Accurate carrier-removal technique based on zero padding in Fourier transform method for carrier interferogram analysis

Optik 125 (2014) 1056–1061

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Accurate carrier-removal technique based on zero padding in Fourier transform method for carrier interferogram analysis Yongzhao Du a,1 , Guoying Feng a,∗ , Hongru Li a , Shouhuan Zhou a,b a b

College of Electronic & Information Engineering, Sichuan University, Chengdu, Sichuan 610064, China North China Research Institute of Electro-Optics, Beijing 100015, China

a r t i c l e

a b s t r a c t

i n f o

Article history: Received 22 March 2013 Accepted 28 July 2013

Based on interferogram zero padding and fast Fourier transform (FFT) methods, an effective, straightforward and stable carrier-removal approach in Fourier transform (FT) based method for carrier interferogram analysis is proposed. The spatial carrier interferogram is firstly extrapolated by zero padding method, and the carrier-frequency values within a small fraction of an integral (or a pixel) are estimated from the extrapolation interferogram with FFT method. Then the carrier-phase component is removed by subtracting a pure carrier-frequency phase constructed by the estimated carrier-frequencies in the spatial domain. Numerical simulations and experiments are given to demonstrate the performance of the proposed method and the results show that the proposed method is effective and stable for suppressing the carrier-removal error in the FT method for carrier interferogram analysis. Crown Copyright © 2013 Published by Elsevier GmbH. All rights reserved.

Keywords: Fringe pattern analysis Linear carrier interferogram Fourier transform Zero padding

1. Introduction The Fourier transform based (FT) method is one of the most popular interference interferogram analysis methods for high accuracy and automatic phase measurement systems. The FT method was originally proposed and demonstrated by Takeda et al. [1], and then many works were published afterward [2–8]. Comparing to temporal phase-shifting (PS) method, the FT method usually requires only one interferogram, which makes it is less sensitive to circumstantial disturbances and vibrations [9]. Therefore, it has been widespread applied to various kinds of interferometric applications, such as optical interferometer measurements [10–12], and holography interferometry and its applications in phase microscopy [13–16]. The major idea of the FT method can be described as follows [1–4,8]. The deformed fringe pattern g(x,y) with linear-carrier is generally expressed as





g(x, y) = a(x, y) + b(x, y) cos 2(f0x x + f0y y) + (x, y)

(1)

where a(x,y), b(x,y) are the background illumination and the modulation intensities, respectively; f0x and f0y are the introduced spatial carrier-frequencies along x and y directions, respectively; (x,y) is the modulating phase. The carrier interferogram, cos(2f0x x + f0y y), serves as an carrier information for recording the measured phase

∗ Corresponding author. E-mail addresses: [email protected] (Y. Du), guoing [email protected] (G. Feng), [email protected] (H. Li), [email protected] (S. Zhou). 1 Tel.: +86 1 949 3378965.

data but it will simultaneously introduce a carrier phase component, 2(f0x x + f0y y), in the phase extraction procedure [7]. Hence the carrier phase component must be subtracted or removed from the overall phase distribution for evaluation of the phase of the measured phase component (x,y). And it is generally achieved using the traditional FT method, as follows. By using the Euler formula to expand the cosine term in Eq. (1), we have





g(x, y) = a(x, y) + c(x, y) exp j2(f0x x + f0y y)





+ c ∗ (x, y) exp −j2(f0x x + f0y y)

(2)

with the definition c(x, y) =

1 b(x, y) exp [j(x, y)] 2

(3)

and superscript * denotes the complex conjugate. Taking the Fourier transform in two-dimension for the carrier interferogram expressed in Eq. (2), we have G(fx , fy ) = A(fx , fy ) + C(fx − f0x , fy − f0y ) + C ∗ (fx + f0x , fy + f0y )

(4)

where fx and fy are the spatial frequency coordinates in the frequency domain. G(fx ,fy ), A(fx ,fy ), C(fx ,fy ) and C*(fx ,fy ) are the Fourier transform of g(x,y), a(x,y), c(x,y) and c*(x,y), respectively. Assuming that the terms, a(x,y), b(x,y), and (x,y), are slow varying functions compared with the spatial carrier-frequency f0x and f0y . The terms A(fx ,fy ), C(fx − f0x ,fy − f0y ) and C*(fx + f0x ,fy + f0y ) in the right hand of Eq. (4) are separate and do not overlap in the frequency domain. Hence the spectrum component C(fx − f0x ,fy − f0y ) can be isolated with a suitable spectral filter. Then the component C(fx − f0x ,fy − f0y )

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Y. Du et al. / Optik 125 (2014) 1056–1061

is shifted to the origin position with a distance of f0x and f0y along x and y direction, respectively, in the frequency domain, and we have C(fx ,fy ). Taking the inverse Fourier transform of C(fx ,fy ), c(x,y) will be obtained then the measured phase component (x,y) is given by (x, y) = tan−1 tan−1 [·]

Re [c(x, y)] Im [c(x, y)]

(5)

where denotes the arctangent operator, Im[·] and Re[·] denote the imaginary and real parts of c(x,y), respectively. However, the fringe patterns are usually recorded by 2D solidstate image sensors, such as CCD camera, in practical applications. As discussed in Ref. [4], the digitization of the interferogram data can seriously distort the retrieved phase (x,y), because the translation is constrained to integer values of the spatial frequency which results in the digitization of sampling frequencies in the frequency domain. The carrier-frequency f0x and f0y usually are not an integer but instead of a small fraction of the frequency interval. In this case, a considerable departure will appear between the discrete carrier-frequency, which is determined by the peak coordinate of the side-lobe, and the real carrier-frequency f0x and f0y . Therefore, a carrier-removal error will be introduced in the traditional spectrum-shifting method as described in Reference [1] and it will result in a seriously tilt error in the retrieved phase (x,y) determined in Eq. (5). In the past two decades, several solutions [17–23] are developed to suppress the carrier-removal error problem. Bone et al. [5] constructs a carrier phase plane from an information-free region in the interferogram by the least-squares fit method, and the measured phase is obtained by subtracting the carrier phase in the spatial domain. However, the accuracy of the retrieved phase is determined by the information-free region in the interferogram and the requirement of an information-free region in the interferogram cannot always be fulfilled [17]. Similarly, Gu and Chen [19] presents a bilinear surface to describe the carrier component. Another carrier-removal approach described in Ref. [20] directly removes the carrier phase in the spatial domain by subtracting a reference carrier phase calculated from an additional reference interferogram. For the same case, Nicola and Ferraro [21] remove the carrier phase in the frequency domain with FT method, and it also requires an additional pure carrier-frequency interferogram. However, recording two individually interferograms, a deformation interferogram and a reference interferogram, are required and negating the advantage of single-shot measurement in the FT method. Moreover, Ge [22] adjusts the carrier frequency values equal to an integral multiple of the sampling frequency by adjusting the inclination angle of the reference mirror with a piezoelectric actuator. However, it makes the setup complicated and consumes more time. Recently, Fan et al. [23] report a spectrum centroid method for suppressing carrier-removal error in the FT method for carrier interferogram analysis, for simplicity, it is shorted as “SC” method. The carrier-frequency values of interferogram are estimated from the spectrum centroid of component C(fx − f0x ,fy − f0y ), and the carrier phase is removed by shifting the C(fx − f0x ,fy − f0y ) to the origin position in the frequency domain by multiplying the original interferogram with a constructed pure carrier phase wave in the spatial domain. However, its accuracy is limited by the spectrum distribution and is unstable when chooses different window sizes of spectral filter. In this paper, we present an effective, straightforward and stable carrier-removal technique for carrier interferogram analysis based on zero padding method. The carrier-frequency values within a small fraction of an integral (or pixel) are estimated from the extrapolation fringe with FFT approach. Then the carrier phase is removed by subtracting a pure carrier-frequency phase constructed by the estimated carrier-frequency f0x and f0y in the spatial domain.

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For brevity and distinction, we refer to the proposed method as “ZP” method. The principle of the proposed method is described in Section 2. Numerical simulations and experiments are carried out to demonstrate performance of the proposed ZP method in Sections 3 and 4, respectively. We conclude all the paper in Section 5. 2. Theory analysis 2.1. Discrete Fourier transform for fringe pattern analysis The intensity of the fringe pattern is usually recorded by a solidstate image sensor such as CCD camera in practical applications thus the fringe pattern described in Eq. (1) usually can be further expressed as discrete form



g(m, n) = a(m, n) + b(m, n) cos 2

u

0

M

m+

v0 N





n + (m, n)

(6)

where m, n are integer; M, N are the sampling points on the x, y directions, respectively; the sampling intervals Tx and Ty both equal to 1; u0 and v0 are integer and the values of them are closed to the true carrier-frequency f0x and f0y , respectively. The discrete Fourier transform (DFT) of Eq. (6) is given by [24] G

u v ,

M N

=

M−1 N−1  



g(m, n) exp −j2

u M

m=0 n=0

m+



v

n

N

(7)

where G(u/M, v/N) is the spectral distribution of Eq. (6); u and v are integer of the sampling frequency interval; and j = (−1)1/2 . Similar to the analysis procedure in Section 1, the spectrum component C(u/M − u0 /M, v/N − v0 /N) in Eq. (7) is extracted with a band filter in frequency domain and its inverse discrete Fourier transform (IDFT) is given by 1  C MN M−1 N−1

c(m, n) =

u=0 v=0

=



1 b(m, n) exp j2 2

u − u

0

M

u

0

M

,

m+

v − v0

v0 N

 

u

e j2

N



M

m+

v N



n



n + (m, n)

(8)

According to Eq. (8), the measured phase can be determined by

(m, n) = tan−1

Im

[c(m, n)] Re [c(m, n)]



− 2

u

0

M

m+

v0 N



n

(9)

According to analysis in Section 1, the carrier-frequency f0x , f0y of carrier interferogram expressed in formula (2) usually are not equal to an integer multiple of the Niquest basic frequency 1/M, 1/N, thus they can be expressed as follow

⎧ u +ı ⎪ ⎨ f0x = 0 x , M

⎪ ⎩ f0y = v0 + ıy , N

− 0.5 < ıx < 0.5 (10) − 0.5 < ıy < 0.5

The deviation between actual carrier-frequency f0x , f0y and the discrete ones u0 , v0 , which caused by the discreteness of the sampling points in the spatial domain of fringe pattern, can be expressed as

⎧ ı u ⎪ ⎨ ıfx = f0x − 0 = x M

M

⎪ ⎩ ı = f0x − v0 = ıy fy N

N

(11)

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Y. Du et al. / Optik 125 (2014) 1056–1061

Thus, the distortion introduced by the spectrum-shifting error in the phase extraction process with traditional FT method is given by

 ı(m, n) = 2



ıy ıx m+ n M N

(12)

According to Eq. (9), it only removes the carrier phase 2(mu0 /M, nv0 /N) when spectrum component C(u/M − u0 /M, v/N − v0 /N) is shifted toward to the origin position in the frequency domain. Therefore, the retrieved phase expressed in Eq. (9) still contains the phase component of the carrier-removal error 2(mıx /M, nıy /N) which caused by the discreteness of the sampling process in the spatial domain, as shown in Eq. (12). Fig. 1. Numerical simulation results: (a) simulated phase map; (b) carrier inrerferogam; (c) the extrapolated interferogram with embed zeros array M × N ( = 10) from (b); the extracted phase maps by the (d) FT, (e) SC, and ZP methods from the simulated carrier interferogram (b), respectively.

2.2. Carrier-frequency estimated from zeros padding fringe pattern

Theorem 1. Zero padding in the spatial domain responding to the bandlimited interpolation in the frequency domain; similarly, zero padding in the frequency domain responding to bandlimited interpolation in the spatial domain [24]. Hence, zero padding in the spatial domain of discretization interferogram will achieve up-sampling in the frequency domain. Hence with the sampling factor  in frequency domain, the carrier-frequency within a small fraction (1/) of an integral (or pixel) can be estimated from extrapolation fringe pattern with FFT approach. With the sampling factor,, the zero padding pattern can be expressed as [24]

 gzp, (m, n) =

g(m, n) ; 0 < m < M − 1, 0 < n < N − 1 0 M < m < M − 1, N < n < N − 1

(13)

The spectral of Eq. (13) is given by

Gzp,

u



,

v

M N





M−1N−1

=

gzp, (m, n)

m=0 n=0



exp −i2

u



M

m+

v N



n

(14)

where u and v are integer of the sampling frequency interval after zero padding. According to Theorem 1 and Fourier theorem, the one-order spectrum component of zero padding fringe patterns can be expressed as Czp,

u



M

−

u0 v v0 , − M N N

 (15)

where u0 , v0 are integer and denote the spectral central of the one-order component, given by



u0 = round[ · (u0 + ıx )],

− 0.5 < ıx < 0.5

v0 = round[ · (v0 + ıy )],

− 0.5 < ıy < 0.5











⎪ ⎩ ı,fy = f0y − Fy N



= f0x −

 = f0y −

mod[ · ıx ] u0 + M M

v0 N

+

mod[ · ıy ] N



=

ıx , M

−

0.5 0.5 < ıx <  

=

ıy , N

−

0.5 0.5 < ıy <  



(18)

As a consequence, the corresponding carrier-removal error is given by



ı˚(m, n) = 2



ıy ıx m+ n M N

(19)

Comparing with Eqs. (11) and (18), it can be seen from that the estimation error of carrier-frequency values are decreased  times with the proposed ZP method, corresponding the carrier-removal error as described in Eq. (19) is reduced to 1/ of the traditional FT method. Once obtaining the carrier-frequency Fx , Fy are estimated from the zero padding interferogram, according to Eq. (9), the measured phase is given by (m, n) = tan−1

Im[c(m, n)]

Re[c(m, n)]

− 2

F

x

M

m+



Fy n N

(20)

Note that just like as describing in Eq. (17), the estimation accuracy of carrier-frequency for the fixed frequency in practical application is uncertain with different sampling factor  but jumps within the range of [−0.5/, 0.5/]. Meanwhile, considering the accuracy requirements in practical application and the memory limitation of the computer, we generally take  = 10 and the estimation accuracy is increased by one order magnitude compared to FT method and it will meet the requirements of the practical applications. Thus we only consider  = 10 in the subsequent discussions and analysis. 3. Numerical simulation experiments and discussions

⎧ u mod[ · ıx ] round[ · (u0 + ıx )] ⎪ = u0 + ⎨ Fx = 0 = 

⎧ Fx ⎪ ⎨ ı,fx = f0x − M

(16)

where round[·] denotes rounding operator, thus the estimated carrier-frequency can be expressed as

⎪ ⎩ Fy = v0 = round[ · (v0 + ıy )] = v0 + mod[ · ıy ]

where mod[·] denotes integer operator. The deviation between actual spatial carrier-frequency f0x , f0y and the estimated carrierfrequency Fx , Fy can be expressed as

(17)

The proposed method was verified by numerical simulations. Fig. 1 shows the simulation results with FT, SC and the proposed ZP methods. The simulated phase, peak-valley value (PV) and root-mean-square value (RMS) of which are 3.664 rad and 0.441 rad, is generated by Peaks function (built-in MATLAB), as shown in Fig. 1(a). Fig. 1(b) shows the simulated carrier interferogram modulated by the simulated phase and the carrier frequencies of f0x = 10.820 pixel−1 , f0y = 16.220 pixel−1 across x and y directions, respectively. The image size is 256 (pixel) × 256 (pixel). For a comparison, the simulated carrier interferogram is firstly analyzed

Y. Du et al. / Optik 125 (2014) 1056–1061

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Table 1 Detailed simulation results of three different methods and corresponding errors. a FTM

f0x (or Fx ) f0y (or Fy ) PV (rad) RMS (rad) Error (rad) a

SCM

ZPM

Real

Caculated

Real

Caculated

Real

Caculated

10.820 16.220 3.664 0.441 PV 2.564

11.000 16.000 3.088 0.611 RMS 0.489

10.820 16.220 3.664 0.441 PV 1.517

11.093 16.114 3.339 0.479 RMS 0.266

10.820 16.220 3.664 0.441 PV 0.344

10.800 16.200 3.696 0.448 RMS 0.0278

PV, RMS denotes the peak-valley value and root-mean-square of the measured phase or the residual phase error, respectively.

0.8

1 FTM

SCM

RMS error (rad)

Error (rad)

0.5 0

-0.5

0.6

0.4

0.2

0

-1 0

50

10 0 15 0 x (pixels)

SCM, F=5 SCM, F=8 ZPM, =10

ZPM

200

250

Fig. 2. Residual errors across x direction with the FT, SC, and the proposed ZP methods.

with the FT method, the carrier-frequency is located at integer multiples of the frequency interval coordinate position (11.000, 16.000), which has a large departure with real carrier-frequency and the extracted phase is shown in Fig. 1(d). Moreover, the estimated carrier-frequency f0x = 11.093 pixel−1 , f0y = 16.114 pixel−1 , are demonstrated by the SC method. The departure between the estimated and actual carrier-frequency values are smaller than ones obtained by FT method, the extracted phase is shown in Fig. 1(e). According to Eq. (13), the extrapolation fringe pattern with zero padding ( = 10) is shown in Fig. 1(c). According to Eq. (17), the estimated carrier-frequency f0x , f0y are equal to f0x = 10.800 pixel−1 , f0y = 16.200 pixel−1 , and the extracted phase is shown in Fig. 1(f). It is evident that the proposed method effectively reduces the carrierremoval error in the retrieved phase compared to ones which are extracted by the FT, and SC methods. On the other hand, the detailed simulation results are given in Table 1, including the real carrier-frequency f0x and f0y , the estimated carrier-frequency (Fx and Fy ), PV and RMS values of the simulation phase and the extracted phase distributions, and the corresponding PV and RMS errors of phase residual error. It shows that the estimated carrier-frequency Fx = 10.800 pixel−1 and Fy = 16.200 pixel−1 by the proposed ZP method, are closer to real values (f0x = 10.820 pixel−1 and f0y = 16.220 pixel−1 ) than ones obtained by the FT method (11.000 pixel−1 and 16.000 pixel−1 ) and SC method (11.093 pixel−1 and 16.114 pixel−1 ) across x and y direction, respectively. The residual error of the proposed ZP method is 0.334 rad (PV) and 0.028 rad (RMS), which are much smaller than that of the FT method (2.564 rad and 0.489 rad) and SC method (1.517 rad and 0.266 rad). For more intuitive comparison of phase residual error by the three different methods, Fig. 2 gives the residual error distribution across x direction of phase residual error. It can be seen from that it has a significant tilt error by the tradition FT and SC methods, while the residual error is almost close to zero by the proposed ZP method. We also find that there is a significant error in the edge region by the all three methods, which caused by spectral leakage effect and boundary effect in FFT analysis

0

0.2

0.4 0.6 RMS (rad)

0.8

1

Fig. 3. Relationship between the carrier-removal error and the RMS of the measured phase.

(Here we only discuss the carrier-removal error, and these problems will be discussed in the other paper). All these results show that the proposed ZP method is effectively for suppressing carrierremoval error in the FT for fringe pattern analysis; it can obtain higher accuracy than FT and SC methods in the phase extraction process. In order to further investigate the performance of the proposed ZP method, and comparing to the SC method, we further give the RMS error residual tilt component in the different distorted of the modulated phase (x,y) and signal to noise ratio (SNR) of fringe pattern, respectively. First, we investigate the performance of the proposed ZP and SC methods with different distorted of the modulated phase (x,y) in carrier interferogram. To compare with the SC method, the common conditions are meet, including the measured phase and the carrier-frequency f0x = 10.826 pixel−1 and f0y = 16.224 pixel−1 across x and y direction, respectively, in each independent simulation experiment. Here, the sampling factor  = 10 in our method, and setting the size of band filter (denoted as F) F = 5,8 in the SC method. The relationship between RMS errors of residual tilt error and the RMS value of the measured phases with different distortion magnificent by the proposed ZP and SC methods are shown in Fig. 3. Obviously, with the increase the distortion magnificent of RMS of the measured phase in the interferogram, the carrierremoval error is increase and shows instability by the SC method. Moreover, with the different size of filter (F = 5,8), the tilt error is great instability. On the contrary, the carrier-removal error of the proposed ZP method is still maintained at a lower level and shows stability. Second, we also evaluate the noise performance of two methods in the presence of different signal to noise ratio (SNR) of the fringe pattern. Fig. 4 presents RMS of the phase residual error as a function of different SNR of fringe pattern. It shows that with the increase SNR of the fringe pattern, the carrier-removal error RMS of the SC method is decrease and the carrier-removal error also shows great instability with different bandwidth of the band filter (F = 5,10). On the contrary, the carrier-removal errors RMS of the

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Y. Du et al. / Optik 125 (2014) 1056–1061 Table 2 Detailed experimental results of three different methods.

0.6 SCM, F=5 SCM, F=10 ZPM, =10

0.4

FTM SCM ZPM

0.3

f0x

f0y

PV

RMS

Tx

Ty

8.000 8.104 7.900

20.000 19.849 20.100

6.038 6.693 5.598

0.878 1.015 0.835

0.196 0. 4849 −0.086

−0.128 −0. 492 0.092

0.2

1

0.1 0

0

20

40

60

SNR (dB) Fig. 4. Residual error as a function of the SNR of fringe pattern.

Zernike coefficients

RMS error (rad)

0.5

0 -1 -2 -3

FTM 1

2

3

SCM

4 5 6 Mode number

ZPM 7

8

Fig. 6. Zernike coefficients of the obtained phase by the FT, SC and ZP methods.

Fig. 5. Experiment results: (a) real interferogram; exracted phase maps with (b) FT, (c) SC, and (d) ZP methods from (a).

proposed ZP method are still maintained at a lower level even SNR close to zero which mean that the proposed ZP method has strong noise immunity. It can be seen from Figs. 3 and 4 that the proposed ZP method for suppressing carrier-removal error is more effectively and more stability than SC method for fringe pattern analysis. This result is determined by the fact that the proposed ZP method estimates the carrier-frequency from the coordinate position corresponding to the maximum value of one-order spectrum component of zero padding fringe patterns, and its accuracy is only relative to the sampling factor . While the SC method estimates the carrier-frequency from the centroid value of one-order spectrum component, thus it shows random and instability due to the spectrum distribution effected by the noise distribution in the fringe pattern and the different bandwidth of filter. 4. Experiment results For further verification of the performance of the proposed carrier-removal method, it is also demonstrated by the optical experiment. The real interferogram is produced by cyclic radial shearing interferometer [12], and the obvious distortion region of interferogram is modulated by the distortion phase caused by a femtosecond laser irradiation to a BK7 sample, as shown in Fig. 5(a).

For a comparison, the phase is simultaneously extracted from interferogram by the FT, SC and the proposed ZP methods, respectively. Fig. 5(b) presents the extracted phase map by the FT method, the PV and RMS of which are 6.038 rad and 0.878 rad. Fig. 5(c) presents the obtained phase with the SC method, the estimated carrierfrequency Fx = 8.104 pixel−1 and Fy = 19.849 pixel−1 across x and y direction, respectively, and the PV and RMS of the obtained phase are 6.693 rad and 1.015 rad. In our method, the interferogram is extrapolated by zero padding with the sampling factor  = 10, the carrier-frequency Fx and Fy are estimated by Eq. (17) and equal to Fx = 7.900 pixel−1 and Fy = 20.100 pixel−1 across x and y direction, respectively. The extracted phase is shown in Fig. 5(d), the PV and RMS of which are 5.598 rad and 0.835 rad. And the detailed experiment results, including the estimated carrier-frequency, PV and RMS of the obtained phase, and the second and third Zernike coefficients Tx and Ty , corresponding the tilt terms along x and y direction, respectively, are given in Table 2. On the other hand, the first eight phase expansion coefficients of Zernikes polynomial of all the three methods are shown in Fig. 6. It can be seen from that the zernikes coefficients of the extracted phase are almost the same expect for the second and third terms, the tilt terms Tx and Ty , which mean that there are different tilt residual error in the retrieved phase with different methods. More for the specific, as shown in Table 2, the tilt coefficients Tx and Ty of the reconstructed phase with our method are Tx = −0.086 and Ty = 0.092, which are much smaller than that the tradition FT method (Tx = −0.196 and Ty = −0.128) and the SC method (Tx = 0.485 and Ty = −0.492). Evidently, it can be concluded from Figs. 5(b)–(d) and fig. 6 and Table 2 that the proposed ZP method is almost removal all tilt error in the extracted phase and it is more effective than the other two methods. 5. Conclusion and discussion In summary, an effective, straightforward and accurate carrierremoval technique in FT method for carrier interferogram analysis is proposed. The original fringe pattern is firstly extrapolated with the zero padding method, and the carrier-frequency values within a small fraction of an integral (or pixel) are estimated from extrapolation pattern with FFT approach. Then carrier phase is removed by subtracting a pure carrier-frequency phase which constructed by the estimated carrier-frequencies in the spatial domain. Therefore, the measured phase can be extracted by the combining the traditional FT and “reference-subtraction approach” from carrier

Y. Du et al. / Optik 125 (2014) 1056–1061

inferferogram. Numerical simulations and actual experiments are carried out to verify the performance of the proposed ZP method and the results show that this method is effective for suppressing the carrier-removal error in the FT for carrier interferogram analysis. It should be note that the improvable of the estimation accuracy by the proposed ZP method is with the expense of the computation. For the interferogram with the size of M × N, the computation complexity is O(MN2 [log (M) + log (N)]) using the proposed method. Moreover, the estimation accuracy of carrier-frequency can be achieved to 1/ pixel−1 in theory, while just like the describing in Eqs. (17) and (18), it cannot conclude that the accuracy is increasing with the sampling factor . Therefore, in order to achieve the balance of the computation and the accuracy, the sampling factor  = 10, corresponding the estimation accuracy is 0.1 order of magnitude of the integral, will meet the accuracy requirements in the practical application and the proposed ZP method is can be easily achieved with an ordinary desktop computer. Moreover, the proposed ZP method can be also applied to other relatively applications, such as determination of the carrier angle of optical configuration [25,26], spatial carrier phase-shifting algorithm [27–29], fringe extrapolation with FT method [30], etc. Acknowledgments This work was supported by Major Program of National Natural Science Foundation of China (60890200) and National Natural Science Foundation of China (10976017). Yongzhao Du is partly supported by the Postgraduate Scholarship Program (Ph.D JointTraining Program) of the China Scholarship Council (CSC) and Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, China. References [1] M. Takeda, H. Ina, S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am. 72 (1982) 156–160. [2] W.W. Macy Jr., Two-dimensional fringe-pattern analysis, Appl. Opt. 22 (1983) 3898–3901. [3] M. Takeda, K. Mutoh, Fourier transform profilometry for the automatic measurement of 3-D object shapes, Appl. Opt. 22 (1983) 3977–3982. [4] K.A. Nugent, Interferogram analysis using an accurate fully automatic algorithm, Appl. Opt. 24 (1985) 3101–3105. [5] D.J. Bone, H.A. Bachor, R.J. Sandeman, Fringe-pattern analysis using a 2-D Fourier transform, Appl. Opt. 25 (1986) 1653–1660. [6] C. Roddier, F. Roddier, Interferogram analysis using Fourier transform techniques, Appl. Opt. 26 (1987) 1668–1673.

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[7] J.B. Liu, P.D. Ronney, Modified Fourier transform method for interferogram fringe pattern analysis, Appl. Opt. 36 (1997) 6231–6241. [8] J.H. Massig, J. Heppner, Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests, Appl. Opt. 40 (2001) 2081–2088. [9] J. Gu, F. Chen, Fourier-transformation phase-iteration, and least-square-fit image processing for Young’s fringe pattern, Appl. Opt. 35 (1996) 232–239. [10] D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing, Marcel Dekker, New York, 1998. [11] D. Li, P. Wang, X. Li, H. Yang, H. Chen, Algorithm for near-field reconstruction based on radial-shearing interferometry, Opt. Lett. 30 (2005) 492–494. [12] D. Liu, Y. Yang, L. Wang, Y. Zhuo, Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer, Appl. Opt. 46 (2007) 8305–8314. [13] P. Ferraro, A. Wax, Z. Zalevsky, Coherent Light Microscopy: Imaging and Quantitative Phase Analysis, Springer, New York, 2010. [14] G. Popescu, L.P. Deflores, K. Badizadega, H. Iwai, R.R. Dasari, M.S. Feld, Fourier phase microscopy for investigation of biological structure and dynamics, Opt. Lett. 29 (2004) 2503–2505. [15] C.J. Mann, L. Yu, C.-M. Lo, M.K. Kim, High-resolution quantitative phase-contrast microscopy by digital holography, Opt. Express. 13 (2005) 8693–8698. [16] N. Lue, W. Choi, G. Popescu, T. Ikeda, R.R. Dasari, K. Badizadegan, M.S. Feld, Quantitative phase imaging of live cells using fast Fourier phase microscopy, Appl. Opt. 46 (2007) 1836–1842. [17] A. Fernandez, G.H. Kaufmann, A.F. Doval, J.B. Garcia, J.L. Fernandez, Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method, Opt. Eng. 37 (1998) 2899–2905. [18] C. Quan, C.J. Tay, L.J. Chen, A study on carrier-removal techniques in fringe projection profilometry, Opt. Laser Technol. 39 (2007) 1155–1161. [19] J. Gu, F. Chen, Fast Fourier transformiteration, and least-squares fit demodulation image processing for analysis of single-carrier fringe pattern, J. Opt. Soc. Am. A 12 (1995) 2159–2164. [20] J.L. Li, X.Y. Su, H.J. Su, S.S. Cha, Removal of carrier frequency in phase-shifting techniques, Opt. Lasers Eng. 30 (1998) 107–115. [21] S. De Nicola, P. Ferraro, Fourier-transform calibration method for phase retrieval of carrier-coded fringe pattern, Opt. Commun. 151 (1998) 217–221. [22] Z. Ge, Fringe analysis method and apparatus using Fourier transform, US Patent, Patent No. US6621579 B2 (2003). [23] Q. Fan, H. Yang, G. Li, J. Zhao, Suppressing carrier removal error in the Fourier transform method for interferogram analysis, J. Opt. 12 (2010) 115401–115404. [24] J.O. Smith, Mathematics of the Discrete Fourier Transform Stanford, CCRMA, 2002, pp. 145–179. [25] P. Gao, B. Yao, J. Min, R. Guo, J. Zheng, T. Ye, I. Harder, V. Nercissian, K. Mantel, Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beamsplitters, Opt. Express. 19 (2011) 1930–1935. [26] T. Tahara, Y. Shimozato, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, T. Kubota, Spatial-carrier phase-shifting digital holography utilizing spatial frequency analysis for the correction of the phase-shift error, Opt. Lett. 37 (2012) 148–150. [27] M. Servin, F.J. Cuevas, A novel technique for spatial phase-shifting interferometry, J. Mod. Opt. 42 (1995) 1853–1862. [28] A. Styk, K. Patorski, Analysis of systematic errors in spatial carrier phase-shifting applied to interferogram intensity contrast determination, Appl. Opt. 46 (2007) 4613–4624. [29] J. Xu, Q. Xu, H. Peng, Spatial carrier phase-shifting algorithm based on leastsquares iteration, Appl. Opt. 47 (2008) 5446–5453. [30] M.J. Dai, Y. Wang, Fringe extrapolation technique based on Fourier transform for interferogram analysis, Opt. Lett. 34 (2009) 956–958.