## Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation |

Optics Express, Vol. 22, Issue 4, pp. 4738-4750 (2014)

http://dx.doi.org/10.1364/OE.22.004738

Acrobat PDF (6670 KB)

### Abstract

A cascade least-squares scheme for wrapped phase extraction using two or more phase-shifted fringe-patterns with unknown and inhomogeneous surface phase shift is proposed. This algorithm is based on the parameter estimation approach to process fringe-patterns where, except for the interest phase distribution that is a function of the space only, all other parameters are functions of both space and time. Computer simulations and experimental results show that phase computing is possible even when an inhomogeneous phase shift is induced by nonlinearity of the piezoelectric materials or miscalibrated phase shifters. The algorithm’s features and its operating conditions will been discussed. Due to the useful properties of this algorithm such as the robustness, computational efficiency, and user-free execution, this proposal could be used in automatic applications.

© 2014 Optical Society of America

## 1. Introduction

1. D. Malacara, M. Servin, and Z. Malacara, *Interferogram Analysis for Optical Testing*, 2nd ed. (Taylor and Francis, 2005). [CrossRef]

3. X. Xu, G. Lu, G. Han, F. Gao, Z. Jiao, and D. Li, “Phase stitching and error correction in aperture synthesis for generalized phase-shifting interferometry,” Appl. Opt. **52**, 4864–4870 (2013). [CrossRef] [PubMed]

4. G. Rajshekhar and P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Lasers Eng. **50**, iii–x (2012). [CrossRef]

*k*= 0, 1,··· ,

*K*− 1, where

*K*is the number of fringe-patterns,

*I*(

_{k}*p*) is a spatial intensity distribution,

*p*= (

*x*,

*y*) is a two-dimensional spatial variable,

*a*(

_{k}*p*) is the background light,

*b*(

_{k}*p*) is the modulation light, Φ

*(*

_{k}*p*) is the encoded phase,

*ϕ*(

*p*) is the phase function of interest to be recovered, and

*δ*(

_{k}*p*) is the phase shift function.

5. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**, 2693–2703 (1974). [CrossRef] [PubMed]

*k*. Second, the phase shift function is commonly homogeneous in space, i.e., the phase shift does not depend on

*p*(the phase shift function is a set of non-tilted planes). And third, the phase shift function must be known

*a priori*.

*K*-step algorithms [5

5. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**, 2693–2703 (1974). [CrossRef] [PubMed]

6. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. **7**, 368–370 (1982). [CrossRef] [PubMed]

7. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. **23**350–352 (1984). [CrossRef]

8. Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. **24**, 3049–3052 (1985). [CrossRef] [PubMed]

10. W. Li and X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. **40**, 761–766 (2001). [CrossRef]

11. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A **8**, 822–827 (1991). [CrossRef]

12. P. Carre, “Installation et utilisation du comparateur photohlectrique et interfhrentiel du bureau international des poids et mesures,” Metrologia **2**, 13–23 (1966). [CrossRef]

13. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. **26**, 2504–2506 (1987). [CrossRef] [PubMed]

*k*. In other words, the phase shift must be non-tilted planes separated one from another by a constant distance as shown in Fig. 1(a).

*k*as shown in Fig. 1(b). The most generalized phase-shifting algorithms fall into this group [11

11. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A **8**, 822–827 (1991). [CrossRef]

14. C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. **5**(6), 648 (1994). [CrossRef]

19. R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. **51**(5), 626–632 (2013). [CrossRef]

20. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. **39**, 3894–3898 (2000). [CrossRef]

22. K. Patorski, A. Styk, L. Bruno, and P. Szwaykowski, “Tilt-shift error detection in phase-shifting interferometry,” Opt. Express **14**, 5232–5249 (2006). [CrossRef] [PubMed]

23. F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express **21**, 17234–17248 (2013). [CrossRef] [PubMed]

26. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express **13**, 3743–3753 (2005). [CrossRef] [PubMed]

27. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. **36**, 2417–2419 (2011). [CrossRef] [PubMed]

28. C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. A. Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express **21**, 17228–17233 (2013). [CrossRef] [PubMed]

*p*and

*k*[29

29. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A **14**, 918–930 (1997). [CrossRef]

*p*and

*k*) [30

30. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. **197**, 37–42 (2001). [CrossRef]

*ϕ*(

*p*) is equal in all fringe-patterns, and the phase shift

*δ*(

_{k}*p*) varies from one to another. In other words,

*ϕ*(

*p*) and

*δ*(

_{k}*p*) are, respectively, the static and dynamic parts of the encoded phase as can be seen in Fig. 2.

*k*, the interest phase distribution is suppressed and the remainder is the phase step

*α*(

_{k}*p*), i.e., Thus, the wrapped phase extraction is possible if

*α*(

_{k}*p*) ≠ 0. This is a basic assumption in the phase-shifting approach. Then, the phase shift is reconstructed by Finally, the interest phase distribution

*ϕ*(

*p*) is recovered by using the phase shift

*δ*(

_{k}*p*) computed by Eq. (3). So the functions

*ϕ*(

*p*) and

*δ*(

_{k}*p*) are clearly distinguished from one another (even when

*δ*(

_{k}*p*) is nonlinear unknown function of both

*p*and

*k*) and the phase computing is possible in the context of phase-shifting. For the discrete case, Eqs. (2) and (3) are the respective finite differences and cumulative sum.

19. R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. **51**(5), 626–632 (2013). [CrossRef]

*k*, Fig. 1(b). For this study, an advanced phase shift estimation stage is developed. Thus, the proposed algorithm can efficiently handle inhomogeneous nonlinear phase shifts of both

*p*and

*k*, Fig. 1(d). The beneficial properties of the algorithm (such as only two or more fringe-pattern requirements, robustness, user-free execution, and computational efficiency) are kept. The results from computer simulation and experimental tests validate this proposal.

## 2. Theoretical principles

*p*are defined in a discretized rectangular domain. Thus, they are considered

*M*×

*N*matrices and the variable

*p*is not written down for brevity. For a

*M*×

*N*matrix 𝔸 with

*M*≥

*N*and rank(𝔸) =

*N*, the notation 𝔸

^{†}= (𝔸

*𝔸)*

^{T}^{−1}𝔸

*denotes the least-squares inverse of the matrix 𝔸.*

^{T}### 2.1. Fringe-pattern normalization

*a*and modulation

_{k}*b*lights from the fringe-patterns

_{k}*I*described by Eq. (1). For this, we used the parameter estimation approach [19

_{k}19. R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. **51**(5), 626–632 (2013). [CrossRef]

*ã*and

_{k}*b̃*are the approximations of the parameters

_{k}*a*and

_{k}*b*, respectively. The columns of the matrices 𝔸

_{k}*and 𝔸*

_{a}*are the basis functions employed.*

_{b}*ã*(a 2nd-order approximation of a Gaussian profile) and, a 4th-degree polynomial to estimate

_{k}**51**(5), 626–632 (2013). [CrossRef]

### 2.2. Phase shift estimation

*A*and subtraction

_{k}*S*of adjacent normalized fringe-patterns defined, respectively, by for

_{k}*k*= 1, 2,··· ,

*K*− 1, where

*α*=

_{k}*δ*−

_{k}*δ*

_{k}_{−1}(a discrete version of Eq. (2) for Δ

*k*= 1) is the

*phase step*between the fringe-patterns

*I*and

_{k}*I*

_{k}_{−1}. From the above equations, by applying some trigonometric identities and a few algebraic operations, we can reach: where

*η*= cos(2

_{k}*ϕ*+

*δ*

_{k}_{−1}+

*δ*). It is possible to estimate the term cos

_{k}*α*from any above equation by considering that the factor

_{k}*η*is noise with zero mean and amplitude of 2cos

_{k}^{2}(

*α*/2) or 2sin

_{k}^{2}(

*α*/2), respectively. However, in some situations, one is more convenient than another as shown in Fig. 3(a). Particularly, for

_{k}*α*∈ [0,

_{k}*π*/2], it is appropriate to choose

*α*∈ [

_{k}*π*/2,

*π*].

*a priori*the data

*c*and

_{Ak}*c*are the fitted polynomials which approximate the term cos

_{Sk}*α*from

_{k}*are the basis functions of the polynomial used. Particularly, in this work, a 4th-degree Taylor polynomial is used.*

_{c}*c*and

_{Ak}*c*, we consider the approximation Γ(·) = [1 + tanh(·)]/2 of the unit step function, Fig. 3(b), as where

_{Sk}*c*= (

_{k}*c*+

_{Ak}*c*)/2, and

_{Sk}*ω*is an appropriate constant (experimentally, we chose

*ω*= 15). Thus, a smooth transition between

*c*and

_{Sk}*c*at

_{Ak}*c*= 0 is obtained.

_{k}*α*, an inverse cosine of Eq. (9) is computed. Then, the phase shift function is constructed by the cumulative sum (a discrete version of Eq. (3) for Δ

_{k}*k*= 1): It is not a loss of generality to consider that

*δ*

_{0}= 0. If additional information over

*δ*

_{0}is available, it can be included in Eq. (10). For example,

*δ*

_{0}=

*d*where

*d*is the last phase shift of a previous evaluation.

### 2.3. Wrapped phase extraction

*p*as where the quantities

*ζ*= sin

*ϕ*and

*ξ*= cos

*ϕ*are obtained by the least-squares method [6

6. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. **7**, 368–370 (1982). [CrossRef] [PubMed]

7. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. **23**350–352 (1984). [CrossRef]

*is, in general, different for each point*

_{ϕ}*p*. Therefore, this last stage is time-consuming because a matrix inversion must be performed for each point to solve Eq. (12). However, the runtime can be reduced by parallel computing because the wrapped phase extraction is pointwise. Particularly, the wrapped phase extraction is faster when the phase shift is homogeneous (non-tilted planes). In this case, the matrix 𝔸

*is the same for all points*

_{ϕ}*p*and a single matrix inversion is necessary.

**51**(5), 626–632 (2013). [CrossRef]

## 3. Computer simulation and experimental results

### 3.1. Computer simulation

*α*

_{1}and the phase distribution

*ϕ*given, respectively, by for

*x*,

*y*∈ [−1, 1]. The command

`peaks(500)`is defined in MATLAB software and returns a linear combination of Gaussian functions. The plots of Eqs. (14) and (15) are shown in the first column of Fig. 5.

*k*= 0, 1, where

*ρ*is a Gaussian noise with zero mean and standard deviation

_{k}*σ*(10%). The respective background and modulation lights are given by The functions

*a*,

_{k}*b*, and

_{k}*I*are shown in the second and third columns of Fig. 5 for

_{k}*k*= 1, 2, respectively.

*I*, Figs. 5(j) and 5(k), are processed by the proposed algorithm. The resulting normalized fringe-patterns

_{k}*Ī*, phase step

_{k}*α*

_{1}, and the recovered wrapped phase are shown in the last column of Fig. 5.

### 3.2. Experimental results

*λ*= 633 nm), a spatial filter (objective microscope and pinhole), and a collimating lens (focal length

*f*= 0.5 m).

*M*

_{1}is driven by two piezo-electric devices,

*PZT*

_{1}and

*PZT*

_{2}. Thus, if the control signals for these devices are equal, Fig. 7(b), a homogeneous phase shift is induced. Otherwise, Fig. 7(c), an inhomogeneous tilted planes phase shift is generated and the tilting is controlled by the difference between the control signals. Moreover, if the control signals are impulsive, Fig. 7(d), an inhomogeneous surface phase shift is induced due to this mechanical perturbation.

*OP*were recorded by using a gray-scale camera sensor (1824 × 1418 pixels with 8-bit pixel depth). Notice that the phase object is not previously characterized. However, it is measured by first phase-shifting with a conventional homogeneous phase shift. Thus, when inhomogeneous phase shifts are considered, the results must be consistent with the first measure.

## 4. Discussion

### 4.1. Reference frame in phase-shifting

*δ*(

_{k}*p*) at

*k*= 0 as is shown below.

33. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. **36**(3), 813–821 (1998). [CrossRef]

*ϕ*

_{1},

*ϕ*

_{2}, and

*ϕ*

_{3}) are shown in Figs. 9(a)–9(c). The differences

*ϕ*

_{2}−

*ϕ*

_{1}and

*ϕ*

_{3}−

*ϕ*

_{1}are shown in Figs. 9(d) and 9(g), respectively. We can see that these differences are planes. This claim is verified by fitting linear functions, Figs. 9(e) and 9(h), where the fitting error, Figs. 9(f) and 9(i), present average values of 4.5 × 10

^{−15}and −3.3 × 10

^{−14}, respectively. Although the difference

*ϕ*

_{3}−

*ϕ*

_{2}is not shown, it can be proved that such difference is also a plane. Accordingly, the experimental wrapped phase maps in Fig. 8 actually describe the same topography.

*δ*

_{0}(

*p*) = 0 was assumed. Then,

- For the first experiment, the wavefront of interest was measured and an additional phase (the non-tilted phase shift) was induced.
- For the second experiment, since
*δ*_{0}(*p*) = 0 was considered, the recovered phase distribution corresponds to the wavefront of interest plus the phase shift from the first experiment. Thus, the difference between the results from this experiment and the first one must be a non-tilted plane. Approximately, this difference is observed in Fig. 9(d). Again, an additional phase (a tilted plane) is induced by the phase-shifting procedure. - For the third experiment, the wavefront of interest plus the cumulated phase shifts was measured by assuming
*δ*_{0}(*p*) = 0. Therefore, the difference between the results from this experiment and the first one is the resultant tilted plane as is observed in Fig. 9(g). - In any future experiment, the wavefront of interest plus the phase added by previous phase-shifting experiments will be sensed.

*δ*

_{0}(

*p*) in Eq. (10). This subtracts this cumulated phase from the final wrapped phase map. In any case, the proposed phase-shifting algorithm can extract the interest wrapped phase distribution up to an initial phase shift

*δ*

_{0}(

*p*) as occurs with any phase-shifting method.

### 4.2. Spatio-temporal visibility fluctuations

### 4.3. Algorithm performance

### 4.4. Operating requirements

*a*,

_{k}*b*and

_{k}*α*where the additive terms cos(

_{k}*ϕ*+

*δ*) and cos(2

_{k}*ϕ*+

*δ*

_{k}_{−1}+

*δ*) (with their respective amplitudes) are seen as noise. Thus, many fringes (open and closed in any combination) across the recorded intensities

_{k}*I*are required in order to satisfy both symmetric distribution and zero mean conditions. The property of additivity of the noise does not allow processing speckled fringe-patterns because speckle noise is multiplicative. However, the method could be applicable if pre-filtering of the fringe-patterns is accomplished.

_{k}*a*,

_{k}*b*and

_{k}*α*requires appropriate basis functions. Fortunately, there are a variety of options from which we can choose for each particular application. For example, the basis functions can be polynomials (such as truncated Taylor and Fourier series, Seidel, Zernike, etc.), splines (piecewise polynomials) of appropriate degree, B-splines, etc. This flexibility allows the proposed algorithm to be implemented in many applications.

_{k}*α*are computed as the argument of a cosine function, Eq (8). Therefore, the phase steps are recovered without ambiguity if

_{k}*α*lies within the interval (0,

_{k}*π*) rad. Nevertheless, the phase shift

*δ*, the cumulative sum of the phase steps, may be greater than 2

_{k}*π*as long as the matrix 𝔸

*of Eq. (13) satisfies rank(𝔸*

_{ϕ}*) = 2. If the phase steps exceed the interval (0,*

_{ϕ}*π*), they are wrapped. We believe that this issue can be overcome by an additional unwrapping procedure; however, this possibility is left as future work.

## 5. Conclusion

## Acknowledgments

## References and links

1. | D. Malacara, M. Servin, and Z. Malacara, |

2. | A. Patil and P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. |

3. | X. Xu, G. Lu, G. Han, F. Gao, Z. Jiao, and D. Li, “Phase stitching and error correction in aperture synthesis for generalized phase-shifting interferometry,” Appl. Opt. |

4. | G. Rajshekhar and P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Lasers Eng. |

5. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. |

6. | C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. |

7. | J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. |

8. | Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. |

9. | X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A |

10. | W. Li and X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. |

11. | G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A |

12. | P. Carre, “Installation et utilisation du comparateur photohlectrique et interfhrentiel du bureau international des poids et mesures,” Metrologia |

13. | P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. |

14. | C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. |

15. | X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. |

16. | K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express |

17. | A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. |

18. | Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. |

19. | R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. |

20. | M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. |

21. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. |

22. | K. Patorski, A. Styk, L. Bruno, and P. Szwaykowski, “Tilt-shift error detection in phase-shifting interferometry,” Opt. Express |

23. | F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express |

24. | J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A Pure Appl. Opt. |

25. | J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. |

26. | O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express |

27. | C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. |

28. | C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. A. Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express |

29. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A |

30. | M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. |

31. | J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. |

32. | H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express |

33. | M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 7, 2013

Revised Manuscript: December 19, 2013

Manuscript Accepted: January 7, 2014

Published: February 21, 2014

**Citation**

Rigoberto Juarez-Salazar, Carlos Robledo-Sanchez, Fermin Guerrero-Sanchez, and A. Rangel-Huerta, "Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation," Opt. Express **22**, 4738-4750 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4738

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### References

- D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor and Francis, 2005). [CrossRef]
- A. Patil, P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253–257 (2007). [CrossRef]
- X. Xu, G. Lu, G. Han, F. Gao, Z. Jiao, D. Li, “Phase stitching and error correction in aperture synthesis for generalized phase-shifting interferometry,” Appl. Opt. 52, 4864–4870 (2013). [CrossRef] [PubMed]
- G. Rajshekhar, P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012). [CrossRef]
- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974). [CrossRef] [PubMed]
- C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982). [CrossRef] [PubMed]
- J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23350–352 (1984). [CrossRef]
- Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985). [CrossRef] [PubMed]
- X. Chen, M. Gramaglia, J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A 17, 2061–2066 (2000). [CrossRef]
- W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001). [CrossRef]
- G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991). [CrossRef]
- P. Carre, “Installation et utilisation du comparateur photohlectrique et interfhrentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966). [CrossRef]
- P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987). [CrossRef] [PubMed]
- C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648 (1994). [CrossRef]
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