## Noise robust linear dynamic system for phase unwrapping and smoothing |

Optics Express, Vol. 19, Issue 6, pp. 5126-5133 (2011)

http://dx.doi.org/10.1364/OE.19.005126

Acrobat PDF (1327 KB)

### Abstract

Phase unwrapping techniques remove the modulus 2*π* ambiguities of wrapped phase maps. The present work shows a first-order feedback system for phase unwrapping and smoothing. This system is a fast phase unwrapping system which also allows filtering some noise since in deed it is an Infinite Impulse Response (IIR) low-pass filter. In other words, our system is capable of low-pass filtering the wrapped phase as the unwrapping process proceeds. We demonstrate the temporal stability of this unwrapping feedback system, as well as its low-pass filtering capabilities. Our system even outperforms the most common and used unwrapping methods that we tested, such as the Flynn’s method, the Goldstain’s method, and the Ghiglia least-squares method (weighted or unweighted). The comparisons with these methods show that our system filters-out some noise while preserving the dynamic range of the phase-data. Its application areas may cover: optical metrology, synthetic aperture radar systems, magnetic resonance, and those imaging systems where information is obtained as a demodulated wrapped phase map.

© 2011 Optical Society of America

## 1. Introduction

3. Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Optics And Lasers In Engineering **45**, 304–317 (2007). [CrossRef]

*π*radians, and this is called the wrapped demodulated phase. The wrapping process occurs because the searched phase information is recovered containing phase jumps (modulo 2

*π*) when the phase information goes below −

*π*radians or above

*π*radians. In optical interferometry most techniques give the recovered phase modulus 2

*π*[1

1. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**, 156 (1982). [CrossRef]

4. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express **17**, 21867–21881 (2009). [CrossRef] [PubMed]

5. L. N. Mertz, “Speckle imaging, photon by photon,” Appl. Opt. **18**, 611–614 (1979). [CrossRef] [PubMed]

6. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. **69**, 393–399 (1979). [CrossRef]

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

10. D. C. Ghiglia and M. D. Pritt, *Two-dimensional Phase Unwrapping; Theory, Algorithms, and Software* (Wiley-Interscience, 1998). [PubMed]

7. K. A. Stetson, J. Wahid, and P. Gauthier , “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. **36**, 4830–4838 (1997). [CrossRef] [PubMed]

11. 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]

18. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. **38**, 1934–1941 (1999). [CrossRef]

10. D. C. Ghiglia and M. D. Pritt, *Two-dimensional Phase Unwrapping; Theory, Algorithms, and Software* (Wiley-Interscience, 1998). [PubMed]

## 2. The phase unwrapping system: analysis, and design

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

10. D. C. Ghiglia and M. D. Pritt, *Two-dimensional Phase Unwrapping; Theory, Algorithms, and Software* (Wiley-Interscience, 1998). [PubMed]

*ϕ̂*(

*t*) is the output (unwrapped phase) obtained from a previously unwrapped phase

*ϕ̂*(

*t*

_{0}), plus the integration of

*ϕ*(

_{w}*t*) the wrapped phase that we want to unwrap. The function

*W*[·] is the modulus 2

*π*operator; the main use of this operator in the phase derivative (or differences in a discreet domain) is to wrap or remove the outliers of the difference operator generated by the 2

*π*phase jumps of the wrapped phase. This 1D line integration may be also stated as the solution of the following continuous system which is a first order differential equation that may be solved numerically in the discrete domain using the following recursive difference system: where

*ϕ̂*(

*n*) is the output unwrapped phase and

*ϕ̂*(

*n*− 1) its unwrapped previous value. However, in 2D we know from previous works that with noisy data the simple line integration (3) fails most of cases. This is because the phase difference operation acts as a high-pass filter, augmenting even more the levels of noise [19].

*ϕ*(

_{w}*n*− 1) at the previous site

*n*− 1 by the unwrapped data

*ϕ̂*(

*n*− 1) at the same site

*n*− 1, and the parameter

*τ*. As will be shown below, the parameter

*τ*is a regularization parameter that filters out most of the noise. In addition to this, we see that the wrapping operator

*W*[·] takes the difference between a noisy data and a noise-reduced estimation. Thus, the probability of generating spurious phase jumps in the unwrapped phase is much lower. A block diagram of our system (Eq. 4) is depicted in Fig. 1.

*π*radians since this is the Nyquist limit, or in other words, the maximum frequency rate that can be sampled without aliasing. Then, just for analysis purposes we can write the following: and the linear system of Eq. (4) can be rewritten as To solve this difference equation we can use the

*z*-transform, and obtain that the response of our dynamic phase unwrapping (4) for any arbitrary input

*ϕ*(

_{w}*n*) is: where

*ϕ*(−1) is the unwrapped phase initial value. This is a causal system, and from its response we have that the system’s impulse response is where

*u*(

*n*) is the unitary step function. Now, to demonstrate the stability of this system we must prove that its impulse response is absolutely summable (see Ref. [19]). Thus, we can proof that the following is true: since this is a geometric series, proving in this way that the system’s impulse response is absolutely summable, and thus we can say that the system (4) is an stable system. To see the frequency response of the system (4) we can take directly the Fourier transform of its impulse response and show that its frequency response is being ℱ{·} the Fourier transform operator. In Fig. 2, panel (a) shows the graph of the system’s impulse response, an panel (b) shows the graph of its frequency response. Note from this figure, that the systems impulse response is attenuated by

*τ*, but its frequency response varies its wideband with

*τ*and it has always a maximum of one in magnitude. Thus, the frequency response of the system behaves like a low-pass filter and its bandwidth is controlled by the parameter

*τ*.

### 2.1. Two-dimensional extension

**r**= (

*x*,

*y*) as the 2D site being unwrapped; being

*x*and

*y*are integers. Now let us denote with

*m*(

**r**) = {

**r**

_{0},

**r**

_{1},...,

**r**

_{N}_{−1}} to the previously unwrapped sites around

**r**. In Fig. 3, we show a graphic representation of the spatial support that we take around the site

**r**. In that figure, Ω is a neighborhood around the site

**r**. There, we show with dark points the set

*m*(

**r**) ⊂ Ω of previously unwrapped sites, and with a low gray, the sites that are not unwrapped yet. Thus, following this scheme, a 2D extension for the 1D dynamic system (4) is Where |

*m*(

**r**)| denote the cardinality of the set

*m*(

**r**) ⊂ Ω. For example, counting the dark points of Fig. 3, we see that for that particular case the dimension of

*m*(

**r**) is |

*m*(

**r**)| = 4. Thus, the 2D dynamic phase unwrapping system (11) can be seen as the mean of the unwrapped phase obtained with the 1D system (4) from all possible directions around

**r**. The unwrapping system (11) must process each site sequentially starting from an initial given site or point. At the starting point, we use the wrapped phase of that point as the initial unwrapped phase. To visit sequentially all sites of the 2D phase domain, one can use any 2D scanning strategies. However, when the phase domain has holes, or the phase domain is not a regular form, we recommend the standard flood-fill scanning used to color sequentially connected regions.

## 3. Tests and results

*Two-dimensional Phase Unwrapping; Theory, Algorithms, and Software* (Wiley-Interscience, 1998). [PubMed]

*π*wrapped phase corresponding to each unwrapped phase surface. The scanning strategy used by our phase unwrapping system was the simple row by row scanning, and we set

*τ*= 0.2. In those results, we see that the Flynn’s method obtains better results than the Goldstein’s and the least-squares method. Also, we can see that our phase unwrapping system obtains a better result than the Flynn’s method, and we see that it reduces some levels of noise. Counting the rings of the iso-phase contours from the projected wrapped phase under each unwrapped phase surface, we can see that only the unwrapped phase of the least-squares method has reduced the dynamic range. This is a drawback of the least-squares method that is already known in this area, and it is important to remark that our phase unwrapping system preserves the dynamic range and reduces the levels of noise. In these tests, the computational time spent for our dynamic phase unwrapping system was of 0.064 seconds, for the Flynn’s method was of 0.858 seconds, for the Goldstein method was of 0.068, and for the least-squares method was of 0.105 seconds (by means of direct transforms [10

*Two-dimensional Phase Unwrapping; Theory, Algorithms, and Software* (Wiley-Interscience, 1998). [PubMed]

*τ*= 0.6, and the second time using

*τ*= 0.2 with the output of the first pass as input. We can see here that our phase unwrapping system is able to reduce considerably the levels of noise without practically affect the dynamic range of the unwrapped phase. This because the projected wrapped phase of Fig. 5.B has the same iso-phase contours as the input wrapped phase used (Fig. 5.A).

## 4. Conclusions

*τ*< 1. The spatial support of the 2D extension allow us to unwrap the phase following arbitrary scanning strategies, however, for the results shown here we used a simple row by row scanning strategy. Nevertheless, we implemented the flood-fill scanning strategy for the phase unwrapping processing obtaining the same results. As reported in the last paragraph of the section 3, our phase unwrapping system takes around 0.065 seconds of computational time to unwrap an filter the wrapped phase, whereas the other phase unwrapping methods that we programmed took approximately the same time but these did not removed any level of noise. Hence, we consider that our phase unwrapping system is a noise tolerant fast phase unwrapping system.

*Important remarks:*

## References and links

1. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

2. | 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. |

3. | Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Optics And Lasers In Engineering |

4. | M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express |

5. | L. N. Mertz, “Speckle imaging, photon by photon,” Appl. Opt. |

6. | B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. |

7. | K. A. Stetson, J. Wahid, and P. Gauthier , “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. |

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

9. | T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Optics and Lasers in Engineering |

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

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

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

13. | 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 |

14. | J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A |

15. | J. L. Marroquin, M. Tapia, R. Rodriguez-Vera, and M. Servin, “Parallel algorithms for phase unwrapping based on Markov random field models,” J. Opt. Soc. Am. A |

16. | K. M. Hung and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Optical Engineering |

17. | V. V. Volkov and Y. Zhu, “Deterministic phase unwrapping in the presence of noise,” Opt. Lett. |

18. | M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. |

19. | J. G. Proakis and D. G. Manolakis, |

**OCIS Codes**

(100.2650) Image processing : Fringe analysis

(100.5088) Image processing : Phase unwrapping

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 11, 2010

Revised Manuscript: January 15, 2011

Manuscript Accepted: January 25, 2011

Published: March 3, 2011

**Citation**

Julio C. Estrada, Manuel Servin, and Juan A. Quiroga, "Noise robust linear dynamic system for phase unwrapping and smoothing," Opt. Express **19**, 5126-5133 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5126

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### References

- M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156 (1982). [CrossRef]
- 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, 2693 (1974). [CrossRef] [PubMed]
- Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007). [CrossRef]
- M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17, 21867–21881 (2009). [CrossRef] [PubMed]
- L. N. Mertz, “Speckle imaging, photon by photon,” Appl. Opt. 18, 611–614 (1979). [CrossRef] [PubMed]
- B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979). [CrossRef]
- K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830–4838 (1997). [CrossRef] [PubMed]
- K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982). [CrossRef] [PubMed]
- T. R. Judge, and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994). [CrossRef]
- D. C. Ghiglia, and M. D. Pritt, Two-dimensional Phase Unwrapping; Theory, Algoritms, and Software (Wiley-Interscience, 1998). [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. 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]
- J. L. Marroquin, and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995). [CrossRef]
- J. L. Marroquin, M. Tapia, R. Rodriguez-Vera, and M. Servin, “Parallel algorithms for phase unwrapping based on markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995). [CrossRef]
- K. M. Hung, and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998). [CrossRef]
- V. V. Volkov, and Y. Zhu, “Deterministic phase unwrapping in the presence of noise,” Opt. Lett. 28, 2156–2158 (2003). [CrossRef] [PubMed]
- M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999). [CrossRef]
- . J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles, Algorothims, ans Applications (Prentice-Hall, October 5, 1995), 3rd ed.

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