- Email: [email protected]
Calibration of the high sensitivity shadow moiré system using random phase-shifting technique
Optics and Lasers in Engineering 63 (2014) 70–75
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Calibration of the high sensitivity shadow moiré system using random phase-shifting technique Hubing Du a,b, Jianhua Wang a, Hong Zhao b,n, Pingping Jia b a b
School of Mechatronic Engineering, Xi'an Technological University, Xi'an, Shaanxi 710032, PR China State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 April 2014 Received in revised form 7 May 2014 Accepted 23 May 2014
In high sensitivity shadow moiré, the small Talbot distance limits the dynamic range. In this case, if the phase shift is introduced by object translation in its own plane, the object may be out of the dynamic range. The result is rapid changes in the period of fringe pattern. So problems arise when Dirckx' way is used to calibrate the fringe distance (or the sensitivity). In the presented paper, we describe a solution to solve the problem. The proposed method based on the idea of random phase shifting technique, which can extract the measurement phase not requiring a previous knowledge of the exact phase shift, and the sensitivity can be calibrated during the process of phase demodulation. Besides, the proposed method can provide an exact close-form result for the sensitivity. Simulations and optical experiments are implemented to verify the effectiveness of this method. The proposed method is suitable for the calibration of the sensitivity in phase shift shadow moiré. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Calibration Shadow moiré Phase Phase-shifting
1. Introduction Shadow moiré topography is an effective optical technique for measuring the three dimensional profile of diffusely reflecting objects. Because its optical arrangement is cheap and easy to implement in industry environment, the technique has been well studied and widely used in various field like structural mechanics, impact testing and vibration measurement etc. Especially in the microelectronic industry, shadow moiré has become a popular choice for sample distortion evaluation under mechanical and/or thermal loading. However as packages continue to grow smaller while the designs grow more diverse, the existing shadow moiré technique must be modified to create a framework for taking measurement that will surpass current existing limits dictated by the traditional moiré setup [1]. To satisfy that end, the phase-shifting technique is usually adopted to improve the sensitivity of shadow moiré system [2–6]. However, due to the nonlinear nature of the height–phase relation, it could be difficult to control the phase variation of all fringes precisely in the field of view [2]. So, if the phase-shifting technique is implemented ignoring the height-phase nonlinear relation, the typical phase shifting algorithms (PSA) fail to produce accurate results [3]. Substantial efforts have been made to overcome this difficulty in the past decades. Some authors proposed that a constant phase shift for every image point can be led by controlling
n
Corresponding author. E-mail address: [email protected] (H. Zhao).
http://dx.doi.org/10.1016/j.optlaseng.2014.05.011 0143-8166/& 2014 Elsevier Ltd. All rights reserved.
accurately the simultaneous variation of two experimental parameters [4,5]. Recently, we proposed a method based on iterative technique [6] to minimize the non-uniform phase-shift error without the addition of the complication of the experimental set-up. A different approach to obtain a relatively high basic measurement sensitivity is the application of the high sensitivity shadow moiré setup using non-zero order Talbot distances [7], where fine gratings (the period of grating in the range 0.2–0.025 mm) are employed. It should be noted that in these techniques a phase-to-height conversion algorithm is usually necessary to reconstruct the 3-D coordinates of the surface to be measured. This algorithm is usually related to the measurement sensitivity (or the period of fringe pattern) of shadow moiré system. Dirckx et al. [8] firstly proposed a numerical analysis based phase-to-height mapping technique which disposes of the use and the need of a calibration object and which only uses the phase shift moiré setup itself. The method is fast and fully automatic and can be implemented easily in conventional shadow moiré techniques, where coarse gratings (the period of grating larger than 0.5 mm) are used. But the technique needs additional calibration experiments prior to the work of measurement and cannot be done simultaneously with the procedure of demodulation of measurement phase. On the other hand, in high-sensitivity shadow moiré [7], the small Talbot distance limits the dynamic range (defined as the useful range of measurement of an instrument). Thus, there is no sufficient dynamic range to allow introducing phase shift by object translation in its own plane. In this case, using the Dirckx' way [8] to determine the period of fringe pattern, the object may be out of dynamic range due to the rapid change of the period of fringe pattern. So the process of sensitivity calibration may be affected.
H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
To overcome the difficulty, here we proposed an analytical method to calibrate the sensitivity of shadow moiré system. In the proposed method, the phase shift is obtained by translating the grating perpendicular to its own plane. The method offers a sufficient dynamic range and helps to capture enough numbers of fringe patterns with constant period for the purpose of sensitivity calibration. We also developed a random phaseshifting based technique to determine the sensitivity. After evaluation of the phase step, we implemented an analytic solution that provides an exact close-form sensitivity to calibrate the shadow moiré system. The correctness of the proposed technique is demonstrated by both simulation and experiment. The details of the proposed algorithm are described below.
Using the condition h⪢zðx; yÞ, the following approximation of e Eq. (4) holds. We obtain the estimated phase step δn as
δen ¼ 2π dΔh=ph;
2.1. The shadow moiré technique In Fig. 1 we show the experimental shadow moiré set-up employed, which is composed of (1) a light source, (2) a CCD camera (3) a Ronchi grating with a pitch p placed over the surface to be measured – which will be described by a function zðx; yÞ. In the set-up, we assume that the optical center of the CCD lens and the light source are located at the same height h from the grating plane, and the distance between them is d. So the intensity distribution at a point ðx; yÞ on the surface is given by Iðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cos ½ϕðx; yÞ;
ð1Þ
where the function Iðx; yÞ is the intensity captured by CCD, Aðx; yÞ is the bias, Bðx; yÞ is the modulation and ϕðx; yÞ is the phase to be measured. The measurement phase ϕðx; yÞ is [6]:
ϕðx; yÞ ¼
2π dzðx; yÞ ; p½h þzðx; yÞ
ð2Þ
When the phase is determined, the height can be expressed by zðx; yÞ ¼
pðh þ zÞ Sðx; yÞ ph s ϕðx; yÞ ¼ ϕðx; yÞ ¼ ϕðx; yÞ; ϕðx; yÞ 2π d 2π 2π d 2π
ð3Þ
We define the parameter Sðx; yÞ ¼ pðh þ zÞ=d as sensitivity. Obviously, due to the height–phase nonlinear relation, the sensitivity in shadow moiré varies with the pixel. But for simplicity, assuming h⪢zðx; yÞ, the approximation value s ¼ ph=d is usually used to replace the true value Sðx; yÞ. In order to introduce the phase shift into the field of view, in each position the grating is translated a constant known step Δh with respect to its last position, then generated phase step is
Δðx; yÞ ¼ 2π dΔh=pðh þ zÞ;
Fig. 1. The optical arrangement of the shadow moiré employed.
ð4Þ
ð5Þ
For a given object, if we change the distance between the Ronchi grating and the object, the surface topography does not vary so; by differentiating Eq. (3) we have 8 > < DSðx; yÞ ¼ Dϕðx; yÞ Sðx; yÞ ϕðx; yÞ ð6Þ > : DSðx; yÞ ¼ pDh d here the operators D½ represent differential operators. Then we obtain Sðx; yÞ ¼
2. Principle
71
ϕðx; yÞ p Dh Dϕðx; yÞ d
ð7Þ
Eq. (7) indicates that the sensitivity can be obtained from two different measurements of the phase of the shadow-moiré pattern obtained with different distance h between the CCD and the grating. This result is the theoretical foundation of the technique proposed. We now pick out a point ðx; yÞ. If we translate the grating over more than the estimated fringe distance, the temporal intensity sequence of the given point can be expressed by 2π dðz þ nΔÞ IðnΔhÞ ¼ A þ B cos pðh þzÞ 2π 2π A þ B cos z þ nΔh ; n ¼ 0; 1; 2; 3::::::; ð8Þ s s where n is the translation number of grating and Δh is the distance of the grating translation. Similarly, the temporal intensity variation can also be implemented by object translation in its own plane [3]; in this case it can be expressed by 2π dðz þ nΔÞ IðnΔhÞ ¼ A þ B cos pðh þ z þ nΔÞ 2π 2π A þ B cos z þ nΔh ; n ¼ 0; 1; 2; 3::::::; ð9Þ s s Eqs. (8) and (9) show that, in both methods the period of the resulting intensity sequences are just the sensitivity of shadow moiré system. But Eq. (8) provides a good approximation to the true intensity values, for its denominator has no the term nΔ. On the other hand, for high basic measurement where the fine gratings are used, the dynamic range is small [7]. In this case, when the second method is used to sample, the object may be out of the dynamic range. Thus in this paper the first method is adopted to sample the intensity sequences. 2.2. Calibration of the sensitivity by analytical method According to the analysis above, the proposed technique firstly obtains two sets of phase phase-shifting shadow-moiré patterns obtained with different the distance h between the CCD and the grating. Then the asynchronous phase-shifting demodulation method is used to extract the measurement phase. In the end, the sensitivity is determined by the derived algebraic equation. We must note that the typical asynchronous phase-shifting demodulation method [9–13] requires that phase variation should be the same for every point of the fringe pattern. However the phase variation across the field of view is not uniform in shadow moiré. This assumption is too strong to be met. Different from the above, the newly developed Principal Component Analysis (PCA) [9,10] determines the phase by obtaining quasi-quadrature signals of the cosine fringes. The feature permits the demodulation process to be insensitive to the non-uniform phase-shift error. So the technique is used to extract the phase in this paper. More information about the application of PCA algorithm into shadow moiré system can be found in reference. More information about
72
H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
the application of PCA algorithm into shadow moiré system can be found in reference [14–16]. In conclusion, the proposed calibration technique can be described as follows: 1) The grating is moved with a constant step Δh with respect to its last position and a set of phase shift images I 1 ; I 2 ; ::::; I m 1 ; I m is captured. 2) Using images I 1 ; I 2 ; ::::; I m 1 , the PCA algorithm can obtain the measurement phase ϕ1 ðx; yÞ. 3) Do the above procedure again, and the measurement phase ϕ2 ðx; yÞ can be obtained by I2 ; ::::; I m 1 ; Im .
4) The introduced phase step is determined by Δðx; yÞ ¼ ϕ1 ðx; yÞ ϕ2 ðx; yÞ. 5) Then the sensitivity is Sðx; yÞ ¼ 2πΔh . Δðx; yÞ Sðx; yÞ 6) In the end the distribution of height is zðx; yÞ ¼ ϕ1 ðx; yÞ. 2π 3. Numerical experiment To verify the feasibility of the proposed algorithm, a numerical experiment has been carried out. We define the actual height map
Fig. 2. The simulation result: (a) the true phase step, (b) the estimated phase step, (c) the error of the estimated phase step, (d) the true sensitivity, (e) the estimated sensitivity, and (f) the error of the estimated sensitivity.
H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
as zðx; yÞ ¼ 0:2x mm, the parameters for simulation as p ¼ 0:05 mm; d ¼ 100 mm; h ¼ 160 mm, and the intensity distributions as I n ¼ 1 þ cos ðϕ þ Δn Þ; ðn ¼ 0; 1; :::Þ, where 0 r x r 2:55; 0 r y r 2:55. The nominal value of the distance of grating translation Δh ¼ 0:02 mm. Then, a set of simulated phase-shifted interferogram are obtained. To illustrate the performance of the proposed calibration algorithm, the error function is defined as eðx; yÞ ¼ absðςðx; yÞ δðx; yÞÞ to test the proposed algorithm accuracy, where the estimated value is denoted as δðx; yÞ, the true one as ςðx; yÞ and absðÞ represents absolute value operation. Fig. 2(a)–(f) respectively shows the true phase step, the estimated phase step, the true sensitivity, the residual errors of the phase step, the estimated sensitivity and the residual errors of the sensitivity. It shows the residual errors of the phase step and the residual errors of the sensitivity are less than 8 10 3 rad and respectively. The simulation results show the correctness of the proposed method.
4. Experiments and discussion To verify the correctness of the proposed method further, we have performed an experiment using the set-up drawn in Fig. 1. We employed LED as light source and a Ronchi grating is used as master grating. The CCD camera was a pike-505B black-and-white
73
CCD camera. The parameters of our setup are list as p ¼ 0:05 mm;
d ¼ 100 mm;
h ¼ 160 mm;
Δh ¼ 0:01 mm
A piece of wafer, which is tilted for a small degree, is selected as the specimen. Then both the proposed method and the Dirckx' way [8] are used to calibrate the sensitivity for comparison. Fig. 3(a) shows the phase shifting images captured by translating the object. It illustrates that in high sensitivity shadow moiré setup, when using the Dirckx' way to determine the fringe distance, the object cannot be translated over more than the estimated fringe distance in the dynamic range. In this case, there is a rapid change in the period of fringe pattern. Thus the calibration result is affected. Fig. 3(b) shows the filtered phaseshifted images captured by the method of translation of the grating. Our method of filter can be found in [13]. Obviously, the method offers a sufficient dynamic range. Thus 35 fringe patterns with constant period are captured for the purpose of calibration. Firstly, using Fig. 3(b), the method proposed by Dirckx et al. [8] is used to calibrate the sensitivity. But the procedure is modified as follows: 1) Move the grating over a predetermined step Δh and record the corresponding calibration image. 2) Repeat the above cycle q times in temporal until the object has been translated over more than the estimated fringe distance and enough images are captured.
Fig. 3. The captured phase shifting images: (a) obtained by translation of object and (b) obtained by translation of grating.
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H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
Fig. 4. The experiment results obtained by the Dirckx' way: (a) the temporal intensity sequence of a given point, (b) the interpolated one, and (c) the determined sensitivity by 50 independent experiments.
Fig. 5. The calibration result of the proposed method: (a) the obtained phase step distribution, and (b) the obtained sensitivity distribution.
3) After necessary filter procedure, pick out a pixel and store its temporal intensity sequence IðnΔhÞ in an array. 4) Interpolate IðnΔhÞ to fitting the true distribution of intensity, then determine the translation distance of two subsequent maxima value of the intensity, i.e. sensitivity.
The calibration result is shown in Fig. 4. Fig. 4(a) is the temporal intensity distribution of a selected point. Fig. 4(b) is its interpolated one. Then the sensitivity is determined. We also do the procedure 50 times independently. The obtained sensitivity is shown in Fig. 4(c). The average of the result is 0.082 mm.
H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
After that, the proposed method is used to calibrate the sensitivity. Fig. 5 is the result processed by the proposed method. The retrieved nonuniform phase step is shown in Fig. 5(a) and the determined pointwise sensitivity is shown in Fig. 5(b). The average of the result is 0.0802 mm. Comparing the result of the proposed method with the result of the Dirckx' way: it proves the correctness of the proposed method. 5. Conclusion To conclude, based on PCA algorithm, an improved version of sensitivity calibration technique is developed for phase shifting shadow moiré. Though it would not be usable for temporal varying surfaces, such as when a surface is experiencing thermal deformations, for the measurement of 3-D shape of static objects, the new algorithm has a stable, simple and fast feature and provides an exact close-form of the sensitivity. Besides, it is insensitive to the height dependent effects, which is the main systematic source of error in phase-shift shadow moiré when reconstructing surfaces from fringe patterns. Numerical simulations and practical experiment demonstrate the effectiveness of the proposed algorithm. Acknowledgments The authors gratefully acknowledge the support from the National Natural Science Foundation of China (No. 50975228), and the Open Research Fund Program of Shaanxi Key Laboratory of Non-Traditional Machining (No. ST-11005).
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References [1] Kwong B. Advanced shadow moire with non-conventional imaging angles [Doctoral dissertation]; (2012). [2] Xie X, Atkinson JT, Lalor MJ, Burton DR. Three-map absolute moiré contouring. Appl Opt 1996;35(35):6990–5. [3] Dirckx JJJ, Decraemer WF, Dielis G. Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moire topograms. Appl Opt 1988;27(6):1164–9. [4] Yoshizawa T, Tomisawa T. Shadow moiré topography by means of the phaseshift method. Opt Eng 1993;32(7):1668–74. [5] Jin L, Kodera Y, Yoshizawa T, Otani Y. Shadow moiré profilometry using the phase-shifting method. Opt Eng 2000;39(8):2119–23. [6] Du H, Zhao H, Li B, Zhao J, Cao S. Phase-shifting shadow moiré based on iterative self-tuning algorithm. Appl Opt 2011;50(36):6708–12. [7] Han CW, Han B. High sensitivity shadow moiré using nonzero-order Talbot distance. Exp Mech 2006;46(4):543–54. [8] Dirckx JJ, Decraemer WF. Automatic calibration method for phase shift shadow moiré interferometry. Appl Opt 1990;29(10):1474–6. [9] Wang Z, Han B. Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms. Opt Lett 2004;29(14):1671–3. [10] Vargas J, Quiroga JA, Belenguer T, Servín M, Estrada JC. Two-step self-tuning phase-shifting interferometry. Opt Express 2011;19(2):638–48. [11] Estrada JC, Servin M, Quiroga JA. A self-tuning phase-shifting algorithm for interferometry. Opt Express 2010;18(3):2632–8. [12] Wang H, Luo C, Zhong L, Ma S, Lu X. Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts. Opt Express 2014;22(5):5147–54. [13] Deng J, Wang H, Zhang D, Zhong L, Fan J, Lu X. Phase shift extraction algorithm based on Euclidean matrix norm. Opt Lett 2013;38(9):1506–8. [14] Vargas J, Quiroga JA, Belenguer T. Phase-shifting interferometry based on principal component analysis. Opt Lett 2011;36(8):1326–8. [15] Vargas J, Quiroga JA, Belenguer T. Analysis of the principal component algorithm in phase-shifting interferometry. Opt Lett 2011;36(12):2215–7. [16] Zhao H, Du H, Li J, Qin Y. Shadow moiré technology based fast method for the measurement of surface topography. Appl Opt 2013;52(33):7874–81.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Calibration of the high sensitivity shadow moiré system using random phase-shifting technique Hubing Du a,b, Jianhua Wang a, Hong Zhao b,n, Pingping Jia b a b
School of Mechatronic Engineering, Xi'an Technological University, Xi'an, Shaanxi 710032, PR China State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 April 2014 Received in revised form 7 May 2014 Accepted 23 May 2014
In high sensitivity shadow moiré, the small Talbot distance limits the dynamic range. In this case, if the phase shift is introduced by object translation in its own plane, the object may be out of the dynamic range. The result is rapid changes in the period of fringe pattern. So problems arise when Dirckx' way is used to calibrate the fringe distance (or the sensitivity). In the presented paper, we describe a solution to solve the problem. The proposed method based on the idea of random phase shifting technique, which can extract the measurement phase not requiring a previous knowledge of the exact phase shift, and the sensitivity can be calibrated during the process of phase demodulation. Besides, the proposed method can provide an exact close-form result for the sensitivity. Simulations and optical experiments are implemented to verify the effectiveness of this method. The proposed method is suitable for the calibration of the sensitivity in phase shift shadow moiré. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Calibration Shadow moiré Phase Phase-shifting
1. Introduction Shadow moiré topography is an effective optical technique for measuring the three dimensional profile of diffusely reflecting objects. Because its optical arrangement is cheap and easy to implement in industry environment, the technique has been well studied and widely used in various field like structural mechanics, impact testing and vibration measurement etc. Especially in the microelectronic industry, shadow moiré has become a popular choice for sample distortion evaluation under mechanical and/or thermal loading. However as packages continue to grow smaller while the designs grow more diverse, the existing shadow moiré technique must be modified to create a framework for taking measurement that will surpass current existing limits dictated by the traditional moiré setup [1]. To satisfy that end, the phase-shifting technique is usually adopted to improve the sensitivity of shadow moiré system [2–6]. However, due to the nonlinear nature of the height–phase relation, it could be difficult to control the phase variation of all fringes precisely in the field of view [2]. So, if the phase-shifting technique is implemented ignoring the height-phase nonlinear relation, the typical phase shifting algorithms (PSA) fail to produce accurate results [3]. Substantial efforts have been made to overcome this difficulty in the past decades. Some authors proposed that a constant phase shift for every image point can be led by controlling
n
Corresponding author. E-mail address: [email protected] (H. Zhao).
http://dx.doi.org/10.1016/j.optlaseng.2014.05.011 0143-8166/& 2014 Elsevier Ltd. All rights reserved.
accurately the simultaneous variation of two experimental parameters [4,5]. Recently, we proposed a method based on iterative technique [6] to minimize the non-uniform phase-shift error without the addition of the complication of the experimental set-up. A different approach to obtain a relatively high basic measurement sensitivity is the application of the high sensitivity shadow moiré setup using non-zero order Talbot distances [7], where fine gratings (the period of grating in the range 0.2–0.025 mm) are employed. It should be noted that in these techniques a phase-to-height conversion algorithm is usually necessary to reconstruct the 3-D coordinates of the surface to be measured. This algorithm is usually related to the measurement sensitivity (or the period of fringe pattern) of shadow moiré system. Dirckx et al. [8] firstly proposed a numerical analysis based phase-to-height mapping technique which disposes of the use and the need of a calibration object and which only uses the phase shift moiré setup itself. The method is fast and fully automatic and can be implemented easily in conventional shadow moiré techniques, where coarse gratings (the period of grating larger than 0.5 mm) are used. But the technique needs additional calibration experiments prior to the work of measurement and cannot be done simultaneously with the procedure of demodulation of measurement phase. On the other hand, in high-sensitivity shadow moiré [7], the small Talbot distance limits the dynamic range (defined as the useful range of measurement of an instrument). Thus, there is no sufficient dynamic range to allow introducing phase shift by object translation in its own plane. In this case, using the Dirckx' way [8] to determine the period of fringe pattern, the object may be out of dynamic range due to the rapid change of the period of fringe pattern. So the process of sensitivity calibration may be affected.
H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
To overcome the difficulty, here we proposed an analytical method to calibrate the sensitivity of shadow moiré system. In the proposed method, the phase shift is obtained by translating the grating perpendicular to its own plane. The method offers a sufficient dynamic range and helps to capture enough numbers of fringe patterns with constant period for the purpose of sensitivity calibration. We also developed a random phaseshifting based technique to determine the sensitivity. After evaluation of the phase step, we implemented an analytic solution that provides an exact close-form sensitivity to calibrate the shadow moiré system. The correctness of the proposed technique is demonstrated by both simulation and experiment. The details of the proposed algorithm are described below.
Using the condition h⪢zðx; yÞ, the following approximation of e Eq. (4) holds. We obtain the estimated phase step δn as
δen ¼ 2π dΔh=ph;
2.1. The shadow moiré technique In Fig. 1 we show the experimental shadow moiré set-up employed, which is composed of (1) a light source, (2) a CCD camera (3) a Ronchi grating with a pitch p placed over the surface to be measured – which will be described by a function zðx; yÞ. In the set-up, we assume that the optical center of the CCD lens and the light source are located at the same height h from the grating plane, and the distance between them is d. So the intensity distribution at a point ðx; yÞ on the surface is given by Iðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cos ½ϕðx; yÞ;
ð1Þ
where the function Iðx; yÞ is the intensity captured by CCD, Aðx; yÞ is the bias, Bðx; yÞ is the modulation and ϕðx; yÞ is the phase to be measured. The measurement phase ϕðx; yÞ is [6]:
ϕðx; yÞ ¼
2π dzðx; yÞ ; p½h þzðx; yÞ
ð2Þ
When the phase is determined, the height can be expressed by zðx; yÞ ¼
pðh þ zÞ Sðx; yÞ ph s ϕðx; yÞ ¼ ϕðx; yÞ ¼ ϕðx; yÞ; ϕðx; yÞ 2π d 2π 2π d 2π
ð3Þ
We define the parameter Sðx; yÞ ¼ pðh þ zÞ=d as sensitivity. Obviously, due to the height–phase nonlinear relation, the sensitivity in shadow moiré varies with the pixel. But for simplicity, assuming h⪢zðx; yÞ, the approximation value s ¼ ph=d is usually used to replace the true value Sðx; yÞ. In order to introduce the phase shift into the field of view, in each position the grating is translated a constant known step Δh with respect to its last position, then generated phase step is
Δðx; yÞ ¼ 2π dΔh=pðh þ zÞ;
Fig. 1. The optical arrangement of the shadow moiré employed.
ð4Þ
ð5Þ
For a given object, if we change the distance between the Ronchi grating and the object, the surface topography does not vary so; by differentiating Eq. (3) we have 8 > < DSðx; yÞ ¼ Dϕðx; yÞ Sðx; yÞ ϕðx; yÞ ð6Þ > : DSðx; yÞ ¼ pDh d here the operators D½ represent differential operators. Then we obtain Sðx; yÞ ¼
2. Principle
71
ϕðx; yÞ p Dh Dϕðx; yÞ d
ð7Þ
Eq. (7) indicates that the sensitivity can be obtained from two different measurements of the phase of the shadow-moiré pattern obtained with different distance h between the CCD and the grating. This result is the theoretical foundation of the technique proposed. We now pick out a point ðx; yÞ. If we translate the grating over more than the estimated fringe distance, the temporal intensity sequence of the given point can be expressed by 2π dðz þ nΔÞ IðnΔhÞ ¼ A þ B cos pðh þzÞ 2π 2π A þ B cos z þ nΔh ; n ¼ 0; 1; 2; 3::::::; ð8Þ s s where n is the translation number of grating and Δh is the distance of the grating translation. Similarly, the temporal intensity variation can also be implemented by object translation in its own plane [3]; in this case it can be expressed by 2π dðz þ nΔÞ IðnΔhÞ ¼ A þ B cos pðh þ z þ nΔÞ 2π 2π A þ B cos z þ nΔh ; n ¼ 0; 1; 2; 3::::::; ð9Þ s s Eqs. (8) and (9) show that, in both methods the period of the resulting intensity sequences are just the sensitivity of shadow moiré system. But Eq. (8) provides a good approximation to the true intensity values, for its denominator has no the term nΔ. On the other hand, for high basic measurement where the fine gratings are used, the dynamic range is small [7]. In this case, when the second method is used to sample, the object may be out of the dynamic range. Thus in this paper the first method is adopted to sample the intensity sequences. 2.2. Calibration of the sensitivity by analytical method According to the analysis above, the proposed technique firstly obtains two sets of phase phase-shifting shadow-moiré patterns obtained with different the distance h between the CCD and the grating. Then the asynchronous phase-shifting demodulation method is used to extract the measurement phase. In the end, the sensitivity is determined by the derived algebraic equation. We must note that the typical asynchronous phase-shifting demodulation method [9–13] requires that phase variation should be the same for every point of the fringe pattern. However the phase variation across the field of view is not uniform in shadow moiré. This assumption is too strong to be met. Different from the above, the newly developed Principal Component Analysis (PCA) [9,10] determines the phase by obtaining quasi-quadrature signals of the cosine fringes. The feature permits the demodulation process to be insensitive to the non-uniform phase-shift error. So the technique is used to extract the phase in this paper. More information about the application of PCA algorithm into shadow moiré system can be found in reference. More information about
72
H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
the application of PCA algorithm into shadow moiré system can be found in reference [14–16]. In conclusion, the proposed calibration technique can be described as follows: 1) The grating is moved with a constant step Δh with respect to its last position and a set of phase shift images I 1 ; I 2 ; ::::; I m 1 ; I m is captured. 2) Using images I 1 ; I 2 ; ::::; I m 1 , the PCA algorithm can obtain the measurement phase ϕ1 ðx; yÞ. 3) Do the above procedure again, and the measurement phase ϕ2 ðx; yÞ can be obtained by I2 ; ::::; I m 1 ; Im .
4) The introduced phase step is determined by Δðx; yÞ ¼ ϕ1 ðx; yÞ ϕ2 ðx; yÞ. 5) Then the sensitivity is Sðx; yÞ ¼ 2πΔh . Δðx; yÞ Sðx; yÞ 6) In the end the distribution of height is zðx; yÞ ¼ ϕ1 ðx; yÞ. 2π 3. Numerical experiment To verify the feasibility of the proposed algorithm, a numerical experiment has been carried out. We define the actual height map
Fig. 2. The simulation result: (a) the true phase step, (b) the estimated phase step, (c) the error of the estimated phase step, (d) the true sensitivity, (e) the estimated sensitivity, and (f) the error of the estimated sensitivity.
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as zðx; yÞ ¼ 0:2x mm, the parameters for simulation as p ¼ 0:05 mm; d ¼ 100 mm; h ¼ 160 mm, and the intensity distributions as I n ¼ 1 þ cos ðϕ þ Δn Þ; ðn ¼ 0; 1; :::Þ, where 0 r x r 2:55; 0 r y r 2:55. The nominal value of the distance of grating translation Δh ¼ 0:02 mm. Then, a set of simulated phase-shifted interferogram are obtained. To illustrate the performance of the proposed calibration algorithm, the error function is defined as eðx; yÞ ¼ absðςðx; yÞ δðx; yÞÞ to test the proposed algorithm accuracy, where the estimated value is denoted as δðx; yÞ, the true one as ςðx; yÞ and absðÞ represents absolute value operation. Fig. 2(a)–(f) respectively shows the true phase step, the estimated phase step, the true sensitivity, the residual errors of the phase step, the estimated sensitivity and the residual errors of the sensitivity. It shows the residual errors of the phase step and the residual errors of the sensitivity are less than 8 10 3 rad and respectively. The simulation results show the correctness of the proposed method.
4. Experiments and discussion To verify the correctness of the proposed method further, we have performed an experiment using the set-up drawn in Fig. 1. We employed LED as light source and a Ronchi grating is used as master grating. The CCD camera was a pike-505B black-and-white
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CCD camera. The parameters of our setup are list as p ¼ 0:05 mm;
d ¼ 100 mm;
h ¼ 160 mm;
Δh ¼ 0:01 mm
A piece of wafer, which is tilted for a small degree, is selected as the specimen. Then both the proposed method and the Dirckx' way [8] are used to calibrate the sensitivity for comparison. Fig. 3(a) shows the phase shifting images captured by translating the object. It illustrates that in high sensitivity shadow moiré setup, when using the Dirckx' way to determine the fringe distance, the object cannot be translated over more than the estimated fringe distance in the dynamic range. In this case, there is a rapid change in the period of fringe pattern. Thus the calibration result is affected. Fig. 3(b) shows the filtered phaseshifted images captured by the method of translation of the grating. Our method of filter can be found in [13]. Obviously, the method offers a sufficient dynamic range. Thus 35 fringe patterns with constant period are captured for the purpose of calibration. Firstly, using Fig. 3(b), the method proposed by Dirckx et al. [8] is used to calibrate the sensitivity. But the procedure is modified as follows: 1) Move the grating over a predetermined step Δh and record the corresponding calibration image. 2) Repeat the above cycle q times in temporal until the object has been translated over more than the estimated fringe distance and enough images are captured.
Fig. 3. The captured phase shifting images: (a) obtained by translation of object and (b) obtained by translation of grating.
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H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
Fig. 4. The experiment results obtained by the Dirckx' way: (a) the temporal intensity sequence of a given point, (b) the interpolated one, and (c) the determined sensitivity by 50 independent experiments.
Fig. 5. The calibration result of the proposed method: (a) the obtained phase step distribution, and (b) the obtained sensitivity distribution.
3) After necessary filter procedure, pick out a pixel and store its temporal intensity sequence IðnΔhÞ in an array. 4) Interpolate IðnΔhÞ to fitting the true distribution of intensity, then determine the translation distance of two subsequent maxima value of the intensity, i.e. sensitivity.
The calibration result is shown in Fig. 4. Fig. 4(a) is the temporal intensity distribution of a selected point. Fig. 4(b) is its interpolated one. Then the sensitivity is determined. We also do the procedure 50 times independently. The obtained sensitivity is shown in Fig. 4(c). The average of the result is 0.082 mm.
H. Du et al. / Optics and Lasers in Engineering 63 (2014) 70–75
After that, the proposed method is used to calibrate the sensitivity. Fig. 5 is the result processed by the proposed method. The retrieved nonuniform phase step is shown in Fig. 5(a) and the determined pointwise sensitivity is shown in Fig. 5(b). The average of the result is 0.0802 mm. Comparing the result of the proposed method with the result of the Dirckx' way: it proves the correctness of the proposed method. 5. Conclusion To conclude, based on PCA algorithm, an improved version of sensitivity calibration technique is developed for phase shifting shadow moiré. Though it would not be usable for temporal varying surfaces, such as when a surface is experiencing thermal deformations, for the measurement of 3-D shape of static objects, the new algorithm has a stable, simple and fast feature and provides an exact close-form of the sensitivity. Besides, it is insensitive to the height dependent effects, which is the main systematic source of error in phase-shift shadow moiré when reconstructing surfaces from fringe patterns. Numerical simulations and practical experiment demonstrate the effectiveness of the proposed algorithm. Acknowledgments The authors gratefully acknowledge the support from the National Natural Science Foundation of China (No. 50975228), and the Open Research Fund Program of Shaanxi Key Laboratory of Non-Traditional Machining (No. ST-11005).
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