Scale factor calibration method for integrating-bucket sinusoidal phase shifting interferometry

Optics and Laser Technology 127 (2020) 106149

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Scale factor calibration method for integrating-bucket sinusoidal phase shifting interferometry

T

⁎

Fajie Duan, Ruijia Bao, Tingting Huang , Xiao Fu, Cong Zhang State Key Lab of Precision Measuring Technology & Instruments, Tianjin University, Tianjin 300072, China

H I GH L IG H T S

demodulation error induced by scale factor miscalibration is analyzed. • Phase calibration method is robust against noise and fringe distortion. • The • The proposed method avoids the need for accurate parameter setting.

A R T I C LE I N FO

A B S T R A C T

Keywords: Phase shifting interferometry Sinusoidal phase modulation Phase shift calibration

In order to recover the phase map φ(x, y) from sinusoidally phase-shifted interferograms obtained with the integrating-bucket method, two images Ψs and Ψc should be calculated first which are proportional to sin(φ) and cos(φ) respectively, and one of them needs to be scaled by a scale factor before they are used to solve the phase with the arctangent function. Because parameters deciding the scale factor such as the sinusoidal phase modulation amplitude and initial phase are difficult to be set accurately, miscalibration of it can always happen and result in significant error in the phase demodulation. Therefore, in this paper we propose a simple scale factor calibration method which performs the calibration directly with the phase-shifted interferograms and avoids the need for accurate parameter setting. First, we introduce high-density fringes to the interferograms; then, we construct a new image variable Γ with the squares of Ψs and Ψc; last, we find the value of the scale factor by minimizing the second harmonic amplitude in Γ’s spatial spectrum. Simulation results show that the proposed method is robust against noise and fringe distortion, and it was tested in a fiber-optic fringe projection profilometer.

1. Introduction Phase shifting technique [1] is frequently employed in interferometers to determine the interference phase for surface profile measurement. Thanks to the progress in computer technology and various phase shifting interferometry (PSI) algorithms, it has seen great development over the past decades in terms of speed and accuracy. Step phase shifting and linear phase shifting are two of the most commonly used phase shifting methods for traditional PSI algorithms. However, the high frequency harmonics resulting from their inherent discontinuity may cause problems. Due to a limited bandwidth of the phase shifter, step phase shifting can bring about ringing effect [2], and the fidelity of linear phase shifting is difficult to retain, especially when high measurement speed is expected. In this regard, sinusoidal phase shifting (SPS) is much friendlier to phase shifters because of the purity and simplicity of sinusoidal wave’s spectrum, and therefore it is more ⁎

suitable for high-speed profile measurement. In general, methods to demodulate the interference phase in SPS interferometry can be divided into two categories, namely point-sampling methods and integrating-bucket methods. The point-sampling methods capture a sequence of fringe images in several modulation periods, and extract the interference phase by utilizing phase-locked detection technique [3,4] or Fourier analysis method [5]. Because short sampling time and high oversampling rate are required, these methods are demanding for image sensors and their measurement speed is limited. The integrating-bucket methods [6–11] divide the whole modulation period into episodes with equal length and integrate the interference pattern over each episode to acquire a number of images. By manipulating these images, two orthometric images Ψs andΨc can be obtained which are proportional respectively to the sine and cosine of the interference phase φ . Consequently, φ can be obtained by calculating

Corresponding author. E-mail address: [email protected] (T. Huang).

https://doi.org/10.1016/j.optlastec.2020.106149 Received 22 October 2019; Received in revised form 3 January 2020; Accepted 22 February 2020 Available online 28 February 2020 0030-3992/ © 2020 Elsevier Ltd. All rights reserved.

Optics and Laser Technology 127 (2020) 106149

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Ψ φ = arctan ⎛K · s ⎞ ⎝ Ψc ⎠ ⎜

⎟

(1)

where K is a scale factor decided by the sinusoidal phase modulation amplitude m , the integration time Δt and the initial phase of the integration ϕ . The image acquisition and phase calculation of the integrating-bucket methods are faster and simpler compared to those of point-sampling methods, so they are more favorable to high-speed applications. For integrating-bucket SPS interferometry, the value of the scale factor K needs to be precisely known first in order to recover φ accurately. A straightforward way to determine K is to calibrate m , Δt and ϕ separately. While m can be calibrated by analyzing the harmonics extracted from a fraction of the interference light [12] and Δt can be set precisely with a high-resolution clock, the calibration of ϕ is difficult because of the integration delay caused by the phase shifter and the camera. If any one of the three parameters deviate from the expected value, the scale factor would change accordingly and further result in nonnegligible error of the interference phase solution. Alternatively, we can also employ generalized phase shifting algorithms such as the advanced iterative algorithm [13] to solve the phase map or determine the phase shift between frames from sinusoidally phase-shifted interferograms. However, as fringe visibility between frames is variant for SPS, demodulation accuracy of these algorithms can be affected. Since interferometric fringe pattern is easy to generate by slightly tilting the reference plane of the interferometer [14] or using a twin fiber fringe projector [15,16], in this paper we propose a method to calibrate the scale factor directly with phase-shifted fringe images captured by the camera. After Ψs and Ψc are obtained by utilizing the integrating-bucket methods, we construct a new image variable Γ with them. By minimizing the second-order harmonic of the fringe frequency in Γ ’s spatial spectrum, the right value of K can be found. This calibration method is rather simple and accurate, and is significant in guaranteeing the accuracy of the interference phase solution.

Fig. 1. Schematic diagram of the four-bucket integration process for SPS interferometry. Shadows in the plot indicate the periods within which the image sensor is exposed.

Ir =

1 Δα

∞

ar = ∑ (−1)n n=1 ∞

br = ∑ (−1)n n=1

∫α

αr + Δα

I (α ) dα,

r

(5)

(6)

J2n sin(nΔα ) 2nΔα

J2n − 1 sin[(n − 0.5)Δα ] (2n − 1)Δα

cos(2nαr + nΔα )

cos[(2n − 1) αr + (n − 0.5)Δα )]

(7)

The values of ar and br are decided by the parameters m , Δt , ϕ . Although the number of frames, namely R , can be any value that is greater than 3 according to the algorithm design method given by [8] for sinusoidal PSI, the most frequently used value of R is four. Therefore, here we assume R = 4 to explain how phase φ can be recovered from ̃ whose integration process has the captured fringe images Ir (r = 1R ), been illustrated in Fig. 1, and how much phase error can be induced by miscalibration of the parameters. When R = 4 , it is easy to demonstrateα3 = α1 + π and α4 = α2 + π , and that

{

(2)

a1 = a3, a2 = a4 b1 = −b3, b2 = −b4

(8)

holds true for any values of m , ϕ and Δα (Δt ). Consequently, we can obtain

where I0 is the average light intensity, v is the interference visibility, φ is the phase to be determined indicating the height of the measurement point, and θ (t ) is the phase modulation needed for solving φ . As the following deduction in this subsection does not involve (x , y ) , the dependence of φ , I0 and I on (x , y ) is ignored temporarily. In sinusoidal PSI, the phase modulation θ (t ) can be written as

⎧ Ψs = −I1 − I2 + I3 + I4 = 4Ωs I0 v sin φ , ⎨ ⎩ Ψc = I1 − I2 + I3 − I4 = 4Ωc I0 v cos φ

(9)

where

⎧ Ωs = −b1 − b2 + b3 + b4 ⎨ ⎩ Ωc = a1 − a2 + a3 − a4

(3)

where m , f andϕ are the amplitude, the frequency and the initial phase of the modulation signal respectively, and ϕ is also the phase of the modulation signal when the camera starts to integrate the first interference image. Makingα = 2πft + ϕ and substituting the time variable with phase α , Eq. (2) can be expanded as

(10)

Define the scale factor as K = Ωc /Ωs , phase φ can be readily recovered by calculating

Ψ φ = arctan ⎛K · s ⎞ Ψ c⎠ ⎝ ⎜

∞

⎟

(11)

Solve the derivative of Eq. (11) and we obtain

I (α ) = I0 + I0 vJ0 cos φ + 2I0 v cos φ ∑ ( −1)nJ2n cos(2nα ) n=1

Δφ =

∞ n=1

I (t ) dt =

where

If we employ a PSI algorithm to demodulate the interference phase for surface profile measurement, the light intensity of the two-dimensional interferogram can be expressed as

+ 2I0 v sin φ ∑ (−1)nJ2n − 1 cos[(2n − 1) α ],

tr + Δt

r

Ir = I0 + I0 vJ0 cos φ + 4ar I0 v cos φ + 4br I0 v sin φ ,

2.1. Integrating-bucket sinusoidal phase shifting interferometry

θ (t ) = m cos(2πft + ϕ),

∫t

where tr = (r − 1) T / R (r = 1R )̃ is the time when integration of the rth frame starts, T = 1/ f is the modulation period, Δt is the integration time, αr = 2πftr + ϕ and Δα = 2πf Δt . Substituting Eq. (4) into Eq. (5), we can obtain

2. Theory and method

I (x , y, t ) = I0 (x , y ){1 + ν cos[θ (t ) + φ (x , y )]},

1 Δt

(4)

1 ΔK sin(2φ) 2 K

(12)

In order to minimize the phase noise induced by additive noise contained in the interference images, the optimum value of K is −1, in which case the value of m is 2.45 and the values of ϕ are 0.96 and 1.5 for Δt = T /4 and Δt = T /8 respectively [11]. We calculated the value of the scale factor according to Eq. (7) and Eq. (10) when m and ϕ deviate

where J∗ stands for J∗ (m) , the ∗th order Bessel function of the first kind. In the integrating-bucket method for sinusoidal PSI, we acquire R interference images in a modulation period, and each of them can be expressed by 2

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Fig. 2. Variation of the scale factor with the variation of (a) modulation amplitude m and (b) initial phase ϕ . When Δt = T /4 , the original values of m and ϕ are 2.45 and 0.96 respectively; when Δt = T /8, the original values of m and ϕ are 2.45 and 1.5 respectively.

from their respective optimum values, and the results are plotted in Fig. 2. As is shown in Fig. 2, when the modulation depth varies by 0.1 rad, the scale factors of Δt = T /4 and Δt = T /8 vary by 0.08 and 0.07 respectively; when the initial phase varies by 0.1 rad, the scale factors of Δt = T /4 and Δt = T /8 vary by 0.2 and 0.29 respectively. Therefore, the value of the scale factor is more sensitive to the variation of the initial phase. Combining Fig. 2 and Eq. (12) we can infer that a miscalibration of 0.1 rad in ϕ can result in as much as 0.15 rad phase error for the solution of φ . While m can be calibrated by analyzing the harmonics extracted from a fraction of the interference light and Δt can be set precisely with a high-resolution clock, the calibration of ϕ is difficult because of the indeterminate integration delay caused by the phase shifter and the camera. Therefore, instead of calibrating the three parameters to decide the right value of K , we propose a method to directly calibrate the scale factor with interferometric images displaying high-spatial-frequency fringes, which will be introduced in the next section.

direction and y direction respectively, and the resultant intensity distribution of Γ is

Γ(x , y ) = +

(13)

(14)

where K ′ is a random real number. Substituting Eq. (13) into Eq. (14) yields

I02 I2 (K ′ 2 + K 2) + 0 (K 2 − K ′ 2) cos 2φ 2 2

(15)

By slightly tilting the reference plane of the interferometer or using a twin fiber fringe projector, we can introduce fringe pattern with high spatial frequency to the interferograms Ir . As a result, when the surface profile to be detected is flat, the two-dimensional distribution of φ can be approximated as

φ (x , y ) = 2πfx x + 2πf y y

(K 2 − K′2) cos(4πfx x + 4πf y y )

(17)

In order to verify the feasibility of the proposed scale factor calibration method, we simulated four phase-shifted fringe images I1 ~I4 according to Eq. (5) by using MATLAB, and then calculated Ψs and Ψc according to Eq. (9). In the simulation, φ (x , y ) was given by Eq. (16); resolution of the fringe image was 480*640; the modulation amplitude, the initial phase and the integration time were set to m = 2.45, ϕ = 1.5 and Δt = T /8 respectively, leading to K = 0.9994 . The average light intensity distribution I0 (x , y ) followed a two-dimensional Gaussian function. The simulation results are shown in Fig. 4. It can be seen from Fig. 4 that the light intensity distributions along a row are both sinusoidal for Ψs and Ψc , and their phase difference is 90 degrees, which means Ψs and Ψc are orthogonal. Besides, when K ′ = 1, which is very close to the value of K , the intensity distribution of Γ is smooth; when K ′ = 1.2 , obvious fringe pattern appears in Γ , whose spatial frequency is twice that of I1 ~I4 . Further, we calculated the value of A2 (K ′) for different values of K ′, and the result is plotted in Fig. 5. It is obvious that the closer K ′is to the value of K , the smaller A2 is. Therefore, it is feasible to estimate the value of K by minimizing A2 (K ′) . Practically, the fringe pattern may distort owing to flatness error of the measurement plane and the reference plane, as a result of which φ (x , y ) would not follow the linear function given by Eq. (16) and spatial frequency of the fringe would not be constant along a certain direction. In order to test the calibration accuracy of the proposed method in this case, we conducted a simulation with distorted fringe pattern shown in Fig. 6. Except for the phase distribution function φ (x , y ) , parameters used this simulation was the same as those of last simulation, and the value of K was estimated by minimizing A2 (K ′) . A

In order to determine the value of the scale factor K , we first define a new image variable Γ :

Γ=

(K′2 + K 2)

3. Simulation

First of all, note that although Eq. (11) is deduced for R=4, it actually applies to any value of R. In other words, for integrating-bucket sinusoidal PSI algorithms, no matter how many interference images are captured in a modulation period, we can always obtain a Ψs proportional to I0 sin φ and a Ψc proportional to I0 cos φ by manipulating these images, and consequently φ can always be solved by calculating Eq. (11). Therefore, after normalizing the proportionality coefficient of Ψs we have

2 2 Γ = (K ′Ψ) s + Ψc

2

2

If K ′ = K , Eq. (17) reduces to Γ(x , y ) = K 2I02 (x , y ) and the spatial frequency of Γ(x , y ) is very low; If K ′ ≠ K , then the coefficient of cos 2φ is not zero and the two-dimensional Fourier transform of the image Γ contains the second harmonic of fringe frequency, namely (2fx , 2f y ). Assume that FΓ (u, v ) is the two-dimensional Fourier transform of Γ , and for any given value of K ′ define A2 (K ′) = |FΓ (2fx , 2f y )|. Then, for any integrating-bucket sinusoidal PSI algorithm, the calibrated scale factor K can be obtained by searching the K ′ that minimizes A2 (K ′) . To this end, we may take advantage of an optimization algorithm such as the golden section method, whose searching process is illustrated by the flowchart shown in Fig. 3. The search scope (K1, K2) and the estimation accuracy e can be set according to the practical situation.

2.2. Scale factor calibration method

⎧ Ψs = I0 sin φ ⎨ ⎩ Ψc = KI0 cos φ

I02 (x , y )

I02 (x , y )

(16)

where fx and f y are the spatial frequencies of the fringe pattern along x 3

Optics and Laser Technology 127 (2020) 106149

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Fig. 5. Variation of the second-order harmonic amplitude in Γ ’s spectrum versus variation of K ′.

Fig. 6. (a) Distorted fringe pattern used in the simulation and (b) its spectrum along the normal direction of the fringe.

the scale factor using the flowchart in Fig. 3. In the simulation, the cutoff condition of the searching was set asK2 − K1 ⩽ 0.0001. Assuming that the r th (r = 1~4) phase-shifted fringe image was contaminated by a white Gaussian noise nr (x , y ) , then the fringe image with noise was generated by Iran (x , y ) = Ir (x , y ) + nr (x , y ) if the noise is additive and by Irmn (x , y ) = Ir (x , y )[1 + nr (x , y )] if it was multiplicative. Power of the noise (NP) was set to different values, and Fig. 7 displays the relative intensity for a row of the fringe images with different NP. For each NP value, the calibration process was repeated 100 times, and the mean and standard deviation (Std) of the results are shown in tables 1 and 2. Two conclusions can be deduced from the table: (1) the influences of additive noise and multiplicative noise on the calibration results are similar when their NPs are equal; (2) the mean of multiple calibration results differs little from the actual value, which indicates that the average calibration error brought by white noise can be ignored. It can be seen from Fig. 7 that the fringe pattern is almost corrupt when NP=-10dB, but the std calibration errors are only 0.25% and 0.21% respectively. Therefore, the proposed is robust against noise.

Fig. 3. Flowchart of the proposed scale factor calibration algorithm by using the golden section optimization method.

4. Experiments 4.1. Experimental setup A fiber-optic fringe projection profilometer was constructed to test the proposed scale factor calibration method. As shown in Fig. 8, light produced by a narrow line-width laser diode (LD, QFBGLD-633-30) is divided into two equal parts by a −3 dB laser coupler (OC). Because endfaces of the OC’s output fibers are analogous to the two pinholes in the famous Young’s double pinhole interference experiment, light emitted from them will interfere and form fringe pattern on the reference plane. A polarization controller (PC) is utilized to maximize the visibility of the interference fringe. If an object is placed on the reference plane, the fringe pattern will be deformed. By analyzing the phase map of the deformed fringe pattern, surface profile of the object can be recovered. Part of the fiber in one arm of the fiber interferometer was wrapped around a piezo-electric transducer (PZT). When sinusoidal voltage is imposed on the PZT, it will stretch the fiber and introduce a sinusoidal

Fig. 4. Simulated fringe images resulting from four-bucket sinusoidal PSI, the images of Ψs , Ψc and the images of Γ obtained with different with K ′. In the simulation, m = 2.45 , ϕ = 1.5 and Δt = T /8, and the resultant K is 0.9994.

Gaussian window was added before the Fourier transform to reduce the effect of spectrum leakage. The simulation result is K = 0.9993, which differs only 0.0001 from the actual K . Therefore, the proposed method is applicable even if the fringe pattern is distorted. In addition, in order to test the robustness of the calibration method against noise, we added additive and multiplicative noises to the simulated fringe image I1 ~I4 in Fig. 4 respectively, and then calibrated 4

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Fig. 7. Relative intensity distribution along a row of the fringe images with different power. (a) Additive noise. (b) Multiplicative noise.

shifting and the image exposure was realized by the PCU. Timing diagram of the image acquisition is shown in Fig. 9. Because of the limited frame rate of the camera, the four fringe images I1 ~I4 needed for one phase modulation were captured in different modulation periods.

Table 1 Scale factor calibration error for additive noise. NP (dB)

free

−40

−30

−20

−10

Mean Std

0.9995 0

0.9995 1 × 10−5

0.9995 2.4 × 10−4

0.9994 6.7 × 10−4

0.9992 0.0025

4.2. Scale factor calibration with flat surface When we calibrate the scale factor using the proposed method, the phase map distribution had better be as close to Eq. (16) as possible, so at first we used a white board as the reference plane and no object was placed on it. Four fringe images captured in the experiment whose resolution is 480*640 have been presented in Fig. 10. It is clear that the visibility of the first and the third images are degraded owing to the SPS [11]. From Eq. (9) we can see that Ψs and Ψc calculated from I1 ~I4 should be orthogonal, so we calculated the inner product of these two images, which is given by the following expression

Table 2 Scale factor calibration error for multiplicative noise. NP (dB)

free

−40

−30

−20

−10

Mean Std

0.9995 0

0.9995 9.2 × 10−5

0.9994 1.9 × 10−4

0.9993 7 × 10−4

0.9989 0.0021

〈Ψs Ψc〉 =

1 480 ∗ 640

480 640

∑ ∑ [Ψs (i, j) i=1 j=1

∗ Ψc (i, j )] (18)

Due to the existence of speckle noise and image sensor noise, the result of Eq. (18) was 11. However, as the maximum value of Ψs (i, j ) ∗ Ψc (i, j ) is 60000, which is far greater than 11, 〈Ψs Ψc〉 can be approximately considered to be 0. Therefore, Ψs and Ψc can truly be deemed orthogonal. We first assumed that the scale factor K was equal to 1 as expected. Setting K ′ = 1, the image of Γ and its spectrum were obtained which are shown in Fig. 11(a) and (c). Although we can hardly see any fringe pattern in Fig. 11(a), the amplitude of the second harmonic is notable in its spectrum. Therefore, the true value of the scale factor was not 1. Then, we calibrated it by using the proposed method,

Fig. 8. Setup of the fiber-optic fringe projection profilometer. LD: laser diode; OC: −3dB optical coupler; PC: polarization controller; PZT: piezo-electric transducer; PCU: phase control unit.

Fig. 9. Timing diagram for the acquisition of the four sinusoidally phase-shifted fringe images in the experiment. The camera was exposed when trigger is high.

modulation θ (t ) to the interference phase. In the experiment, the sinusoidal signal applied to the PZT was generated by a phase control unit (PCU) built with FPGA. The modulation frequency and the integration time were set to f = 1kHz and Δt = T /4 respectively. The modulation amplitude and the initial phase were supposed to be m = 2.45 and ϕ = 0.98 respectively in order to obtain K = 1. Nevertheless, the true scale factor might deviate from 1 because m and ϕ were uncalibrated. The fringe patterns were captured by a CMOS camera (DALSA, G3-GM10-M0640), and synchronization of the phase

Fig. 10. Four fringe patterns captured for fringe phase demodulation using the integrating-bucket SPS algorithm. Parameters in the algorithm are: m = 2.45 , ϕ = 0.98 and Δt = T /4 . 5

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spatial frequency is twice that of the captured fringe image. Although the recovered phase maps seem to be the same for K = 0.953 and K = 1, we can see slight fringe pattern in the phase residual map of K = 1 but none in that of K = 0.953, which proves that the scale factor calibration is successful. By comparison, we also recovered the phase map from the four fringe images shown in Fig. 10 by using the advanced iterative algorithm (AIA), which has been demonstrated to be an effective algorithm for demodulating the phase map from arbitrarily phase-shifted interferograms. Results of the AIA method are presented by Fig. 12(c) and (f). In the residual phase map of AIA we can see apparent fringe pattern similar to that of K = 1, so AIA is not applicable to the phase demodulation of sinusoidally phase-shifted fringe images, which is because it assumes that fringe visibility and light intensity are invariant for the phase-shifted fringe images. 4.3. Scale factor calibration with real object surface Fig. 11. (a) Image of Γ for K ′ = 1 and (c) its two-dimensional spatial spectrum. (b) Image of Γ for K ′ = 0.953 (the calibrated value) and (d) its two-dimensional spatial spectrum.

As is deduced in Section 2.1, the scale factor K is decided by the modulation amplitude, the initial phase and the integration time, which generally can be considered constant within a certain time duration, so it needs not to be calibrated for every measurement. Normally, we recommend that K be calibrated with a flat surface every time when we turn on the instrument, following the procedure in Section 4.1. After that, we can then use the calibrated value to calculate the phase map and recover the 3D shape of any objects by using Eq. (11). However, sometimes there is a need for in-situ calibration of the scale factor, so we need to investigate the applicability of the proposed method further with fringe images of real objects. Fig. 13(b)–(e) are four fringe images of a slow rebound toy (Fig. 13(a)) captured in SPS interferometry with the system shown in Fig. 8. Parameters used were: m = 2.45, ϕ = 1.47 and Δt = T /8, which are supposed to lead to K = 1. Similarly, we calibrated the scale factor with the four fringe images by utilizing the proposed method, and demodulated the phase map of the toy with the AIA algorithm and the SPS integrating-bucket method respectively. Consequently, 3D shape of the toy was obtained, and results of different methods have been displayed in Fig. 14. We can see from Fig. 14(d) that there is obvious fringe pattern in the result of the AIA algorithm, so its demodulation again fails. Besides, comparing results (Fig. 14(e) and (f)) obtained by using the integrating-bucket SPS method with uncalibrated and calibrated scale factors, we find that the calibration successfully removes the slight fringe pattern caused by a wrong scale factor. Therefore, we can conclude that the proposed method is also applicable to scale factor calibration with real object surface.

Fig. 12. (a) The demodulated phase map by the integrating-bucket SPS algorithm whose scale factor has been calibrated. (b) The demodulated phase map with K = 1. (c) The demodulated phase map by the AIA algorithm. (d) Phase residual map after fitting the phase map in (a) to a plane. (e) Phase residual map after fitting the phase map in (b) to a plane. (f) Phase residual map after fitting the phase map in (c) to a plane.

and the result was K = 0.953. In this case, the image of Γ and its spectrum are shown in Fig. 11(b) and (d) respectively. There is little difference between the images of Γ before and after the calibration, but clearly the peak in the second harmonic of the spectrum disappears after the calibration. In order to verify the calibrated scale factor, we further recovered the phase map of the reference plane by using Eq. (11) with different scale factors, and the results are displayed in Fig. 12(a)–(b). Because no object was placed on the reference plane, the phase distribution of the fringe image is nearly planar. By fitting the phase maps in Fig. 12(a)–(b) to a plane, we obtained the residual error maps which are shown in Fig. 12(d)–(e). From Eq. (12) we can infer that if the scale factor is miscalibrated, the phase demodulation error map will exhibit fringe pattern whose

5. Conclusions In this paper, we first analyzed the phase demodulation error caused by scale factor miscalibration in integrating-bucket SPS interferometry. Then, we proposed a simple scale factor calibration method which performs the calibration directly with the phase-shifted interferograms and avoids the need for accurate parameter setting, and the calibration flowchart was given. Simulation results showed that the proposed

Fig. 13. (a) Color image of a slow rebound toy. (b–e) Four fringe images I1 ~I4 captured for shape measurement of the toy shown in (a) by using the integrating-bucket SPS method. Parameters in the algorithm are: m = 2.45 , ϕ = 1.47 and Δt = T /8. 6

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Fig. 14. 3D shapes of the toy shown in Fig. 13(a) recovered with the four fringe images shown in Fig. 13(b)–(e) by using different methods: (a) the AIA algorithm; (b) the integrating bucket SPS with uncalibrated scale factor, K = 1; (c) the integrating bucket SPS method with calibrated scale factor, K = 1.097 . (d–e) are the X-Y views of (a–c) respectively.

method could achieve a standard deviation accuracy of 0.25% even if the signal-to-noise ratio was only −10 dB. In the experiment, we utilized the proposed method to calibrate the scale factor of a sinusoidally phase-modulated fiber-optic fringe projection profilometer, and phase maps of the captured fringe images were successfully recovered with the calibrated scale factor. By comparison, we also recovered the phase map with the AIA algorithm, and as expected, the result contained obvious phase error due to variation of the fringe visibility caused by sinusoidal phase shifting.

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Funding information National Key Research and Development Plan (2017YFF0204800), National Natural Science Foundations of China (No. 61501319, 51775377 and 61505140), Tianjin Natural Science Foundations of China (No. 17JCQNJC01100), Young Elite Scientists Sponsorship Program by China Association for Science and Technology (No. 2016QNRC001). CRediT authorship contribution statement Fajie Duan: Conceptualization, Methodology, Supervision, Funding acquisition, Writing - review & editing. Ruijia Bao: Validation, Software, Writing - original draft. Tingting Huang: Writing - review & editing, Formal analysis, Investigation. Xiao Fu: Project administration. Cong Zhang: Resources. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] H. Schreiber, J.H. Bruning, Phase shifting interferometry, in: D. Malacara (Ed.), Opt. Shop Test, John Wiley & Sons, Hoboken, 2007, pp. 563 (Chapter 14).

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