3D profilometry based on digital sinusoidal fringeprojection techniques has been playing an increas-ingly important role in optical metrology due to therapid advancement of digital video display technol-ogy [1]. However, challenges remain to perform high-quality 3D shape measurement with off-the-shelfdigital video projectors. One of the major issues isto generate ideal sinusoidal fringe patterns becauseof the projector’s nonlinear gamma effect.
To circumvent this problem, we recently reported atechnique that only needs binary structured patterns[2]. The sinusoidal fringe patterns were generated byproperly defocusing the binary structured ones. Suet al. has also used the defocusing technique to gen-erate ideal sinusoidal fringe patterns by defocusingwith a Ronchi grating [3]. However, because it usesa mechanical grating, the phase-shift error is domi-nant. In contrast, because the digital binary fringeprojection technique does not have a phase-shift er-ror, it turned out to have more advantages besideseliminating the nonlinear gamma problem: it allowsfor an unprecedentedly high-speed 3D profilometrywith a phase-shifting technique using the digital-light-processing (DLP) Discovery projection platform[4], and permits the 3D profilometry speed bottle-neck elimination of an off-the-shelf DLP projector
[5]. However, this technique is not trouble-free. Be-cause the projector must be properly defocused togenerate high-quality sinusoidal fringe patterns,there are two major problems: (1) the smaller mea-surement range, and (2) the challenge of calibratingthe defocused projector [2].
This paper proposes a novel method that onlyrequires binary structured patterns to realize pixel-level spatial resolution 3D profilometry. This tech-nique is based on our theoretical analysis andexperimental findings. Our theoretical analysisshows that the low frequency (less than eleventh or-der) harmonics of a square wave only introduce 6Xphase error for a three-step phase-shifting techniquewith a phase-shift of 1=3 period (or 2π=3), and thisphase error can be significantly eliminated by aver-aging two sets of fringe patterns with a phase-shift of1=12 period (or π=6). If the projector is slightly de-focused, meaning that the frequency componentsbeyond tenth order harmonics are suppressed to anegligible level, the phase error caused by the binarypatterns could be reduced to be less than RMS 0.2%,which is less than the quantization error of an 8 bitcamera (1=28 ≈ 0:4%).
Our further analysis shows that when the projec-tor is close to being in focus, the next dominant errorfrequency doubled, i.e., 12X. This type of phase errorcan be eliminated by introducing another two sets offringe patterns with a phase-shift of 1=24 period (orπ=12) from the first two sets. Averaging the phases
obtained from these four sets will bring down theerror to be less than 0.2%. Because under most mea-surement scenarios, the projector or the cameraare not perfectly in focus, using these 12 binarypatterns is sufficient to achieve high-quality 3Dprofilometry.
Of course, if higher accuracy is required, additionalsets of binary phase-shifted patterns with similaranalysis can be adopted to further improve themeasurement quality. By this means, only binarypatterns are necessary to achieve the spatial resolu-tion of the conventional sinusoidal fringe patternsbased method. Therefore, this technique allows forachieving high measurement accuracy with binarypatterns instead of sinusoidal ones, which might in-troduce better means for 3D profilometry because itis significantly easier to generate binary patternsthan to generate ideal sinusoidal ones.
Section 2 introduces the principle sinusoidal andbinary phase-shifting algorithms. Section 3 showssome simulations to verify the proposed binaryphase-shifting techniques. Section 4 presents someexperimental results, and Sec. 5 summarizes thispaper.
2. Principle
A. Single Three-Step Phase-Shifting Algorithm
Phase-shifting methods are widely used in opticalmetrology because of their speed and accuracy [6].A single three-step phase-shifting (SPS) algorithmwith a phase-shift of 2π=3 can be described as
Where I0ðx; yÞ is the average intensity, I00ðx; yÞ the in-tensity modulation, and ϕðx; yÞ the phase to be solvedfor. Simultaneously solving these three equationsgives the phase
ϕðx; yÞ ¼ tan−1
� ffiffiffi3
pðI1 − I3Þ=ð2I2 − I1 − I3Þ
�: ð4Þ
The phase obtained in Eq. (4) ranges from −π to þπwith 2π discontinuities. A phase unwrapping algo-rithm can be adopted to obtain the continuous phase[7]. The phase unwrapping is to locate the 2π discon-tinuity positions and remove them by adding or sub-tracting multiples of 2π. In other words, the phaseunwrapping step is to find an integer numberkðx; yÞ for each point ðx; yÞ so that the continuousphase can be obtained as
Φðx; yÞ ¼ 2π × kðx; yÞ: ð5ÞHere, Φðx; yÞ is the unwrapped phase. Once the con-tinuous phase map is obtained, 3D shape can berecovered if the system is calibrated [8].
B. Dual Three-Step Phase-Shifting Algorithm
The SPS works well if the fringe patterns are ideallysinusoidal in profile. However, for binary phase-shifted fringe patterns, if the projector is not properlydefocused, some binary structures will appear, andthe phase error will be significant. To learn how tocompensate for this type of phase error, the binarystructured patterns are analyzed.
The cross section of a binary structured pattern isa square wave, thus, understanding the effect of abinary structured pattern can be simplified to studya square wave. A normalized square wave with a per-iod of 2π can be written as
yðxÞ ¼�0 x ∈ ½ð2n − 1Þπ; 2nπÞ1 x ∈ ½2nπ; ð2nþ 1ÞπÞ : ð6Þ
Here, n is an integer number. The square wave can beexpanded as a Fourier series
yðxÞ ¼ 0:5þX∞k¼0
2ð2kþ 1Þπ sin½ð2kþ 1Þx�: ð7Þ
To understand how each harmonics affects the mea-surement quality, we analyzed the phase error byeach frequency component. The phase error is ob-tained by finding the difference between the basephase Φbðx; yÞ obtained from the fundamental fre-quency and the phase obtained from the combinationof the fundamental frequency and the particularhigh-frequency harmonics, Φkðx; yÞ. The base phasecan be theoretically computed by applying Eq. (4).The fringe patterns with (2kþ 1)-th order harmonicsfrequencies can be written as
Similarly, the wrapped phase can obtained by using asimilar equation as Eq. (4)
ϕkðx; yÞ ¼ tan−1
� ffiffiffi3
pðIk1 − Ik3Þ=ð2Ik2 − Ik1 − Ik3Þ
�: ð11Þ
Phase ϕkðx; yÞ can be unwrapped to determine theΦkðx; yÞ. The phase error is thus defined as
ΔΦkðx; yÞ ¼ Φkðx; yÞ −Φðx; yÞ: ð12ÞFigure 1 shows the phase error for each harmonics ifa three-step phase-shifting algorithm is utilized. Itshows that the third, ninth, and fifteenth harmonics
(k ¼ 1, 4, 7) will not bring phase error because aphase-shift of 2π=3 is utilized. It also indicates thatthe low frequency harmonics (less than eleventh) in-troduces 6X phase error. If another phase map with aphase-shift of 1=12 period (or π=6) is obtained, it canbe used to compensate for this error by averaging itwith the other phase map. Because two sets of fringepatterns are used, this technique is called the dualthree-step phase-shifting (DPS) method.
C. Quadratic Three-Step Phase-Shifting Algorithm
Figure 1 also shows that the secondary phase error is12X. This means that if another two sets of fringepatterns have a phase-shift of 1=24 period (orπ=12) between the first two sets, the 12X phase errorcan be eliminated. Because four sets of fringe pat-terns are used to eliminate both 6X and 12X phaseerror, this technique is called the quadratic three-step phase-shifting (QPS) method. After applyingQPS, the residual phase errors are induced by har-monics higher than fifteenth order, which can be neg-ligible if the patterns are not perfectly focused (orsquared). It is important to notice that the 12X phaseerror is induced by over tenth-order harmonics. If thepatterns are slightly defocused, this type of error canbe suppressed to a negligible level, thus, the DPSmethod might be sufficient.
3. Simulations
Simulations were performed to verify the perfor-mance of the proposed methods. In this simulation,we simulated a square wave with a fringe period of96 pixels. The defocusing is realized by applying aGaussian smoothing filter, and the different degreesof defocusing are achieved by using different breathof filters. Larger size of Gaussian filters are realizedby applying smaller size ones multiple times [9]. Inthis research, we utilize a 9-pixel Gaussian filter
with a standard deviation of 1.5 pixels. If this filteris applied once, the square wave will be deformed asshown in Fig. 2(a). Because the filter size is verysmall in comparison with the square wave period(96 pixels), the shape of the square wave is well pre-served. If an SPS algorithm is applied, the phase er-ror is very large (RMS 0:22 rad or 3.46%) as shown inFig. 2(b). The DPS method reduces the error to be1.07% (3.5 times smaller). The phase error obtainedfrom the QPS method makes is negligible (0.10%) incomparison with the quantization error of an 8 bitcamera (1=28 ≈ 0:4%). This simulation confirmedthat the QPS algorithm can generate satisfactory re-sult even when the binary patterns are close to ideal.
If the same filter applies four times, the squarewave will be further deformed but still has clearbinary structures, as shown in Fig. 2(c). Figure 2(d)shows that the phase errors are RMS 1.61%, 0.11%,and 0.02% for the SPS, DPS, and QPS methods,respectively. It should be noticed that the DPS canreduce the phase error to be approximately 0.11%,which is negligible. This simulation confirmed thatwhen the binary patterns are slightly defocused, theDPS is sufficient to provide high-quality 3D shapemeasurement.
4. Experiments
Experiments were also conducted to test the pro-posed method. In the experiment, we used a USBCCD camera (The Imaging Source DMK 21BU04)and the LED digital-light-processing projector (DellM109S). The camera is attached with a 12mm focallength Megapixel lens (Computar M1214-MP). Theresolution of the camera is 640 × 480. The projectorhas a resolution of 858 × 600 with a lens of F=2:0and f ¼ 16:67mm.
We first measured a uniform white surface withthe proposed technique. Figure 3 shows the measure-ment results. In this experiment, we used a very widefringe pattern, where the fringe pitch (number ofpixels per period) is 96 projector’s pixels. The firstrow shows the results when the fringe patterns areclose to being in focus. The binary structures areclearly shown in the fringe patterns as illustrated inFig. 3(b). Figure 3(c) shows that even on the wrap-ped phase map, the phase error is very obvious.Figures 3(d)–3(f) show the phase error maps for theSPS, DPS, and QPS, respectively. It can be seen from
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Fig. 2. (Color online) Phase errors with using different phase-shifting methods under different degrees of defocusing. (a) Close to being anideal square wave; (b) Phase errors for the SPS, DPS, and QPS methods are RMS 3.46%, 1.07%, and 0.10%, respectively; (c) Slightlyblurred square wave; (d) Phase errors for the SPS, DPS, and QPS methods are RMS 1.61%, 0.11%, and 0.02%, respectively.
the error map that the SPS method generates signif-icant error, while the QPS method reduces the errorto a negligible level. The second row of Fig. 3 showsthat when the fringe patterns are slightly defocused,the DPS method is sufficient to perform high-qualitymeasurement.
Figure 4 shows the cross sections of the phase errormaps shown in Fig. 3. Figure 4(a) shows the phaseerrors generated by different methods when the pro-jector is close to being in focus. It can be seen thatwhen the projector is close to being in focus, theSPS method clearly does not generate reasonablequality of measurement, while the DPS method im-proves its quality dramatically. And the phase errorcaused by the QPS method is less than 0.21%, whichis very low. Figure 4(b) shows the results when theprojector is slightly defocused. For this case, theDPS and the QPS does not make much difference,thus, a DPS method is sufficient to perform high-quality measurement. These experiments demon-
strated that the real measurements conform to oursimulation results.
A more complex 3D sculpture was also measured.Figure 5 shows the results with different SPS, DPS,and QPS algorithms under the same defocusing de-gree. These experiments indicated that when theprojector is close to being in focus, a QPS method canperform high-quality 3D profilometry, while only theDPS method is needed for slightly defocused binarypatterns. It should be noted that the binary patternsare very wide: the fringe pitch is 96 projector’s pixel.
5. Summary
This paper has presented binary phase-shift meth-ods for high-resolution 3D profilometry. Because thistechnique allows the binary method to perform pixel-level spatial resolution when the projector is close tobeing in focus, it solved the two very challengingproblems (smaller depth range and difficulty ofdefocused projector calibration) for the technique
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Fig. 3. (Color online) Phase errors with using different phase-shifting methods. (a) Fringe pattern is close to being in focus; (b) 320th rowcross section; (c) Wrapped phase map; (d)–(f) Phase error maps for the SPS, DPS, and QPS methods, respectively; (g) Fringe pattern isslightly defocused; (h) 320th row cross section; (i) Wrapped phase map; (j)–(l) Phase error maps for the SPS, DPS, and QPS methods,respectively.
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Fig. 4. (Color online) Phase errors with using different phase-shiftingmethods. (a) Cross sections of the phase error shown in the first rowof Fig. 3. The phase errors are 2.67%, 0.46%, and 0.17% for the SPS, DPS, and QPS methods, respectively; (b) Cross sections of the phaseerror shown in the second row of Fig. 3. The phase errors are 1.48%, 0.21%, and 0.14% for the SPS, DPS, and QPS methods, respectively.
to achieve sinusoidal phase-shifting methods by de-focusing binary structured ones. This technique,however, is at the cost of increasing the number offringe patterns used, which will reduce the measure-ment speed. Nevertheless, this proposed method hasgreat value in 3D profilometry when only binary pat-terns can be used (e.g., grating), and it simplifies thedigital fringe projection system development withoutworrying about the nonlinearity of the projector.
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Fig. 5. (Color online) Experimental results with different binary phase-shifting methods. The top row shows the results when the pro-jector is close to being in focus, and the bottom row shows the results when the projector is slightly defocused. (a) One of the binary fringepatterns; (b) 3D result with the SPS method; (c) 3D result with the DPS method; (d) 3D result with the QPS method; (e) One of the binaryfringe patterns; (f) 3D result with the SPS method; (g) 3D result with the DPS method; (h) 3D result with the QPS method.
