## An integral approach to phase shifting interferometry using a super-resolution frequency estimation method

Optics Express, Vol. 12, Issue 20, pp. 4681-4697 (2004)

http://dx.doi.org/10.1364/OPEX.12.004681

Acrobat PDF (1247 KB)

### Abstract

The objective of this paper is to describe an integral approach -based on the use of a super-resolution frequency estimation method - to phase shifting interferometry, starting from phase step estimation to phase evaluation at each point on the object surface. Denoising is also taken into consideration for the case of a signal contaminated with white Gaussian noise. The other significant features of the proposal are that it caters to the presence of multiple PZTs in an optical configuration, is capable of determining the harmonic content in the signal and effectively eliminating their influence on measurement, is insensitive to errors arising from PZT miscalibration, is applicable to spherical beams, and is a robust performer even in the presence of white Gaussian intensity noise.

© 2004 Optical Society of America

## 1. Introduction

1. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia **2**, 13–23 (1966). [CrossRef]

6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. **22**, 3421–3432 (1983). [CrossRef] [PubMed]

19. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A **12**, 354–365 (1995). [CrossRef]

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

1. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia **2**, 13–23 (1966). [CrossRef]

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

12. J. van Wingerden, H. J. Frankena, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. **30**, 2718–2729 (1991). [CrossRef] [PubMed]

22. P. L. Wizinowich, “Phase-shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. **29**, 3271–3279 (1990). [CrossRef] [PubMed]

27. Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser diode phase-shifting interferometry,” Opt. Lett. **20**, 1883–1885 (1995). [CrossRef] [PubMed]

18. R. Józwicki, M. Kujawinska, and M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing, Opt. Engg. **31**, 422–433 (1992). [CrossRef]

6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. **22**, 3421–3432 (1983). [CrossRef] [PubMed]

28. Ch. Ai and J. C. Wyant, “Effect of spurious reflection on phase shift interferometry,” Appl. Opt. **27**, 3039–3045 (1988). [CrossRef] [PubMed]

29. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. **34**, 3610–3619 (1995). [CrossRef] [PubMed]

30. B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. **10**, R33–R55 (1999). [CrossRef]

17. B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. **36**, 2070–2075 (1997). [CrossRef] [PubMed]

31. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A **7**, 537–541 (1990). [CrossRef]

21. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A **12**, 1997–2008 (1995). [CrossRef]

30. B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. **10**, R33–R55 (1999). [CrossRef]

19. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A **12**, 354–365 (1995). [CrossRef]

22. P. L. Wizinowich, “Phase-shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. **29**, 3271–3279 (1990). [CrossRef] [PubMed]

32. S. Ellingsrud and G. O. Rosvold, “Analysis of data-based TV-holography system used to measure small vibration amplitudes,” J. Opt. Soc. Am. A **9**, 237–251 (1992). [CrossRef]

33. G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. and Lasers Engg. **7**, 37–68 (1986). [CrossRef]

21. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A **12**, 1997–2008 (1995). [CrossRef]

14. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A **12**, 761–768 (1995). [CrossRef]

14. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A **12**, 761–768 (1995). [CrossRef]

11. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**, 51–60 (1996). [CrossRef] [PubMed]

*κ*+ 3 and 2

*κ*+ 2 samples are respectively necessary to minimize the effect of the

*κ*

*order harmonic, with the phase step 2*

^{th}*π*/(

*κ*+2) between samples. The restriction on the free use of phase steps represents a drawback and a possible limitation for the methods wide spread application. The other important limitations of the algorithm reside in its inability to function in configurations requiring multiple PZTs.

1. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia **2**, 13–23 (1966). [CrossRef]

34. Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. **11**, 1220–1223 (2000). [CrossRef]

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

36. K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Exp. **9**, 236–253 (2001). [CrossRef]

37. C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. **3**, 953–958 (1992). [CrossRef]

39. F. L. Bookstein, “Fitting conic sections to scattered data,” Com. Graphics and Image Process. **9**, 56–71 (1979). [CrossRef]

41. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Let. **28**, 1808–1810 (2003). [CrossRef]

42. X Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. **39**, 585–591 (2000). [CrossRef]

43. G. -S. Han and S. -W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. **33**, 7321–7325 (1994). [CrossRef] [PubMed]

45. G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. and Lasers in Eng. **28**, 61–69 (1997). [CrossRef]

46. B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” App. Mathematics & Computation **146**/2–3, 729–758 (2003). [CrossRef]

47. A. Patil, B. Raphael, and P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. **29**, 1381–1383 (2004). [CrossRef] [PubMed]

*annihilating filter*[50] is designed which has zeroes at the frequencies that are associated with the polynomial for the intensity fringes. The discrete convolution of the filter and the intensity fringe yields zero. The spectral information embedded in the signal corresponds to the phase steps imparted by the multiple PZTs in the interferometric setup. Hence, the parametric estimation of this annihilating filter yields the desired spectral information embedded in the signal, which in our case are the phase steps. Although the discrete Fourier transform is an efficient tool for the estimation of well separated frequencies, the separation of closely spaced frequencies in the presence of noise and less number of samples can be handled efficiently with high resolution techniques [51

51. J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing **36**, 1846–1853 (1988). [CrossRef]

*f*=

*ω*/2

*π*by less than 1/

*N*cycles per sampling interval (here,

*ω*and

*N*refers to angular frequency and number of samples, respectively), which is the resolution limit for the classical periodogram-based methods, such as Fourier transform. As a case study, we show the effectiveness of our algorithm to extract the dual phase steps in holographic moiré. Since, the proposed technique functions by extracting the phase step at each pixel location of the acquired frames, this method allows the use of diverging as well as converging beams. The advantage of using any arbitrary phase step between 0 and

*π*radians overcomes the limitation exhibited by previously reported methods. We also show that the robustness of our algorithm to additive white Gaussian noise can be enhanced by incorporating a denoising step using the concept of truncated singular value decomposition [52–53

52. M. Dendrinos, S. Bakamidis, and G. Carayannis, “Speech enhancement from noise: A regenerative approach,” Speech Communication **10**, 45–57 (1991). [CrossRef]

## 2. Holographic moiré

54. P. K. Rastogi and E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” Appl. Opt. **31**, 2402–2404 (1992). [CrossRef] [PubMed]

55. P. K. Rastogi, M. Barillot, and G. Kaufmann, “Comparative phase shifting holographic interferometry,” Appl. Opt. **30**, 722–728 (1991). [CrossRef] [PubMed]

56. P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. **38**, 1251–1263 (1991). [CrossRef]

48. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” App. Opt. **31**, 1680–1681 (1992). [CrossRef]

*x*,

*y*) for

*m*phase step is given by

^{th}## 3. Theory of estimation of phase steps using a super-resolution method

*=*

_{k}*a*exp(

_{k}*ikφ*

_{1}),

*u*= exp(

_{k}*ikα*),

*℘*=

_{k}*b*exp(

_{k}*ikφ*

_{2}),

*v*= exp(

_{k}*ikβ*); superscript * denotes the complex conjugate;

*φ*

_{1}and

*φ*

_{2}, and,

*α*and

*β*, are the unknown parameters;

*I*(

_{n}*x*,

*y*;

*m*) corresponds to the

*n*frame, where

^{th}*n*= 0 refers to the data frame corresponding to the first phase shifted intensity pattern. Equation (4) represents the complex valued sinusoidal signals where

*α*,….,

*κα*; -

*α*,….,-

*κα*;

*β*,….,

*κβ*; -

*β*,….,-

*κβ*represent the frequencies embedded in the signal, and by estimating them, the phase steps

*α*and

*β*, imparted by the PZTs can be determined. Frequency estimation of sinusoids is a classical problem in spectral estimation and as mentioned in Section 1 can be better handled using a super-resolution method. In this method we first transform the discrete time domain signal

*I*in Eq. (4) into a complex frequency domain by taking its Z-transform. Let the Z-transform of

_{n}*I*be denoted by I(

_{n}*z*). The objective here is to design another polynomial P(

*z*) termed as

*annihilating filter*which has zeros at frequencies associated with I(

*z*), which in turn would result in I(

*z*)P(

*z*) = 0. Since the frequencies translate into zeros, spectral estimates feature high resolution. In what follows we will derive the expression for P(

*z*) and explore the condition for which the multiplication of P(

*z*) and I(

*z*) is zero.

*z*) represents the numerator and P(

*z*) the denominator in the right hand side of Eq. (7), with the latter being given by

*z*), let us consider for the sake of simplicity writing only its first term,

_{1}(

*z*), specifically,

*C*1

_{0},

*C*1

_{1},

*C*1

_{2}……

*C*1

_{4κ}, and

*C*1

*,*

_{N}*C*1

_{N+1},

*C*1

_{N+2}……

*C*1

_{N+4κ}depend upon the unknown amplitude and frequency. We observe that the coefficients

*C*1

_{4κ+1},

*C*1

_{4κ+2},

*C*1

_{4κ+3}……,

*C*1

_{N-1}are identically zero for

*N*≥ 4

*κ*+ 1. This condition is at the essence of the design of annihilating filter P(

*z*). In what follows we show that the (4

*κ*+1)

*degree polynomial P(*

^{th}*z*), when discretely convolved with I(

*z*) yields zero.

*p*is the inverse Z-transform of P(

_{n}*z*) defined by

*N*-4

*κ*-2 rows equations corresponding to zero row values in the matrix on the right hand side, and form a new matrix as follows:

*z*) is an annihilating filter which when convolved with the moiré intensity signal yields zero. It can further be deduced that at least

*N*≥ 8

*κ*+2 samples are required to extract the roots of the polynomial P(

*z*). This enables us to find the unknown values

*u*and

_{k}*v*. The phase steps

_{k}*α*and

*β*, can hence be computed using

*α*= ℜ(ln

*u*

_{1}/

*i*) and

*β*= ℜ(ln

*v*

_{1}/

*i*).

## 4. Detection of nonsinusoidal waveform and the corresponding harmonic content

*κ*present in the signal, which can subsequently be applied in the design of the Vandermonde system of equations in Section 6 for the determination of phase values

*φ*

_{1}and

*φ*

_{2}. Let us rewrite Eq. (1), for the case

*κ*= 1 (i.e. pure sinusoidal wave) and noiseless signal, as

*I*(

*x*,

*y*;

*m*) in Eq. (17) corresponds to two wave interferometry. Fourier transform of

*I*(

*x*,

*y*;

*m*) for

*m*= 1,2,……,

*N*should result in three peaks in frequency domain corresponding to

*I*,

_{dc}*α*and -

*α*. Similarly for T1≠0 and T2≠0, a case in which the intensity

*I*(

*x*,

*y*;

**m**) in Eq. (17) corresponds to holographic moiré interferometry, Fourier transform of

*I*(

*x*,

*y*;

*m*) for

*m*= 1,2,……,

*N*should result in five peaks in frequency domain corresponding to

*I*,

_{dc}*α*, -

*α*,

*β*, and -

*β*. However, with limited data samples, the resolution of closely spaced frequencies is troublesome in the presence of noise. Also because of the “leakage” phenomenon, energy in the main lobe can leak into the side lobes obscuring and distorting other spectral responses.

**R**in Eq. (16), represented as

**R**=

**USV**

^{T}, yields more reliable information than the Fourier transform method regarding the number of harmonics present in the signal, where

**S**is a diagonal matrix with

*M*nonzero and

*N*-

*M*zero singular values;

**U**and

**V**are unitary matrices [57

57. R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems **AES-19**, 134–139 (1983). [CrossRef]

**S**is

*M*= 3 (corresponding to

*I*,

_{dc}*α*, and -

*α*). On the other hand, for T1 ≠ 0 and T2 ≠ 0, the number of nonzero diagonal entries is

*M*= 5 (corresponding to

*I*,

_{dc}*α*, -

*α*,

*β*, and -

*β*). Hence, if H is the number of PZTs used in the optical setup, the number of harmonics

*κ*in the signal can be determined using

*M*= 2H

*κ*+1. In the presence of noise,

*M*principal values of

**S**would still tend to be larger than the

*N*-

*M*values which were originally zero. In addition, the

*M*eigenvectors corresponding to the

*M*eigen values of

**R**

^{T}

**R**are less susceptible to noise perturbations in comparison to the remaining

*N*-

*M*eigenvectors.

**S**when the number of samples (data frames) used are

*N*= 25. In this example we assume the presence of second order harmonics in the signal,

*κ*= 2, and the presence of two PZTs in the optical configuration, H = 2. In this case and for a noiseless signal we should expect

*M*= 2H

*κ*+ 1 = 9 diagonal entries of

**S**in Fig. 1 to be significantly larger in magnitude; the

*N*- 9 diagonal entries being zero. Even in presence of noisy data for the case with (SNR = 10 dB),

*M*= 9 principal values are still larger in magnitude than the

*N*- 9 diagonal values. Hence by selecting only those significant values of

**S**(which in this example, corresponds to

*M*= 9), the number of harmonics

*κ*is determined to be

*κ*= 2. The plot thus allows for a reliable estimation of the number of harmonics present in the signal.

## 5. Denoising the signal

58. K. B. Hill, S. A. Basinger, R. A. Stack, and D. J. Brady, “Noise and information in interferometric cross correlators,” Appl. Opt. **36**, 3948–3958 (1997). [CrossRef] [PubMed]

21. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A **12**, 1997–2008 (1995). [CrossRef]

*κ*+2 are needed for applying the denoising procedure.

52. M. Dendrinos, S. Bakamidis, and G. Carayannis, “Speech enhancement from noise: A regenerative approach,” Speech Communication **10**, 45–57 (1991). [CrossRef]

**R**is first written in Hankel matrix form, say

**R̂**, and its singular value decomposition

**R̂**=

**ÛS̄V̂**

^{T}shows the nonsingular principal values of

**S̄**which are significantly different from zero. After setting the non significant

*N*-

*M*singular values of

**S̄**to zero, a matrix

**Ŝ**is formed. A denoised matrix

**Z**

*=*

_{M}**ÛŜV̂**

^{T}which approximates

**R̂**in the least squares sense, is then obtained by using the first

*M*columns of

**Û**,

**Ŝ**and

**V̂**;

**Û**and

**V̂**being unitary matrices. Finally, a denoised signal

*Ī*is retrieved by arithmetic averaging along the anti-diagonals (or diagonals) of

_{n}**Z**

*using*

_{M}*r*= max[1,

*n*- number of rows (

**R̂**)+1) and

*q*= min[number of rows (

**R̂**),

*n*]. The denoised signal

*Ī*is subsequently applied in Eq. (16).

_{n}## 6. Evaluation of phase steps and phase distributions in presence of noise

*x*

_{0},

*y*

_{0}) is the origin of the

*X*×

*Y*pixels for fringe pattern with pitch

*λ*

_{1}; (

*p*

_{0},

*y*

_{0}) is the origin of

*X*×

*Y*pixels for fringe pattern with pitch

*λ*

_{2};

*ξ*is some arbitrary constant;

*κ*= 2 and the phase steps are selected as

*α*=

*π*/4 and

*β*=

*π*/3. A white Gaussian noise with SNR from 0 to 100 dB is added to test the robustness of the proposed concept. Fringes shown in Fig. 2(a)–(c) correspond to Eq. (19) for

*κ*= 1 and for three different noise levels. On the other hand, fringes shown in Figs. 3(a)–(c) corresponds to Eq. (19) for second order harmonic,

*κ*= 2, but for same noise levels as in Figs. 2(a)–(c), respectively. These fringes have been generated under the assumption

*a*

_{0, ± 1, ± 2}=

*b*

_{0, ± 1, ± 2}= 0.5.

*and*

_{k}*℘*can be solved using the linear Vandermonde system of equations obtained from Eq. (4). The Vandermonde system of equations always has full rank as long as

_{k}*α*’

_{i}*s*and

*β*’

_{i}*s*are distinct. The matrix thus obtained can be written as

*α*

_{1},

*β*

_{1}), (

*α*

_{2},

*β*

_{2}), (

*α*

_{3},

*β*

_{3}), .. and (

*α*,

_{N}*β*) are the phase steps for frames

_{N}*I*

_{0},

*I*

_{1},

*I*

_{2}, .., and

*I*

_{N-1}, respectively. The phases

*φ*

_{1}and

*φ*

_{2}are subsequently computed from the argument of ℓ

_{1}and

*℘*

_{1}. Figures 5(a) and 5(b) show typical errors in the computation of phase

*φ*

_{1}without and with the application of the denoising procedure, respectively.

*φ*

_{2}without and with the application of the denoising step, respectively, all the other parameters remaining the same. Figure 7 shows the wrapped phases

*φ*

_{1}and

*φ*

_{2}obtained with the denoising step.

## 7. Conclusion

## Acknowledgments

## References and Links

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2. | 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, App. Opt. |

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10. | Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. |

11. | Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. |

12. | J. van Wingerden, H. J. Frankena, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. |

13. | K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A |

14. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A |

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

16. | B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. |

17. | B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. |

18. | R. Józwicki, M. Kujawinska, and M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing, Opt. Engg. |

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20. | P. de Groot and L. L. deck, “Numerical simulations of vibration in phase-shifting interferometry,“ App. Opt. |

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28. | Ch. Ai and J. C. Wyant, “Effect of spurious reflection on phase shift interferometry,” Appl. Opt. |

29. | J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. |

30. | B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. |

31. | C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A |

32. | S. Ellingsrud and G. O. Rosvold, “Analysis of data-based TV-holography system used to measure small vibration amplitudes,” J. Opt. Soc. Am. A |

33. | G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. and Lasers Engg. |

34. | Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. |

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

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

37. | C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. |

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

39. | F. L. Bookstein, “Fitting conic sections to scattered data,” Com. Graphics and Image Process. |

40. | A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” Pattern Anal. and Machine Intelligence. |

41. | L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Let. |

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

43. | G. -S. Han and S. -W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. |

44. | W. Jüptner, T. Kreis, and H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” In W. F. Fagan, ed., |

45. | G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. and Lasers in Eng. |

46. | B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” App. Mathematics & Computation |

47. | A. Patil, B. Raphael, and P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. |

48. | P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” App. Opt. |

49. | P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” App. Opt. |

50. | P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, New Jersey, 1997). |

51. | J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing |

52. | M. Dendrinos, S. Bakamidis, and G. Carayannis, “Speech enhancement from noise: A regenerative approach,” Speech Communication |

53. | P. C. Hansen and S. H. Jensen, “FIR filter representations of reduced-rank noise reduction,” IEEE Transactions on Signal Processing |

54. | P. K. Rastogi and E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” Appl. Opt. |

55. | P. K. Rastogi, M. Barillot, and G. Kaufmann, “Comparative phase shifting holographic interferometry,” Appl. Opt. |

56. | P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. |

57. | R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems |

58. | K. B. Hill, S. A. Basinger, R. A. Stack, and D. J. Brady, “Noise and information in interferometric cross correlators,” Appl. Opt. |

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.4120) Instrumentation, measurement, and metrology : Moire' techniques

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

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 17, 2004

Revised Manuscript: September 15, 2004

Published: October 4, 2004

**Citation**

Abhijit Patil, Rajesh Langoju, and Pramod Rastogi, "An integral approach to phase shifting interferometry using a super-resolution frequency estimation method," Opt. Express **12**, 4681-4697 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4681

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