Gamma correction for digital fringe projection profilometry

Transcription

1 Gamma correction for digital fringe projection profilometry Hongwei Guo, Haitao He, and Mingyi Chen Digital fringe projection profilometry utilizes a digital video projector as a structured light source and thus gains great flexibility. However, the gamma nonlinearity of the video projector inevitably decreases the accuracy and resolution of the measurement. We propose a gamma-correction technique based on statistical analysis of the fringe images. The technique allows one to estimate the value of gamma from the normalized cumulative histogram of the fringe images. By iterating the two steps, gamma estimation and phase evaluation, the actual gamma value can be calculated. At the same time the phase distribution of the fringe pattern can be solved with higher accuracy. In so doing, neither photometric calibration nor knowledge of the device is required. Both computer simulation and experiment are carried out to demonstrate the validity of this technique Optical Society of America OCIS codes: , , , , Introduction Fringe projection profilometry, a well-known noncontact three-dimensional measurement technique, has been extensively developed to meet the demands of various applications. The fundamental principle of this technique is that periodic fringe patterns are projected on the object surface, and the distorted patterns caused by the depth variation of the surface are recorded from different directions. The phase distributions of the distorted fringe patterns are often recovered by phase-shifting techniques 1,2 or the methods based on Fourier transformation analysis, 3 7 and then the depth map of the object surface is further reconstructed. For this purpose the projection of a grating slide has been popularly used to obtain a periodic fringe pattern. 2 7 The main limitation of this method is that it is difficult to fabricate a sinusoidal grating, and it is inconvenient to vary the phase and frequency of fringe pattern. Another available choice is use of a laser interferometer to generate sinusoidal fringes in the measurement space. 1,8 10 With this approach the phase and frequency of the fringe pattern can be manipu- The authors are with the Laboratory of Applied Optics and Metrology, Shanghai University, Shanghai 20072, China for H. Guo, hw-guo@yeah.net. Received 5 September 2003; revised manuscript received 20 January 2004; accepted 13 February $ Optical Society of America lated either by a mechanical or by an electronic actuator, but the stringent requirement for environmental stability degrades the practicability of this technique in many manufacturing environments. To overcome these shortcomings, recently a substitute technique termed as digital fringe projection profilometry has created great interest, which uses a digital video projector, such as a LCD or a digital light processing projector, as an illuminating device and offers the following advantages over conventional methods: arbitrary grating shape and profile, high speed for phase shifting, variable grating frequency, and insensitivity to circumstance disturbance However, the problem is that the luminance nonlinearity of the video projector significantly decreases the accuracy and resolution of the measurement. As a matter of fact the inherent luminance nonlinearity of a display device can often be described with a simple pointwise operation of the form w u, (1) where u 0, 1 denotes the normalized image pixel value, w is the normalized actual output intensity, and is a constant particular to the device. In the first case the function comes from the physics of the cathode-ray-tube monitor in which the brightness of the screen is proportional to the voltage applied to the electron gun raised to the power gamma. The process used to correct this power-law response phenomenon is referred to as a gamma correction APPLIED OPTICS Vol. 43, No May 2004

2 Although there are many disputes about the veracity of the simple function, newly developed display technologies emulate the behavior not only for compatibility but also for reasons based on human vision science. Typically a display device has a gamma greater than 1.0, and the standard of the National Television System Committee recommends a gamma of 2.2. However, the actual gamma of a digital video projector depends not only on its own factory preadjustment but also on the computer system. In addition, some graphics cards dynamically vary the amount of gamma correction for a balanced visual effect on the display. The standard approach to determining the actual gamma of an imaging device or a display is to perform a photometric calibration. In the existing techniques for digital fringe projection a similar calibration procedure is frequently used to quantify the nonlinearity of a fringe projection system. 13,15 A target plane is set and illuminated by the projector with a full range of known luminance values, and the intensities are recorded with the camera. As a result the response curve of the overall system is plotted and employed to compensate for the influence of the nonlinearity. Because the curve depends not only on the behavior of the projector but also on the factors including the response of the camera, the reflectivity and slope of the target plane, as well as the environmental brightness, one has to interpret and use it carefully. In fact there are some techniques for eliminating the effect of luminance nonlinearity in the absence of any calibration information and knowledge of the device. For example, in Ref. 18 the amount of gamma correction imposed on a natural image can be blindly estimated by using the tools based on polyspectral analysis. Obviously the polyspectral analysis can also be used in our measurement system to estimate the gamma value of the video projector directly from the distorted fringe patterns; the principal drawback is its prohibitive computational complexity. In practical measurements the phase-shifting technique is recognized as one of the most popularly used solutions without photometric calibrations. This technique allows extracting the phase of fundamental order from a temporal intensity signal and removing the effect of harmonics caused by the luminance nonlinearity. However, its success in the elimination of higher-order harmonics usually involves sampling a relatively large number of fringe patterns. Detailed discussions of this issue are found in Refs. 19 and 20. Another possibility is to project black and white halftone fringe patterns on the object surface. Because the intensity the average intensity of a halftone pattern depends on an area of dots instead of gray levels, this approach is immune to the luminance nonlinearity in which only a lower resolution can be achieved. In this paper we propose a gamma-correction technique for digital fringe projection profilometry. The technique is based on the statistical analysis of fringe images. The cumulative distribution functions CDFs of sinusoidal signals homologically have fixed forms independent of phases and frequencies of the signals. When a sinusoidal intensity signal passes through a system with gamma nonlinearity, only a nonsinusoidal output can be obtained whose CDF is strongly dependent on the gamma of the system. This means that, by analyzing the CDF, the actual gamma of the system can be estimated. With our technique the estimation procedure can be summarized as follows: First, we evaluate the initial phase distribution by employing a conventional phaseshifting algorithm. Second, we eliminate the ununiformities of background brightness and modulation of the fringe images and then, according to the intensity statistic of the images, we estimate the value of gamma. Third, by using the estimate of the gamma value, we evaluate the phase distribution once again to enhance its accuracy. By iterating the two steps, the gamma estimation and phase evaluation, we can calculate the actual gamma value. At the same time the phase distribution of the fringe pattern can be solved with higher accuracy. With this technique the effect of luminance nonlinearity caused by gamma is eliminated without photometric calibrations and knowledge of the device. Although the phase-shifting technique is used in this approach, the number of required fringe images notably decreases. Both computer simulations and the experiment are carried out and give support to the technique. 2. Principle A. Gamma-Estimation Principle Insight is gained into the proposed technique by considering a simple one-dimensional sinusoidal signal as follows: u x a b cos 2 fx, x,, (2) where a denotes the bias of the signal, b is the amplitude of the waveform, f represents the frequency, and represents the phase. Without losing generality, we let u x satisfy 0 a b u x a b 1. When the signal is passed through a system with gamma nonlinearity, the output is w x a b cos 2 fx, x,, (3) where is an unknown positive used to describe the nonlinearity of the system. Obviously w x satisfies 0 a b w x a b 1. Let P indicate probability. The CDF can be deduced from Eq. 3 according to its definition, that is, F w P w x w 1 1 arccos w1 a b, a b w a b. (4) Clearly the CDF is not relative to the frequency and phase of the input signal but depends strongly on the value of gamma as shown in Fig. 1. Note that 10 May 2004 Vol. 43, No. 14 APPLIED OPTICS 2907

3 Fig. 2. Values of LUTs with, solid curve, parameters a b 0.5 and, dashed curve, parameters a and b the estimate of CDF. We obtain Let ˆ denote the estimate of. H w w 1 1 arccos w1 ˆ a b. (8) Fig. 1. Signal waveform and CDF curve as changed by gamma nonlinearity: a response curves of system with different gamma values, b input sinusoidal signal, c output waveforms, d curves of CDFs. the dependence relationship between and the CDF allows one to estimate the gamma value from the CDF. In the measurement practice the output signal usually suffers from a linear operation in addition to gamma nonlinearity and is often discrete. Consider the output signal given by s n, where n 0, 1,..., N 1 is an integer-valued vector. Let s max and s min denote their maximum and minimum, respectively. Then the normalized cumulative histogram can be calculated as H s 1 N 1 n, where n 0 if s n s N 1 if s n s. (5) n 0 In Eq. 5 s s min, s max. Noting that the mapping relationship between s and w is linear and is given by w s s min a b a b s max s min a b, (6) we have H s H w a b s max s min a b a b s min H w w. (7) H w w also represents a normalized cumulative histogram where w is the parameter to appear on the horizontal axis. In fact H w w can be considered as By replacing Eq. 6 with Eq. 8 and using Eq. 7,we obtain an equation with unique unknown ˆ as H s 1 1 arccos ( 1 b s s min a b ˆ a b ˆ s max s min ˆ 1 ˆ a b a b). (9) If we explicitly specify a value for s, ˆ can be solved from Eq. 9. To simplify the computation, we specify s max s min 2, and then the equation is restated as H s max s min ( arccos 1 b a b ˆ a b ˆ 1 ˆ 2 a b). (10) To solve ˆ from Eq. 10, an approach based on the look-up table LUT is employed. First, a range of possible gamma values from 0.1 to 4.0 is sampled in increments of Afterward, by substituting in sequence the gamma values into the right-hand side of Eq. 10, we calculate a corresponding vector and use it to establish the LUT. In Fig. 2 we plot the curves of the LUT values versus gamma values with different parameters. If the normalized cumulative histogram of an output signal is calculated, the value of the left-hand side of Eq. 10 is determined. Thus, by searching the approximation of this value in the LUT, the corresponding gamma can be found. In this procedure a linear interpolation can be used to enhance the resolution of the gamma estimation when the value searched for lies between two adjacent LUT values APPLIED OPTICS Vol. 43, No May 2004

4 Step 1. Because parameters a and b are known, a LUT can be established by using the approach described in Subsection 2.A. Step 2. Let ˆ i, j, Rˆ i, j, and Bˆ i, j denote the estimated values of i, j, R i, j, and B i, j, respectively. Ignoring the influence of nonlinearity assuming that ˆ 0 1 and according to the conventional phase-shifting algorithm, 1 we have Fig. 3. Measurement system. B. Gamma Correction for Digital Fringe Projection Figure 3 shows the measurement system commonly used in digital fringe projection, which consists of a digital projector, a CCD camera, an image board, and a computer system. First, the total N frames of sinusoidal fringe patterns are generated in the computer with a constant phase increment 2 N between two consecutive frames. The nth frame can be represented as g n x, y a b cos 2 x p 2 n N, (11) where n 0, 1,..., N 1 is an integer-valued vector, x, y denotes the coordinates of the point in the fringe pattern, and then g n x, y is the gray level at point x, y. On the left-hand side of Eq. 11 a is the bias, b is the amplitude, and p is the fringe pitch. Second, the fringe patterns are cast on the object surface in sequence by the digital projector, and then a series of distorted patterns is captured by the CCD camera. After analog-to-digital conversion the nth distorted pattern can be denoted as I n i, j R i, j a b cos i, j 2 n N B i, j, (12) where i, j are the pixel coordinates in the captured image, I n i, j is the intensity at point i, j, denotes the phase in which the depth information of the object surface is included, R is a parameter relative to the reflectivity and slope of the object surface, and B is the additional background brightness caused by the environmental illumination. If gamma is not equal to unity 1, luminance nonlinearity is introduced into the images. Parameters R and B both are functions of i, j. This means that both the background brightness and the modulation are not uniform across the images and hence spoil the cumulative histogram of the images. Consequently the approach described in Subsection 2.A cannot be accessed directly, and the nonuniformities of background brightness and modulation have to be eliminated before the gamma estimation. In this case an iterative procedure is proposed, and the techniques adopted in each step can be summarized as follows, in which the parenthesized superscript k is used to denote the number of iterations: N 1 ˆ 0 i, j arctan I n i, j sin 2 n N n 0 N 1, (13) I n i, j cos 2 n N Rˆ 0 i, j 2 N 1 Nb n 0 N 1 n 0 N 1 n 0 2 I n i, j sin 2 n N 2 I n i, j cos 2 n N 1 2, (14) Bˆ 0 i, j 1 I n i, j arˆ i, j. (15) N n 0 Then the images are processed by J n 0 i, j I n i, j Bˆ 0 i, j Rˆ 0 i, j, (16) where J 0 n i, j denotes the image in which the ununiformities of background brightness and modulation are eliminated. Step 3. If J k n i, j is calculated, from it the maximum and minimum can be determined and are denoted J max and J k min, respectively. Then k the value of H J max k J min k 2 is given by H J max k J min k 2 1 M i, j where n i, j 0 1 N 1 n i, N j, n 0 (17) if J k n i, j J k max J k min 2 1 if J k n i, j J k max J k min 2 and M is the number of points involved in computation. According to the approach in Subsection 2.A, by searching for the value in the LUT, ˆ k 1 is estimated. Note that in this step the unreliable points, such as low-modulation points and luminance saturation points, must be excluded from the computation, because these points induce great errors in determining the values of J k max and J k min. In Ref. 21 a method to remove the unreliable points from fringe images is provided. Step 4. From Eq. 12 we see that the presence of gamma makes the phase distribution difficult to calculate, since the nonlinear equation system has to be 10 May 2004 Vol. 43, No. 14 APPLIED OPTICS 2909

5 solved. To simplify the computation, Eq. 12 is rewritten in terms of its Taylor series expansion: I n i, j Î n i, j Î n i, j R i, j R i, j Î n i, j B i, j B i, j Î n i, j i, j i, j,, (18) where Î n i, j Rˆ i, j a b cos ˆ i, j 2 n N ˆ Bˆ i, j, R i, j R i, j Rˆ i, j, B i, j B i, j Bˆ i, j, and i, j i, j ˆ i, j. Ignoring the high-order nonlinear terms and replacing ˆ i, j, Rˆ i, j, and Bˆ i, j with ˆ k i, j, Rˆ k i, j, and Bˆ k i, j, respectively, and replacing i, j, R i, j, and B i, j with k 1 i, j, R k 1 i, j, and B k 1 i, j, respectively, yield R k 1 i, j a b cos ˆ k ˆ k i, j 2 n N B k 1 i, j k 1 i, j Rˆ k i, j ˆ k b a b cos ˆ k i, j 2 n N ˆ k 1 sin ˆ k i, j 2 n N I n i, j Rˆ k i, j a b cos ˆ k i, j 2 n N ˆ k Bˆ k i, j. (19) For a certain pixel i, j, let n equal 0, 1,..., N 1 in sequence. A linear system with N equations is obtained based on Eq. 19, and the three unknowns, k 1 i, j, R k 1 i, j, and B k 1 i, j, can be solved. Consequently ˆ k i, j, Rˆ k i, j, and Bˆ k i, j are further corrected as ˆ k 1 i, j ˆ k i, j k 1 i, j, (20) Rˆ k 1 i, j Rˆ k i, j R k 1 i, j, (21) Bˆ k 1 i, j Bˆ k i, j B k 1 i, j. (22) Then the ununiformities of background brightness and modulation of the images are eliminated once again, that is, J k 1 n i, j I n i, j Bˆ k 1 i, j Rˆ k 1 i, j. (23) By iterating Steps 3 and 4 until the algorithm converges, we estimate the value of gamma and at the same time solve the phase distribution with higher accuracy. 3. Computer Simulations To verify the validity of this technique, computer simulations have been performed. Assuming that three fringe patterns are generated with the parameters a 0.5 and b 0.5, the phase increment between two consecutive frames is 2 3. The LUT is established, and its values are shown in Fig. 2 by the solid curve. A predefined phase distribution is given in Fig. 4. Accordingly, the synthetic distorted patterns with pixels can be calculated by Eq. 12,in Fig. 4. Predefined phase distribution a with the carrier frequency and b without the carrier frequency. which the ununiformities of background brightness and modulation are purposely introduced by varying the values of R i, j and B i, j across the images. Different gamma values with 0.45, 0.80, 1.0, 1.5, 2.2, 3.0 are used to introduce luminance nonlinearity. For example, the fringe images with a gamma of 2.2 are shown in Fig. 5. In the absence of noise the gamma values are estimated by using the suggested approach and are listed in Table 1, in which a threshold of is set for the increment of the estimated gamma to automatically control the number of iterations. At the same time the phase distributions are recovered. By checking the differences between the recovered phase distributions and their predefined prototype, the accuracies can be investigated for both the suggested technique and the conventional phase-shifting algorithm. As a result the peak-to-valley PV errors are also listed in Table 1. In addition, when 2.2, the phase-error curves of both approaches along a horizontal section of image are graphically compared in Fig. 6, and the results demonstrate that by the suggested technique the measurement accuracy is remarkably enhanced. To investigate the convergence of this approach, we illustrate in Fig. 7 the relationship between estimated gamma values and the number of iterations. If the actual gamma is 2.2, 15 iterations simply give an estimate of 2.15, and after 43 iterations the result is Also from Table 1 we see that the convergence speed of this approach is relative to the value of the actual gamma. The larger deviation of the actual gamma from unity leads to an increased number of iterations. Table 1 shows the examples where, if the actual gamma is 0.45 or 3.0, the iterative procedure is stopped by the preset threshold before the algorithm fully converges. With this approach, in Fig. 5. Synthetic fringe patterns with a phase shift of a 0, b 120, and c APPLIED OPTICS Vol. 43, No May 2004

6 Table 1. Results of Simulations in the Absence of Noise Actual Gamma Estimated Gamma Suggested Approach Iteration Number PV Phase Error PV Phase Error of Conventional Phase-Shifting Algorithm rad Estimated Gamma with the Method of Polyspectral Analysis addition to the number of iterations, the computational time depends on available computer sources. Here a personal computer with a 1.7-GHz processor is used, and it takes several minutes to complete one overall computation. To further evaluate the performances of the proposed technique, the method based on polyspectral analysis as a comparison is also used to estimate the gamma values, 22 and the results are also reported in Table 1, from which we can see, in the absence of noise, that the gamma values can also be accurately estimated by using the tools base in the polyspectral analysis. As mentioned in Section 1, the principal drawback of this method is its prohibitive computational complexity. In the simulation, to process a pixel image, several hours are necessary. Another simulation is for investigating the robustness of the proposed technique in the presence of noise. In practical measurements the noise attached to the image is usually modeled as independent additive noise and of a zero-mean Gaussian distribution, and in a normalized image the standard deviation SD of the noise is generally not greater than the level of In the simulation the actual gamma is assumed to be 2.2, and the Gaussian noises with different SDs are added to the fringe images. Figure 8 shows the horizontal sections of the images corrupted by the noises. The SDs of the noises are 0.005, 0.01, 0.02, and Correspondingly, the gamma values are estimated with the proposed technique, and the results are , , , and , respectively. The satisfying results demonstrate that the proposed technique is robust for a gamma estimation in practical measurement. At the same time we failed to estimate the gamma by using the method based on the polyspectral analysis when the SD of noise is greater than the level of 10 4, and the sensitivity to the noise makes this approach unpractical in measurement. In addition, another attractive feature of the proposed technique is its Fig. 6. Phase errors rad of, solid line, the suggested approach and, dotted lines, the conventional phase-shifting algorithm, when the actual gamma is 2.2. Fig. 7. Relationship between the estimated gamma and number of iterations. (The actual gamma is 2.2. Fig. 8. Sections of the fringe images corrupted by additive Gaussian noises. The actual gamma is 2.2, and the noise SDs are a 0.005, b 0.01, c 0.02, and d The estimated gamma values are , , , and , respectively. 10 May 2004 Vol. 43, No. 14 APPLIED OPTICS 2911

7 Fig. 10. surface. Three-dimensional reconstruction result of the measured Fig. 9. Fringe image processing: a a deformed fringe pattern, b a wrapped phase map, c an unwrapped phase map, d the phase difference. repeatability. In view of its statistical nature the insensitivity to many unknown stochastic factors is guaranteed by the strong law of large numbers. 4. Experiment and Discussion Here the proposed approach is used to measure a plaster statue. The measurement system is shown schematically in Fig. 3. An LCD projector Toshiba TLP660 is used as a structured light source. The deformed fringe patterns are captured by a CCD camera Mintron MTV-1881EX and are digitized by an 8-bit image board Joinhope OK-M10M. A series of sinusoidal fringe patterns is generated in the computer with the parameters a and b Accordingly the LUT is established and shown in Fig. 2 by a dashed curve. The number of phase steps is 4, and the phase increment between two consecutive frames is 2. Figure 9 shows the procedure for fringe image processing: a one of the deformed fringe patterns, b the wrapped phase map obtained with the suggested method, c the unwrapped phase distribution, and d the phase difference between the measured object and a reference plane. According to the mapping relationship between the phase difference and the depth of the object surface, the threedimensional shape of the plaster statue is reconstructed as shown in Fig. 10. With this procedure after five iterations the algorithm converges and the value of gamma is estimated to be In addition the relationship between the estimated gamma values and the number of iterations is illustrated in Fig. 11. According to the estimated gamma value, the nonlinear luminance of the LCD projector is shown in Fig. 12 by the solid curve. The photometric calibration method described in Section 1 is also used to obtain the response curve of the LCD projector, and its result is shown in Fig. 12 by circles. Comparing the two curves, we find that they are in accord with each other in a large range of gray levels. This means that the nonlinearity of the projector can be effectively estimated by the proposed technique. An issue worth discussion is the effect of temporal luminance fluctuations on this technique, which may be caused by projector instability. In Ref. 23 the authors addressed the fact that, if a large number of phase steps are used, then the time for capturing the images is increased and the noticeable measurement error may be induced by luminance fluctuations. In fact the digital video projectors can perform highspeed phase shifting, and when the proposed technique is used, the number of required fringe images is notably decreased. Therefore with this technique the adverse effect of temporal luminance fluctuations can be restrained to a minimal degree. However, note that there are certain limitations and restrictions on these results. First, the camera is thought of as a linear device in this technique, but in fact this precondition is not always satisfied. For example, to precompensate for the nonlinearity of the displays, the gamma correction can be accomplished by analog circuits at the CCD camera, and typically a 0.45-power function as a standard option is often used for this purpose w u u. In this Fig. 11. Relationship between the estimated gamma and the number of iterations APPLIED OPTICS Vol. 43, No May 2004

8 The authors gratefully acknowledge the support of the Science & Technology Commission of Shanghai Municipality STCSM, China, project , and Shanghai Municipal Education Commission SMEC, China, project 03AQ82. Fig. 12. Luminance nonlinearity of the LCD projector: solid curve, suggested algorithm gamma ; circles, photometric calibration result. case the nonlinearity of the camera should be removed by its inverse function before subsequent processing stages. Second, a simple one-parameter gamma function is used in this technique, but it is unlikely to accurately model the full range of the device. In this experiment, for example, there is a toe region and a shoulder region on the calibration curve as shown in Fig. 12, and with this phenomenon the curve cannot be described perfectly by a simple one-parameter gamma function. Even so, the suggested technique can still be used to improve the accuracy of the phase recovery, but its effectiveness is remarkably reduced. In addition, we are currently investigating a higher-parameter gamma-correction approach for digital fringe projection profilometry, in which a more sophisticated algorithm is required. 5. Conclusion Digital fringe projection profilometry utilizes a digital video projector as a structured light source and thus gains great flexibility. However, the gamma nonlinearity of the video projector inevitably decreases the accuracy and resolution of the measurement. In this paper we have proposed a gamma-correction technique for digital fringe projection profilometry on the basis of the statistical analysis of the fringe images. The technique allows one to estimate the value of gamma from the normalized cumulative histogram. By iterating the two steps, the gamma estimation and the phase evaluation, the actual gamma value can be calculated. At the same time the phase distribution of the fringe pattern can be solved with higher accuracy. Both computer simulation and the experiment are carried out and demonstrate that this technique offers several advantages over conventional methods. First, prior knowledge of the projector is not required, and the time-consuming photometric calibration is avoided. Second, in addition to the simplicity of the computation, high accuracy can be achieved and good repeatability and robustness are offered. References and Notes 1. V. Srinivasan, H. C. Liu, and M. Halioua, Automated phasemeasuring profilometry of a 3-D diffuse object, Appl. Opt. 23, V. Srinivasan, H. C. Liu, and M. Halioua, Automated phasemeasuring profilometry: a phase mapping approach, Appl. Opt. 24, M. Takeda and K. Mutoh, Fourier transform profilometry for the automatic measurement of 3-D object shapes, Appl. Opt. 22, K. H. Womack, Interferometric phase measurement using spatial synchronous detection, Opt. Eng. 23, S. Tang and Y. Y. Hung, Fast profilometer for the automatic measurement of 3-D object shapes, Appl. Opt. 29, M. Pirga and M. Kujawińska, Modified procedure for automatic surface topography, Measurement 13, J.-F. Lin and X.-Y. Su, Two-dimensional Fourier transform profilometry for the automatic measurement of threedimensional object shapes, Opt. Eng. 34, M. S. Mermelstein, D. L. Feldhun, and L. G. Shirley, Videorate surface profiling with acousto-optic accordion fringe interferometry, Opt. Eng. 39, G. S. Spagnolo and D. Ambrosini, Diffractive optical elementbased profilometer for surface inspection, Opt. Eng. 40, F. Wu, H. Zhang, M. J. Lalor, and D. R. Burton, A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry, Opt. Commun. 187, P. S. Huang, F. Jin, and F.-P. Chiang, Quantitative evaluation of corrosion by a digital fringe projection technique, Opt. Lasers Eng. 31, P. S. Huang, Q. Hu, and F.-P. Chiang, Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring, Opt. Eng. 38, C. R. Coggrave and J. M. Huntley, High-speed surface profilometer based on a spatial light modulator and pipeline image processor, Opt. Eng. 38, G. Sansoni, M. Carocci, and R. Rodella, Three-dimensional vision based on a combination of gray-code and phase-shift light projection: analysis and compensation of the systematic errors, Appl. Opt. 38, S. Kakunai, T. Sakamoto, and K. Iwata, Profile measurement taken with liquid-crystal grating, Appl. Opt. 38, Y. Y. Hung, L. Lin, H. M. Shang, and B. G. Park, Practical three-dimensional computer vision techniques for full-field surface measurement, Opt. Eng. 39, C. A. Poynton, Gamma and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video, SMPTE J. 102, H. Farid, Blind inverse gamma correction, IEEE Trans. Image Process. 10, K. A. Stetson and W. R. Brohinsky, Electro-optic holography and its application to hologram interferometry, Appl. Opt. 24, Y. Surrel, Design of algorithms for phase measurements by the use of phase stepping, Appl. Opt. 35, May 2004 Vol. 43, No. 14 APPLIED OPTICS 2913

9 21. W.-S. Li and X.-Y. Su, Phase unwrapping algorithm based on phase fitting reliability in structured light projection, Opt. Eng. 41, The fringe patterns are typically not the natural images described in Ref. 18, and the processing operation is different and can be described as follows: First, we measure and remove the additional background brightness from the images. Then we sample the full range of possible gamma values and calculate the high-order correlations by using the method described in Ref. 18. Last, we search the gamma value corresponding to the maximum of the high-order correlations. 23. M. Rivera, J. L. Marroquin, S. Botello, and M. Servin, Robust spatiotemporal quadrature filter for multiphase stepping, Appl. Opt. 39, APPLIED OPTICS Vol. 43, No May 2004