## Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities-theory and experiment |

Optics Express, Vol. 19, Issue 4, pp. 3086-3105 (2011)

http://dx.doi.org/10.1364/OE.19.003086

Acrobat PDF (1719 KB)

### Abstract

This paper proposes a robust noise and phase jump detection scheme for noisy phase maps containing height discontinuities. The detection scheme has two primary functions, namely to detect the positions of noise and to locate the positions of the phase jumps. Generally speaking, the removal of noise from a wrapped phase map causes a smearing of the phase jumps and therefore leads to a loss of definition in the unwrapped phase map. However, in the proposed scheme, the boundaries of the phase jump regions are preserved during the noise detection process. The validity of the proposed approach is demonstrated using the simulated and experimental wrapped phase maps of a 3D object containing height discontinuities, respectively. It is shown that the noise and phase jump detection scheme enables the precise and efficient detection of three different types of noise, namely speckle noise, residual noise, and noise at the lateral surfaces of the height discontinuities. Therefore, the proposed scheme represents an ideal solution for the pre-processing of noisy wrapped phase maps prior to their treatment using a filtering algorithm and phase unwrapping algorithm.

© 2011 OSA

## 1. Introduction

1. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. **24**(18), 3053–3058 (1985). [CrossRef] [PubMed]

2. 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. B. F. Pouet and S. Krishnaswamy, “Technique for the removal of speckle phase in electronic speckle interferometry,” Opt. Lett. **20**(3), 318–320 (1995). [CrossRef] [PubMed]

4. I. Moon and B. Javidi, “Three-dimensional speckle-noise reduction by using coherent integral imaging,” Opt. Lett. **34**(8), 1246–1248 (2009). [CrossRef] [PubMed]

5. M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain **44**(3), 239–247 (2008). [CrossRef]

6. E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express **17**(10), 7818–7830 (2009). [CrossRef] [PubMed]

10. A. Wada, M. Kato, and Y. Ishii, “Large step-height measurements using multiple-wavelength holographic interferometry with tunable laser diodes,” J. Opt. Soc. Am. A **25**(12), 3013–3020 (2008). [CrossRef]

11. R. Yamaki and A. Hirose, “Singularity-Spreading Phase Unwrapping,” IEEE Trans. Geosci. Rem. Sens. **45**(10), 3240–3251 (2007). [CrossRef]

12. H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. **162**(4–6), 205–210 (1999). [CrossRef]

9. H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. **36**(13), 2770–2775 (1997). [CrossRef] [PubMed]

13. J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. **32**(17), 3047–3052 (1993). [CrossRef] [PubMed]

14. D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. **44**(35), 7515–7521 (2005). [CrossRef] [PubMed]

15. S. Zhang, X. L. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. **46**(1), 50–57 (2007). [CrossRef]

23. A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to InSAR time series,” J. Opt. Soc. Am. A **24**(9), 2737–2747 (2007). [CrossRef]

24. K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express **18**(5), 5229–5244 (2010). [CrossRef] [PubMed]

16. W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. **22**(23), 3898–3901 (1983). [CrossRef] [PubMed]

17. D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A **4**(1), 267–280 (1987). [CrossRef]

19. H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. **30**(6), 487–502 (1998). [CrossRef]

15. S. Zhang, X. L. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. **46**(1), 50–57 (2007). [CrossRef]

20. B. Marendic, Y. Yang, and H. Stark, “Phase unwrapping using an extrapolation-projection algorithm,” J. Opt. Soc. Am. A **23**(8), 1846–1855 (2006). [CrossRef]

23. A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to InSAR time series,” J. Opt. Soc. Am. A **24**(9), 2737–2747 (2007). [CrossRef]

## 2. Principles of proposed noise and phase jump detection scheme

12. H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. **162**(4–6), 205–210 (1999). [CrossRef]

*i, j*) is the pixel position;

*ϕ*is the corresponding phase value in the phase map; [ ] indicates a rounding operation; and

### 2.1 Characteristics of Condition I pixels

*ϕ*(

*i, j*),

*ϕ*(

*i + 1, j*),

*ϕ*(

*i, j + 1*), and

*ϕ*(

*i + 1, j + 1*) in the region of the 2π phase jump are treated as continuous phase value pixels (i.e. non-noisy pixels) in the proposed noise and phase jump detection scheme.

### 2.2 Characteristics of Condition II pixels

*ϕ*(

*i, j*),

*ϕ*(

*i + 1, j*),

*ϕ*(

*i, j + 1*), and

*ϕ*(

*i + 1, j + 1*), are sufficiently close. That is, given the condition

*i, j*) as noisy pixel if any one of S1, S2, S3 or S4 has a value other than zero.

*PD*denote the phase difference between any two neighboring continuous pixels at any position in the 2 × 2 area. In other words,

*=*

*i, j*) is defined as a “good pixel”. Conversely, if the absolute phase difference of any two neighboring pixels falls outside the range

*i, j*) is defined as a “bad pixel”.

## 3. Simulation results

*imnoise (each of five interferograms, 'speckle', 0.08).”*And the speckle noise (i.e. Noise A) is confined to a rectangular area with a size of 100 x 246 pixels. Meanwhile, the residual noise (Noise B) was produced by the “imnoise” function with the salt and pepper noise. In order to mimic the effects of dust or some other form of environmental contamination in the experimental interferometry process, the intensity parameter was setting of 0.35, written as “

*imnoise (each of five interferograms, 'salt & pepper', 0.35).”*Finally, the noise at the lateral surfaces of the height discontinuities was produced by the written program which considers not only the signals at the low and high positions of discontinuity but also the “imnoise” function of Noise B with the intensity parameter of 0.01. As a result, it generates the effects of depth of field and diffraction limit constraints in the experimental process. Figure 2(b) presents a 3D cross-sectional view of the phase map shown in Fig. 2(a) at a position corresponding to the 125th pixel column (denoted by the horizontal blue dotted line in Fig. 2(a)). In Figs. 2(a) and 2(b), the ellipses indicate the region of a significant phase jump in the wrapped phase map. In the following discussions, it is shown that the proposed noise and phase jump detection scheme successfully identifies all the noisy pixels within the wrapped phase map irrespective of their origin whilst simultaneously preserving the boundaries of the 2π phase jumps, thereby improving the performance of the subsequent phase unwrapping algorithm of filtering algorithm.

### 3.1 Simulation Results on the 1st, 2nd, and 3rd positions by detection scheme

#### 3.1.1 1st position (no turbulence)

#### 3.1.2 2nd position (turbulent speckle noise)

*i, j*) as the noisy pixel. It should be noted here that if the phase difference values of the four neighboring pixels are very close to

#### 3.1.3 3rd position (containing phase jump)

### 3.2 Noise map obtained from detection scheme

### 3.3 Phase jump map obtained from detection scheme

5. M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain **44**(3), 239–247 (2008). [CrossRef]

12. H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. **162**(4–6), 205–210 (1999). [CrossRef]

### 3.4 Choosing Suitable parameter of σ Α from the noise and phase jump maps

25. A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. **36**(9), 2466–2472 (1997). [CrossRef]

*i, j*) when applied to the wrapped phase map shown in Fig. 2. Note that for convenience, the positions of the pixels which return a value of + 1 are marked as “O” while those of the pixels which return a value of −1 are marked as “X”. It is evident that Eq. (12) fails to detect all the noisy pixels in the unwrapped phase map; particularly those in the speckle noise region. By contrast, the results presented in Fig. 5(a) show that the proposed noise and phase jump detection scheme provides a far better approximation of the noise distribution provided that an appropriate value of the threshold parameter is assigned (i.e.

## 4. Experimental setup and results

### 4.1 Noise and phase jump map results

### 4.2 Phase unwrapping results

26. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. **28**(16), 3268–3270 (1989). [CrossRef] [PubMed]

*N*x

*N*mask centered on the noisy pixel. The work of Filters A and B is as follows. First, the detection scheme detects the positions of noise and phase jumps. Then, Filter A is used to pre-process the wrapped phase map, which removes most noise. During the phase unwrapping process with the MACY algorithm [16

16. W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. **22**(23), 3898–3901 (1983). [CrossRef] [PubMed]

### 4.3 Oblique angle with phase jump line and lateral surface line

### 4.4 Parallel with phase jump line and lateral surface line

## 4. Conclusions

## Acknowledgements

## References and links

1. | K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. |

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

3. | B. F. Pouet and S. Krishnaswamy, “Technique for the removal of speckle phase in electronic speckle interferometry,” Opt. Lett. |

4. | I. Moon and B. Javidi, “Three-dimensional speckle-noise reduction by using coherent integral imaging,” Opt. Lett. |

5. | M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain |

6. | E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express |

7. | W. H. Su, K. Shi, Z. Liu, B. Wang, K. Reichard, and S. Yin, “A large-depth-of-field projected fringe profilometry using supercontinuum light illumination,” Opt. Express |

8. | P. Potuluri, M. Fetterman, and D. Brady, “High depth of field microscopic imaging using an interferometric camera,” Opt. Express |

9. | H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. |

10. | A. Wada, M. Kato, and Y. Ishii, “Large step-height measurements using multiple-wavelength holographic interferometry with tunable laser diodes,” J. Opt. Soc. Am. A |

11. | R. Yamaki and A. Hirose, “Singularity-Spreading Phase Unwrapping,” IEEE Trans. Geosci. Rem. Sens. |

12. | H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. |

13. | J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. |

14. | D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. |

15. | S. Zhang, X. L. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. |

16. | W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. |

17. | D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A |

18. | A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. |

19. | H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. |

20. | B. Marendic, Y. Yang, and H. Stark, “Phase unwrapping using an extrapolation-projection algorithm,” J. Opt. Soc. Am. A |

21. | S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express |

22. | O. S. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single closed-fringe pattern,” J. Opt. Soc. Am. A |

23. | A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to InSAR time series,” J. Opt. Soc. Am. A |

24. | K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express |

25. | A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. |

26. | J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.5088) Image processing : Phase unwrapping

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 3, 2010

Revised Manuscript: January 25, 2011

Manuscript Accepted: January 30, 2011

Published: February 2, 2011

**Citation**

Jing-Feng Weng and Yu-Lung Lo, "Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities-theory and experiment," Opt. Express **19**, 3086-3105 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3086

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### References

- K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24(18), 3053–3058 (1985). [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]
- B. F. Pouet and S. Krishnaswamy, “Technique for the removal of speckle phase in electronic speckle interferometry,” Opt. Lett. 20(3), 318–320 (1995). [CrossRef] [PubMed]
- I. Moon and B. Javidi, “Three-dimensional speckle-noise reduction by using coherent integral imaging,” Opt. Lett. 34(8), 1246–1248 (2009). [CrossRef] [PubMed]
- M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008). [CrossRef]
- E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express 17(10), 7818–7830 (2009). [CrossRef] [PubMed]
- W. H. Su, K. Shi, Z. Liu, B. Wang, K. Reichard, and S. Yin, “A large-depth-of-field projected fringe profilometry using supercontinuum light illumination,” Opt. Express 13(3), 1025–1032 (2005). [CrossRef] [PubMed]
- P. Potuluri, M. Fetterman, and D. Brady, “High depth of field microscopic imaging using an interferometric camera,” Opt. Express 8(11), 624–630 (2001). [CrossRef] [PubMed]
- H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997). [CrossRef] [PubMed]
- A. Wada, M. Kato, and Y. Ishii, “Large step-height measurements using multiple-wavelength holographic interferometry with tunable laser diodes,” J. Opt. Soc. Am. A 25(12), 3013–3020 (2008). [CrossRef]
- R. Yamaki and A. Hirose, “Singularity-Spreading Phase Unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007). [CrossRef]
- H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999). [CrossRef]
- J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993). [CrossRef] [PubMed]
- D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44(35), 7515–7521 (2005). [CrossRef] [PubMed]
- S. Zhang, X. L. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46(1), 50–57 (2007). [CrossRef]
- W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [PubMed]
- D. C. Ghiglia, G. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987). [CrossRef]
- A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991). [CrossRef]
- H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998). [CrossRef]
- B. Marendic, Y. Yang, and H. Stark, “Phase unwrapping using an extrapolation-projection algorithm,” J. Opt. Soc. Am. A 23(8), 1846–1855 (2006). [CrossRef]
- S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007). [CrossRef] [PubMed]
- O. S. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single closed-fringe pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008). [CrossRef]
- A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to InSAR time series,” J. Opt. Soc. Am. A 24(9), 2737–2747 (2007). [CrossRef]
- K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010). [CrossRef] [PubMed]
- A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997). [CrossRef]
- J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28(16), 3268–3270 (1989). [CrossRef] [PubMed]

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