Abstract

This paper presents a computationally efficient two-dimensional phase-unwrapping method based on a multichannel least-mean-square algorithm. The performance of the proposed method is evaluated by applying phase unwrapping to several simulated very noisy images and to a genuine noisy interferometrical image taken from a five-step phase-shift interferogram obtained from a surface plasmon resonance imaging biosensing experiment. The results confirm that the proposed method is more widely applicable, more computationally efficient, and more robust in the presence of noise than the representative methods presented in this paper.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
  8. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  9. Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, 254–263 (1993).
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
    [CrossRef]
  13. J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
    [CrossRef]
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1999 (1)

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

1997 (1)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[CrossRef]

1996 (2)

1995 (3)

1994 (2)

1991 (1)

1989 (1)

1988 (1)

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

1987 (1)

1982 (2)

Ai, C.

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, 254–263 (1993).

Bernabeu, E.

Bertani, D.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[CrossRef]

Bone, D. J.

Buckland, J. R.

Capanni, A.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[CrossRef]

Cetica, M.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[CrossRef]

Cusack, R.

Derauw, D.

D. Derauw, “Phase unwrapping using coherence measurements,” in Synthetic Aperture Radar and Passive Microwave Sensing, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, eds., Proc. SPIE2584, 319–324 (1995).
[CrossRef]

Francini, F.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[CrossRef]

Galizzi, G. E.

Ghiglia, D. C.

Goldrein, H. T.

Goldstein, R. M.

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

González-Cano, A.

Huntley, J. M.

Ina, H.

Itoh, K.

Jain, A. K.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Kaufmann, G. H.

Kerr, D.

Kobayashi, S.

Mastin, G. A.

Pezzati, L.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998).

Quiroga, J. A.

Romero, L. A.

Strand, J.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Strarns, S.

B. Widrow, S. Strarns, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Takeda, M.

Taxt, T.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Turner, S. T. E.

Werner, C. L.

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

Widrow, B.

B. Widrow, S. Strarns, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Xu, Y.

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, 254–263 (1993).

Zebker, H. A.

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

Appl. Opt. (8)

IEEE Trans. Image Process. (1)

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[CrossRef]

Radio Sci. (1)

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

Other (4)

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, 254–263 (1993).

D. Derauw, “Phase unwrapping using coherence measurements,” in Synthetic Aperture Radar and Passive Microwave Sensing, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, eds., Proc. SPIE2584, 319–324 (1995).
[CrossRef]

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998).

B. Widrow, S. Strarns, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1995).

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Figures (5)

Fig. 1
Fig. 1

System block diagram for adaptive signal prediction.

Fig. 2
Fig. 2

Results of phase unwrapping for single-channel LMS algorithm.

Fig. 3
Fig. 3

Processing scheme of multichannel LMS 2-D phase unwrapping with K channels and filter length L.

Fig. 4
Fig. 4

Unwrapping process: (a) the 2-D synthetic wrapped phase without noise addition, (b) the wrapped phase with noise addition, (c) the unwrapped phase without multichannel LMS filter, and (d) the unwrapped phase with multichannel LMS filter.

Fig. 5
Fig. 5

Real interferometrical phase unwrapping process: (a) original wrapped image obtained from five-step phase-shift interferogram and (b) restored image from multichannel LMS phase unwrapping method.

Tables (1)

Tables Icon

Table 1 Performance Comparison for Different Levels of Noise Added to Original Phase Map

Equations (22)

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Xn=xn-1, xn-2,, xn-LT.
Bn=b1n, b2n,, bLnT.
en=dn-i=1L binxn-i=dn-BTnXn=dn-dˆn,
Bn+1=Bn+μXnen,
0<μ<1Lsignal power.
en=en-2πxi=xi-2π,for i=n, n+1,, N,
en=en+2πxi=xi+2π,for i=n, n+1,, N.
dˆn=dn-en=sn+wn-en,
dˆn=sn+ŵn,
original phase SNR in decibels=10 log10VarsnVarwn,
restored phase SNR in decibels=10 log10VarsnVarŵn.
SNR improvement in decibels=10 log10VarwnVarŵn.
Xn=xn-1, xn-2,, xn-LˆT.
Bin=bi1n, bi2n,, biLˆnT, i=1, 2,, K.
ein=din-BiTnXn=din-dˆin.
Bin+1=Bin+μeinXTn, i=1, 2,, K.
ein=ein-2π, for i=1, 2,, K, xip=xip-2π, for p=n, n+1,, N.
ein=ein+2π, for i=1, 2,, K, xip=xip+2π, for p=n, n+1,, N.
krm, n=ijint|pm, n-pm-i, n-j|π×|pm, n-pm-i, n-j|pm, n-pm-i, n-j,
Lr=2π|kr|kr,
Pr+1m, n=Prm, n-Lr,
Gj=Ir+Ir+1/2.

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