Abstract

Phase unwrapping continues to be an important step in those techniques that obtain the phase from Fourier transforms. We propose a fast two-dimensional phase-unwrapping algorithm that has been specially designed to be used as part of an iterative algorithm. It can be used also as a final step of a phase retrieval process with other unwrapping techniques. The algorithm consists of a modal least-squares estimation of the wrapped phase by using as inputs to the linear estimation the derivative of the wrapped phase. A theoretical description of the method, simulations, and experimental validations are presented.

© 2003 Optical Society of America

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References

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  1. J. H. Massing, J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: Theory and experimental test,” Appl. Opt. 40, 2081–2088 (2001).
    [CrossRef]
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    [CrossRef]
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  4. V. V. Voitsekhovich, S. Bará, S. Rios, E. Costa, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
    [CrossRef]
  5. M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  6. X. Yuan et al., “Proposed algorithm for phase unwrapping,” Appl. Opt. 41, 7422–7428 (2002).
    [CrossRef]
  7. R. C. Gonzalez, R. E. Woods, Tratamiento Computacional de Imágenes (Addison-Wesley, Reading, Mass., 1996).

2002 (1)

2001 (1)

1998 (1)

V. V. Voitsekhovich, S. Bará, S. Rios, E. Costa, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

1990 (1)

1982 (1)

Bará, S.

V. V. Voitsekhovich, S. Bará, S. Rios, E. Costa, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Costa, E.

V. V. Voitsekhovich, S. Bará, S. Rios, E. Costa, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Fienup, J. R.

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Tratamiento Computacional de Imágenes (Addison-Wesley, Reading, Mass., 1996).

Heppner, J.

Marron, J. C.

Massing, J. H.

Rios, S.

V. V. Voitsekhovich, S. Bará, S. Rios, E. Costa, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Sanchez, P. P.

Sullivan, R. C.

Voitsekhovich, V. V.

V. V. Voitsekhovich, S. Bará, S. Rios, E. Costa, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Tratamiento Computacional de Imágenes (Addison-Wesley, Reading, Mass., 1996).

Yuan, X.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

V. V. Voitsekhovich, S. Bará, S. Rios, E. Costa, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Other (2)

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

R. C. Gonzalez, R. E. Woods, Tratamiento Computacional de Imágenes (Addison-Wesley, Reading, Mass., 1996).

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Figures (7)

Fig. 1
Fig. 1

(a) Three-dimensional vision of the continuous phase, (b) profiles of the continuous phase (solid curve), the corresponding wrapped phase (dotted and dashed curve), and the introduced 2π jumps (dotted curve).

Fig. 2
Fig. 2

Block diagram of the unwrapping algorithm.

Fig. 3
Fig. 3

(a) Three-dimensional vision of the 2π jumps, (b) profiles of the wrapped (dotted and dashed curve), unwrapped (solid curve) and detected 2π jumps (dotted curve), (c) three-dimensional vision of the unwrapped phase.

Fig. 4
Fig. 4

(a) Three-dimensional vision of the continuous phase, (b) three-dimensional vision of the 2π jumps, (c) profiles of the wrapped (dotted and dashed curve) and unwrapped (solid curve), and detected 2π jumps (dotted curve), (d) three-dimensional vision of the unwrapped phase.

Fig. 5
Fig. 5

(a) Simulated wrapped phase, (b) unwrapped phase map with the traditional algorithm, (c) with the proposed algorithm.

Fig. 6
Fig. 6

(a) Experimentally obtained wrapped phase, (b) unwrapped phase map with the traditional algorithm, (c) with the proposed algorithm.

Fig. 7
Fig. 7

Evolution of the phase-retrieval algorithm: the noiseless case (solid curve), and the noisy case (dashed curve).

Equations (9)

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ϕ=l=1J alΩl,
φi,j=ϕi,j+2kπ,
ddζ φ=ddζϕ+2kπ=ddζ ϕ,
ddζ φi,j=ddζ ϕi,j=l=1J alddζ Ωi,jl.
ms=ddζ φi,j,
a=Ams,
Li,j=14ui+1,j+ui-1,j+ui,j+1+ui,j-1-ui,j,
mi,jx=φi-1,j-φi,jΔxi,j mi,jy=φi,j-1-φi,jΔyi,j,
merit= Is-Ir2 Is,

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