High-speed 3D shape measurement with fiber interference

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1 Mechanical Engineering Conference Presentations, Papers, and Proceedings Mechanical Engineering High-speed 3D shape measurement with fiber interference Beiwen Li Iowa State University, Pan Ou Beihang University Song Zhang Iowa State University, Follow this and additional works at: Part of the Computer-Aided Engineering and Design Commons, Graphics and Human Computer Interfaces Commons, and the Optics Commons Recommended Citation Li, Beiwen; Ou, Pan; and Zhang, Song, "High-speed 3D shape measurement with fiber interference" (214). Mechanical Engineering Conference Presentations, Papers, and Proceedings This Conference Proceeding is brought to you for free and open access by the Mechanical Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Mechanical Engineering Conference Presentations, Papers, and Proceedings by an authorized administrator of Iowa State University Digital Repository. For more information, please contact

2 High-speed 3D shape measurement with fiber interference Abstract This paper presents a miniaturized fringe projection system that only uses two fibers to potentially achieve superfast (e.g., MHz to GHz) 3D shape measurement speeds. The proposed method uses two optical fibers that carry the same wavelength of laser light with polarization and phase information properly modulated to generate high-quality sinusoidal fringe patterns through interference. The high-speed phase shifting is achieved by employing a high-speed Lithium iobate (L) electrooptic phase modulator. Since only two optical fibers are used to generate sinusoidal patterns, the system has a great potential of miniaturization for applications where the sensor size is critical (e.g., 3D endoscopy). Principle of the proposed techniques will be introduced, and preliminary experimental results will be presented in this paper to prove the success of the proposed method Keywords Finge projection, high-speed, fiber interfeerence, phase shift, miniaturization Disciplines Computer-Aided Engineering and Design Graphics and Human Computer Interfaces Optics Comments This is a conference proceeding from Interferometry XVII: Techniques and Analysis 923 (214): 1, doi:1.1117/ Posted with permission. Copyright 214 Society of Photo-Optical Instrumentation Engineers. One print or electronic copy may be made for personal use only. Systematic electronic or print reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. This conference proceeding is available at Iowa State University Digital Repository:

3 Invited Paper High-speed 3D shape measurement with fiber interference Beiwen Li 1, Pan Ou 2, and Song Zhang 1+ 1 Department of Mechanical Engineering, Iowa State University, Ames, IA, USA School of Instrumentation Science and Opto-electronics Engineering, Beihang University, Beijing, China ABSTRACT This paper presents a miniaturized fringe projection system that only uses two fibers to potentially achieve superfast (e.g., MHz to GHz) 3D shape measurement speeds. The proposed method uses two optical fibers that carry the same wavelength of laser light with polarization and phase information properly modulated to generate high-quality sinusoidal fringe patterns through interference. The high-speed phase shifting is achieved by employing a high-speed Lithium iobate (L) electrooptic phase modulator. Since only two optical fibers are used to generate sinusoidal patterns, the system has a great potential of miniaturization for applications where the sensor size is critical (e.g., 3D endoscopy). Principle of the proposed techniques will be introduced, and preliminary experimental results will be presented in this paper to prove the success of the proposed method. Keywords: Fringe projection, high-speed, fiber interference, phase shift, miniaturization 1. ITRODUCTIO Three-dimensional (3D) shape measurement is enjoying a wide range of applications from manufacturing to biomedical science. Among the existing technologies, fringe projection techniques have drawn attention from scientific research and industrial practices because of the high speeds and high accuracy which can be achieved. Over the years, researchers have been endeavoring to increase the measurement speeds while working to decrease the physical footprint since highspeed, ultra-compact, and flexible 3D shape measurement systems are greatly needed in applications where there are space constraints and where motion is involved. In medicine, for example, it is of great challenge for any existing 3D endoscope to measure dynamically deformable organs accurately and quickly. The state-of-the-art digital fringe projection (DFP) technique can achieve high-quality 3D shape measurement with high speeds. 1 However, DFP systems are usually rather bulky and the miniaturization is nontrivial due to the complicated optical system design, albeit researchers have endeavored to bring such methods to 3D endoscopic sensing. 2 In general, it is still not easy for conventional DFP methods to achieve real time (e.g., 3 Hz or better) 3D shape measurement speeds with a reasonably high resolution (e.g., 64 48). Our recently developed binary-defocusing method could simplify the DFP system development and simultaneously achieve unprecedented high speeds (e.g., khz), 3, 4 yet its speed bottleneck has been reached due the mechanical motion of the digital micro mirrors and its transient effect cannot be ignored. 5 Interference methods using coherent light could also generate phase-shifted sinusoidal fringe patterns, and such methods have been well-studied and extensively utilized in the field of optical metrology with numerous commercial products on the market. Fiber optic interferometric fringe projection has the great potential for system miniaturization since the fiber is very small (e.g., in µm) and only two fibers are required to generate a sinusoidal pattern. 6 It is well known that to achieve high-quality 3D shape measurement, phase-shifting methods are usually adopted. To introduce phase shifts, one of the most popular approaches is to use a piezoelectric transducer (PZT) In such methods, the phase shifts are generated by wrapping the optical fibers around the PZT tube. Alternatively, phase shifts can be introduced by modulating the frequency of a tunable laser through varying injection current These approaches have demonstrated their success in implementation and for compact system design. However, their measurement speeds are either limited by the modulation rate of the mechanical moving part (i.e. PZT), or by the tuning rate of the injection current. To the best of our knowledge, none of them have been able to achieve khz rate 3D shape measurement which the DFP systems can achieve. This research endeavors to simultaneously address the speed bottleneck as well as the system miniaturization issues related to the existing high-speed 3D shape measurement. In the proposed system, we use the well-developed fiber optic + song@iastate.edu; phone ; fax ; song. Interferometry XVII: Techniques and Analysis, edited by Katherine Creath, Jan Burke, Joanna Schmit, Proc. of SPIE Vol. 923, SPIE CCC code: X/14/$18 doi: / Proc. of SPIE Vol

4 interferometric method to generate fringe patterns through interference, and introduce the Lithium iobate (L) electrooptic phase modulator to achieve superfast phase shifts. Since the L electro-optic phase modulator can change the phase of the laser at MHz, even GHz, rates by properly synchronizing with the camera and the projector, the proposed technique could achieve unprecedented high speeds. Furthermore, since the proposed system only uses two optical fibers to generate sinusoidal patterns, the projection head is much smaller than that of the digital video projectors; this enables the possibility to be embedded into an endoscopic 3D sensing system for miniaturization. In the prototype system we developed, due to the limited power output of the laser source used and the large range of measurements, we achieved 1 Hz 3D shape measurement speeds. Even though 1 Hz is not a very high speed to our standard, the same principle can be applied to develop MHz, even GHz, 3D shape measurement systems. Therefore, this paper serves as a proof-of-concept rather than practical implementation. This paper will present the principle of the proposed technique and show some preliminary experimental results to verify the success of the proposed method. Section 2 explains the theoretical foundation of the proposed method. Section 3 shows simulation results. Section 4 presents some preliminary experimental rests, and finally, Section 5 summarizes this paper. 2. PRICIPLE In this research, we use two fibers to generate interference fringe patterns and phase-shifting algorithms to analyze the phase. Therefore, this section will briefly explain the principle of fringe pattern generation with two fibers and will also introduce the phase-shifting algorithms we use. 2.1 Fringe pattern generation with interference It is well know that, from physical optics, the wavefront of a light source can be described as w(x,y,t) = a(x,y)e i[φ(x,y)], (1) where x and y are spatial coordinates, a(x,y) the wavefront amplitude, and Φ(x,y) = 2πr(x,y)/λthe wavefront phase. Here λ is the wavelength, r(x,y) the distance that the light travels from its origin. Interference plane Figure 1: Illustration of wavefront interference in y z plane. Assume two wavefronts with the same wavelength λ, separated by a distance d along the y axis (as illustrated in Fig. 1), travel along the z direction; also assume the origin of the coordinate system is aligned with one point light source L 1. At any given point (x,y,z) in space, the wavefronts of these two point light sources are where w 1 (x,y,z) = a 1 (x,y,z)e i[φ 1(x,y,z)], (2) w 2 (x,y,z) = a 2 (x,y,z)e i[φ 2(x,y,z)], (3) Φ 1 (x,y,z) = 2πr 1 (x,y,z)/λ = 2π x 2 + y 2 + z 2 /λ, (4) Φ 2 (x,y,z) = 2πr 2 (x,y,z)/λ = 2π x 2 + (y d) 2 + z 2 /λ. (5) When these two wavefronts meet at a point (x, y, z), the intensity of the wavefront, which represents the resultant interference fringe pattern, can be written as I(x,y,z) = w 1 (x,y,z) + w 2 (x,y,z) 2, (6) = I (x,y) + I (x,y)cos[φ 1 (x,y) Φ 2 (x,y)], (7) Proc. of SPIE Vol

5 where I (x,y,z) = a 2 1 (x,y,z) + a2 2 (x,y,z) is the average intensity, and I (x,y) = 2a 1 (x,y)a 2 (x,y) is the fringe, or intensity modulation. If we define the phase difference as then we obtain the fundamental equation of the resultant fringe pattern: ϕ(x,y,z) = Φ 1 (x,y,z) Φ 2 (x,y,z), (8) I(x,y,z) = I (x,y,z) + I (x,y,z)cos[ϕ(x,y,z)], (9) where ϕ(x, y, z) is the unknown phase related to the temporal phase difference of this sinusoidal variation relative to the reference wavefront. Obviously, if Φ(x,y) = 2kπ, (k =,1,2,...), the interference pattern is at its peaks. This is equivalent to the condition where the travel distance between the reference wavefront and the testing wavefront is an integer number of λ. The interference pattern is at its valleys when the travel distance is (2n + 1)λ (n =,1,2,...). All previous analyses are based on the assumption that both light sources leave with phase and in phase. However, in reality, the starting phase may vary which will change the resultant fringe pattern slightly to I(x,y,z,t) = I (x,y,z) + I (x,y,z)cos[ϕ(x,y,z) + δ(t)], (1) here δ(t) is the initial phase difference that could vary over time. If δ(t) is modulated on purpose, phase-shifting algorithms can be applied to retrieve the phase for 3D shape measurement. 2.2 Least-squares phase-shifting algorithm Phase-shifting algorithms are extensively employed in optical metrology due to their high speed and high accuracy. 17 A variety of phase-shifting algorithms have been developed including three-step, four-step, least squares, etc. Typically, the more steps used, the better the measurement accuracy which can be achieved. For the least-squares phase-shifting algorithm 18, 19 with equal phase shifts, the k-th fringe image can be represented as I k (x,y) = I (x,y) + I (x,y)cos(φ + 2kπ/), (11) where δ(t) = 2kπ/ is the phase shift, and φ(x,y) the phase to be solved for, [ φ(x,y) = tan 1 ] k=1 I k sin(2kπ/) k=1 I. (12) k cos(2kπ/) This equation provides the wrapped phase ranging [ π, +π) with 2π discontinuities. A continuous phase map can be obtained by adopting a spatial 2 or temporal phase unwrapping algorithm. In this research, we used the spatial phase unwrapping framework introduced in SIMULATIOS We first simulated the interference pattern by varying d with a fixed z, where z is the distance of the imaging plane from the emitting plane, as shown in Fig. 2. These simulation results demonstrated that the larger the distance d, the denser the interference pattern. Figure 3 shows the interference pattern by varying z with a fixed d. It is obvious that as the interference plane moves away from the light source, the interference fringe stripes become wider and wider. In practice, reasonably dense fringe stripes are desirable for high accuracy 3D shape measurement. Therefore, for applications where the space is limited (e.g., for endoscopic applications), these simulation results indicate that controlling the distance between two fibers is critical. As introduced in Sec. 1, we used a phase modulator to achieve high-speed phase shifting, and thus high-speed 3D shape measurement. Theoretically, there should not be any phase shift error because the phase modulations are controlled electronically and precisely by a function generator. However, in reality, the single-mode fibers, used in our research system, carry random phase drift which will deteriorate the phase quality substantially if a phase-shifting algorithm is applied. To alleviate the phase error caused by inaccurate phase shifts in an interferometric system, researchers have developed numerous methods over the past few decades. Carré 22 proposed an algorithm that modestly compensates for phase shift Proc. of SPIE Vol

6 (a) (b) (c) (d) Figure 2: Simulation results of interference pattern by varying d with fixed z (λ = 82 nm). (a) d = 1 µm, z = 2 mm; (b) d = 2 µm, z = 2 mm; (c) d = 5 µm, z = 2 mm; (d) d = 8 µm, z = 2 mm. (a) (b) (c) (d) Figure 3: Simulation results of interference pattern by varying z with fixed d (λ = 82 nm). (a) d = 8 µm, z = 2 mm; (b) d = 8 µm, z = 3 mm; (c) d = 8 µm, z = 5 mm; (d) d = 8 µm, z = 8 mm. errors by using four fringe images with equal phase separation for a given phase shift. Hariharan 23 developed an algorithm that can better tolerate phase shift errors by using five fringe patterns. Larkin and Oreb 24 introduced an elegant symmetrical phase-shifting method that can suppress a specific harmonic. Hibino et al. 25 found that 2 j + 3 fringe images are necessary to eliminate jth-order harmonic. Phillion 26 introduced a generic and sophisticated method for phase-shift algorithm design. de Groot 27 derived an error-compensation algorithm for the phase-shifting interferometry using data-sampling windows, which was further analyzed by Schmit and Creath. 28 Researchers also found that the averaging technique 29 and the extended averaging technique 3 are also effective to compensate for phase-shift error. Freischlad and Koliopoulos 31 proposed an analytical technique which regards phase evaluation as a filtering process in the Fourier domain. The least squares algorithm 18, 19 can also be used to reduce error introduced by inaccurate phase shifts if a large number of samples are used. 32 In this research, we first simulate the influence of phase shift error on the resultant phase when least square phaseshifting algorithms with equal phase shifts are adopted. In the case where random phase shift error is present, the fringe images in Eq. (11) can be re-modeled as I k (x,y) = I (x,y) + I (x,y)cos[φ + (2kπ + ε k )/], (13) where ε k is the phase shift error of the k-th fringe image. The phase shift error will consequently introduce errors in phase obtained by Eq. (12). To determine the phase errors, we generated an ideal sinusoidal pattern with fringe period T = 3 pixels and then computed the phase error after applying 4-step and 1-step phase-shifting algorithms, respectively. In this simulation, we assigned the phase shift error ε k with random numbers within ±.1 rad. The simulation result is shown in Fig. 4. Fig. 4(a) and 4(b) respectively show the fringe pattern that we used and its cross section. Figure 4(c) and 4(d) respectively show the phase error of one cross section of the unwrapped phase maps after applying 4-step and 1-step phase-shifting algorithms. From these simulation results, we can observe that the inaccurate phase shifts induce a periodical phase error structure with the frequency being double of the pattern frequency; after applying a 1-step phaseshifting algorithm, the phase error is substantially smaller compared to a 4-step algorithm. These simulations indeed confirmed that by increasing the number of steps, the phase error caused by random phase shifts can be greatly alleviated, and that the 1-step phase-shifting seems to be sufficient for us. Proc. of SPIE Vol

7 Intensity Phase error (rad).1.1 Phase error (rad).1.1 (a) (b) (c) Figure 4: Simulation result of phase errors caused by inaccurate phase shift. (a) One fringe patten; (b) one cross-section of (a); (c) phase error of using a 4-step phase-shifting algorithm (rms error:.1 rad); (d) phase error of using a 1-step phase-shifting algorithm (rms error:.1 rad). (d) 4.1 System setup 4. EXPERIMETS Figure 5 shows the actual system we developed to verify the viability of the proposed method. It is composed of a laser fringe projection unit, a high-speed CMOS camera (Phantom V9.1) and a function generator (Tektronix AFG322B). The laser fringe projection unit includes a laser diode (LD) (QFBGLD-85-1) with an operating wavelength of 85 nm and a maximum output power of 1 mw. A coupler with a coupling ratio of 7/3 was used to separate the laser light into two fiber paths. Two polarization controllers (Thorlabs FPC3) were used to alter the polarization state of the laser in two light paths to achieve good fringe contrast. The L electro-optic phase modulator (Photline IR-MPX8-L-.1-P-P-FA-FA) is used to generate the phase shifts. The phase modulator has an operating wavelength of 85 nm and an electro-optical bandwidth of 15 MHz; its typical voltage V π for generating π phase shift is 2 V. LD#controller# Coupler# LD#, Polariza8on# controller# Phase# modulator# Emi<ng# fibers# Func8on## generator# High/speed# Camera# Figure 5: Photograph of the real measurement system. Figure 6 shows signals generated by the function generator to precisely synchronize the projection unit and the camera. In this system, the function generator supplies stair-type voltage signals to the phase modulator to introduce phase shifts for phase-shifted fringe patterns generation. In the meantime, the camera also takes the trigger signal from the function generator to ensure its precise synchronization with the projected phase-shifted fringe patterns. On this figure, V 2π is the required voltage to generate 2π phase shift, T is the period for projecting one set of phase-shifted patterns, and V open (= 5 V) is the required voltage to trigger the camera. For the stair wave coming from Channel 1, each stair corresponds to one of the -step phase-shifted fringe patterns. Channel 2 provides a 5 V pulse signal whenever a new phase-shifted fringe pattern starts its projection. By these means, the camera can precisely capture each of the phase-shifted fringe patterns. Since the particular phase modulator we employed can modulate the phase at 15 MHz, the maximum frame rate for the phaseshifted fringe patterns generated by the projection unit is also 15 MHz, and thus MHz 3D shape measurement speeds could be achieved. However, due to the low power of the laser diode (1 mw) we used and the large area (approximately 5 mm 5 mm) that we were interested in sensing, all our experiments were performed at 1 Hz (i.e., T =.1 sec). Proc. of SPIE Vol

8 Voltage ( 1)V 2 ( 2)V 2... Channel 1 2V 2 V 2 Voltage Channel 2 Time V open T 2T 3T ( 2) T ( 1) T T Figure 6: Design of waveform output from the function generator. Time 4.2 Experimental results To test the performance of our system, we carried out some experiments. Figure 7 illustrates the procedure of obtaining a continuous phase map. Figure 7(a) - 7(b) show two of the phase-shifted fringe images. The wrapped phase map, shown in Fig. 7(c), is determined by applying the phase-shifting algorithm introduced in Sec. 2.2, and the continuous phase map can be obtained by applying a spatial phase unwrapping algorithm. 21 The unwrapped phase map is shown in Fig. 7(d). Once the continuous phase is obtained, 3D geometry can be reconstructed by applying a phase-to-height conversion algorithm introduced in. 33 (a) (b) (c) (d) Figure 7: Example of generating continuous phase map using phase-shifting algorithms. (a)-(b) Two of the phase-shifted fringe images; (c) wrapped phase map; (d) unwrapped phase map. We first measured a white board to examine the influence of random phase shift error. In this research, we adopted both 4-step and 1-step phase-shifting algorithms to demonstrate their differences. Fig. 8 shows the experimental results. Figure 8(a) and 8(b) respectively show the cross-sections of the unwrapped phase obtained from 4-step and 1-step phaseshifting algorithms after removing their gross slopes. These figures show that the phase obtained from 4-step depicts a clear periodic structural error, while this is not obvious when the 1-step phase-shifting algorithm is adopted. To better visualize the periodical structural error, we applied a large Gaussian filter to the phase data and took the difference between the filtered phase and the original phase. Similar to the simulation results depicted in Fig. 4, the periodical structure error is substantial if the 4-step phase-shifting algorithm is adopted; such structural error is less obvious when the 1-step phase-shifting algorithm is used. This experiment confirmed that using more steps indeed could alleviate phase error caused by the fiber interferometric system we developed. Lastly, we measured a complex 3D statue, and Fig. 9 shows the results. Figures 9(a)-9(b) shows the result using a 4-step and 1-step phase-shifting algorithm, respectively. The corresponding zoom-in views are shown in Fig. 9(c)-9(d). These experimental results, once again, illustrate that the result obtained from the 1-step phase-shifting algorithm is much better than that from the 4-step algorithm. Similarly, the 4-step phase-shifting algorithm gives horizontal structural error Proc. of SPIE Vol

9 Phase (rad) (a) Phase (rad) (b) Phase (rad) (c) Phase (rad) Figure 8: Experimental results of measuring a white board. (a) Cross section of the unwrapped phase after removing gross slope for a 4-step phase-shifting algorithm; (b) cross section of the unwrapped phase after removing gross slope for a 1-step phase-shifting algorithm; (c) phase error for the 4-step phase-shifting algorithm; (d) phase noise for the 1-step phase-shifting algorithm. (d) that is caused by the phase shift error; structure error is not clearly shown when the 1-step algorithm is used. All our experimental results demonstrated that using a 1-step phase-shifting algorithm and a 1 Hz sampling rate is sufficient to reduce the phase error induced by random phase shift, albeit at the cost of slowing down the measurement speed. (a) (b) (c) (d) Figure 9: 3D measurement results of a small statue with complex geometry (Please refer to online version for color map visualization, and the color scale used for all figures is -3 mm to 1 mm presenting blue to red). (a) 3D results using 4-step phase-shifting algorithm; (b) 3D results using 1-step phase-shifting algorithm; (c) zoom-in view of (a); (d) zoom-in view of (b). 5. SUMMARY This paper has presented an optical fiber interferometric system that could simultaneously achieve unprecedented high speed (e.g., MHz even GHz) 3D shape measurement and system miniaturization. This proposed method used two optical fibers that carry the same wavelength of laser light with polarization and phase information properly modulated to generate high-quality sinusoidal fringe patterns through interferences. High-speed (i.e., MHz to GHz) phase shifts are realized by employing a Lithium iobate (L) electro-optic phase modulator. Since only two optical fibers are used to generate sinusoidal patterns, the projection head could be miniaturized (e.g., a few µm) for applications where the space is confined (e.g., 3D endoscopy). Because of the low power of the laser diode used and the rather large sensing area of interest, the prototype system realized 1 Hz 3D shape measurement, which is not very high to our standard. However, the same principle can be applied to develop MHz, even GHz, 3D shape measurement systems should a more powerful laser diode be used and or if a smaller sensing area is desired. Principle of the proposed techniques were introduced, and preliminary experimental results demonstrated the success of the proposed method. Proc. of SPIE Vol

10 Acknowledgments We thank Tyler Bell for polishing the writing of this paper. We also thank our sponsor, the ational Science Foundation (SF) under grant numbers: CMMI and CMMI-13376, for supporting this research. It is important to note that the views expressed in this chapter are those of the authors and not necessarily those of the SF. REFERECES [1] Zhang, S., Recent progresses on real-time 3-d shape measurement using digital fringe projection techniques, Opt. Laser Eng. 48(2), (21). [2] Geng, J. and Xie, J., Review of 3-d endoscopic surface imaging techniques, IEEE Sensors Journal 14(4), (214). [3] Zhang, S., van der Weide, D., and Oliver, J., Superfast phase-shifting method for 3-d shape measurement, Opt. Express 18(9), (21). [4] Li, B., Wang, Y., Dai, J., Lohry, W., and Zhang, S., Some recent advances on superfast 3d shape measurement with digital binary defocusing techniques, Opt. Laser Eng. 54, (214). [5] Wang, Y., Bhattacharya, B., Winer, E. H., Kosmicki, P., El-Ratal, W. H., and Zhang, S., Digital micromirror transient response influence on superfast 3d shape measurement, Opt. Laser Eng. 58, (214). [6] Pennington, T. L., Xiao, H., May, R., and Wang, A., Miniaturized 3-d surface profilometer using a fiber optic coupler, Opt. & Laser Tech. 33(5), (21). [7] Mercer, C. and Beheim, G., Projected-fringe, phase-stepping profilometer, Appl. Opt. (1993). [8] Valera, J. and Jones, J., Phase stepping in projected-fringe fibre-based moiré interferometry, Electron. Lett. 29(2), (1993). [9] Fan, H., Zhao, H., and Tan, Y., Automated three-dimensional surface profilometry using dual-frequency optic fiber phase-shifting method, Opt. Eng. 36(11), (1997). [1] Zhang, H., Wu, F., Lalor, M. J., and Burton, D. R., Spatiotemporal phase unwrapping and its application in fringe projection fiber optic phase-shifting profilometry, Opt. Eng. 39(7), (2). [11] Lv, C., Duan, F., Zhang, F., Duan, X., Bo, E., and Feng, F., fiber optic interferometer fringe projector using sinusoidal phase-modulating, in [Proc. SPIE], 8916, 8916V 8916V (213). [12] Chao, Z. and Fa-Jie, D., Phase stepping methods based on ptdc for fiber-optic projected-fringe digital interferometry, Opt. & Laser Tech. 44(4), (212). [13] Wu, F., Zhang, H., Lalor, M. J., and Burton, D. R., A novel design for fiber optic interferometric fringe projection phase-shifting 3-d profilometry, Opt. Commun. 187(4), (21). [14] Wang, X., Wang, X., Liu, Y., Zhang, C., and Yu, D., A sinusoidal phase-modulating fiber-optic interferometer insensitive to the intensity change of the light source, Opt. and Laser Tech. 35(3), (23). [15] Wang, B., Wang, X., Sasaki, O., and Li, Z., Sinusoidal phase-modulating interferometer insensitive to intensity modulation of a laser diode for displacement measurement, Appl. opt. 51(12), (212). [16] Zhu, R., Song, Q., Zhu, R., and Li, J., Three-dimensional shape measurement with wavelength-modulated laser base on fiber optic interferometer projection, in [Proc. SPIE], 8769, (213). [17] Malacara, D., ed., [Optical Shop Testing], John Wiley and Sons, ew York, Y, 3rd ed. (27). [18] Greivenkamp, J. E., Generalized data reduction for heterodyne interferometry, Opt. Eng. 23(4), (1984). [19] Brunning, J. H., Herriott, D. R., Gallagher, J. E., Rosenfeld, D. P., White, A. D., and Brangaccio, D. J., Digital wavefront measuring interferometer for testing optical surfaces, lenses, App. Opt. 13(11), (1974). [2] Ghiglia, D. C. and Pritt, M. D., [Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software], John Wiley and Sons, Inc, ew York, Y (1998). [21] Zhang, S., Li, X., and Yau, S.-T., Multilevel quality-guided phase unwrapping algorithm for real-time threedimensional shape reconstruction, Appl. Opt. 46(1), 5 57 (27). [22] Carré, P., Installation et utilisation du comparateur photoélectrigue et interferentiel du bureau international des poids ek measures, Metrologia 2(1), (1966). [23] Hariharan, P., Oreb, B. F., and Eiju, T., Digital phase-shifting interferometry: A simple error-compensation phase calculation algorithm, Appl. Opt. 26(13), (1987). Proc. of SPIE Vol

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