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.

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References

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  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]

2011

1999

1997

1995

1994

1988

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

1982

1961

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

Antoniou, A.

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

Bucy, R. S.

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

Cuevas, F. J.

Estrada, J. C.

Flynn, T. J.

Ghihlia, D. C.

Goldstein, R.

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

Itoh, K.

Jähne, B.

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

Judge, T.

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

Kalman, R. E.

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

Lu, W.-S.

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

Malacara, D.

Manolakis, D. G.

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

Marroquin, J. L.

Pritt, M. D.

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

Proakis, J. G.

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

Quiroga, J. A.

Rivera, M.

Rodriguez-Vera, R.

Romero, L. A.

Servin, M.

Werner, C.

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

Zebker, H.

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

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lasers Eng.

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

Radio Sci.

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

Trans. ASME, Ser. D

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

Other

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

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).

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Figures (2)

Fig. 1
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.

Fig. 2
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.

Equations (15)

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ϕ ^ ( n ) = ( 1 τ ) ϕ ^ ( n 1 ) + τ ϕ ( n )
H ( ω ) = τ 1 ( 1 τ ) e i ω , and h ( n ) = τ ( 1 τ ) n , for n 0 .
ϕ ^ ( n ) = ϕ ^ ( n 1 ) + τ [ ϕ ( n ) ϕ ^ ( n 1 ) ] .
ϕ ^ ( n ) = ϕ ^ ( n 1 ) + W [ ϕ ( n ) ϕ ( n 1 ) ] .
ϕ ^ ( n ) = ϕ ^ ( n 1 ) + τ W [ ϕ ( n ) ϕ ^ ( n 1 ) ] .
W [ ϕ ( n ) ϕ ^ ( n 1 ) ] = ϕ ( n ) ϕ ^ ( n 1 ) for | ϕ ( n ) ϕ ^ ( n 1 ) | π .
ϕ ^ ( x , y ) = ϕ ^ p ( x , y ) + τ ϕ ^ c ( x , y ) ,
ϕ ^ p ( x , y ) = 1 S 1 m = 1 1 m = 1 1 ϕ ^ ( x m , y n ) s ( m , n ) .
S = ( 1 0 0 0 0 0 0 0 0 ) , S = ( 1 1 1 1 0 0 0 0 0 ) and S = ( 1 1 1 1 0 1 1 1 1 ) ,
ϕ c ( x , y ) = m = 1 1 n = 1 1 W [ ϕ ( x m , y n ) ϕ ^ p ( x , y ) ] s ¯ ( m , n ) } ,
U ¯ 2 = { ( z x 1 , z y 1 ) : | z x 1 | 1 | z y 1 | 1 } .
H ( z x , z y ) = τ m = 1 1 n = 1 1 z x m z y n s ¯ ( m , n ) 1 ( 1 τ S ¯ 1 ) S 1 m = 1 1 n = 1 1 z x m z y n s ( m , n ) .
| 1 ( 1 τ S ¯ 1 ) S 1 m = 1 1 n = 1 1 z x m z y n s ( m , n ) | ( 1 | 1 τ S ¯ 1 | ) .
2 S 1 > τ > 0 .
H ( u , v ) = τ ( e i u + e i ( u + v ) + e i v + e i ( u v ) + 1 ) 1 1 4 ( 1 5 τ ) ( e i u + e i ( u + v ) + e i v + e i ( v u ) ) .

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