## Fast two-dimensional simultaneous phase unwrapping and low-pass filtering |

Optics Express, Vol. 20, Issue 3, pp. 2556-2561 (2012)

http://dx.doi.org/10.1364/OE.20.002556

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### Abstract

Here, we present a fast algorithm for two-dimensional (2D) phase unwrapping which behaves as a recursive linear filter. This linear behavior allows us to easily find its frequency response and stability conditions. Previously, we published a robust to noise recursive 2D phase unwrapping system with smoothing capabilities. But our previous approach was rather heuristic in the sense that not general 2D theory was given. Here an improved and better understood version of our previous 2D recursive phase unwrapper is presented. In addition, a full characterization of it is shown in terms of its frequency response and stability. The objective here is to extend our previous unwrapping algorithm and give a more solid theoretical foundation to it.

© 2012 OSA

## 1. Introduction

*π*discontinuities of wrapped phase maps. Itoh provided a simple analysis of the one-dimensional phase unwrapping algorithm and showed that the true phase can be recovered by line integration of its wrapping differences [1

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

*π*discontinuities of the wrapped phase map in the spatial domain [2]. The simplest unwrapping system integrates the wrapped phase differences on a line by line basis. However, in noisy conditions, erroneous phase unwrapped values may propagate outside from high noise regions corrupting the rest of the unwrapped phase [3

3. T. Judge, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. **21**, 199–239 (1994). [CrossRef]

4. D. C. Ghihlia 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]

5. J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A **12**, 2393–2400 (1995). [CrossRef]

6. 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]

4. D. C. Ghihlia 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]

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

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

9. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express **19**, 5126–5133 (2011). [CrossRef] [PubMed]

9. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express **19**, 5126–5133 (2011). [CrossRef] [PubMed]

## 2. One-dimensional recursive filters for phase unwrapping

9. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express **19**, 5126–5133 (2011). [CrossRef] [PubMed]

*ϕ̂*(

*n*) is the filters output at the current

*n*-site,

*ϕ̂*(

*n*– 1) is the previous output (or estimate), and

*τ*is a parameter that controls the cut-off frequency of the filter [10–12]. The frequency and impulse response of this linear low-pass filter are: This linear system is stable if its impulse response is absolutely summable. Thus, the system in Eq. (1) is stable only for

*τ*∈ (0,2) [10,11]. To transform Eq. (1) as a phase unwrapping system, let us rewrite it as: Apart from

*τ*, and the previous estimation

*ϕ̂*(

*n*– 1) in the phase difference, Eq. (3) looks very similar to the following basic line unwrapping algorithm [2]: where

*ϕ*(

*n*) is the input (wrapped phase) and

*ϕ̂*(

*n*) the output (unwrapped phase). This important observation was firstly given in Ref. [9

**19**, 5126–5133 (2011). [CrossRef] [PubMed]

*W*[·] is the wrapping modulus 2

*π*operator. As the input is wrapped, the phase difference has a principal value plus a residue multiple of 2

*π*[1

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

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

*ϕ*(

*n*) in Eq. (3), it has a principal value plus a residue multiple of 2

*π*. Then, using the wrapping operator in Eq. (3) we can obtain the unwrapping phase as the low-pass filtering of its principal values. Thus, the recursive first order unwrapper system is: In brief, the transition from a recursive linear low-pass filter and a low-pass filtered phase unwrapper is based on the following property of the unwrapping operator: Thus, for synthesis and analysis purposes, the wrapping operator is not taken into account and the unwrapping system is treated as a linear Infinite Impulse Response (IIR) recursive filter.

## 3. Two-dimensional recursive filter construction for phase unwrapping

**19**, 5126–5133 (2011). [CrossRef] [PubMed]

13. R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME, Ser. D **83**, 95–107 (1961). [CrossRef]

*ϕ̂*(

*x,y*) is the 2D unwrapped pixel at (

*x,y*),

*ϕ̂*(

_{p}*x, y*) is the 2D phase predictor, and

*ϕ̂*(

_{c}*x, y*) is the 2D corrector. Parameter

*τ*controls the bandwidth of the system.

### 3.1. The predictor

*ϕ̂*(

_{p}*x, y*), we propose the mean of the 3 × 3 neighborhood of unwrapped-only pixels around pixel (

*x,y*), marked by the indicators

*s*(

*m,n*) as follows: This is a convolution of

*ϕ̂*(

*x,y*) with an adapting kernel

*S*whose elements are

*s*(

*m,n*). The elements

*s*(

*m,n*) indicates with ones the unwrapped pixels and with zeros the wrapped ones. Finally, ‖

*S*‖

_{1}is the

*L*

^{1}norm of

*S; i.e.*the sum of its elements. The kernel

*S*is adapted for each visited pixel (

*x,y*) being unwrapped, as illustrated in Fig. 1. The Panels (a), (b) and (c) shows three possible neighborhood configurations found in a sequential scanning around (

*x,y*), Fig. 1(a) presents a single previously unwrapped pixel, Fig. 1(b) presents 4 unwrapped pixels and Fig. 1(c) presents 8. For these cases, the kernel is adapted as and their

*L*

^{1}norm is ‖

*S*‖

_{1}= 1, ‖

*S*‖

_{1}= 4 and ‖

*S*‖

_{1}= 8, respectively.

### 3.2. The corrector

*π*from phase differences as the 1D system in Eq. (5) does. Taking all these criteria into account, our 2D corrector is defined as: Where kernel

*S̄*is the complement of

*S*whose elements are

*s̄*(

*m,n*), in such a way that ‖

*S*‖

_{1}+ ‖

*S̄*‖

_{1}= 9. Then,

*s̄*(

*m, n*) is the complement of

*s*(

*m, n*).

## 4. System stability

*τ*, and visiting each pixel following a predefined scanning strategy. For any bounded input, an stable recursive filter must obtain a bounded output. For this analysis we consider the system’s corrector in Eq. (10) without the wrapping operator, as shown in Eq. (6). In 2D, one uses the stability Shanks stability theorem which says that a 2D recursive filter is stable if the denominator of its

*z*-transform is non zero for

*Ū*

^{2}is the unit bidisc defined as Taking the

*z*-transfer function of the linear system in Eq. (7) (which includes Eq. (8) and Eq. (10)) we obtain the following From here, we can demonstrate that the denominator into the unit bidisc

*Ū*

^{2}is Then, the denominator of Eq. (12) is not zero only if |1 –

*τ*‖

*S̄*‖

_{1}| < 1, giving the following stability condition for

*τ*: To guarantee that our 2D unwrapping system is stable in all neighborhood cases found in a sequential scanning, we choose our worst configuration case which is ‖

*S̄*‖

_{1}= 8 (see Fig. 1(a)). Then, in this worst case scenario the stability range is

## 5. Frequency response

*z*-transform of our recursive phase unwrapping system. Using the

*z*-transform, the frequency response is obtained by substituting

*z*=

_{x}*e*and

^{iu}*z*=

_{y}*e*, being

^{iv}*u*and

*v*the spatial frequencies along the

*x*and

*y*axis. Figure 1 shows three possible neighborhood cases found in a sequentially scanning strategy. As example, we are going to obtain the frequency response of the most probable case, which is the one shown in panel 1(b). For this case, its 2D frequency response is: For illustration purposes, Figs. 1(d), 1(e) and 1(f), shows the 2D graphic of the power spectrum of the frequency response corresponding to each presented case. To obtain those graphics, first obtain its corresponding frequency response from its

*z*-transform Eq. (12), using

*τ*= 0.13.

## 6. Test and results

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

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

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

**19**, 5126–5133 (2011). [CrossRef] [PubMed]

*τ*= 0.013. As we can see in Fig. 2(c), our proposed 2D recursive phase unwrapping system obtains a cleaner unwrapped phase, whereas the Flynn’s unwrapped phase is fare more noised and damaged.

*S̄*‖

_{1}+ 9 arithmetic operations (without taking into account the operations needed by the wrapping modulus 2

*π*operator). Therefore, for ‖

*S̄*‖

_{1}= 1 (Fig. 1(c)) it requires 10 operations and for ‖

*S̄*‖

_{1}= 8 (Fig. 1(a)) it requires 17 operations. However our most probable case (Fig. 1(b)) requires 14 operations. Thus, taking the most probable case as the average, to unwrap a phase map of

*n*=

*M*×

*N*pixels one needs an average of 14

*n*operations. In this way, compared with the Flynn’s method (and the least-squares methods), our recursive phase unwrapping system is faster than the Flynn’s method, because the Flynn’s method requires far more operations to unwrap a 2D phase map, as you can see in Ref. [8

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

## 7. Conclusions and commentaries

**19**, 5126–5133 (2011). [CrossRef] [PubMed]

**19**, 5126–5133 (2011). [CrossRef] [PubMed]

**19**, 5126–5133 (2011). [CrossRef] [PubMed]

*π*/2 radians. However, in practice our recursive phase unwrapping system has not this limit due to the recursive inertia of the previously unwrapped pixels already processed.

## References and links

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

2. | D. C. Ghihlia and M. D. Pritt, |

3. | T. Judge, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. |

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

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

6. | 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. |

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

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

9. | J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express |

10. | B. Jähne, |

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

12. | W.-S. Lu and A. Antoniou, |

13. | R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME, Ser. D |

**OCIS Codes**

(100.2650) Image processing : Fringe analysis

(100.5088) Image processing : Phase unwrapping

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 9, 2011

Revised Manuscript: December 2, 2011

Manuscript Accepted: January 4, 2012

Published: January 20, 2012

**Citation**

Miguel A. Navarro, Julio C. Estrada, M. Servin, Juan A. Quiroga, and Javier Vargas, "Fast two-dimensional simultaneous phase unwrapping and low-pass filtering," Opt. Express **20**, 2556-2561 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2556

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### References

- K. Itoh, “Analysis of the phase unwrapping algorithm.” Appl. Opt.21, 2470 (1982). [CrossRef] [PubMed]
- D. C. Ghihlia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
- T. Judge, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng.21, 199–239 (1994). [CrossRef]
- D. C. Ghihlia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A11107–117 (1994). [CrossRef]
- J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A12, 2393–2400 (1995). [CrossRef]
- 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]
- R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci.23, 713–720 (1988). [CrossRef]
- T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A14, 2692–2701 (1997). [CrossRef]
- J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express19, 5126–5133 (2011). [CrossRef] [PubMed]
- B. Jähne, Digital Image Processing (Springer, 2005).
- J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles, Algorothims, and Applications, 3rd ed. (Prentice-Hall, October5, 1995).
- W.-S. Lu and A. Antoniou, Two-Dimensional Digital Filters (Marcel Dekker, Inc., 1992).
- R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME, Ser. D83, 95–107 (1961). [CrossRef]

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