Abstract

Phase unwrapping techniques remove the modulus 2π ambiguities of wrapped phase maps. The present work shows a first-order feedback system for phase unwrapping and smoothing. This system is a fast phase unwrapping system which also allows filtering some noise since in deed it is an Infinite Impulse Response (IIR) low-pass filter. In other words, our system is capable of low-pass filtering the wrapped phase as the unwrapping process proceeds. We demonstrate the temporal stability of this unwrapping feedback system, as well as its low-pass filtering capabilities. Our system even outperforms the most common and used unwrapping methods that we tested, such as the Flynn’s method, the Goldstain’s method, and the Ghiglia least-squares method (weighted or unweighted). The comparisons with these methods show that our system filters-out some noise while preserving the dynamic range of the phase-data. Its application areas may cover: optical metrology, synthetic aperture radar systems, magnetic resonance, and those imaging systems where information is obtained as a demodulated wrapped phase map.

© 2011 Optical Society of America

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References

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2009 (1)

2007 (1)

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

2003 (1)

1999 (1)

1998 (1)

K. M. Hung, and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

1997 (1)

1995 (2)

1994 (2)

D. C. Ghiglia, 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]

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

1989 (1)

1987 (1)

1982 (2)

1979 (2)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Bryanston-Cross, P. J.

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

Cuevas, F. J.

Estrada, J. C.

Gallagher, J. E.

Gauthier, P.

Ghiglia, D. C.

Herriott, D. R.

Hung, K. M.

K. M. Hung, and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

Hunt, B. R.

Huntley, J. M.

Ina, H.

Itoh, K.

Judge, T. R.

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

Kemao, Q.

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

Kobayashi, S.

Malacara, D.

Marroquin, J. L.

Mastin, G. A.

Mertz, L. N.

Quiroga, J. A.

Rivera, M.

Rodriguez-Vera, R.

Romero, L. A.

Rosenfeld, D. P.

Servin, M.

Stetson, K. A.

Takeda, M.

Tapia, M.

Volkov, V. V.

Wahid, J.

White, A. D.

Yamada, T.

K. M. Hung, and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

Zhu, Y.

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

K. M. Hung, and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (2)

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

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

Opt. Lett. (1)

Other (2)

. J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles, Algorothims, ans Applications (Prentice-Hall, October 5, 1995), 3rd ed.

D. C. Ghiglia, and M. D. Pritt, Two-dimensional Phase Unwrapping; Theory, Algoritms, and Software (Wiley-Interscience, 1998).
[PubMed]

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Figures (5)

Fig. 1
Fig. 1

Block diagram of the first order phase unwrapping system (4). The block system z−1 delays its input one sample.

Fig. 2
Fig. 2

Impulse response, and frequency response of the dynamic phase unwrapping system (4), respectively.

Fig. 3
Fig. 3

Particular 2D graphic representation of neighborhood Ω around the site r that is going to be unwrapped.

Fig. 4
Fig. 4

Phase unwrapping of simulated data. A) is the simulated wrapped phase, B) the simulated wrapped phase plus noise with normal distribution, mean zero and variance of 1.3 radians, C) is the unwrapped surface with the herein proposed method, D) with the Flynn’s method, E) with the Goldstein’s method, and F) with the leas-squares method. To locate the dynamic range, bellow each unwrapped surface is projected the image corresponding to the modulus 2π wrapped phase of each unwrapped phase surface.

Fig. 5
Fig. 5

Test using an experimentally obtained phase map. The phase map dimensions are 512 × 512. A) shows the experimental phase map image, B) the unwrapped phase surface obtained with our dynamic phase unwrapping system, and C) with the Flynn’s method.

Equations (11)

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ϕ ^ ( t ) = ϕ ^ ( t 0 ) + t 0 t ϕ w ( x ) d x .
d ϕ ^ ( t ) d t = W [ d ϕ w ( t ) d t ] ,
ϕ ^ ( n ) = ϕ ^ ( n 1 ) + W [ ϕ w ( n ) ϕ w ( n 1 ) ] , n = 0 , 1 , 2 ,
ϕ ^ ( n ) = ϕ ^ ( n 1 ) + τ W [ ϕ w ( n ) ϕ ^ ( n 1 ) ] ,
W [ ϕ w ( n ) ϕ ^ ( n ) ] = ϕ w ( n ) ϕ ^ ( n ) , for | ϕ w ( n ) ϕ ^ ( n ) | < π
ϕ ^ ( n ) ( 1 τ ) ϕ ^ ( n 1 ) τ ϕ w ( n ) = 0 .
ϕ ^ ( n ) = ( 1 τ ) n + 1 ϕ ^ ( 1 ) + τ k = 0 n ( 1 τ ) k ϕ w ( k )
h ( n ) = τ ( 1 τ ) n u ( n ) ,
n = | h ( n ) | = τ n = 0 ( 1 τ ) n = 1 , τ < 1 .
H ( ω ) = { h ( n ) } = τ e i ω e i ω ( 1 τ ) ,
ϕ ^ ( r ) = 1 | m ( r ) | i = 0 | m ( r ) | 1 { ϕ ^ ( r i ) + τ W [ ϕ w ( r ) ϕ ^ ( r i ) ] } , r i m ( r ) Ω .

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