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 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156 (1982).
    [Crossref]
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693 (1974).
    [Crossref] [PubMed]
  3. Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Optics And Lasers In Engineering 45, 304–317 (2007).
    [Crossref]
  4. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17, 21867–21881 (2009).
    [Crossref] [PubMed]
  5. L. N. Mertz, “Speckle imaging, photon by photon,” Appl. Opt. 18, 611–614 (1979).
    [Crossref] [PubMed]
  6. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [Crossref]
  7. K. A. Stetson, J. Wahid, and P. Gauthier , “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830–4838 (1997).
    [Crossref] [PubMed]
  8. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470–2470 (1982).
    [Crossref] [PubMed]
  9. T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Optics and Lasers in Engineering 21, 199 – 239 (1994).
    [Crossref]
  10. D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping; Theory, Algorithms, and Software (Wiley-Interscience, 1998).
    [PubMed]
  11. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [Crossref]
  12. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [Crossref] [PubMed]
  13. 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]
  14. J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [Crossref]
  15. J. L. Marroquin, M. Tapia, R. Rodriguez-Vera, and M. Servin, “Parallel algorithms for phase unwrapping based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
    [Crossref]
  16. K. M. Hung and T. Yamada, “Phase unwrapping by regions using least-squares approach,” Optical Engineering 37, 2965–2970 (1998).
    [Crossref]
  17. V. V. Volkov and Y. Zhu, “Deterministic phase unwrapping in the presence of noise,” Opt. Lett. 28, 2156–2158 (2003).
    [Crossref] [PubMed]
  18. 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]
  19. J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles, Algorithms, and Applications (Prentice-Hall, October5, 1995), 3rd ed.

2009 (1)

2007 (1)

Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Optics And Lasers In Engineering 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,” Optical Engineering 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,” Optics and Lasers in Engineering 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,” Optics and Lasers in Engineering 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,” Optical Engineering 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,” Optics and Lasers in Engineering 21, 199 – 239 (1994).
[Crossref]

Kemao, Q.

Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Optics And Lasers In Engineering 45, 304–317 (2007).
[Crossref]

Kobayashi, S.

Malacara, D.

Manolakis, D. G.

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

Marroquin, J. L.

Mastin, G. A.

Mertz, L. N.

Pritt, M. D.

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

Proakis, J. G.

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

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,” Optical Engineering 37, 2965–2970 (1998).
[Crossref]

Zhu, Y.

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

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

Opt. Express (1)

Opt. Lett. (1)

Optical Engineering (1)

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

Optics And Lasers In Engineering (1)

Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Optics And Lasers In Engineering 45, 304–317 (2007).
[Crossref]

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

Other (2)

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

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

Equations on this page are rendered with MathJax. Learn more.

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 .

Metrics