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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2556–2561
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Fast two-dimensional simultaneous phase unwrapping and low-pass filtering

Miguel A. Navarro, Julio C. Estrada, M. Servin, Juan A. Quiroga, and Javier Vargas  »View Author Affiliations


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

2. One-dimensional recursive filters for phase unwrapping

3. Two-dimensional recursive filter construction for phase unwrapping

Previously [9

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]

], we used sets to describe our recursive phase unwrapping system. For convenience, here we change the notation for an easier to read. Thus, our recursive system in Eq. (5) may be seeing as a sum of two terms, where the first term will be a predictor and the second a corrector. This Predictor and corrector concepts are actually very used in estimation theory, for example, in Kalman processing systems [13

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]

]. The phase predictor is an estimation based on the previously unwrapped values, while the corrector adds input data to correct the current estimation. Thus, our recursive system in 2D has the following general form:
ϕ^(x,y)=ϕ^p(x,y)+τϕ^c(x,y),
(7)
where ϕ̂(x,y) is the 2D unwrapped pixel at (x,y), ϕ̂p(x, y) is the 2D phase predictor, and ϕ̂c(x, y) is the 2D corrector. Parameter τ controls the bandwidth of the system.

3.1. The predictor

A recursive filter takes information only from its previously estimated values as predictor. Therefore, for our predictor ϕ̂p(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:
ϕ^p(x,y)=1S1m=11m=11ϕ^(xm,yn)s(m,n).
(8)
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, ‖S1 is the L1 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
S=(100000000),S=(111100000)andS=(111101111),
(9)
and their L1 norm is ‖S1 = 1, ‖S1 = 4 and ‖S1 = 8, respectively.

Fig. 1 In (a), (b) and (c), it is shown three different neighborhoods that we can find following a sequential scanning strategy. In (d), (e) and (f), we show their 2D power spectrums of its frequency responses for τ = 0.13. The pixel (x,y) visited is at the center, and the power spectrums are shown between the range (−π,π) in both frequency directions.

3.2. The corrector

The corrector must include previously unwrapped phase and new wrapped input to correct our estimated prediction. Besides, the corrector must remove the residues modulus 2π from phase differences as the 1D system in Eq. (5) does. Taking all these criteria into account, our 2D corrector is defined as:
ϕc(x,y)=m=11n=11W[ϕ(xm,yn)ϕ^p(x,y)]s¯(m,n)},
(10)
Where kernel is the complement of S whose elements are (m,n), in such a way that ‖S1 + ‖1 = 9. Then, (m, n) is the complement of s(m, n).

4. System stability

The recursive filter shown in Eq. (7) is applied by setting an stable value for the parameter 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 (zx1,zy1)U¯2, where Ū2 is the unit bidisc defined as
U¯2={(zx1,zy1):|zx1|1|zy1|1}.
(11)
Taking the z-transfer function of the linear system in Eq. (7) (which includes Eq. (8) and Eq. (10)) we obtain the following
H(zx,zy)=τm=11n=11zxmzyns¯(m,n)1(1τS¯1)S1m=11n=11zxmzyns(m,n).
(12)
From here, we can demonstrate that the denominator into the unit bidisc Ū2 is
|1(1τS¯1)S1m=11n=11zxmzyns(m,n)|(1|1τS¯1|).
(13)
Then, the denominator of Eq. (12) is not zero only if |1 – τ1| < 1, giving the following stability condition for τ:
2S1>τ>0.
(14)
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 ‖1 = 8 (see Fig. 1(a)). Then, in this worst case scenario the stability range is 14>τ>0.

5. Frequency response

In Eq. (12) we have shown the 2D z-transform of our recursive phase unwrapping system. Using the z-transform, the frequency response is obtained by substituting zx = eiu and zy = eiv, being 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:
H(u,v)=τ(eiu+ei(u+v)+eiv+ei(uv)+1)114(15τ)(eiu+ei(u+v)+eiv+ei(vu)).
(15)
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

Fig. 2 In (a) we have the experimental wrapped phase used as input for the phase unwrapping systems, in (b) shows the recovery phase using the Flynn’s phase unwrapping method (c) shows the recovery unwrapped phase using our proposed phase unwrapping recursive system.

The computation load to unwrap a single pixel using our recursive system requires ‖1 + 9 arithmetic operations (without taking into account the operations needed by the wrapping modulus 2π operator). Therefore, for ‖1 = 1 (Fig. 1(c)) it requires 10 operations and for ‖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 14n 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]

]. The computational time taken for our recursive phase unwrapping system was 0.01 seconds, while the computational time taken by the Flynn’s algorithm was 4.48 seconds. Both algorithms were programmed in C -language and compiled in a 64bit CPU.

7. Conclusions and commentaries

Because some phase differences in Eq. (10) are taken at a distances of 2 pixels, our system is theoretically limited to spatial frequencies lower than 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. 21, 2470 (1982). [CrossRef] [PubMed]

2.

D. C. Ghihlia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

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 11107–117 (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]

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]

10.

B. Jähne, Digital Image Processing (Springer, 2005).

11.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles, Algorothims, and Applications, 3rd ed. (Prentice-Hall, October5, 1995).

12.

W.-S. Lu and A. Antoniou, Two-Dimensional Digital Filters (Marcel Dekker, Inc., 1992).

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]

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

  1. K. Itoh, “Analysis of the phase unwrapping algorithm.” Appl. Opt.21, 2470 (1982). [CrossRef] [PubMed]
  2. D. C. Ghihlia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  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. A11107–117 (1994). [CrossRef]
  5. J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A12, 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]
  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. A14, 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. Express19, 5126–5133 (2011). [CrossRef] [PubMed]
  10. B. Jähne, Digital Image Processing (Springer, 2005).
  11. J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles, Algorothims, and Applications, 3rd ed. (Prentice-Hall, October5, 1995).
  12. W.-S. Lu and A. Antoniou, Two-Dimensional Digital Filters (Marcel Dekker, Inc., 1992).
  13. 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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