − cos kα1 � cos kα2 − cos kα3 �…� cos kαn�:

(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 magni-tude for the fundamental frequency), and solve forthe angles for all notches [15]. For instance, to elim-inate fifth, seventh, and 11th-order harmonics, twonotches could be set, and the following equationscould be formulated

b1 � 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)

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: (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 be-come 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

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, evalu-ating 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 it-self from the defocused binary patterns because themeasured surface property may play a vital role, andthe lens distortion complicates the problem. To cir-cumvent 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 measure-ment, 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 er-rors 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 ; �18�

Fig. 3. Quarter-wave symmetric OPWM waveform.

Fig. 4. (Color online) Example of three different patterns. (a) SBM pattern; (b) SPWM pattern; (c) OPWM pattern.


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. How-ever, 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. Influence of High-Order Harmonics on Phase

Since each harmonic will contribute to the profile of abinary structured pattern, it would be desirable tounderstand the influence of each harmonic’s contri-bution 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 as

y�x� �( 0 x ∈ ��2n − 1�π; 2nπ�1 x ∈ �2nπ; �2n� 1�π� . (21)

Here, n is an integer number. The square wave can beexpanded as a Fourier series:

y�x� � 0.5�X∞k�0

2�2k� 1�π sin��2k� 1�π�. (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 as

Ih1�x; y� � I0�x; y� � I00�x; y� cos�ϕ − 2π∕3��…Ik�x; y� cos��2k� 1��ϕ − 2π∕3��; (23)

Ih2�x; y� � I0�x; y� � I00�x; y� cos�ϕ��…Ik�x; y� cos��2k� 1�ϕ�; (24)

Ih3�x; y� � I0�x; y� � I00�x; y� cos�ϕ� 2π∕3��…Ik�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 as

ϕ�x; y� � tan−1��3

p�Ih1 − Ih3�∕�2Ih2 − Ih1 − Ih3�� (26)

� tan−1��3

p�I1 − I3�∕�2I2 − I1 − I3��: (27)

From this equation, we can see that what matters isthe difference values of I1 − I3 and 2I2 − I1 − I3. 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 Criteria

A. SPWM

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 fre-quencies. In addition, to maintain the fundamentaldifference between SPWM and SBM, the modulationfrequency must be much higher than the fundamen-tal frequency, i.e., f c > f 0. The performance is evalu-ated based upon the criteria of the introduced phaseerror instead of the sinusoidality appearance.

B. OPWM

Because the OPWM technique has the ability to se-lectively 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 op-timization problem, which often leads to multiple so-lutions. 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 simpli-fies the problem since there are fewer harmonics toconsider.

In other words, the most dominant phase error iscaused by fifth- and seventh-order harmonics. There-fore, if we could remove these two frequency compo-nents, 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. Simulations

A. Simulation on Influence of High-Order Harmonics

Our first simulation is to verify the influence of high-order harmonics on phase error. This simulation wascarried out by analyzing the phase error introducedby each frequency harmonics. Equation (22) indi-cates that the magnitudes of high-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 the-oretical 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 as

Ik�x; y� � 0.5� 23π sin�f 0x�

� 2�2k� 1�π sin��2k� 1�f 0x�: �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 indi-vidual harmonic on phase error. This confirms that

no phase error is introduced by 3nth-order harmo-nics, and the phase error decreases when the otherharmonic order increases. This indicates that it isnot sufficient to look at the appearance of fringe pat-terns nor their frequency spectra [18]. Example inFig. 5 clearly shows that the less sinusoidal patternscould have less phase errors (thus better measure-ment quality) than those seemingly better sinusoidalpatterns.

B. Simulation on SPWM Optimization

The defocusing effect can be approximated as a Gaus-sian smoothing filter. A 2D Gaussian filter is usuallydefined as

G�x; y� � 1

2πσ2 e−�x−�x�2��y−�y�2

2σ2 . (29)

Here σ is standard deviation and �x and �y are meanvalues of the x and y axis, respectively.

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Fig. 5. (Color online) Influence of high-order harmonics on phase error. (a) Fringe pattern containing � 11th-order harmonics compo-nents. (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.

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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).


For our case, because the structured stripes areeither vertical or horizontal, only one cross sectionperpendicular to the fringe stripes needs to be con-sidered, which means the problem is reduced to1D. A 1D Gaussian filter is defined as

G�x� � 1��2π

pσe−

�x−�x�22σ2 . (30)

We first simulate the defocusing effect by applyinga Gaussian filter to the signal. The degree of defo-cusing 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 devia-tion 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. In

the simulation shown in Fig. 6(b), we determinedhow the modulation frequency affects the measure-ment 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 in-crement of 2 pixels is used for the modulation pulsewidth to ensure the symmetry of a triangular wave-form. 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 qual-ity of 3D shape measurement by reducing measure-ment error. Figure 6(c) shows the results. This

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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).

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Fig. 8. (Color online) OPWMoptimization example. The first row shows a bad OPWMpattern, 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.


simulation is to compare the phase error between theoriginal binary patterns and the SPWM patterns un-der 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 modu-lated 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 pat-terns increases, the phase error does not signifi-cantly 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 simu-lation shows that when the patterns are significantlydefocused, a modulation period of 6 pixels stillgives close to the minimum errors for different fre-quency of fringe patterns, while between 6 and 12

pixels, the results are decent. However, in comparisonwith the phase error for the traditional binary meth-od, 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 simula-tion 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 optimization

The 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 compo-nents should not be considered during the opti-mization 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 OPWMpat-terns 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 deviation

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Fig. 9. (Color online) Validation of the ideal sinusoidal fringe pat-terns utilized as reference. (a) The unwrapped phase map. (b) Theactual phase error that will be coupled into the real measurement.

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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.


of 5 pixels). Figures 8(c) and 8(g) show the Fourierspectra after applying the smoothing filter. Most ofthe high-order harmonics were completely sup-pressed by the smoothing filter for the better de-signed OPWMpattern shown in Fig. 8(e). Finally, thephase errors shown in Figs. 8(d) and 8(h) confirmthat if the fifth and seventh harmonics are well elimi-nated, the phase error would be very small.

5. Experiments

A. Experimental System Setup

Experiments 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 16mm fo-cal lengthMega-pixel lens (Model: ComputarM1614-MP) 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 dis-tance of 0.49–2.80 m.

B. Validation of Ideal Sinusoidal Fringe PatternGeneration

This experiment is to verify that the adopted conven-tional sinusoidal method does not bring significantphase error, since we used the phase obtained fromthese patterns as our reference. In the experiments,auniformwhite platewasused to quantitatively showthe introduced phase errors by thismethod. Since theprojector is a nonlinear device, the projection nonli-nearity needs to be corrected. In this research, weused the method proposed in [4] to actively changethe patterns before projection. Figure 9(a) shows

one cross section of the unwrapped phase map. Thelarger profile was introduced by 3D shape measure-ment 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 gener-al profile of the unwrapped phase, which is shown inFig. 9(b). It can be seen that the projection nonlinearinfluence was effectively alleviated and the intro-duced random error is negligible in comparison withthe digitization error.

C. Validation of Optimal Modulation Frequency Selectionfor SPWM

The 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. Figure10(c) shows the experimental

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Fig. 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 third-order 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).

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results for SPWM patterns with different pitcheswhen the projector is nearly focused. It indeedverifies that in most cases, the optimal modulationperiod is 6 pixels. When the projector is significantlydefocused, the corresponding results are shownin Fig. 10(d)–10(f). Again, it agrees with our simula-tion very well: the optimal modulation period ranges6–10 pixels.

D. Validation of OPWM

To verify the performance of different OPWM pat-terns, we designed two different OPWM patterns

to compare with the SBM patterns. Figure 11(a)shows the SBM pattern and the Fourier spectra ofthe cross section is shown in Fig. 11(d). Figure 11(b)shows the OPWM patterns with fifth and seventh-order harmonics well eliminated but with the third-order harmonics, and its Fourier spectra is shown inFig. 11(e). Figure 11(c) shows another OPWM pat-tern with third-order harmonics eliminated butkeeping the fifth- and seventh-order harmonics,and Fig. 11(f) shows its Fourier spectra. The phaseerrors of these three sets of patterns are shown inFig. 12. From this experiment, we can see that the

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Fig. 13. (Color online) Comparisons among SBM, SPWM, and OPWM under their respective optimal conditions. (a) The projector isnearly focused. (b) The projector is significantly defocused.

Fig. 14. (Color online) 3D complex shape measurement under different amounts of defocusing. The first row shows the result when theprojector is nearly focused, and the second row shows the results when the projector is significantly defocused. (a) Binary pattern whenprojector is nearly focused. (b) 3D result using SBM patterns. (c) 3D result using SPWM patterns. (d) 3D result using OPWM patterns.(e) Binary pattern when projector is more defocused. (f) 3D result using SBM patterns. (g) 3D result using SPWM patterns. (h) 3D resultusing OPWM patterns.


fifth and seventh harmonics could cause severephase errors. On the other hand, the phase error isreally small when the magnitude of the third-orderharmonic is very large. This finding matches wellwith our simulation. This means if those undesiredharmonics are well eliminated, the phase errors willbecome small enough for measurement. However,this experiment also shows that if the OPWM pat-tern (as shown in the OPWM pattern 2) was poorlydesigned, it could drastically deteriorate the mea-surement quality.

E. Comparison among SBM, SPWM, and OPWM

Experimentswerealso carried out to compare theper-formance of different techniques under their respec-tive optimal conditions. In this set of experiments,againwe considered two cases, the nearly focused pro-jector and the significantly defocused projector. TheSPWM and OPWM patterns were optimized accord-ing to the findings discussed above. Figure 13(a)shows the error comparison results with a nearly fo-cused projector. Figure 13(b) shows the results whentheprojector is significantlydefocused.Fromtheseex-periments, we can see that when the projector isnearly focused, the OPWM performs the best whilethe SPWM performs better than SBM if the fringepitch is large. When the projector is significantly de-focused, the SBM performs similarly as the OPWMtechnique when the fringe is dense, while the SBMperforms theworstwhen the fringe stripe is verywide.

F. Complex Shape Measurement

Additional experiments were conducted to measurecomplex 3D shape using these techniques and to com-pare their performance visually. In this experiment,we used P � 60 pixels fringe patterns for all threemethods and performed the measurement when theprojector is nearly focused and significantly defo-cused. Figure 14(a) shows one of the fringe patternsfor the SBM patterns when the projector is nearlyfocused; it clearly shows the binary structure.Figures 14(b)–14(d) respectively show the recon-structed 3D results from SBM, SPWM, and OPWMpatternswhen the projector is nearly focused. This ex-periment indeed shows that the SPWMperforms bet-ter than theSBM,while theOPWMperforms thebest.

Figure 14(e) shows one of the fringe patterns forthe SBM patterns when the projector is significantlydefocused, and the patterns are close to being sinu-soidal. 3D shape measurement results under thisdegree of defocusing are shown in Figs. 14(f)–14(h).Under this condition, all methods, as expected, resultin reasonable good-quality 3D shape measurement,with the OPWM delivering the best result again.It should be noted that this system was calibratedusing the reference-plane based method describedin [19], and the phase was unwrapped using the qual-ity-guided phase-unwrapping algorithm presentedin [20]. Even though the calibration method usedis not very accurate, it helps to present the 3D datafor visual comparison.

6. Conclusion

This paper has presented some comparisons amongthe recently proposed three binary defocusing tech-niques: SBM, SPWM, and OPWM. To achieve this,different breaths of fringe patterns were projectedfor test. The projector was set to be nearly focusedand be significantly defocused. The phase errorswere obtained by comparing phase maps from thosethree patterns with sinusoidal pattern. In this paper,the optimization for designing SPWM and OPWMpatterns were also presented. Finally, our experi-ments have found that: (1) the OPWM performsthe best if it was well optimized, (2) the SBM per-forms well when the fringe stripe is narrow or theprojector is well defocused, and (3) the SPWM per-forms better than the SBM in most cases when thefringe stripe is wide enough. We hope this presenta-tion would be helpful for those who would like to usethe binary defocusing techniques for 3D shape mea-surement, as this paper can guide them to selectthe proper method under their measurementconditions.

The authors would like to thank Leah Merner forher proofreading this paper.

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Comparison of the squared binary, sinusoidal pulse width ... · X(pixels) Intensity 20 40 60 80 100 120 20 40 60 80 100 120 140 X(pixels) Intensity Fig. 1. (Color online) Influences