## Phase-shifting algorithm by use of Hough transform |

Optics Express, Vol. 20, Issue 23, pp. 26037-26049 (2012)

http://dx.doi.org/10.1364/OE.20.026037

Acrobat PDF (1296 KB)

### Abstract

This paper presents a phase-shifting algorithm based on Hough transform for sinusoidal curves. Firstly, the background intensities of phase-shifting fringe patterns are removed by calculating their differences, thus we get purely sinusoidal intensity data for each pixel; and then we implement Hough transform to the intensity difference data of each pixel. As a result, the sinusoidal parameters, including phase and amplitude, of each pixel are extracted. The simulation and experimental results demonstrate that this algorithm enables eliminating the impacts of some gross errors such as saturation of camera and impulse noise in fringe patterns, and exactly recovering the phase map from fringe patterns.

© 2012 OSA

## 1. Introduction

3. 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]

6. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express **13**(10), 3743–3753 (2005). [CrossRef] [PubMed]

8. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. **24**(2), 185–188 (1985). [CrossRef] [PubMed]

9. H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. **44**(3), 033603 (2005). [CrossRef]

10. J. J. J. Dirckx and W. F. Decraemer, “Automatic calibration method for phase shift shadow moire interferometry,” Appl. Opt. **29**(10), 1474–1476 (1990). [CrossRef] [PubMed]

11. P. Gao, I. Harder, V. Nercissian, K. Mantel, and B. Yao, “Phase-shifting point-diffraction interferometry with common-path and in-line configuration for microscopy,” Opt. Lett. **35**(5), 712–714 (2010). [CrossRef] [PubMed]

13. T. Yatagai, K. Ohmura, S. Iwasaki, S. Hasegawa, J. Endo, and A. Tonomura, “Quantitative phase analysis in electron holographic interferometry,” Appl. Opt. **26**(2), 377–382 (1987). [CrossRef] [PubMed]

18. N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth **237–239**, 1818–1824 (2002). [CrossRef]

6. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express **13**(10), 3743–3753 (2005). [CrossRef] [PubMed]

## 2. Principle

### 2.1 Hough transform of a sinusoid curve

*xy-*plane as those shown in Fig. 1(a) . Our task is to find a sinusoidal curve passing through them. This curve should have a function of the formBecause this sinusoid is determined by the amplitude

*A*and the phase

*φ*, we have a two-dimensional parameter space. A sinusoid in the

*xy*-plane corresponds to a point (

*φ*,

*A*) in this parameter space. Whereas for a fixed point (

*x*,

_{k}*y*) in the

_{k}*xy*-plane, there exist infinitely many sinusoids passing through it, corresponding to a curve

*A*=

*y*/cos(

_{k}*φ + x*) in the

_{k}*φA-*plane with

*φ*and

*A*being variables. The problem with using

*A*and

*φ*is that

*A*sometimes approaches infinity, for example when

*φ*= π/2

*−x*, leading to a difficulty in implementation.

_{k}*u*=

*A*cos

*φ*and

*v*=

*A*sin

*φ*, and then rewrite Eq. (1) asby which we have new parameters

*u*and

*v*, instead of

*A*and

*φ*, within finite expected ranges. As a result, the point (

*x*,

_{k}*y*) corresponds to a straight line in the

_{k}*vu-*plane. This phenomenon is exemplarily illustrated in Fig. 1(b), where each line corresponds to a single point in Fig. 1(a). Note that all the lines intersect at the same pair (

*v*,

*u*). This fact means that, in Fig. 1(a), all the points lie on the same sinusoid, whose shape is determined by this intersection pair (

*v*,

*u*). In Eq. (2), we do not use an explicit function form that

*u*depends on

*v*, because it cannot denote a vertical line with infinite slope.

*vu-*plane into accumulator cells, and set them to 0’s. In the example of Fig. 1, the expected ranges of

*u*and

*v*are [-1, 1] since we presume that the amplitude of the sinusoid is smaller than 1. The accumulator is subdivided into 256 × 256 cells. Second, for each point in the

*xy*-plane, we calculate the locus of its corresponding straight line in the

*vu-*plane. Third, if the locus passes through an accumulator cell once, the value at this cell increases by 1. Finally, we search the accumulator for the maximum value, whose position gives the sinusoidal parameters

*v*and

*u*.

*v*,

*u*) = (0.6094, 0.3516), corresponding to

*A*= 0.7036 and

*φ =*0.3334π. The corresponding sinusoid is plotted in Fig. 1(a) with a dot curve. Note that these results are obtained under the noise free condition. We will discuss the case in the presence of noise (or other errors) in the next subsection.

### 2.2 Hough transform in the presence of disturbance

*vu*-plane no longer intersect at a single point like the case in Fig. 1(b), and that several local peaks appear over the accumulator. These facts mean that there is no sinusoid passing through all the points in Fig. 2(a) simultaneously. Even so, we find that, over the

*vu-*plane, there is a cluster of small peaks associated with the points A, B, and E, implying a sinusoid that can fit these three points closely. The arisen problem is how its parameters are extracted.

### 2.3 Fringe phase retrieving using Hough transform

*k*’th frame of these fringe patterns is usually represented bywhere

*I*(

_{k}*i*,

*j*),

*a*(

*i*,

*j*), and

*b*(

*i*,

*j*) are the recorded intensity, background intensity, and modulation at the pixel (

*i*,

*j*), respectively.

*φ*is the phase to be measured.

*α*denotes the

_{k}*k*’th phase shift, and

*K*is the number of phase shifts. Comparing Eq. (5) with Eq. (1), we find that Eq. (5) is not a purely sinusoidal function because of the presence of a direct current. Directly using Hough transform can retrieve its parameters, but a three-dimensional accumulator has to be established thus making the computational complexity prohibitive. We overcome this problem by using differences of fringe patterns, that isin which and in the following text the pixel coordinates (

*i*,

*j*) are omitted because only pointwise operations are carried out in this paper. For each pixel, the number of the possible intensity differences between fringe patterns, under the condition

*k*>

*l*, is given by

*u*=

*b*cos

*φ*and

*v*=

*b*sin

*φ*, we get an equation following from Eq. (6) thatAlthough the form of Eq. (8) is somewhat different from that of Eq. (2), it is a purely sinusoidal function when

*α*or

_{k}*α*varies. A fixed data set (

_{l}*ΔI*

_{k}_{,}

*,*

_{l}*α*, and

_{k}*α*) corresponds to a straight line in the

_{l}*vu-*plane, so we can determine the sinusoidal parameters

*u*and

*v*, and further

*b*and

*φ*, by using the Hough transform method just presented.

### 2.4 Algorithm implementation

**. Use least-squares method to estimate the parameters of sinusoidal intensity signal for each pixel. By using the definitions of**

*Step 1**u*and

*v*in Section 2.3, rewrite Eq. (5) asBecause

*k*= 0, 1…

*K*-1, we have a linear system with

*K*equations for each pixel. Solve it for the unknowns

*a*,

*u*, and

*v*in the least-squares sense [5], and then calculate the sum of square of errors

**. Define an error threshold**

*Step 2**T*by which we can control what algorithm will be implemented in the following steps. (In this step, the error threshold

*T*is empirically determined). Compare

*E*with

*T*. If

*E*≤

*T*, directly calculate the phase withand skip the following steps. Whereas if

*E*>

*T*, go to the next step. By the first two steps, we identify the pixels whose intensity sequence contains large noise, but we do not know in what frame the large noise appears, so the filtering method cannot be used to remove the noise. The following steps will use Hough transform to exclude these noisy intensities automatically.

**. Subdivide the**

*Step 3**vu*-plane into accumulator cells, and set them to be 0’s. In this step the ranges of

*u*and

*v*in both positive and negative directions should be greater than the expected amplitude of the sinusoid.

**. Calculate the differences of fringe patterns, getting totally**

*Step 4**K*(

*K*-1)/2 sets of data (

*ΔI*

_{k}_{,}

*,*

_{l}*α*, and

_{k}*α*) for each pixel. Perform Hough transform, calculating the corresponding line for each data set according to Eq. (8). If the line passes through an accumulator cell once, the value at this cell increases by 1.

_{l}**. Smooth the accumulator values with a spatial filter. Set a threshold for peak height. Search for the cell (**

*Step 5**v*,

*u*) where the maximum peak appears. Calculate the phase with Eq. (11). In this step, if there is no maximum peak in the accumulator (i.e. all peaks are below the peak height threshold), the measurement data for this pixel are invalid.

## 3. Results and discussions

### 3.1 Phase retrieving in the presence of impulse noise

20. M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express **17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

22. K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. **36**(10), 2084–2093 (1997). [CrossRef] [PubMed]

3. 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]

13. T. Yatagai, K. Ohmura, S. Iwasaki, S. Hasegawa, J. Endo, and A. Tonomura, “Quantitative phase analysis in electron holographic interferometry,” Appl. Opt. **26**(2), 377–382 (1987). [CrossRef] [PubMed]

14. W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear **261**(2), 164–171 (2006). [CrossRef]

15. S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) **122**(18), 1666–1671 (2011). [CrossRef]

17. J.-R. Lee, “Spatial resolution and resolution in phase-shifting laser interferometry,” Meas. Sci. Technol. **16**(12), 2525–2533 (2005). [CrossRef]

18. N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth **237–239**, 1818–1824 (2002). [CrossRef]

*v*and

*u*are limited within [-0.5, 0.5], and the

*vu*-plane is subdivided into 256 × 256 accumulator cells. The error threshold

*T*is empirically set as 0.001. The accumulator is smoothed by using an 11 × 11 Gaussian filter with its standard deviation being 6. In Table 1, the maximum error with this technique decreases to 0.0529 radians, because it is insensitive to the discontinuities in fringe patterns. The RMS error decreases to 0.0130 radians, demonstrating that the proposed method is also effective in removing the impact of salt-and-pepper noise in the continuous area of fringe patterns.

*p*, and that the number of phase shifts is

*K*, the probability that

*N*or more intensities in this sequence are corrupted by impulse noise is given byAccording to this equation, Fig. 4(a) shows that, as the number of phase shifts increases, the probability of that at least one value in the intensity sequence contains impulse noise becomes higher. In this case, the least-squares algorithm cannot give an accurate result. Returning to the example of Fig. 3(a),

*K*is 8 and

*p*is 1%, so that the probability of large errors in the phase map is 7.73%.

*p*, and decline rapidly as

*K*increases. This fact implies that increasing the number of phase shifts is helpful for improving the reliability of the proposed algorithm. We shall further investigate this issue through the following results.

8. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. **24**(2), 185–188 (1985). [CrossRef] [PubMed]

9. H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. **44**(3), 033603 (2005). [CrossRef]

25. H. Guo, Q. Yang, and M. Chen, “Local Frequency Estimation for the Fringe Pattern with a Spatial Carrier: Principle and Applications,” Appl. Opt. **46**(7), 1057–1065 (2007). [CrossRef] [PubMed]

### 3.2 Phase retrieving in the presence of camera saturation

*N*and the relative phase shift is a constant 2π/

*N*radians, the phase shifts are equally spaced over a 2π period and the least-squares algorithm is known as the synchronous detection algorithm [3

3. 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]

*N*-2 order can be eliminated [26

26. K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. **24**(21), 3631–3637 (1985). [CrossRef] [PubMed]

### 3.3 Discussions

*a*,

*u*and

*v*being variables. Performing a 3-D Hough transform allows us to determine these variables directly, but the computational time with it is prohibitive. Therefore, we employ a 2-D Hough transform by removing the background intensities from fringe patterns first. In some traditional phase-shifting algorithms, the backgrounds of fringe patterns are removed by subtracting from them a fixed frame. In so doing, if this fixed frame contains large errors, these errors will transfer into all the fringe pattern differences. Another adverse effect with using fringe differences is that, if the fringe patterns have Gaussian noise, their differences have Gaussian noise with a double higher variance. For this reason, we calculate all possible differences between any couple of fringe patterns, and the data redundancy is helpful for improving the reliability and correctness in locating the peaks in Hough transform.

28. H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express **19**(8), 7807–7815 (2011). [CrossRef] [PubMed]

30. P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. **34**(22), 3553–3555 (2009). [CrossRef] [PubMed]

*v*,

*u*) in the accumulator corresponds to a vector, which has a length

*A*and phase

*φ*. When

*φ*varies, a fixed

*A*plots a circle in the accumulator. When

*A*becomes smaller, the number of the accumulator cells along the circumference decreases. This fact means that, the value of the phase resolution is inversely proportional to the amplitude

*A*. In Section 3.1, we use an accumulator with 256 × 256 cells, which corresponds to the range [-0.5, 0.5] for

*v*and

*u*. If the amplitude is 0.5, we have the highest phase resolution 1/128 radians. In Section 3.2, the accumulator size is 512 × 512, the highest phase resolution is 1/256 radians.

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | H. Schreiber and J. H. Bruning, “Phase Shifting Interferometry,” in |

3. | 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. |

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

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

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

7. | I. Yamaguchi, “Phase-Shifting Digital Holography,” in |

8. | V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. |

9. | H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. |

10. | J. J. J. Dirckx and W. F. Decraemer, “Automatic calibration method for phase shift shadow moire interferometry,” Appl. Opt. |

11. | P. Gao, I. Harder, V. Nercissian, K. Mantel, and B. Yao, “Phase-shifting point-diffraction interferometry with common-path and in-line configuration for microscopy,” Opt. Lett. |

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

13. | T. Yatagai, K. Ohmura, S. Iwasaki, S. Hasegawa, J. Endo, and A. Tonomura, “Quantitative phase analysis in electron holographic interferometry,” Appl. Opt. |

14. | W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear |

15. | S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.) |

16. | J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A |

17. | J.-R. Lee, “Spatial resolution and resolution in phase-shifting laser interferometry,” Meas. Sci. Technol. |

18. | N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth |

19. | R. C. Gonzalez and R. E. Woods, |

20. | M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express |

21. | Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. |

22. | K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. |

23. | D. C. Ghiglia and M. D. Pritt, |

24. | M. Kujawinska, “Spatial phase measurement methods,” in |

25. | H. Guo, Q. Yang, and M. Chen, “Local Frequency Estimation for the Fringe Pattern with a Spatial Carrier: Principle and Applications,” Appl. Opt. |

26. | K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. |

27. | H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE |

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

29. | H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. |

30. | P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. |

**OCIS Codes**

(100.2650) Image processing : Fringe analysis

(100.5070) Image processing : Phase retrieval

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

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

**ToC Category:**

Image Processing

**History**

Original Manuscript: August 27, 2012

Revised Manuscript: October 14, 2012

Manuscript Accepted: October 23, 2012

Published: November 2, 2012

**Citation**

Hongwei Guo and Beiting Lü, "Phase-shifting algorithm by use of Hough transform," Opt. Express **20**, 26037-26049 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-26037

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### References

- K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, 1993), pp. 94–140.
- H. Schreiber and J. H. Bruning, “Phase Shifting Interferometry,” in Optical Shop Test, D. Malacara, ed. (Wiley-Interscience, 2007), 547–666.
- 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]
- C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett.7(8), 368–370 (1982). [CrossRef] [PubMed]
- J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng.23, 350–352 (1984).
- O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express13(10), 3743–3753 (2005). [CrossRef] [PubMed]
- I. Yamaguchi, “Phase-Shifting Digital Holography,” in Digital Holography and Three-Dimensional Display, T-C. Poon, ed. (Springer, 2006), 145–171.
- V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt.24(2), 185–188 (1985). [CrossRef] [PubMed]
- H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng.44(3), 033603 (2005). [CrossRef]
- J. J. J. Dirckx and W. F. Decraemer, “Automatic calibration method for phase shift shadow moire interferometry,” Appl. Opt.29(10), 1474–1476 (1990). [CrossRef] [PubMed]
- P. Gao, I. Harder, V. Nercissian, K. Mantel, and B. Yao, “Phase-shifting point-diffraction interferometry with common-path and in-line configuration for microscopy,” Opt. Lett.35(5), 712–714 (2010). [CrossRef] [PubMed]
- D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis Group, 2005).
- T. Yatagai, K. Ohmura, S. Iwasaki, S. Hasegawa, J. Endo, and A. Tonomura, “Quantitative phase analysis in electron holographic interferometry,” Appl. Opt.26(2), 377–382 (1987). [CrossRef] [PubMed]
- W. H. Wang, Y. S. Wong, and G. S. Hong, “3D measurement of crater wear by phase shifting method,” Wear261(2), 164–171 (2006). [CrossRef]
- S. Singh, S. Rana, S. Prakash, and O. Sasaki, “Application of wavelet filtering techniques to Lau interferometric fringe analysis for measurement of small tilt angles,” Optik (Stuttg.)122(18), 1666–1671 (2011). [CrossRef]
- J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Composites: Part A35(7-8), 849–859 (2004). [CrossRef]
- J.-R. Lee, “Spatial resolution and resolution in phase-shifting laser interferometry,” Meas. Sci. Technol.16(12), 2525–2533 (2005). [CrossRef]
- N. A. Booth, A. A. Chernov, and P. G. Vekilov, “Characteristic lengthscales of step bunching in KDP crystal growth: in situ differential phase-shifting interferometry study,” J. Cryst. Growth237–239, 1818–1824 (2002). [CrossRef]
- R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007), Chap. 10.
- M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express17(11), 8789–8794 (2009). [CrossRef] [PubMed]
- Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt.36(1), 271–276 (1997). [CrossRef] [PubMed]
- K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt.36(10), 2084–2093 (1997). [CrossRef] [PubMed]
- D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley-Interscience, 1998).
- M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, 1993), pp. 141–193.
- H. Guo, Q. Yang, and M. Chen, “Local Frequency Estimation for the Fringe Pattern with a Spatial Carrier: Principle and Applications,” Appl. Opt.46(7), 1057–1065 (2007). [CrossRef] [PubMed]
- K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt.24(21), 3631–3637 (1985). [CrossRef] [PubMed]
- H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE5180, 437–444 (2003).
- H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express19(8), 7807–7815 (2011). [CrossRef] [PubMed]
- H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng.45(2), 281–292 (2007). [CrossRef]
- P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett.34(22), 3553–3555 (2009). [CrossRef] [PubMed]

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