## Path-independent phase unwrapping using phase gradient and total-variation (TV) denoising |

Optics Express, Vol. 20, Issue 13, pp. 14075-14089 (2012)

http://dx.doi.org/10.1364/OE.20.014075

Acrobat PDF (2265 KB)

### Abstract

Phase unwrapping is a challenging task for interferometry based techniques in the presence of noise. The majority of existing phase unwrapping techniques are path-following methods, which explicitly or implicitly define an intelligent path and integrate phase difference along the path to mitigate the effect of erroneous pixels. In this paper, a path-independent unwrapping method is proposed where the unwrapped phase gradient is determined from the wrapped phase and subsequently denoised by a TV minimization based method. Unlike the wrapped phase map where _{$2\pi $}phase jumps are present, the gradient of the unwrapped phase map is smooth and slowly-varying at noise-free areas. On the other hand, the noise is greatly amplified by the differentiation process, which makes it easier to separate from the smooth phase gradient. Thus an approximate unwrapped phase can be obtained by integrating the denoised phase gradient. The final unwrapped phase map is subsequently determined by adding the first few modes of the unwrapped phase. The proposed method is most suitable for unwrapping phase maps without abrupt phase changes. Its capability has been demonstrated both numerically and by experimental data from shearography and electronic speckle pattern interferometry (ESPI).

© 2012 OSA

## 1. Introduction

1. Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. **36**(4), 526–528 (2011). [CrossRef] [PubMed]

3. B. Osmanoglu, T. H. Dixon, S. Wdowinski, and E. Cabral-Cano, “On the importance of path for phase unwrapping in synthetic aperture radar interferometry,” Appl. Opt. **50**(19), 3205–3220 (2011). [CrossRef] [PubMed]

4. J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging **27**(9), 1293–1301 (2009). [CrossRef] [PubMed]

5. C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. **43**(5), 1152–1159 (2004). [CrossRef]

_{2π}to each pixel in the wrapped phase map as necessary to confine the phase change of adjacent pixels to less than

_{π}. However, real experiment data always contain noise, which considerably inhibits the phase unwrapping task [6].

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

13. M. Gdeisat, M. Arevalillo-Herráez, D. Burton, and F. Lilley, “Three-dimensional phase unwrapping using the Hungarian algorithm,” Opt. Lett. **34**(19), 2994–2996 (2009). [CrossRef] [PubMed]

14. S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express **18**(2), 560–565 (2010). [CrossRef] [PubMed]

15. J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt. **47**(29), 5358–5369 (2008). [CrossRef] [PubMed]

_{Lp}norm. This turns the problem of phase unwrapping into solving a discretized partial differential equation [6,16

16. D. C. Ghiglia and L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. **14**(20), 1107–1109 (1989). [CrossRef] [PubMed]

17. M. A. Schofield and Y. M. Zhu, “Fast phase unwrapping algorithm for interferometric applications,” Opt. Lett. **28**(14), 1194–1196 (2003). [CrossRef] [PubMed]

_{L2}norm is employed. Methods in this category seem to underestimate the magnitude of the resultant unwrapped phase due to the strong smoothing effect of Poisson solvers [6], thus post-processing techniques [18

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

19. S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. **49**(25), 4735–4745 (2010). [CrossRef] [PubMed]

21. J. Strand and T. Taxt, “Performance evaluation of two-dimensional phase unwrapping algorithms,” Appl. Opt. **38**(20), 4333–4344 (1999). [CrossRef] [PubMed]

23. J. Parkhurst, G. Price, P. Sharrock, and C. Moore, “Phase unwrapping algorithms for use in a true real-time optical body sensor system for use during radiotherapy,” Appl. Opt. **50**(35), 6430–6439 (2011). [CrossRef] [PubMed]

24. J. F. Weng and Y. L. Lo, “Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities,” Opt. Express **20**(10), 10896–10920 (2012). [CrossRef] [PubMed]

25. 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]

26. Y. H. Huang, F. Janabi-Sharifi, Y. Liu, and Y. Y. Hung, “Dynamic phase measurement in shearography by clustering method and Fourier filtering,” Opt. Express **19**(2), 606–615 (2011). [CrossRef] [PubMed]

_{2π}jumps in the wrapped phase map. Thus, the noise can be successfully separated from the genuine phase gradient map; that is, TV effectively acts as a denoising process in the phase gradient domain.

_{2π}jumps are detected in the residual wrapped phase map. In our experiments, 1-2 iterations were typically sufficient to determine the final correct unwrapped phase map.

## 2. Problem description and path-following solutions

_{ϕw}from an extended arctangent function which confines the wrapped phase within

_{(−π,π]}and makes it different from the genuine unwrapped phase

_{ϕ}representing the physical quantity by an integer multiple of

_{2π}as:The task of phase unwrapping is thus to add a correct

_{2nπ}to each pixel of the wrapped phase such that a continuous unwrapped phase map is reconstructed.

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

8. Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE **2003**, 254–263 (1993). [CrossRef]

10. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A **14**(10), 2692–2702 (1997). [CrossRef]

_{2π}phase jumps (“jump cuts”) and extend them to form close loops. By iteratively adding or subtracting

_{2π}to one of the part separated by the loop, the phase jumps number is gradually minimized, and an unwrapped phase is finally generated. Flynn’s algorithm is most suitable for unwrapping smooth phase maps as the algorithm tends to minimize the discontinuity in the unwrapped result.

27. Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. **47**(5), 054301 (2008). [CrossRef]

28. Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. **36**(4), 526–528 (2011). [CrossRef] [PubMed]

27. Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. **47**(5), 054301 (2008). [CrossRef]

28. Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. **36**(4), 526–528 (2011). [CrossRef] [PubMed]

27. Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. **47**(5), 054301 (2008). [CrossRef]

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

8. Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE **2003**, 254–263 (1993). [CrossRef]

10. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A **14**(10), 2692–2702 (1997). [CrossRef]

## 3. Phase unwrapping using TV prior

### 3.1 Definition of the wrapping function

_{(−π,π]}by adding an integer multiple of

_{2π}to each pixel can be defined as:where

_{arctan[⋅]}is a normal arctangent function with range

_{(−π/2, π/2)}and the wrapping function

_{W{⋅}}has an extended range of

_{(−π, π]}. This wrapping function enables comparison between the wrapped and unwrapped phase maps and will be frequently used in the following derivations.

### 3.2 Determination of the phase gradient

_{2π}jumps, the discretized unwrapped phase derivatives can then be obtained from the wrapped phase map as

_{2π}jumps due to wrapping have been removed by the wrapping function

_{W{⋅}}, making the derivative map admissible to noise-removal filters. The second merit is that the differentiation process amplifies the noise. Intuition usually suggests that such action is to be avoided; however, the core concept of our denoising is that the amplified noise will be easier to be picked up and removed by the TV solver. Thus, surprisingly, we can turn noise amplification to our advantage.

_{2π}phase jumps in the wrapped phase map have been removed and phase derivatives are smooth for most areas except the central part where noise is amplified. The next section describes how TV removes this amplified noise.

### 3.3 TV minimization for denoising

19. S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. **49**(25), 4735–4745 (2010). [CrossRef] [PubMed]

_{ϕx}and subsequently calculate a denoised estimate

_{ϕ^x}according towhere

_{ϕx}is the input noisy phase derivative,

_{||⋅||}is the Euclidean norm,

_{μ}is a regularization parameter, and

_{TV(⋅)}is a functional that determines the total variation of its argument. If we define the TV functional as

_{TV(ψ)=∑||∇ψ||},

*i.e.*the sum of the gradient magnitude over all pixels, it has been proved [29

29. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. **20**(1/2), 89–97 (2004). [CrossRef]

_{πμκ(ϕx)}is the nonlinear projection of

_{ϕx}. Computing the nonlinear projection

_{πμκ(ϕx)}is identical to solving the following problemand

_{πμκ(ϕx)}can then be determined as

29. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. **20**(1/2), 89–97 (2004). [CrossRef]

_{p→}as the convergent value of a vector sequence

_{p→n}ofwhere the iteration is initiated at

_{p→1=(0, 0)}and the parameter

_{τ}is routinely set to

_{τ=0.25}.

_{μ}and a stopping criterion for the iteration in Eq. (9) still need to be defined. We found that the quality of reconstruction is quite tolerant to any value of

_{μ}within the range of

_{[0.5, 5]}so we routinely used

_{μ=2}in our simulations and experiments. We also applied consistently a stopping criterion of

_{||∇⋅p→n+1−∇⋅p→n||∞<0.002}. The results of applying TV denoising on the partial derivatives maps in Fig. 2(a) and 2(b) are shown in Fig. 2(c) and 2(d), respectively. It can be visually judged that the noise caused by phase residues has been totally removed at the cost of slight distortion in the phase derivative information; we will show how to remove this distortion finally in sections 3.5-3.6.

### 3.4 Integration of phase derivative

_{ϕ^x,ϕ^y}, a simple integration gives a good approximation of the unwrapped phase map. However, the unwrapped phase values on the top and left lines of the image should be given as starting values. They can be obtained by setting the top-left corner unwrapped phase as the same as the wrapped phase, and integrating from left to right using

_{ϕ^x}to obtain the unwrapped phase values for the top line while integrating from top to bottom using

_{ϕ^y}to obtain the unwrapped phase values for the left line. The integration over the whole image can then be conducted based on the unwrapped boundary information. Figure 2(e) shows the unwrapped phase map

_{ϕ^}thus obtained. To visually evaluate the fidelity of the approximated unwrapped phase map to the original wrapped phase map, we rewrap the result in Fig. 2(e) into Fig. 2(f). By comparing Fig. 1(d) and Fig. 2(f), it can be observed that the rewrapped phase map is similar to the original wrapped phase, but not identical, which means that TV denoising has slightly modify the true phase gradient, and further method for the error correction may be required.

### 3.5 Error metric

_{ϕ^}and

_{ϕ}is within

_{2π}for every pixel, then the residual wrapped phase map

_{ϕr}will only contain phase jumps from the phase residues. On the other hand, if the approximation

_{ϕ^}is underestimated or overestimated for more than

_{2π}for some areas, apparent

_{2π}phase jumps will present and form loops in the residual wrapped phase

_{ϕr}. Thus a simple algorithm can be designed to count the total number of

_{2π}phase jumps and denote it as

_{N}. If

_{N}is less than a preset criterion

_{Nc}(set to 0.1% of the total pixel number by default), we can conclude that the approximated unwrapped phase is within

_{2π}from the correct unwrapped phase, and no further processing on the residual wrapped phase

_{ϕr}is required. On the other hand, if

_{N}exceeds

_{Nc}, the residual wrapped phase still contains loops of

_{2π}jumps and these

_{2π}jumps should be further eliminated by adding more modes to the approximated unwrapped phase

_{ϕ^}, which will be described in the following section.

_{2π}jumps due to insufficient unwrapping can be clearly observed and quantified. This residual wrapped phase map will be further processed to extract its approximated unwrapped phase.

### 3.6 Final unwrapping result from iteration

_{2π}phase jumps are present in the residual wrapped phase map, the process described in sections 3.2-3.4 needs to be applied to this residual wrapped phase map again to extract an approximated unwrapped phase map for it. In such case, the 1st mode is assigned as

_{ϕ^1}and the 2nd, 3rd

_{ϕ^2,ϕ^3⋯}until no apparent

_{2π}jumps remain in the final residual wrapped phase

_{ϕrf}. The final unwrapped phase can then be obtained by adding up all the modes and the final residual wrapped phase as

_{N}<

_{Nc}is never satisfied, a forced stopping criterion is implemented which stops the cycling when the total iteration number exceeds a certain number like 500. In our shearography example shown in Fig. 2, two iterations were required. Figure 2(h) shows the final residual wrapped phase

_{ϕrf}obtained by subtracting 2(g) to its approximated unwrapped phase. It can be observed and quantified that only phase jumps due to residues but no apparent

_{2π}phase jumps are present. Thus the final unwrapped phase map is obtained by Eq. (11) and shown in Fig. 2(i) where a satisfying result similar to Flynn’s minimum discontinuity method (Fig. 1(i)) has been obtained but the need to define any implicit unwrapping path has been eliminated.

### 3.7 Flowchart of the algorithm

## 4. Experiment evaluation

30. Y. H. Huang, S. Y. Hung, F. Janabi-Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Lasers Eng. **50**(4), 534–539 (2012). [CrossRef]

30. Y. H. Huang, S. Y. Hung, F. Janabi-Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Lasers Eng. **50**(4), 534–539 (2012). [CrossRef]

## 5. Simulation evaluation using wrapped phase with height discontinuity

24. J. F. Weng and Y. L. Lo, “Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities,” Opt. Express **20**(10), 10896–10920 (2012). [CrossRef] [PubMed]

*i.e. imnoise (each specklegram, ‘speckle’, 0.18)*in Matlab code). The second wrapped phase, as shown in the second row of Fig. 5, contains speckle noise of 0.08, residual noise of 0.35 and edge noise of 0.01 (

*i.e. imnoise (each specklegram, ‘speckle’, 0.08), imnoise (each specklegram, ‘salt & pepper’, 0.35)*and

*imnoise(edges at each specklegram, ‘salt & pepper’, 0.01)*in Matlab code), which is the same as the simulation in reference [24

24. J. F. Weng and Y. L. Lo, “Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities,” Opt. Express **20**(10), 10896–10920 (2012). [CrossRef] [PubMed]

## 6. Simulation evaluation of effect of noise and fringe density

_{201×201}pixels with the formwhere

_{φ0}is the “height” information which controls the magnitude of phase,

_{x,y∈[−1, 1]}is the normalized distance units,

_{a}= 0.3 is the phase map spread factor, and the wavelength

_{λ}is set to 1.

_{i=1,2,3,4}are the image numbers and

_{δi=(i−1)π/2}are the phase-shift amounts for each image. Standard four-step phase shifting method is then used to recover the wrapped phase map from the four fringe maps with different statistics for

_{noise(x,y)}. Before applying the TV unwrapping algorithm, the wrapped phase filtering process is applied three times to obtain a denoised wrapped phase.

_{noise~N(0,σ2)}with varying standard deviation

_{σ}. We then unwrap the noisy wrapped phase map using the proposed TV unwrapping method. The results are shown in Fig. 6 where the signal to noise ratio (SNR) is defined as the ratio of the average intensity in Eq. (13) to the standard deviation of noise

_{σ}. It can be seen that the algorithm is able to deal with very noisy wrapped phase map. However, the unwrapping for the case of SNR = 0.3 is incomplete due to the severe noise and the compulsory stopping criteria.

_{φ0}approach 25, this is partially due to the fact that the fringe density in the central area is approaching 4 pixels per fringe, which is merely two times of the sampling limit. At such high fringe density, the wrapped phase filtering process has begun to produce abundant phase residues at central area and cause both computation and reconstruction error to increase.

_{576×768 pixels}image shown in Fig. 1(d) (maximum phase gradient is

_{0.5 rad/pixel}) needs 38 iterations and costs 2.04 seconds in an i7 2.8G laptop, and the simulated

_{201×201 pixels}image with Gaussian noise and

_{φ0}= 20 as shown in 2nd column of Fig. 9 (maximum phase gradient is about

_{1.25 rad/pixel}) costs only 0.12 seconds. The algorithm can still be optimized for more efficient computation, especially for the TV iteration part.

## 7. Conclusion and future works

_{2π}loops. Based on the residual wrapped phase, the unwrapping task is much easier for other kind of unwrapping methods. Thus the TV unwrapping method can also been combined with existing unwrapping algorithms to tackle more complex problem. Future work will include extending the proposed TV minimization scheme for unwrapping other complex wrapped phase such as that from InSAR, and investigation of alternative compressive sensing methods for phase unwrapping.

## Acknowledgments

## References and links

1. | Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. |

2. | E. Volkl, L. F. Allard, and D. C. Joy, eds., |

3. | B. Osmanoglu, T. H. Dixon, S. Wdowinski, and E. Cabral-Cano, “On the importance of path for phase unwrapping in synthetic aperture radar interferometry,” Appl. Opt. |

4. | J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging |

5. | C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. |

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

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

8. | Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE |

9. | C. Prati, M. Giani, and N. Leuratti, “SAR interferometry: a 2-D phase unwrapping technique based on phase and absolute values information,” in Proceedings of the 1990 International Geoscience and Remote Sensing Symposium (IEEE, 1990), pp. 2043–2046. |

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

11. | I. Shalem and I. Yavneh, “A multilevel graph algorithm for two dimensional phase unwrapping,” Comput. Vis. Sci. |

12. | T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express |

13. | M. Gdeisat, M. Arevalillo-Herráez, D. Burton, and F. Lilley, “Three-dimensional phase unwrapping using the Hungarian algorithm,” Opt. Lett. |

14. | S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express |

15. | J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt. |

16. | D. C. Ghiglia and L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. |

17. | M. A. Schofield and Y. M. Zhu, “Fast phase unwrapping algorithm for interferometric applications,” Opt. Lett. |

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

19. | S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. |

20. | M. D. Pritt, “Congruence in least-squares phase unwrapping,” in Proceedings Vol. II: Remote Sensing - a Scientific Vision for Sustainable Development, IGARSS '97 - 1997 International Geoscience and Remote Sensing Symposium, 1997), pp.875–877. |

21. | J. Strand and T. Taxt, “Performance evaluation of two-dimensional phase unwrapping algorithms,” Appl. Opt. |

22. | E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. |

23. | J. Parkhurst, G. Price, P. Sharrock, and C. Moore, “Phase unwrapping algorithms for use in a true real-time optical body sensor system for use during radiotherapy,” Appl. Opt. |

24. | J. F. Weng and Y. L. Lo, “Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities,” Opt. Express |

25. | 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 |

26. | Y. H. Huang, F. Janabi-Sharifi, Y. Liu, and Y. Y. Hung, “Dynamic phase measurement in shearography by clustering method and Fourier filtering,” Opt. Express |

27. | Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. |

28. | Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. |

29. | A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. |

30. | Y. H. Huang, S. Y. Hung, F. Janabi-Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Lasers Eng. |

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

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

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

(280.6730) Remote sensing and sensors : Synthetic aperture radar

(100.5088) Image processing : Phase unwrapping

(120.6165) Instrumentation, measurement, and metrology : Speckle interferometry, metrology

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 23, 2012

Revised Manuscript: May 23, 2012

Manuscript Accepted: June 1, 2012

Published: June 11, 2012

**Citation**

Howard Y. H. Huang, L. Tian, Z. Zhang, Y. Liu, Z. Chen, and G. Barbastathis, "Path-independent phase unwrapping using phase gradient and total-variation (TV) denoising," Opt. Express **20**, 14075-14089 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14075

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### References

- Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett.36(4), 526–528 (2011). [CrossRef] [PubMed]
- E. Volkl, L. F. Allard, and D. C. Joy, eds., Introduction to Electron holography (Plenum, 1999).
- B. Osmanoglu, T. H. Dixon, S. Wdowinski, and E. Cabral-Cano, “On the importance of path for phase unwrapping in synthetic aperture radar interferometry,” Appl. Opt.50(19), 3205–3220 (2011). [CrossRef] [PubMed]
- J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging27(9), 1293–1301 (2009). [CrossRef] [PubMed]
- C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng.43(5), 1152–1159 (2004). [CrossRef]
- D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, (John Wiley & Sons, 1998).
- R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988). [CrossRef]
- Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE2003, 254–263 (1993). [CrossRef]
- C. Prati, M. Giani, and N. Leuratti, “SAR interferometry: a 2-D phase unwrapping technique based on phase and absolute values information,” in Proceedings of the 1990 International Geoscience and Remote Sensing Symposium (IEEE, 1990), pp. 2043–2046.
- T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A14(10), 2692–2702 (1997). [CrossRef]
- I. Shalem and I. Yavneh, “A multilevel graph algorithm for two dimensional phase unwrapping,” Comput. Vis. Sci.11(2), 89–100 (2008). [CrossRef]
- T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express16(10), 6985–6998 (2008). [CrossRef] [PubMed]
- M. Gdeisat, M. Arevalillo-Herráez, D. Burton, and F. Lilley, “Three-dimensional phase unwrapping using the Hungarian algorithm,” Opt. Lett.34(19), 2994–2996 (2009). [CrossRef] [PubMed]
- S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express18(2), 560–565 (2010). [CrossRef] [PubMed]
- J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt.47(29), 5358–5369 (2008). [CrossRef] [PubMed]
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