## Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities |

Optics Express, Vol. 20, Issue 10, pp. 10896-10920 (2012)

http://dx.doi.org/10.1364/OE.20.010896

Acrobat PDF (2758 KB)

### Abstract

For 3D objects with height discontinuities, the image reconstruction performance of interferometric systems is adversely affected by the presence of noise in the wrapped phase map. Various schemes have been proposed for detecting residual noise, speckle noise and noise at the lateral surfaces of the discontinuities. However, in most schemes, some noisy pixels are missed and noise detection errors occur. Accordingly, this paper proposes two robust filters (designated as Filters A and B, respectively) for improving the performance of the phase unwrapping process for objects with height discontinuities. Filter A comprises a noise and phase jump detection scheme and an adaptive median filter, while Filter B replaces the detected noise with the median phase value of an *N* × *N* mask centered on the noisy pixel. Filter A enables most of the noise and detection errors in the wrapped phase map to be removed. Filter B then detects and corrects any remaining noise or detection errors during the phase unwrapping process. Three reconstruction paths are proposed, Path I, Path II and Path III. Path I combines the path-dependent MACY algorithm with Filters A and B, while Paths II and III combine the path-independent cellular automata (CA) algorithm with Filters A and B. In Path II, the CA algorithm operates on the whole wrapped phase map, while in Path III, the CA algorithm operates on multiple sub-maps of the wrapped phase map. The simulation and experimental results confirm that the three reconstruction paths provide a robust and precise reconstruction performance given appropriate values of the parameters used in the detection scheme and filters, respectively. However, the CA algorithm used in Paths II and III is relatively inefficient in identifying the most suitable unwrapping paths. Thus, of the three paths, Path I yields the lowest runtime.

© 2012 OSA

## 1. Introduction

1. R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. **45**(10), 3240–3251 (2007). [CrossRef]

2. J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express **19**(4), 3086–3105 (2011). [CrossRef] [PubMed]

1. R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. **45**(10), 3240–3251 (2007). [CrossRef]

4. A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. **40**(3), 699–709 (2002). [CrossRef]

5. 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]

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

11. 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. 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]

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

27. H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. **50**(6), 063602–063608 (2011). [CrossRef]

28. 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]

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

16. 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]

18. 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]

19. R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. **23**(4), 713–720 (1988). [CrossRef]

27. H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. **50**(6), 063602–063608 (2011). [CrossRef]

19. R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. **23**(4), 713–720 (1988). [CrossRef]

27. H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. **50**(6), 063602–063608 (2011). [CrossRef]

19. R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. **23**(4), 713–720 (1988). [CrossRef]

20. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A **14**(10), 2692–2701 (1997). [CrossRef]

21. M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express **20**(3), 2556–2561 (2012). [CrossRef] [PubMed]

**50**(6), 063602–063608 (2011). [CrossRef]

21. M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express **20**(3), 2556–2561 (2012). [CrossRef] [PubMed]

24. J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. **5**(2), 147–151 (2008). [CrossRef]

25. S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express **15**(13), 8059–8064 (2007). [CrossRef] [PubMed]

26. L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. **50**(4), 043605–043612 (2011). [CrossRef]

**50**(6), 063602–063608 (2011). [CrossRef]

2. J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express **19**(4), 3086–3105 (2011). [CrossRef] [PubMed]

12. 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]

16. 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]

**23**(4), 713–720 (1988). [CrossRef]

20. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A **14**(10), 2692–2701 (1997). [CrossRef]

25. S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express **15**(13), 8059–8064 (2007). [CrossRef] [PubMed]

17. 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]

## 2. Underlying principles of detection scheme and filtering operations

### 2.1 Noise and phase jump detection scheme

2. J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express **19**(4), 3086–3105 (2011). [CrossRef] [PubMed]

*i*,

*j*)) is designated as a “good” pixel. Conversely, if the absolute phase difference between any two neighboring pixels is greater than

### 2.2 Filter A – detection scheme combined with adaptive median filter

29. 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]

29. 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]

29. 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]

### 2.3 Filter B – detection scheme combined with noisy pixel replacement mechanism

*N*×

*N*mask centered at the pixel of interest, i.e., pixel (i

_{c}, j

_{c}). The phase unwrapping procedure commences by marking the pixels within the phase map as either “good” or “bad” using the detection scheme given in Eq. (1). In the subsequent filtering operation, the phase of any pixel identified as a good pixel and having a deviation of less than

### 2.4 Implementation of CA algorithm using array processor and additional sub-map area (using in Path III)

17. 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]

## 3. Integration of filtering and phase unwrapping algorithms for image reconstruction

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

16. 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]

_{1}is applied to remove any noise missed by Filter A and any row-unwrapping errors. The MACY algorithm is then re-applied to unwrap the phase map in the column direction. Filter B

_{2}is then applied once again to remove any noise missed by Filter A and any column-unwrapping errors. Finally, the unwrapped phase map (i.e., the image reconstruction result) is obtained. In Path II, the wrapped phase map obtained from Filter A is unwrapped by the CA algorithm. As shown in Fig. 3, the CA algorithm acts on the uncut wrapped phase map and involves a cycle of local and global iterations separated by Filter B

_{1}. Once the local / global iteration procedure terminates, Filter B

_{2}is applied and the unwrapped phase map is obtained. In Path III, the noise-reduced phase map obtained from Filter A is partitioned into several sub-maps. As described in Section 2.4, an additional border area is added to each sub-map, and each map is then unwrapped using the CA algorithm. As in Path III, the unwrapping procedure involves a cycle of local and global iterations integrated with Filter B

_{1}. On completion of the iterative cycle, the border areas are cropped from the unwrapped sub-maps, and the maps are then meshed in order to construct the full unwrapped phase map. Finally, Filter B

_{2}is applied to obtain the final image reconstruction results.

## 4. Simulation results

*“imnoise (each of five interferograms, 'speckle', 0.08)”*). Residual noise (Noise B) was produced using the “imnoise” salt and pepper noise function with an intensity parameter setting of 0.35 (i.e.,

*“imnoise (each of five interferograms, 'salt & pepper', 0.35)”*). Finally, noise at the lateral surface of the height discontinuities (Noise C) was produced using a self-written program based on the phase signal corresponding to the low and high positions of the discontinuity and the “imnoise” salt and pepper noise function with an intensity parameter setting of 0.01.

*G*= 20; the size of the sliding window in Filter A was specified as 5 × 5 pixels; and the size of the sliding window in Filter B was set as 9 × 9 pixels. In the Path III reconstruction process, the wrapped sub-maps were extended by an additional 3 pixels in both the row and the column directions, respectively. Finally, Filter A was applied twice in order to remove most of the noise and detection errors from the wrapped phase map prior to unwrapping.

### 4.1 Application of Filter A to noisy wrapped phase map

### 4.2 Application of Filter B to noise-reduced wrapped phase map

#### 4.2.1 Path I

#### 4.2.2 Path II

#### 4.2.3 Path III

#### 4.2.4 Summary of simulation results for Paths I, II and III

**19**(4), 3086–3105 (2011). [CrossRef] [PubMed]

#### 4.2.5 Effect of Filter B on Path III reconstruction performance

## 5. Experimental setup and results

### 5.1 Precision evaluation of three reconstruction paths using sample with perpendicular phase jump lines

*5.2* Robustness evaluation of three reconstruction paths using sample with non-straight phase jump lines

### 5.3 Sensitivity evaluation of Path I and Path III reconstruction paths using sample with two different height discontinuities

## 6. Conclusions

**22**(23), 3898–3901 (1983). [CrossRef] [PubMed]

**4**(1), 267–280 (1987). [CrossRef]

**19**(4), 3086–3105 (2011). [CrossRef] [PubMed]

**36**(9), 2466–2472 (1997). [CrossRef]

*N*×

*N*mask centered on the noisy pixel. In all three reconstruction routes, Filter A is applied repeatedly prior to unwrapping in order to remove the majority of the noise within the wrapped phase map and to correct the detection errors generated by the detection scheme [2

**19**(4), 3086–3105 (2011). [CrossRef] [PubMed]

30. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. **38**(34), 6994–7001 (1999). [CrossRef] [PubMed]

31. T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. **25**(3), 232–244 (1985). [CrossRef]

## Acknowledgments

## References and links

1. | R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. |

2. | J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express |

3. | R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process. |

4. | A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. |

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

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

7. | J. W. Goodman, “Statistical properties of laser speckle patterns,” in |

8. | R. Jones and C. Wykes, |

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

10. | I. Pitas and A. N. Venetsanopoulos, |

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

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

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

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

17. | 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. |

18. | 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. |

19. | R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. |

20. | T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A |

21. | M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express |

22. | J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012). |

23. | X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. |

24. | J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. |

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

26. | L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. |

27. | H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. |

28. | 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 |

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

30. | E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. |

31. | T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.6890) Image processing : Three-dimensional image processing

(100.5088) Image processing : Phase unwrapping

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 10, 2012

Revised Manuscript: April 5, 2012

Manuscript Accepted: April 23, 2012

Published: April 26, 2012

**Virtual Issues**

Vol. 7, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Jing-Feng Weng and Yu-Lung Lo, "Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities," Opt. Express **20**, 10896-10920 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-10-10896

Sort: Year | Journal | Reset

### References

- R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens.45(10), 3240–3251 (2007). [CrossRef]
- J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express19(4), 3086–3105 (2011). [CrossRef] [PubMed]
- R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process.3(5), 325–333 (1995). [CrossRef]
- A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens.40(3), 699–709 (2002). [CrossRef]
- 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]
- J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, 1984).
- R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge Univ. Press, 1989).
- H. A. Aebischer 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. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Springer, 1990).
- M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain44(3), 239–247 (2008). [CrossRef]
- 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]
- 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]
- 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. A4(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]
- R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988). [CrossRef]
- T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A14(10), 2692–2701 (1997). [CrossRef]
- M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express20(3), 2556–2561 (2012). [CrossRef] [PubMed]
- J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).
- X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig.5(3), 296–304 (2011). [CrossRef]
- J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens.5(2), 147–151 (2008). [CrossRef]
- S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express15(13), 8059–8064 (2007). [CrossRef] [PubMed]
- L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng.50(4), 043605–043612 (2011). [CrossRef]
- H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng.50(6), 063602–063608 (2011). [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. Express18(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]
- E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt.38(34), 6994–7001 (1999). [CrossRef] [PubMed]
- T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech.25(3), 232–244 (1985). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

« Previous Article | Next Article »

OSA is a member of CrossRef.