## Phase extraction from three and more interferograms registered with different unknown wavefront tilts

Optics Express, Vol. 13, Issue 10, pp. 3743-3753 (2005)

http://dx.doi.org/10.1364/OPEX.13.003743

Acrobat PDF (1939 KB)

### Abstract

We propose phase retrieval from three or more interferograms corresponding to different tilts of an object wavefront. The algorithm uses the information contained in the interferogram differences to reduce the problem to phase shifting. Three interferograms is the minimum for restoring the phase over most of the image. Four or more interferograms are needed to restore the phase over the whole image. The method works with images including open and closed fringes in any combination.

© 2005 Optical Society of America

## 1. Introduction

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

5. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A **8(5)**, 822–827 (1991). [CrossRef]

*et al*[6

6. K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. **84(3,4)**, 118–124 (1991). [CrossRef]

7. I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. **34**, 183–187 (1995). [CrossRef]

*et al*[8

8. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. **39(22)**, 3894–3898 (2000). [CrossRef]

*et al*[9

9. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. **41(13)**, 2435–2439 (2002). [CrossRef]

10. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A **3(6)**, 847–855 (1986). [CrossRef]

11. J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. **197**, 43–51 (2001). [CrossRef]

12. C. Roddier and F. Roddier, “Interferogram Analysis Using Fourier Transform Techniques,” Appl. Opt. **26**, 1668–1673 (1987). [CrossRef] [PubMed]

13. Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. **40(10)**, 1649–1657 (2001). [CrossRef]

14. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. **40(17)**, 2886–2893 (2001). [CrossRef]

*a priori*information; while time-domain based phase shifting provides ease of implementation. Both phase-shifting and Fourier transform-based methods calculate the phase wrapped in the interval (-

*π,π*]. Some methods — such as the local phase-tracking technique and genetic algorithms [15

15. F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. **203**, 213–223 (2002). [CrossRef]

16. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A **20(5)**, 925–934 (2003). [CrossRef]

17. E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. **37(8)**, 1370–1376 (1998). [CrossRef]

## 2. The algorithm

**x**=(

*x,y*) denotes the position in the recorded image, and square brackets are used to emphasize its discrete nature. We assume the functions

*a*and

*b*to be dependent only on the detector position

**x**. Sometimes we will omit the argument

**x**for simplicity.

*I*

_{0},

*I*

_{1},

*I*

_{2}, which differ from each other only by a small linear term in the phase

*τ*[x]=

_{i}**t**

*i*·

**x**are the tilt terms and

*σ*the phase shifts. The interferograms can be obtained by, for instance, tilting the reference mirror in the interferometer by a small random angle. Again, we assume that

_{i}*a*and

*b*can be considered the same for all three interferograms.

*a,b,ϕ0,δ*=

_{1}*τ*

_{1}+

*σ*

_{1},

*δ*

_{2}=

*τ*

_{2}+

*σ*

_{2}. In phase shifting interferometry,

*δ*

_{1}and

*δ*

_{2}with

*τ*

_{1},

*τ*

_{2}=0, are known, and equations (2) can be solved for every pixel to obtain the (wrapped) phase

*ϕ*

_{0}. For instance, the following identity:

*I*

_{1}-

*I*

_{2})cos

*ϕ*+(

*I*

_{2}-

*I*)cos(

_{0}*δ*

_{1}+ϕ)+(

*I*

_{0}-

*I*

_{1})cos(

*δ*

_{2}+

*ϕ*)=0

*δ*

_{1}and

*δ*

_{2}are supposed to be unknown and different for every pixel. Moreover, for those pixels where one or both of the phase shifts are equal or close to integer numbers of 2

*π*, the number of equations in system (2) is reduced from three to two, or one. However, as will be shown later, for the rest of the object area, the set of three interferograms contains enough information to determine the tilt and piston terms, and thus to make system (2) solvable.

*I*and

_{d,1}*I*:

_{d,2}*I*

_{d,1}=I_{1}-I_{0}=b(cos(ϕ+τ_{1}+σ_{1})-cos(ϕ))*I*=

_{d,2}*I*-

_{2}*I*

_{0}=

*b*(cos(

*ϕ*+

*τ*

_{2}+

*σ*

_{2})-cos(

*ϕ*)),

*τ*+

_{i}*σ*=2

_{i}*kπ*, and the second contains the points where

*k*is an integer. The first group is not dependent on

*a,b,ϕ*and forms parallel lines. It can be fully described by three parameters:

*θ*is the common normal,

*λ*is the separation distance, and s is the distance from the origin. The algorithm proceeds by finding these parameters. Then

**t**can be found as

*δ*for every pixel, which can be substituted in equations (2) or directly in the phase formula (3).

_{i}## 3. Practical implementation and examples

*x,y*) onto a sinusoidal function

*ρ*=

*x*cos

*θ*+

*y*sin

*θ*. The key property of the Hough transform is that sinusoidals in Hough space associated with points lying on the same line have a common point of intersection, say (

*ρ*

_{0},

*θ*

_{0}). This line has a normal (cos

*θ*

_{0}, sin

*θ*

_{0}) and is shifted by distance

*ρ*

_{0}from the origin.

*ρ,θ*)∊[-

*R,R*]×[0,

*π*) for some

*R*(the maximum distance of any image point from the origin). Then it is quantized with finite steps

*m*+1)×

*n*

**A**=(

*Ai,j*),

*i*=-

*m*,…,

*m*,

*j*=0,

*n*-1,

**A**represent the angle

*θ*; the columns, the radius vector

*ρ*. The sinusoidals for every feature pixel in Fig. 3 are “drawn” in the accumulator as follows. Initially, every cell of the accumulator is set to zero. Then, for every row number

*j*, the column number

*i*

_{(x,y)}(

*j*) is calculated by

*M*in the

**A**and use

*θ*and

_{M}*ρ*corresponding to its position as lines normal

_{M}*θ*and shift from origin

*s*. To find the separation distance

*λ*, consider the row of the accumulator corresponding to

*θ*(Fig. 5) and find the average distance between its local maxima, which have values greater than

_{M}*α*for some threshold

_{M}*α*(typical values are 0,3–0, 7, see Fig. 7). To reduce influence of the noise, the algorithm replaces the values of nearby positioned maxima with their average.

*ϕ*can be found. Replacing the formula (3), one can use a simpler identity

*ϕ*+

*α*)=cos(

*ϕ*+

*β*) sin(

*α*-

*β*)+cos(

*α*-

*β*) sin(

*ϕ*+

*β*)

*ϕ*+

*δ*/2 as

_{2}## 4. Discussion

### 4.1. Artifacts in the extracted phase

*δ*

_{1}, or

*δ*

_{2}, or

*δ*

_{1}–

*δ*

_{2}is equal to 2

*kπ, k∊ℤ*, that is, where the system (2) is badly defined. Although these artifacts introduce a small

*rms*error, they can seriously affect the unwrapped phase. They can be removed either by introducing a fourth interferogram with tilt

*δ*

_{3}in the algorithm and calculating the phase as the median of the results of three possible tilt combinations (see Fig. 10), or by obtaining

*a*and

*b*from the system (2) and then calculating the phase

*ϕ*by substituting their smoothed by low-pass filter values in equation (1) or just by using robust and noise-immune phase unwrapping algorithms.

### 4.2. Initial phase

### 4.3. Optimal tilt range

### 4.4. Ambiguity in the detection of phase-shift signs

*θ*only in the range from 0 to

*π*. It is clear from the equations (4), that four different pairs of phase shifts {±

*δ*

_{1},±

*δ*

_{2}} produce parallel lines with identical parameters (5). Thus the algorithm does not see the difference between interferograms with “positive” and “negative” tilts, and formula (8) provides, generally speaking, four different phase distributions. Two of them are the “real” phase accurate to the sign, and two the “faulty” phase, which can be recognized by its almost binary shape (see Fig. 13). The algorithm cannot determine the sign of the real phase, as the interferograms (2) are the same for

*ϕ*

_{0},

*δ*

_{1},

*δ*

_{2}and -

*ϕ*

_{0},-

*δ*

_{1},-

*δ*

_{2}, and some additional information is necessary. In practice, one can limit all possible tilts only to those increasing in y-direction, which correspond to

*θ*∊(0,

*π*) and exclude the tilt signs ambiguity.

### 4.5. Computational effectiveness and accumulator resolution

*δ*, one needs

_{θ}=π/n*O*(

*n*×

*N*) operations, where N is the number of points in the zero-level set. To speed up the calculation one can first estimate the tilt with low resolution, and then fill only the small region of the accumulator near the maximum with high resolution. Note that too high resolution in

*ρ*makes the algorithm too sensitive to noise and to quantization of the interferograms. For the zero-level points from Fig. 12, for instance, low resolution in

*ρ*and high resolution in

*θ*is necessary.

### 4.6. Robustness

*a*and

*b*as long as they are the same for all the interferograms. In regions where

*b*is close to zero or in cases with high noise level of the detector, spurious zero-level points appear and decrease the accuracy. The algorithm also fails on overexposed interferograms or on interferograms recorded with a non-linear camera.

## 5. Conclusions

## Acknowledgments

## References and links

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

2. | Y. Surrel, “Fringe analysis,” in |

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

4. | D. Malacara, ed., |

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

6. | K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. |

7. | I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. |

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

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

10. | T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A |

11. | J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. |

12. | C. Roddier and F. Roddier, “Interferogram Analysis Using Fourier Transform Techniques,” Appl. Opt. |

13. | Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. |

14. | K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. |

15. | F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. |

16. | M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A |

17. | E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. |

18. | G. X. Ritter and J. N. Wilson, |

**OCIS Codes**

(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:**

Research Papers

**History**

Original Manuscript: April 12, 2005

Revised Manuscript: May 4, 2005

Published: May 16, 2005

**Citation**

Oleg Soloviev and Gleb Vdovin, "Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Express **13**, 3743-3753 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3743

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### References

- D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).
- Y. Surrel, �??Fringe analysis,�?? in Photomechanics, P. K. Rastogi, ed. (Springer, 1999).
- Y. Surrel, �??Design of algorithms for phase measurements by the use of phase stepping,�?? Appl. Opt. 35(1), 51 �??60 (1996). [CrossRef]
- D. Malacara, ed., Optical Shop Testing, 2nd ed. (John Wiley & Sons, Inc., 1992).
- G. Lai and T. Yatagai, �??Generalized phase-shifting interferometry,�?? J. Opt. Soc. Am. A 8(5), 822�??827 (1991). [CrossRef]
- K. Okada, A. Sato, and J. Tujiuchi, �??Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,�?? Opt. Commun. 84(3,4), 118�??124 (1991). [CrossRef]
- I.-B. Kong and S.-W. Kim, �??General algorithm of phase-shifting interferometry by iterative least-squares fitting,�?? Opt. Eng. 34, 183�??187 (1995). [CrossRef]
- M. Chen, H. Guo, and C.Wei, �??Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,�?? Appl. Opt. 39(22), 3894�??3898 (2000). [CrossRef]
- A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, �??Tilt-compensating algorithm for phase-shift interferometry,�?? Appl. Opt. 41(13), 2435�??2439 (2002). [CrossRef]
- T. Kreis, �??Digital holographic interference-phase measurement using the Fourier-transform method,�?? J. Opt. Soc. Am. A 3(6), 847�??855 (1986). [CrossRef]
- J. A. Quiroga, J. A. Gómez-Pedrero, and �?. García-Botella, �??Algorithm for fringe pattern normalisation,�?? Opt. Commun. 197, 43�??51 (2001). [CrossRef]
- C. Roddier and F. Roddier, �??Interferogram Analysis Using Fourier Transform Techniques,�?? Appl. Opt. 26, 1668�??1673 (1987). [CrossRef] [PubMed]
- Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, �??Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,�?? Appl. Opt. 40(10), 1649�??1657 (2001). [CrossRef]
- K. A. Goldberg and J. Bokor, �??Fourier-transform method of phase-shift determination,�?? Appl. Opt. 40(17), 2886�??2893 (2001). [CrossRef]
- F. Cuevas, J. Sossa-Azuela, and M. Servin, �??A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,�?? Opt. Commun. 203, 213�??223 (2002). [CrossRef]
- M. Servin, J. A. Quiroga, and J. L. Marroquin, �??General n-dimensional quadrature transform and its application to interferogram demodulation,�?? J. Opt. Soc. Am. A 20(5), 925�??934 (2003). [CrossRef]
- E. Yu and S. S. Cha, �??Two-dimensional regression for interferometric phase extraction,�?? Appl. Opt. 37(8), 1370�??1376 (1998). [CrossRef]
- G. X. Ritter and J. N.Wilson, Handbook of computer vision algoritm in image algebra (CRC Press, Boca Raton, New York, London, Tokyo, 1996).

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