## Practical eight-frame algorithms for fringe projection profilometry |

Optics Express, Vol. 21, Issue 1, pp. 903-917 (2013)

http://dx.doi.org/10.1364/OE.21.000903

Acrobat PDF (1388 KB)

### Abstract

In this paper we present several eight-frame algorithms for their use in phase shifting profilometry and their application for the analysis of semi-fossilized materials. All algorithms are obtained from a set of two-frame algorithms and designed to compensate common errors such as phase shift detuning and bias errors.

© 2013 OSA

## 1. Introduction

## 2. Description of some phase shifting algorithms (PSA)

## 3. Designing an eight-frame phase shifting algorithm

*M*of each interferogram recorded by a CCD detector can be expressed as [6

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

8. 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]

*x*and

*y*denote the pixel position;

5. T. M. Kaiser and H. Katterwe, “The application of 3D-microprofilometry as a tool in the surface diagnosis of fossil and sub-fossil vertebrate hard tissue. An example from the pliocene upper laetolil beds, Tanzania,” Int. J. Osteoarchaeol. **11**(5), 350–356 (2001). [CrossRef]

- I.
*M*images are captured with several phase shifting among them. - II. To choose or design a specific M-frames phase shifting algorithm (PSA) to process the set of M images to obtain the wrapped phase.
- III. An unwrapping algorithm to recover the desired phase is designed.
- IV. A texture is applied to the obtained phase to exhibit the desired target.

14. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A **12**(9), 1997–2008 (1995). [CrossRef]

15. C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. **27**(19), 1687–1689 (2002). [CrossRef] [PubMed]

13. 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]

16. 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]

17. P. D. Ruiz, J. M. Huntley, and G. H. Kaufmann, “Adaptive phase-shifting algorithm for temporal phase evaluation,” J. Opt. Soc. Am. A **20**(2), 325–332 (2003). [CrossRef] [PubMed]

^{th}order is given by [6

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

7. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. **36**(31), 8098–8115 (1997). [CrossRef] [PubMed]

16. 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. 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**(6), 4908–4923 (2011). [CrossRef] [PubMed]

*N*and

*D*are the desired numerator and denominator row vectors. In a previous work [1

1. 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]

16. 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. 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**(6), 4908–4923 (2011). [CrossRef] [PubMed]

*M-1*frequencies that are the necessary conditions to be a specific filter [16

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

18. 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**(6), 4908–4923 (2011). [CrossRef] [PubMed]

*M*= 8, the corresponding eight-frame algorithm is,Then, an option to obtain the required ratio

*N/D*with symmetric coefficients is obtained from the expression [16

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

**19**(6), 4908–4923 (2011). [CrossRef] [PubMed]

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

**19**(6), 4908–4923 (2011). [CrossRef] [PubMed]

*π/4,*it is well known that a filter that eliminates harmonics corresponds to the cut-off frequencies

*0, π/4, π/2, 3π/4, π, 5π/4*and

*3π/2*[1

1. 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]

11. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. **32**(19), 3598–3600 (1993). [CrossRef] [PubMed]

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

**19**(6), 4908–4923 (2011). [CrossRef] [PubMed]

1. 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]

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

**19**(6), 4908–4923 (2011). [CrossRef] [PubMed]

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

**19**(6), 4908–4923 (2011). [CrossRef] [PubMed]

*M = 8*, the result equivalent to Eq. (10) becomes,And the equivalent phase shifted filter is,This case is equivalent to the commonly named least-squares filter that can be used for this application with certain restrictions, because it is sensible to bias and detuning errors. On the other hand, for the analysis of an eight frame series of interferograms we can use auto tuning methods like the Carré’s algorithm, or the algorithm with immunity to systematic errors, as the Surrel technique based on (N + 1) bucket [6

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

11. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. **32**(19), 3598–3600 (1993). [CrossRef] [PubMed]

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

*α*as the ideal phase step to tune the filter, then a new (

*n + m*-1) frame filter is obtained from two individual filters as shown in Eq. (7). As mentioned before, the design of a tunable filter allows this case to be extended further to an eight-frame filter, which allows the selection of the data to be removed. Furthermore, we select polynomial roots implying that the filter must suppress frequencies in Fig. 1 . To assure that the filter eliminates harmonics, undesirable frequencies and the systematic errors involved, we first propose a filter to deal with harmonics, mainly as in Eq. (13). Also, we define a filter that is able to handle an optimal SNR and linear detuning errors as

*0, π/4, π/4, π/2, 3π/4, 3π/4, π*(from now known as mainly detuning error filter or MDE filter). Applying the same method to design a filter that compensates mainly bias errors (MBE filter) associated to the system, we calculate a filter considering cut off frequencies in

*0, 0, π/4, π/2, 3π/4, π, π,*and finally a filter centered in

*0, 0, π/4, π/4, π/2, 3π/4, π*to compensate detuning and bias errors (DBE filter). Graphic representation of these filters with their cut off frequencies is shown in Fig. 1 according to [6

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

## 4. Evaluation of the algorithms

### 4.1 Simulation

11. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. **32**(19), 3598–3600 (1993). [CrossRef] [PubMed]

13. 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]

19. 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]

### 4.2 Experiment

20. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. **48**(2), 133–140 (2010). [CrossRef]

21. S. Ma, C. Quan, R. Zhu, and C. J. Tay, “Investigation of phase error correction for digital sinusoidal phase-shifting fringe projection profilometry,” Opt. Lasers Eng. **50**(8), 1107–1118 (2012). [CrossRef]

5. T. M. Kaiser and H. Katterwe, “The application of 3D-microprofilometry as a tool in the surface diagnosis of fossil and sub-fossil vertebrate hard tissue. An example from the pliocene upper laetolil beds, Tanzania,” Int. J. Osteoarchaeol. **11**(5), 350–356 (2001). [CrossRef]

## 5. Results and discussion

17. P. D. Ruiz, J. M. Huntley, and G. H. Kaufmann, “Adaptive phase-shifting algorithm for temporal phase evaluation,” J. Opt. Soc. Am. A **20**(2), 325–332 (2003). [CrossRef] [PubMed]

12. K. Creath and J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng. **24**(5-6), 365–379 (1996). [CrossRef]

*0, π/4, π/4, π/2, 3π/4, 3π/4, π*has the better immunity to experimental errors in this FPP arrangement and these samples in particular.

22. J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. **40**(3), 114–131 (2011). [CrossRef]

## 6. Conclusions

## Acknowledgments

## References and links

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

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

3. | P. D. Ruiz, J. M. Huntley, and G. H. Kaufmann, “Adaptive phase-shifting algorithm for temporal phase evaluation,” J. Opt. Soc. Am. A |

4. | H. Katterwe, “Modern approaches for the examination of toolmarks and other surface marks,” Forensic Sci. Rev. |

5. | T. M. Kaiser and H. Katterwe, “The application of 3D-microprofilometry as a tool in the surface diagnosis of fossil and sub-fossil vertebrate hard tissue. An example from the pliocene upper laetolil beds, Tanzania,” Int. J. Osteoarchaeol. |

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

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

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

9. | K. Creath, “Temporal phase measurement methods,” in |

10. | J. M. Huntley, “Automated Analysis of Speckle Interferograms,” in |

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

12. | K. Creath and J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng. |

13. | 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. | C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A |

15. | C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. |

16. | 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 |

17. | P. D. Ruiz, J. M. Huntley, and G. H. Kaufmann, “Adaptive phase-shifting algorithm for temporal phase evaluation,” J. Opt. Soc. Am. A |

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

20. | S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. |

21. | S. Ma, C. Quan, R. Zhu, and C. J. Tay, “Investigation of phase error correction for digital sinusoidal phase-shifting fringe projection profilometry,” Opt. Lasers Eng. |

22. | J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. |

**OCIS Codes**

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

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

(170.1420) Medical optics and biotechnology : Biology

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: September 13, 2012

Revised Manuscript: October 26, 2012

Manuscript Accepted: November 14, 2012

Published: January 9, 2013

**Virtual Issues**

Vol. 8, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Juan C. Gutiérrez-García, J. F. Mosiño, Amalia Martínez, Tania A. Gutiérrez-García, Ella Vázquez-Domínguez, and Joaquín Arroyo-Cabrales, "Practical eight-frame algorithms for fringe projection profilometry," Opt. Express **21**, 903-917 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-903

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### References

- J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express17(18), 15772–15777 (2009). [CrossRef] [PubMed]
- K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express9(5), 236–253 (2001). [CrossRef] [PubMed]
- P. D. Ruiz, J. M. Huntley, and G. H. Kaufmann, “Adaptive phase-shifting algorithm for temporal phase evaluation,” J. Opt. Soc. Am. A20(2), 325–332 (2003). [CrossRef] [PubMed]
- H. Katterwe, “Modern approaches for the examination of toolmarks and other surface marks,” Forensic Sci. Rev.8, 45–72 (1996).
- T. M. Kaiser and H. Katterwe, “The application of 3D-microprofilometry as a tool in the surface diagnosis of fossil and sub-fossil vertebrate hard tissue. An example from the pliocene upper laetolil beds, Tanzania,” Int. J. Osteoarchaeol.11(5), 350–356 (2001). [CrossRef]
- 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] [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]
- K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993).
- J. M. Huntley, “Automated Analysis of Speckle Interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, 2001).
- Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt.32(19), 3598–3600 (1993). [CrossRef] [PubMed]
- K. Creath and J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng.24(5-6), 365–379 (1996). [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. A14(4), 918–930 (1997). [CrossRef]
- C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A12(9), 1997–2008 (1995). [CrossRef]
- C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett.27(19), 1687–1689 (2002). [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. Express18(24), 24405–24411 (2010). [CrossRef] [PubMed]
- P. D. Ruiz, J. M. Huntley, and G. H. Kaufmann, “Adaptive phase-shifting algorithm for temporal phase evaluation,” J. Opt. Soc. Am. A20(2), 325–332 (2003). [CrossRef] [PubMed]
- 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. Express19(6), 4908–4923 (2011). [CrossRef] [PubMed]
- 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]
- S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng.48(2), 133–140 (2010). [CrossRef]
- S. Ma, C. Quan, R. Zhu, and C. J. Tay, “Investigation of phase error correction for digital sinusoidal phase-shifting fringe projection profilometry,” Opt. Lasers Eng.50(8), 1107–1118 (2012). [CrossRef]
- J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt.40(3), 114–131 (2011). [CrossRef]

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