generic nonsinusoidal phase error correction for three ...€¦ · method using a simple...

Generic nonsinusoidal phase error correction forthree-dimensional shape measurement using a digitalvideo projector

Song Zhang and Shing-Tung Yau

A structured light system using a digital video projector is widely used for 3D shape measurement.However, the nonlinear � of the projector causes the projected fringe patterns to be nonsinusoidal, whichresults in phase error and therefore measurement error. It has been shown that, by using a small look-uptable (LUT), this type of phase error can be reduced significantly for a three-step phase-shifting algo-rithm. We prove that this algorithm is generic for any phase-shifting algorithm. Moreover, we proposea new LUT generation method by analyzing the captured fringe image of a flat board directly.Experiments show that this error compensation algorithm can reduce the phase error to at least 13times smaller. © 2007 Optical Society of America

OCIS codes: 120.0120, 120.2650, 120.2650, 120.5050.

1. Introduction

Optical noncontact 3D surface profile methods, in-cluding stereovision, laser scanning, and time orcolor-coded structured light, have been developed toobtain a 3D contour.1,2 Structured-light-based meth-ods have the advantage of fast 3D shape measurementand high accuracy, among which various phase-shifting methods have been widely used due to theiraccuracy and efficiency.3 For phase-shifting methods,the nonsinusoidal waveform error and the phase-shifterror are two major error sources.4

With the development of digital display technolo-gies, commercial digital video projectors are more andmore extensively utilized in 3D shape measurementsystems. A structured-light-based system differs froma stereo-based system by replacing one of the camerasof the stereo system with a projector, and projectscoded structured patterns, which help to identify thecorrespondence. The structured-light system based ona binary-coding method has the disadvantages of spa-

tial resolution and measurement speed since the stripewidth must be larger than one pixel. But it is robust tothe nonlinear � of the projector since only two levels (0sand 1s) are used for this method. In contrast, a systembased on a phase-shifting method has the advantagesof measurement speed and resolution because it canobtain the coordinates of the object pixel by pixel. Thedrawback of such a system is that it is sensitive to thenonlinear � of the projector; any nonlinear effect de-forms the ideal sinusoidal fringe pattern to be nonsi-nusoidal and introduces error. The phase error causedby this nonsinusoidal waveform is the single dominanterror source of the measurement because the phase-shift error does not appear in this method due to thedigital fringe pattern generation nature. Since anymeasurement error of a phase-shifting-based systemwill appear in the phase, similarly, the error in phasewill appear in the final measurement. In this research,the measurement error due to the nonsinusoidal wave-form is simply called the phase error. For accuratemeasurement, the nonlinear � of the projector is notdesirable. However, the � of any commercial projectoris purposely distorted to be nonlinear to have bettervisual effect. To obtain better measurement accuracy,this type of phase error has to be reduced. Previouslyproposed methods, including the double three-stepphase-shifting algorithm,5 3 � 3 phase-shiftingalgorithm,6,7 and direct correction of the nonlinearityof the projector’s �,8 demonstrated significant phaseerror reduction; however, the residual error remainsnonnegligible. Guo et al. proposed a � correction

The authors are with the Department of Mathematics, HarvardUniversity, Cambridge, Massachusetts 02138. S. Zhang’s e-mailaddress is [email protected]. S.-T. Yau’s e-mail address [email protected].

Received 30 May 2006; revised 3 September 2006; accepted 7September 2006; posted 11 September 2006 (Doc. ID 71457); pub-lished 15 December 2006.

0003-6935/06/010036-08$15.00/0© 2007 Optical Society of America

36 APPLIED OPTICS � Vol. 46, No. 1 � 1 January 2007

method using a simple one-parameter � function tech-nique by statistically analyzing the fringe images.9This technique significantly reduces the phase errordue to the nonlinear �. However, the one-parameterassumption for the � function of the projector is toostrong. The actual � of the projector is complicated.Therefore this method cannot completely remove theerror due to the projector’s nonlinear �. Skocaj andLeonardis proposed an algorithm to compensate forthe nonlinear effect of the projector–camera system forrange image acquisition of the objects with nonuniformalbedo.10 The algorithm works well for their purpose.However, it is time consuming to calibrate the nonlin-ear function, and the accuracy is not high since theydid not account for the ambient light effect.

Zhang and Huang proposed a method by calibratingthe � of the projector, precomputing the phase errorthat is stored in a look-up table (LUT), which was usedto reduce the phase error significantly for a three-stepphase-shifting algorithm.11 In this research we provethat this type of phase error compensation method isnot limited to the three-step phase-shifting method. Itis generic for any phase-shifting method. The phaseerror compensation algorithm is able to theoreticallyeliminate the phase error caused by the projector’s �completely. It is based on our finding that, in the phasedomain, the phase error due to the � of the projector ispreserved for the arbitrary object’s surface reflectivityunder an arbitrary ambient lighting condition. How-ever, the method introduced by Zhang and Huangrequires calibration of the projector’s �, which is atime-consuming procedure. Moreover, it requires theprojectors � to be monotonic, so that the phase errorLUT can be created. In this research, we propose amethod that does not require calibration of the � ofthe projector directly and does not require the � to bemonotonic. Instead, we use a calibration board, imageit, and analyze the fringe images to obtain the phaseerror LUT. A uniform flat surface is utilized for such apurpose. We captured a set of fringe images of a uni-form flat surface board. The phase error of the cap-tured fringe images is analyzed and stored in a LUTfor phase error compensation. This method is also suit-able for use in other phase-shifting-based systemswhen the nonlinearity of the fringe generator is diffi-cult to obtain. Therefore it is more general than thepreviously proposed method. Our experimental resultsshow that, by using this method, the measurementerror can be reduced by at least 13 times for any phase-shifting algorithm.

In Section 2 we explain the theoretical backgroundof this error compensation method. In Section 3 weexplain the LUT creation procedures. Section 4 showsthe experimental results. Section 5 discusses the ad-vantages and limitations of the method, and Section6 concludes the paper.

2. Principle

A. Least-Squares Algorithms

The phase-shifting method has been adopted exten-sively in optical metrology to measure 3D shapes of

objects at various scales. Many different sinusoidalphase-shifting algorithms have been developed; thegeneral form of phase-shifting algorithms is the least-squares algorithms. The intensity of the ith imageswith a phase shift of �i is as follows:

Ii�x, y� � I��x, y� � I ��x, y�cos��x, y� � �i�, (1)

where I� is the average intensity, I� is the intensitymodulation, and � is the phase to be solved.

Solving Eq. (1) simultaneously by the least-squaresalgorithm, we obtain

��x, y� � tan�1��a2�x, y�a1�x, y� �, (2)

��x, y� �I ��x, y�I��x, y�

��a1�x, y�2 � a2�x, y�2�1�2

a0�x, y�, (3)

where

�a0�x, y�a1�x, y�a2�x, y�

� A�1��i�B�x, y, �i�. (4)

Here,

A��i� � N � cos��i� � sin��i�

� cos��i� � cos2��i� � cos��i�sin��i�� sin��i� � cos��i�sin��i� � sin2��i�

�,(5)

B�x, y, �i� � � Ii

� Ii cos��i�� Ii sin��i�

�, (6)

where I��x, y� � a0�x, y� can be used to obtain the flatimage of the measured object. Equation (2) provides theso-called modulo 2� phase at each pixel, whose valuesrange from 0 to 2�. If a multiple fringe pattern is used, acontinuous phase map can be obtained by phase unwrap-ping.12 The continuous phase map can be further con-verted to coordinates through calibration.13,14

B. Phase Error Correction

The images captured by the camera are formedthrough the procedures illustrated in Fig. 1. Let us

Fig. 1. Camera image generation procedure.

1 January 2007 � Vol. 46, No. 1 � APPLIED OPTICS 37

assume that the projector’s input sinusoidal fringeimages generated by a computer have intensity

Iis�x, y� � b1 1 � cos��x, y� � �i�� b0, (7)

where b1 is the dynamic range of the fringe imagesand b0 is the bias. After being projected by the pro-jector, the output intensity of the fringe images be-comes

Iip�x, y� � fi�Ii

s�, (8)

where fi�Iis� is a function of Ii

s, which represents thereal projection response of the projector to the inputintensity for the ith image, the �. If we assume thatthe projector projects light onto a surface with reflec-tivity r(x, y) and the ambient light a1�x, y�, the re-flected light intensity is

Iio�x, y� � r�x, y��Ii

p�x, y� � a1�x, y��, (9)

The reflected image is captured by a camera with asensitivity of �, here we assume the camera is thelinear response to the input light intensity, namely, �is a constant. Then, the intensity of the image cap-tured by the camera is

Iic�x, y� � ��Ii

o � a2�x, y��, (10)

� �r�x, y�Iip � �r�x, y�a1�x, y� � �a2�x, y�,

(11)

� c1Iip � c2, (12)

where a2�x, y� represents ambient light enteringthe camera. c1 � �r�x, y� and c2 � �r�x, y�a1�x, y� ��a2�x, y�.

If we assume

C � A�1 ��C00 C01 C02

C10 C11 C12

C20 C21 C22, (13)

then

a2�x, y� � C20 � Iic � C21 � Ii

c cos��i�� C22 � Ii

c sin��i�, (14)

� C20 � �c1Iip � c2� � C21 � �c1Ii

p � c2� cos��i� � C22 � �c1Ii

p � c2�sin��i�, (15)

� c1�C20 � Iip � C21 � Ii

p cos��i�� C22 � Ii

p sin��i�� C2�C20N

� C21 � cos��i� � C22 � sin��i��. (16)

From C � A�1, i.e., CA � identity. From Eqs. (5) and(13) for definition of matrices A and C, the third row,first column element in the resultant matrix is zero,we have

C20N � C21 � cos��i� � C22 � sin��i� � 0.

If we substitute these constraints into Eq. (16), weobtain

a2�x, y� � c1�C20 � Iip � C21 � Ii


p sin��i��. (17)

Similarly, we can prove that

a1�x, y� � c1�C10 � Iip � C11 � Ii


p sin��i��. (18)

From Eq. (2), phase ��x, y� is

Fig. 2. (Color online) Phase error LUT generation. (a) Real wrapped phase and the ideal wrapped phase. (b) Phase error relative to thereal wrapped phase.


From Eqs. (19)–(21) we can see that phase ��x, y� isindependent of the response of the camera, the re-flectivity of the object, and the intensity of the ambi-ent light. This indicates that the phase error due tononsinusoidal waveforms depends only on the non-linearity of the projector’s �. Therefore if the projec-tor’s � is calibrated, and the phase error due to thenonlinearity of the � is calculated, a LUT that storesthe phase error can be constructed for error compen-sation, since the wrapped phase �̂�x, y� for the idealsinusoidal fringe images with vertical stripes increaseslinearly from 0 to 2� horizontally within one period.

However, once the fringe images are distorted to benonsinusoidal due to the nonlinear effects of the sys-tem, e.g., the � of the projector, the real wrappedphase ��x, y� is nonlinear. The difference between thewrapped phase and the ideal linear one is

��x, y�� x, y� � kx. (22)

Here k � 2��N is the slope of the line ranging from0 to 2�; N is the number of points sampled in onefringe period. � is a function of the computed realphase value ��x, y�. The error LUT is therefore con-structed as a map ��, �. Figure 2 shows the phaseerror creation method. In this figure, we used 312sampling points to show the concept. Figure 2(a) plotsthe real wrapped phase values and the ideal phase

values for each sampling point. We subtract the realphase value by the ideal phase value for each pointand obtain its phase error. The phase error corre-sponding to the real wrapped phase value is plottedin Fig. 2(b). Once the discrete values of phase errorsare obtained for the corresponding phase, the contin-uous phase error map can be obtained by fitting thedata with splines. The error LUT can then be createdby sampling the spline curve and storing the value ina LUT.

��x, y� � tan�1��a2�x, y�a1�x, y� �, (19)

� tan�1��c1�C20 � Iip � C21 � Ii

p cos��i� � C22 � Iip sin��i��

c1�C10 � Iip � C11 � Ii

p cos��i� � C12 � Iip sin��i�� , (20)

� tan�1��C20 � Iip � C21 � Ii

p cos��i� � C22 � Iip sin��i�

C10 � Iip � C11 � Ii

p cos��i� � C12 � Iip sin��i� �. (21)

Fig. 3. (Color online) Phase error with a fringe pitch of 120.

Fig. 4. (Color online) Phase error for fringe pitches 60 and 120.Fig. 5. (Color online) Phase error for multiple fringe pitches�Pitch � 60 � 30 k, k � 1, 2, . . . , 6�.


3. Look-Up-Table Generation

Zhang and Huang proposed a phase error LUT gen-eration method by calibrating the � of the projector,precomputing the phase error value for the corre-sponding phase value, and storing them in a LUT.11

They demonstrated that the proposed method canreduce the error to ten times smaller. However, thismethod requires calibration of the projector’s �,which is a time-consuming procedure. Moreover, itrequires the projector’s � to be monotonic, so that thephase error LUT can be created. In this research, wepropose a method that does not require the calibra-tion of the � of the projector directly and does notrequire the � to be monotonic. Instead, we use a cal-ibration board, image it, and analyze the fringe im-ages to obtain the phase error LUT. A uniform flatsurface is utilized for such a purpose.

To study the characteristics of the phase error, weproject four phase-shifted fringe patterns with an ar-bitrary phase shift of �1 � 0°, �2 � 270°, �3 � 130°,and �4 � 220° onto the calibration board, capture theimages using the camera, and compute the phaseerror using the method introduced in the previoussection. The phase error is shown in Fig. 3. In thisfigure, the fringe pitch, the number of pixels perfringe period, used is 120. We used ten-row points inthe center of the image. It can be clearly seen that thephase error is a pattern similar to the phase.

We further analyze the phase error using two dif-ferent fringe pitches, 60 and 120, respectively. The

phase errors of ten rows on different points from thecenter of the image are shown in Fig. 4. It can be seenthat for different fringe pitches, the phase error pat-tern is also similar. From Fig. 4 we can also see thatthe phase error is independent of the projected fringepitch. To verify this, we image the same board andobtain the phase error from a series of fringe imagesets for different fringe pitches. The result is shownin Fig. 5. The fringe pitch numbers used in this ex-periment range from 60 to 240 within intervals of 30.This figure shows that the phase error pattern is wellmaintained for different fringe pitches. These ex-periments demonstrated that the phase error is in-dependent of the fringe pitch number used in themeasurement. Therefore the phase error LUT can bebuilt based on one pitch fringe image set.

In this research we use the fringe pitch of 120 as anexample to create the LUT. We project four fringeimages with the same phase shift as previously spec-ified onto a uniform flat board, capture them, andcompute the phase error. Figure 6 plots the phaseerror for 101 row points. The phase error map canthen be quantized and stored in a LUT. Let us as-sume that we want to create a 256-element errorLUT. We then divide the 2� period into 256 regions.For the kth region, the phase value ranges from�k � 1�2��256 to k2��256. The average error of thedata points that belong to this region is stored in thekth element of the LUT. The solid curve in Fig. 6 plotsthe errors stored in the LUT. Once the phase errorLUT is created, it can be used for future error compen-sation. For real captured fringe images, the phasecomputed from the fringe images can then be com-pensated for by using this LUT. Assume the phasevalue for one point is �0, which belongs to regionk0 � �256�0�2��, here the operator [x] provides therounded integer value of x. Then, the compensatedphase is �� 0 � LUT�k0�.

4. Experimental Results

To demonstrate the performance of the proposedphase error compensation algorithm, we built a 3Dshape measurement system using a digital videoprojector (Optoma EP739) and a CCD digital cam-era (Dalsa CA-D6-0512). The digital video projec-tor projects four phase-shifted fringe images withphase shifts of �1 � 0°, �2 � 270°, �3 � 130°, and�4 � 220°, the CCD camera captures the reflected

Fig. 6. (Color online) Phase error LUT creation.

Fig. 7. Phase-shifted fringe images for the flat board: (a) I1 ��1 � 0°�, (b) I2 ��2 � 270°�, (c) I3 ��3 � 130°�, (d) I4 ��4 � 220°�.


fringe images by the object. Figure 7 shows all thefringe images of the flat board. Figure 8 shows the3D measurement results before and after phase er-ror correction. The phase error before error reduc-tion is approximately rms 0.16 rad and reduces torms 0.012 rad once the phase error compensation al-gorithm is applied. The error is approximately 13times smaller.

We also implemented the error compensation algo-rithm in our real-time 3D shape system using a three-step phase-shifting method.15–17 Figure 9 shows themeasurement results of the flat board using the al-gorithm proposed in this paper and the one developedin Ref. 11. In comparison with the previously pro-posed method, the error compensation method pro-posed here is much simpler and faster, since no

Fig. 8. (Color online) Three-dimensional measurement result of flat board before and after error compensation. (a) Three-dimensionalgeometry without correction. (b) Three-dimensional geometry after correcting the phase. (c) Cross section of the 250th row of the aboveimage (rms: 0.16 rad). (d) Cross section of the 250th row of the above image (rms: 0.02 rad).

Fig. 9. (Color online) Comparison of the proposed method and the method developed in Ref. 11. (a) Phase error before compensation(rms: 0.16 rad). (b) Phase error after error compensation with the algorithm proposed in this paper (rms: 0.012 rad). (c) Phase error aftererror compensation using the previously proposed algorithm (rms: 0.006 rad).


projector’s � calibration is required, although its ac-curacy is mitigated.

In addition, we measured a more complex plastermodel, Zeus’s head. Figure 10 shows the reconstructed3D models before and after error compensation. Thereconstructed 3D geometric surface after error com-pensation is much smoother and has better visualeffects. These experimental results confirm that theerror correction algorithm successfully improved theaccuracy of measurement significantly.

5. Discussions

The phase error compensation method discussed inthis paper has the following advantages:

Y Simple. The compensation algorithm is simplesince the phase error can be easily corrected using asmall LUT.

Y Accurate. In theory, the phase error due to anonlinear � curve can be completely eliminated aslong as the projection response functions can be de-termined accurately by calibration.

Y Universal. The method discussed can correctthe phase errors due to the nonsinusoidal waveformfor any phase-shifting algorithm and for any systemusing a phase-shifting-based method.

On the other hand, the algorithm is based on theassumption that the camera is a linear device. Thisassumption is true for most of the cameras, althoughsome cameras may be nonlinear. If the camera isnonlinear, it has to be calibrated before applying thisalgorithm.

6. Conclusions

We have introduced an error compensation methodfor the 3D shape measurement system using a digitalvideo projector. We proved that the phase errorcaused by the nonlinear � of the projector can betheoretically completely eliminated for any phase-

shifting algorithm. It is based on our finding that thephase error caused by the projector’s nonlinear � ispreserved for the arbitrary object’s surface reflectiv-ity under an arbitrary ambient lighting condition. Asimple phase error LUT creation method was intro-duced that uses a uniform calibration board andfringe image analysis technique. The experimentalresults demonstrated that by utilizing a small LUT(256 elements in this research), the phase error canbe reduced to at least 13 times smaller. Theoreticalanalysis and experimental findings were also pre-sented.

This work was supported by Geometric Informatics,Inc., funded by the Advanced Program Project (ATP)of the National Institute of Standards and Technol-ogy (NIST).

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Fig. 10. (Color online) Three-dimensional measurement results of a sculpture (a) before and (b) after error compensation.


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generic nonsinusoidal phase error correction for three ...€¦ · method using a simple...

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