## Robust phase unwrapping by spinning iteration

Optics Express, Vol. 15, Issue 13, pp. 8059-8064 (2007)

http://dx.doi.org/10.1364/OE.15.008059

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### Abstract

This work describes a rapid-phase unwrapping algorithm that combines the rapidity and simplicity of a path-dependent algorithm and the robustness of a path-independent algorithm by rotating the phase map or the unwrapping direction 90° after a scan in one direction. It offers a solution for noise-contaminated phase data, which includes artifacts, complex-shaped borders, or regions of holes. The algorithm can be used in real-time processing. In addition, a phase-dislocation masking method is presented that can be used to detect and clean inconsistent data and improve the rms values of signal-to-noise ratio in unwrapped phase maps.

© 2007 Optical Society of America

## 1. Introduction

1. B. Wang, Y. Shi, T. Pfeifer, and H. Mischo, “Phase unwrapping by blocks,” Measurement **25**, 285–290 (1999). [CrossRef]

1. B. Wang, Y. Shi, T. Pfeifer, and H. Mischo, “Phase unwrapping by blocks,” Measurement **25**, 285–290 (1999). [CrossRef]

2. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. **21**, 2470–2470 (1982). [CrossRef] [PubMed]

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

## 2. Phase dislocation masking algorithm

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

1. B. Wang, Y. Shi, T. Pfeifer, and H. Mischo, “Phase unwrapping by blocks,” Measurement **25**, 285–290 (1999). [CrossRef]

- Wang's method is used for tracking the 2π phase jumps [Type 3 in Fig. 1(b)] by counting their length.
- If the length of these 2π phase jumps is shorter than the threshold, they are surely pseudo-2π phase jumps and can be masked by Wang's method. The result is the elimination of the pair of opposite residues if the two end points of pseudo-2π phase jumps are in the map or the elimination of a residue if only one end point of a pseudo-2π phase jump is in the map [Fig. 2(b)].
- The inconsistent data left are masked as part of the true 2π phase jumps submerged in noises by Goldstein's method. The residues of the two end points of the inconsistent 2π phase jumps are always opposite. We can connect the nearest plus and minus residues with a branch cut and can mask all points on it [Fig. 2(b)].

## 3. Phase unwrapping by spinning iteration

- Appoint a new starting point in the phase map and establish an image bitmap file. Mark the point as an unwrapped point at the corresponding point in the image file.
- Scan line by line from left to right using the path-dependent algorithm. During the process of unwrapping a line, if an unwrapped point is found and the point to the right is a wrapped point, take the unwrapped point as the new starting point and unwrap the line from left to right. Generally, the unwrapping part is to the right of the unwrapped point. If the masked point or border is met, stop unwrapping [Shown in Fig. 3(a)].
- Scan column by column from top to bottom using the path-dependent algorithm. During the process of unwrapping a column, if an unwrapped point is found and the point below is a wrapped point, take the unwrapped point as the new starting point and unwrap the column from top to bottom. Generally, the unwrapping part is below the unwrapped point. If the masked point or border is met, stop unwrapping [Shown in Fig. 3(b)].
- Scan line by line from right to left using the path-dependent algorithm. During the process of unwrapping a line, if an unwrapped point is found and the point to the left is a wrapped point, take the unwrapped point as the new starting point and unwrap the line from right to left. Generally, the unwrapping part is to the left of the unwrapped point. If the masked point or border is met, stop unwrapping [Shown in Fig. 3(c)].
- Scan column by column from bottom to top using the path-dependent algorithm. During the process of unwrapping a column, if an unwrapped point is found and the point above is a wrapped point, take the unwrapped point as the new starting point and unwrap the column from bottom to top. Generally, the unwrapping part is above the unwrapped point. If the masked point or border is met, stop unwrapping [Shown in Fig. 3(d)]. A round of iteration is done.
- If there are still wrapped parts, repeat the spinning iteration processes 2-5 clockwise until the whole phase map is unwrapped [Shown in Fig. 3(e)].

- Appoint a new starting point in the phase map and establish an image bitmap file. Mark the point as an unwrapped point at the corresponding point in the image file.
- Scan line by line from left to right by the path-dependent algorithm. During the process of unwrapping a line, if an unwrapped point is found and the point to the right is a wrapped point, take the unwrapped point as the new starting point and unwrap the line from left to right. Generally, the unwrapping part is to the right of the unwrapped point. If the masked point or border is met, stop unwrapping.
- Rotate the whole phase map 90° counterclockwise.
- Repeat 2-3 three times to finish one iteration.
- If there are still wrapped parts, repeat 2-4 until the whole phase map is unwrapped.

## 4. Application and results

## 5. Conclusion

## References and links

1. | B. Wang, Y. Shi, T. Pfeifer, and H. Mischo, “Phase unwrapping by blocks,” Measurement |

2. | K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. |

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

4. | J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE |

5. | K. Andresen and Q. Yu, “Robust phase unwrapping by spin filtering combined with a phase direction map,” Optik |

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

7. | D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A |

8. | 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. |

9. | R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 30, 2007

Revised Manuscript: May 9, 2007

Manuscript Accepted: May 14, 2007

Published: June 13, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Shi Yuqing, "Robust phase unwrapping by spinning iteration," Opt. Express **15**, 8059-8064 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-13-8059

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### References

- B. Wang, Y. Shi, T. Pfeifer, and H. Mischo, "Phase unwrapping by blocks," Measurement 25, 285-290 (1999). [CrossRef]
- K. Itoh, "Analysis of the phase unwrapping algorithm," Appl. Opt. 21, 2470-2470 (1982). [CrossRef] [PubMed]
- D. C. Ghiglia, G. A. Mastin, and L. A. Romero, "Cellular-automata method for phase unwrapping," J. Opt. Soc. Am. A 4, 267-280 (1987). [CrossRef]
- J. J. Gierloff, "Phase unwrapping by regions," Proc. SPIE 818, 2-9 (1987).
- K. Andresen and Q. Yu, "Robust phase unwrapping by spin filtering combined with a phase direction map," Optik 4, 145-149 (1994).
- J. M. Huntley, "Noise immune phase unwrapping algorithm," Appl. Opt. 28, 3268-3270 (1989). [CrossRef] [PubMed]
- D. C. Ghiglia and L. A. Romero, "Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods," J. Opt. Soc. Am. A 11, 107-117 (1994). [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, 25-37 (1991). [CrossRef]
- R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988). [CrossRef]

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