## Algorithm for phase extraction from a set of interferograms with arbitrary phase shifts |

Optics Express, Vol. 19, Issue 6, pp. 4908-4923 (2011)

http://dx.doi.org/10.1364/OE.19.004908

Acrobat PDF (1244 KB)

### Abstract

The generalized analytical quadrature filter from a set of interferograms with arbitrary phase shifts is obtained. Both symmetrical and non symmetrical algorithms for any order are reported. The analytic expression is obtained through the convolution of a set of two-frame algorithms and expressed in terms of the combinatorial theory. Finally, the solution is applied to obtain several generalized tunable quadrature filters.

© 2011 OSA

## 1. Introduction

1. 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**(11), 2693–2703 (1974). [CrossRef] [PubMed]

14. A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. **49**(32), 6224–6231 (2010). [CrossRef] [PubMed]

12. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express **17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

13. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express **17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

14. A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. **49**(32), 6224–6231 (2010). [CrossRef] [PubMed]

15. J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express **18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

16. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. **29**(14), 1671–1673 (2004). [CrossRef] [PubMed]

## 2. The two-frame algorithm in phase shifting interferometry (PSI)

### 2.1 The symmetric two-frame filter

*α*is the frequency carrier or usually named phase shift,

*x,y*) and

1. 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**(11), 2693–2703 (1974). [CrossRef] [PubMed]

15. J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express **18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

12. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express **17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

12. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express **17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

15. J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express **18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

*α*over the frequency axes because it Eq. (3) satisfies the quadrature condition

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

### 2.2 The discrete temporal convolution

*n*and

*m*frames respectively. On the other hand,

*nx1*] row vectors, likewise

*mx1*] row vectors too. Therefore, the temporal impulse responses

*nx1*] and [

*mx1*] column vectors expressed in terms of the impulse functions such as:Therefore, the combined filter between

*h*(

*t*)=

*h*

_{1}(

*t*)*

*h*

_{2}(

*t*), where ∗ denotes the temporal discrete convolution. Then, from Eq. (6) the temporal response

*n + m-*1) resultant filter is obtained, and from Eq. (8) the desired phase

*X*and

*Y*vectors with

*n*and

*m*orders is obtained from

*k*= 1,2 ...

*m + n-*1. The coefficients of the product are given by the convolution of the original coefficient sequences

*X*and

*Y*, extended with zeros where necessary to avoid undefined terms and this is the well known Cauchy product of two polynomials.

### 2.3 The tunable rotated two-frame algorithm

*θ*is the rotation angle. That implies that a filter

*N/D*rotated and angle

*–D/N.*Then, from Eqs. (4),11), the rotated two-frame filter for an arbitrary

*θ*is given by

*α*. Then, Eq. (13) for a specific angle

## 3. The design of an *M* order quadrature filter

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

*M*can be obtained from a set of

*M-1*frequencies

### 3.1 The general theory of phase shifting interferometry

*ϕ*of a quadrature filter order

*M*is expressed as [1

1. 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**(11), 2693–2703 (1974). [CrossRef] [PubMed]

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

8. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**(1), 51–60 (1996). [CrossRef] [PubMed]

*M-1*values such that

*M-1*roots

*k*= 1, 2, …

*M-1*. Thus, for each roots an angle

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

*M-1*interceptions

*M*is expressed as the product of a set of

*M-1*individual filters

*M-1*individual filters

*N*and

*D*are the vectors defined in Eq. (18). Then, from Eq. (10) the numerator and the denominator of an algorithm is expressed asBy using Eq. (13) into Eq. (26) the filter is expressed asIn other words, in the same way that a polynomial with order

*M-1*is expressed as the product of

*M-1*first order polynomials, the Fourier transform of a quadrature filter order

*M*is expressed as the product of

*M-1*individual filters. Where for each filter a two-frame filter is associated. Therefore, through the convolution of this set of

*M-1*two-frame filters showed above, the desired filter is obtained. Additionally, from Eqs. (15),23) the Fourier transform of the quadrature filter can be recovered asFor

*σ*becomes the real quantityIn this work, this case is so called symmetric algorithm and from Eq. (27) resultsOtherwise, for any real value

*M*order filter from a set of

*M*-1 arbitrary frequencies (interceptions). The second goal is to analyze the behavior of any given algorithm. Finally, the last goal is to improve a known algorithm adding new properties through the convolution algorithm.

### 3.2 The tunable non symmetric three-frame algorithm

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

### 3.3 The tunable non symmetric four frame algorithm

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

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

### 3.4 The tunable non symmetric five-frame algorithm

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

2. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. **14**(11), 2622–2626 (1975). [CrossRef] [PubMed]

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

*α*[15

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

### 3.5 The symmetric tunable three-frame algorithm

### 3.6 The symmetric tunable four-frame algorithm

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

7. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. **34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

**18**(24), 24405–24411 (2010). [CrossRef] [PubMed]

### 3.7 The symmetric and tunable five-frame algorithm

7. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. **34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

7. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. **34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

## 4. The general non symmetric quadrature filter *M* order

### 4.1 Combinatorial theory

*W*= {

*r*objects chosen from

*W*. For a trivial example,

*a, b, c,*and

*d*. From this set

*W*, only four combinations with three objects chosen from

*W*can be made, and they are:In the same way, six different arrangements for two objects chosen from

*W*are:The other cases are,

*W*is the empty set and it is written as:

*r*objects taken from

*n*possibilities.

*Σ*be the sum operator, that over a set gives a new set where each element is the sum of each combination in the set. Then, by applying this operator to Eq. (79) the result isThus, applying a function over the expression above the following set is obtained:The operations with a scalar value

*σ*obeys the following algebra ruleCombining Eq. (82) into Eq. (84) the set obtained is:Finally, combining all the rules above the scalar value obtained is:That is, through this notation, a filter is expressed analytically in terms of combinatory theory in a simple expression that can easily be evaluated computationally.

### 4.2 A general tunable non symmetric M-frame algorithm

*M*order rotated filter is expressed as:Through the combinatory theory, each element of the numerator is expressed as: and the last element can be expressed aswhere the scalar

*σ*is simply the sum of all cut off frequencies (interceptions) and given byThat is, each element of the numerator

*π*/2 filter is,

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

### 4.3 A general symmetric and tunable M-frame algorithm

*M*order symmetric algorithm is expressed byHowever, two cases for odd and even order are presented. From section 3, for any

*M*odd order symmetric algorithm, each element of the numerator and the denominator become: Therefore, each element of the numerator and the denominator are generalized as: As expected, for an

*M*odd order filter the following symmetries

*M*order case can also be obtained from the expression above, but the obtained filter will be a shifted

*π/2*aliased algorithm. However, from Eq. (11) the symmetric algorithm to recover the phase is obtained as:

*M*even order filter the symmetries

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

## 5. Some applications

### 5.1 Improving a known algorithm

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

8. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**(1), 51–60 (1996). [CrossRef] [PubMed]

10. 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**(4), 918–930 (1997). [CrossRef]

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

8. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**(1), 51–60 (1996). [CrossRef] [PubMed]

### 5.2 Example of a tunable six-frame algorithm

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

### 5.3 An example of a tunable eleven-frame algorithm

**35**(1), 51–60 (1996). [CrossRef] [PubMed]

10. 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**(4), 918–930 (1997). [CrossRef]

## 6. Conclusions

## Acknowledgments

## References and links

1. | 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. |

2. | J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. |

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

4. | J. Schwider, “Advanced evaluation techniques in interferometry,” in |

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

6. | P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. |

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

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

9. | D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. |

10. | 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 |

11. | H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in |

12. | J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express |

13. | M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express |

14. | A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. |

15. | J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express |

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

**OCIS Codes**

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

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 18, 2010

Revised Manuscript: January 14, 2011

Manuscript Accepted: February 8, 2011

Published: March 1, 2011

**Citation**

J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, F. Castillo, M. A. García-González, and V. A. Gutiérrez-García, "Algorithm for phase extraction from a set of interferograms with arbitrary phase shifts," Opt. Express **19**, 4908-4923 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-4908

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### References

- 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(11), 2693–2703 (1974). [CrossRef] [PubMed]
- J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14(11), 2622–2626 (1975). [CrossRef] [PubMed]
- P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]
- J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (Elsevier, 1990).
- K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]
- P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef] [PubMed]
- J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]
- Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]
- D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]
- 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(4), 918–930 (1997). [CrossRef]
- H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., 2007).
- J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]
- M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]
- A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. 49(32), 6224–6231 (2010). [CrossRef] [PubMed]
- J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express 18(24), 24405–24411 (2010). [CrossRef] [PubMed]
- Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef] [PubMed]

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