## Frequency guided methods for demodulation of a single fringe pattern

Optics Express, Vol. 17, Issue 17, pp. 15118-15127 (2009)

http://dx.doi.org/10.1364/OE.17.015118

Acrobat PDF (647 KB)

### Abstract

Phase demodulation from a single fringe pattern is a challenging task but of interest. A frequency-guided regularized phase tracker and a frequency-guided sequential demodulation method with Levenberg-Marquardt optimization are proposed to demodulate a single fringe pattern. Demodulation path guided by the local frequency from the highest to the lowest is applied in both methods. Since critical points have low local frequency values, they are processed last so that the spurious sign problem caused by these points is avoided. These two methods can be considered as alternatives to the effective fringe follower regularized phase tracker. Demodulation results from one computer-simulated and two experimental fringe patterns using the proposed methods will be demonstrated.

© 2009 OSA

## 1. Introduction

*x, y*) is the pixel coordinate,

*f*(

*x, y*),

*a*(

*x, y*),

*b*(

*x, y*) and

*ϕ*(

*x, y*) are the recorded image intensity, background intensity, fringe amplitude and phase distribution, respectively [1]. The three unknowns,

*a*(

*x, y*),

*b*(

*x, y*) and

*ϕ*(

*x, y*), make the retrieval of phase information difficult. Phase shifting technique [1] is often used for phase measurement, which requires several phase-shifted fringe patterns. Carrier method with Fourier transform is also frequently used [1] by introducing a carrier frequency into a fringe pattern. However, there are always cases where multiple fringe patterns or carrier frequencies are hard to obtain, and consequently phase demodulation of a single fringe pattern is important. Usually a high-pass filter is used to remove the background intensity [2

2. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. **36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

3. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A **18**(3), 689–695 (2001). [CrossRef]

2. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. **36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

3. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A **18**(3), 689–695 (2001). [CrossRef]

4. J. A. Quiroga and J. A. Gomez-Pedrero, “Algorithm for fringe pattern normalization,” Opt. Commun. **197**(1-3), 43–51 (2001). [CrossRef]

5. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. **45**(2), 304–317 (2007). [CrossRef]

6. J. Weickert, “Coherence-Enhancing Diffusion Filtering,” Int. J. Comput. Vis. **31**(2/3), 111–127 (1999). [CrossRef]

7. H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. **34**(8), 1141–1143 (2009). [CrossRef] [PubMed]

2. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. **36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

3. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A **18**(3), 689–695 (2001). [CrossRef]

8. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A **14**(8), 1742–1753 (1997). [CrossRef]

15. O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A **25**(6), 1361–1370 (2008). [CrossRef]

*et al*. proposed a popular demodulation method for closed fringe patterns, called the regularized phase tracker (RPT) [2

**36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

**18**(3), 689–695 (2001). [CrossRef]

*et al*. proposed an adaptive quadrature filter based on Bayesian estimation theory and complex-valued Markov random-field prior models [8

8. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A **14**(8), 1742–1753 (1997). [CrossRef]

9. J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A **15**(6), 1536–1544 (1998). [CrossRef]

10. J. C. Estrada, M. Servin, and J. L. Marroquín, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express **15**(5), 2288–2298 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2288. [CrossRef] [PubMed]

*et al*. for closed fringe patterns, which has interesting similarity with windowed Fourier ridges algorithm [5

5. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. **45**(2), 304–317 (2007). [CrossRef]

11. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A **22**(6), 1170–1172 (2005). [CrossRef]

*et al*. proposed a two dimensional Hilbert transform for fringe patterns which turns the demodulation problem into orientation estimation problem [12

12. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A **18**(8), 1862–1870 (2001). [CrossRef]

*et al*. [13

13. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A **20**(5), 925–934 (2003). [CrossRef]

14. Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. **32**(2), 127–129 (2007). [CrossRef]

*et al*. proposed a demodulation method by dividing the fringe patterns into tiles which are analyzed by an open-fringe-analysis algorithm. The phase sign is propagated and determined tile by tile so that the algorithm is more robust and faster [15

15. O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A **25**(6), 1361–1370 (2008). [CrossRef]

**18**(3), 689–695 (2001). [CrossRef]

14. Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. **32**(2), 127–129 (2007). [CrossRef]

14. Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. **32**(2), 127–129 (2007). [CrossRef]

## 2. Fringe-follower regularized phase tracker and frequency-guided sequential demodulation

### 2.1 Fringe-follower regularized phase tracker

*N*is the neighborhood region centering at (

_{xy}*x, y*) and the coordinates of the pixels in this region are denoted as (

*ε, η*);

*ϕ*(

_{0}*x, y*) is the phase estimation;

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) are the local frequency estimation in

*x*and

*y*direction, respectively;

*ϕ*(

_{e}*x, y, ε, η*) indicates the phase values of pixels in

*N*that are calculated from the phase and frequency information of pixel (

_{xy}*x, y*);

*m*(

*ε, η*) is defined to be one if a pixel has already been demodulated and zero otherwise. The first term indicates the fidelity of

*ϕ*(

_{0}*x, y*),

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*). The second term controls the smoothness of phase with its neighbors. It is often called as a regularization term, with

*λ*as the regularizing parameter. The energy function is minimized on pixel

*-*by

*-*pixel basis with respect to

*ϕ*(

_{0}*x, y*),

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*). A gradient descent method is adopted for optimization in [2

**36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

**18**(3), 689–695 (2001). [CrossRef]

*ϕ*(

_{0}*x, y*),

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) are needed for optimization.

**18**(3), 689–695 (2001). [CrossRef]

**18**(3), 689–695 (2001). [CrossRef]

5. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. **45**(2), 304–317 (2007). [CrossRef]

**36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

**18**(3), 689–695 (2001). [CrossRef]

4. J. A. Quiroga and J. A. Gomez-Pedrero, “Algorithm for fringe pattern normalization,” Opt. Commun. **197**(1-3), 43–51 (2001). [CrossRef]

### 2.2 Frequency-guided sequential demodulation

*ϕ*(

_{0}*x, y*) ∈ [0, π]. Since cosine function does not have unique inverse, the true phase, if wrapped, can be

*ϕ*(

*x, y*) =

*ϕ*(

_{0}*x, y*) or

*ϕ*(

*x, y*) = -

*ϕ*(

_{0}*x, y*), which raises the phase ambiguity problem.

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) are calculated by minimizing the following cost function:which is essentially the first term in the right hand of Eq. (3). Since

*ϕ*(

_{0}*x, y*) has already been calculated by Eq. (5), regularization term in Eq. (3) is not used in this method; only

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) need to be estimated. In the original FSD,

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) are obtained by exhaustive search within intervals

*ω*(

_{xl}*x, y*) ≤

*ω*(

_{x}*x, y*) ≤

*ω*(

_{xh}*x, y*) and

*ω*(

_{yl}*x, y*) ≤

*ω*(

_{y}*x, y*) ≤

*ω*(

_{yh}*x, y*), respectively. Candidates of

*ω*(

_{x}*x, y*) are uniformly sampled from

*ω*(lower bound) to

_{xl}*ω*(upper bound) with a sampling interval

_{xh}*ω*. Candidates of

_{xi}*ω*(

_{y}*x, y*) are generated similarly. Once

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) have been obtained, the local frequency density is calculated as

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) are spatially continuous, the sign of the phase can be determined. Details can be found in [14

**32**(2), 127–129 (2007). [CrossRef]

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*), need to be optimized, it is feasible to use exhaustive search algorithm which guarantees a global minimum. However it increases the computation cost. A faster algorithm is nevertheless expected if possible, which inspires us to adopt Levenberg-Marquardt optimization into FSD and is introduced in subsection 3.2.

## 3. Proposed methods

### 3.1 Frequency-guided regularized phase tracker

*ω*(

*x, y*) to guide the demodulation path is naturally a suitable alternative to the fringe follower. A frequency-guided regularized phase tracker (FGRPT) is proposed. As critical points have low local frequencies, high frequency points are processed first so that the critical points will be guaranteed to be processed last and therefore reduce the possibility of error propagating from critical points.

- 1) Randomly choose a pixel as a seed pixel;
- 2) As no initial values are provided for the seed pixel, FSD is used to demodulate the seed pixel: obtain the phase from Eq. (5), then obtain
*ω*(_{x}*x, y*) and*ω*(_{y}*x, y*) from Eq. (6) by exhaustive search. Compute*ω*(*x, y*) as in Eq. (7). If it is smaller than a pre-defined threshold, go back to step 1; otherwise, push the seed pixel into a demodulation register. Only one register is needed for FGRPT. - 3) Select the pixel with the highest local frequency
*ω*(*x, y*) in the demodulation register; demodulate its four adjacent pixels by estimating*ϕ*(_{0}*x, y*),*ω*(_{x}*x, y*) and*ω*(_{y}*x, y*) that minimize energy function in Eq. (3). Levenberg-Marquardt instead of gradient descent is used in this paper. Details on the Levenberg-Marquardt optimization method and its initial value setting will be discussed later. - 4) Push the processed pixels into the register and sort the pixels according to their local frequencies
*ω*(*x, y*). - 5) Repeat step 3 and 4 until all pixels are processed.

**36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

**18**(3), 689–695 (2001). [CrossRef]

*N'*is 3×3 neighbors centering at (

_{xy}*x, y*);

*m*(

*ε, η*) is the same as in Eq. (3) indicating whether this pixel has been processed;

*U*(

*ε, η*) is the energy of the pixel defined by Eq. (3). Low energy value of a pixel embodies accurate and reliable estimation of demodulation parameters. Thus low energy can be treated as high quality of demodulation and thus a high weight is assigned to this pixel. The same idea is applied to

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*). The denominator in Eq. (8) is used for normalization purpose.

*m*(

*x, y*) = 1 for all pixels, a better estimation of initial condition can be made, which may lead to better optimization results. The partial refinement is applicable to both FFRPT and FGRPT. However, in all our examples presented in Section 4, the refinement is not applied.

### 3.2 Frequency-guided sequential demodulation - Levenberg Marquardt

*ϕ*(

_{0}*x, y*),

*ω*(

_{x}*x, y*) and

*ω*(

_{y}*x, y*) to two for optimization. The regularization term is not included in the energy function in Eq. (6). Thus the optimization problem is simpler. As mentioned earlier, an exhaustive search is used in original FSD. In order to increase the optimization speed, the success of using the Levenberg-Marquardt optimization in FGRPT in Sec. 3.1 suggests that the Levenberg-Marquardt optimization can also be adopted for FSD to replace the exhaustive search, which becomes the proposed frequency-guided sequential demodulation - Levenberg Marquardt (FSD-LM) method.

**45**(2), 304–317 (2007). [CrossRef]

## 4. Results and discussions

*N*and a value of 0.3 for

_{xy}*λ*are used for all the three examples.

*ω*(

*x, y*) estimated using FGRPT, which is used to guide the demodulation process. The frequency is estimated along with phase demodulation, as indicated in the algorithm outline given in Subsection 3.1. Figures 2(b) and 2(c) are two snapshots showing the sequences of the FGRPT demodulation. From the snapshots, the critical points are well avoided during the processing and guaranteed to be processed later than their surroundings. Figure 2(d) is the phase obtained by FGRPT. Figure 2(e) is the estimated frequency using FSD-LM method. Figure 2(f) is the phase result of FSD-LM. The demodulation sequence for FSD-LM is almost the same as Fig. 2(b) and 2(c). Both methods guarantee that the critical points are processed last and they successfully demodulate the fringe pattern. Thus they can be alternative methods to FFRPT. FGRPT consumes similar computation time as FFRPT while FSD-LM is faster than FFRPT and FGRPT. In detail, for the image of Fig. 1(a) of size 256×256 processed in Intel® Xeon® quad-core CPUs of 2.5GHz main frequency using C++ programming, FSD-LM needs about 13s while FSD needs 276s. Addition 7s is needed if windowed Fourier filtering is applied to post-process the obtained phase for both FSD and FSD-LM methods. FFRPT needs about 32s and FGRPT needs about 34s. The mean square errors for the cosine function of the retrieved phase are 0.0125, 0.0097, and 0.0095 for FGRPT, FSD-LM and FSD, respectively. The phase result of FSD is very similar to FSD-LM therefore not shown here. Though FSD-LM does not use exhaustive search, it yields very similar results as FSD and performs 21 times faster.

6. J. Weickert, “Coherence-Enhancing Diffusion Filtering,” Int. J. Comput. Vis. **31**(2/3), 111–127 (1999). [CrossRef]

7. H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. **34**(8), 1141–1143 (2009). [CrossRef] [PubMed]

4. J. A. Quiroga and J. A. Gomez-Pedrero, “Algorithm for fringe pattern normalization,” Opt. Commun. **197**(1-3), 43–51 (2001). [CrossRef]

*ω*(

*x, y*) estimated by FGRPT which is used to guide the demodulation process. The phase result is shown in Fig. 3(d). Figure 3(e) is the cosine value of Fig. 3(d). Figure 3(f) shows the frequency value obtained by FSD-LM. Figure 3(g) shows the phase result using FSD-LM method and Fig. 3(h) is the cosine value of Fig. 3(g). The results by both FGRPT and FSD-LM are satisfactory. The quantitative data are presented in Table 1 .

6. J. Weickert, “Coherence-Enhancing Diffusion Filtering,” Int. J. Comput. Vis. **31**(2/3), 111–127 (1999). [CrossRef]

7. H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. **34**(8), 1141–1143 (2009). [CrossRef] [PubMed]

**36**(19), 4540–4548 (1997). [CrossRef] [PubMed]

**18**(3), 689–695 (2001). [CrossRef]

*ω*(

*x, y*) estimated by FGRPT. The phase result is shown in Fig. 4(d). Figure 4(e) is the cosine value of Fig. 4(d). Figure 4(f) shows the frequency value obtained by FSD-LM applied to Fig. 4(b). Figure 4(g) shows the phase result using FSD-LM method and Fig. 4(h) is the cosine value of Fig. 4(g). Again, the results by both methods are satisfactory. The quantitative data are also presented in Table 1.

## 5. Conclusion

## References and links

1. | D. W. Robinson, and G. T. Reid, eds., in |

2. | M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. |

3. | M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A |

4. | J. A. Quiroga and J. A. Gomez-Pedrero, “Algorithm for fringe pattern normalization,” Opt. Commun. |

5. | Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. |

6. | J. Weickert, “Coherence-Enhancing Diffusion Filtering,” Int. J. Comput. Vis. |

7. | H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. |

8. | J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A |

9. | J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A |

10. | J. C. Estrada, M. Servin, and J. L. Marroquín, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express |

11. | M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A |

12. | K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A |

13. | M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A |

14. | Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. |

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

16. | W. H. Press, S. A. Teukolsky, and W. T. Vetterling, B. P. Flannery, in |

17. | D. C. Ghiglia, and M. D. Pritt, in |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Image Processing

**History**

Original Manuscript: May 28, 2009

Revised Manuscript: August 6, 2009

Manuscript Accepted: August 6, 2009

Published: August 11, 2009

**Citation**

Haixia Wang and Qian Kemao, "Frequency guided methods for demodulation of a single fringe pattern," Opt. Express **17**, 15118-15127 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-15118

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### References

- D. W. Robinson, and G. T. Reid, eds., in Interferogram analysis: digital fringe pattern measurement techniques, (Bristol, England: Institute of Physics1993).
- M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997). [CrossRef] [PubMed]
- M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001). [CrossRef]
- J. A. Quiroga and J. A. Gomez-Pedrero, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001). [CrossRef]
- Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007). [CrossRef]
- J. Weickert, “Coherence-Enhancing Diffusion Filtering,” Int. J. Comput. Vis. 31(2/3), 111–127 (1999). [CrossRef]
- H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. 34(8), 1141–1143 (2009). [CrossRef] [PubMed]
- J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14(8), 1742–1753 (1997). [CrossRef]
- J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15(6), 1536–1544 (1998). [CrossRef]
- J. C. Estrada, M. Servin, and J. L. Marroquín, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express 15(5), 2288–2298 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2288 . [CrossRef] [PubMed]
- M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22(6), 1170–1172 (2005). [CrossRef]
- K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001). [CrossRef]
- M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003). [CrossRef]
- Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32(2), 127–129 (2007). [CrossRef]
- O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008). [CrossRef]
- W. H. Press, S. A. Teukolsky, and W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C: The Art of Scientific Computing (Second Edition), (Cambridge University Press, 2002), pp. 683–685.
- D. C. Ghiglia, and M. D. Pritt, in Two-Dimensional Phase Unwrapping: Theory, Algorithm and Software, (John Wiley & Sons, Inc, 1998).

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