Abstract

The search for fast and robust phase-unwrapping algorithms remains an important problem in the development of real-time interferometric systems. Our phase-unwrapping approach uses a spatial binary-tree image decomposition to permit maximum parallelism in implementation. At each node in the tree structure, a single unwrapping decision is made between two image blocks. The unwrapping rule is derived from a statistical-estimation framework. Specifically, a maximum-likelihood estimate of the demodulation term is used. This term can be viewed as that which minimizes a discontinuity-penalizing cost function. We show that the algorithm exhibits a high level of robustness. Quantitative measures of performance are provided, and many phase maps are shown for subjective evaluation.

© 1998 Optical Society of America

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References

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  1. T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
    [CrossRef]
  2. P. K. Rastogi, Holographic Interferometry Principles and Methods (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  3. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
    [CrossRef]
  4. S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of mr phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–675 (1995).
    [CrossRef]
  5. M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase-unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sensing 32, 706–708 (1994).
    [CrossRef]
  6. U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
    [CrossRef]
  7. D. C. Ghiglia, 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]
  8. D. C. Ghiglia, L. A. Romero, “Minimum 1(p)-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
    [CrossRef]
  9. B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3006 (1996).
    [CrossRef]
  10. Z.-P. Liang, “A model-based method for phase unwrapping of 2-D signals,” IEEE Trans. Med. Imag. 15, 893–897 (1996).
    [CrossRef]
  11. M. A. Harraez, D. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
    [CrossRef]
  12. P. R. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
    [CrossRef]
  13. J. M. Huntly, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef]
  14. J. Geirloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).
  15. W. D. Pauw, “Multitrees with internal storage,” IEEE Trans. Comput. Aided Design Integr. Circuits Sys. 12, 1428–1436 (1993).
    [CrossRef]
  16. P. Strobach, “Quadtree-structured recursive plane decomposition coding of images,” IEEE Trans. Signal Process. 39, 1380–1397 (1991).
    [CrossRef]
  17. R. Szeliski, H.-Y. Shum, “Motion estimation with quadtree lines,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 1199–1210 (1996).
    [CrossRef]
  18. C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).

1996 (6)

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
[CrossRef]

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3006 (1996).
[CrossRef]

Z.-P. Liang, “A model-based method for phase unwrapping of 2-D signals,” IEEE Trans. Med. Imag. 15, 893–897 (1996).
[CrossRef]

R. Szeliski, H.-Y. Shum, “Motion estimation with quadtree lines,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 1199–1210 (1996).
[CrossRef]

M. A. Harraez, D. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

D. C. Ghiglia, L. A. Romero, “Minimum 1(p)-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
[CrossRef]

1995 (2)

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of mr phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–675 (1995).
[CrossRef]

U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
[CrossRef]

1994 (4)

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

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase-unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sensing 32, 706–708 (1994).
[CrossRef]

P. R. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

D. C. Ghiglia, 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]

1993 (1)

W. D. Pauw, “Multitrees with internal storage,” IEEE Trans. Comput. Aided Design Integr. Circuits Sys. 12, 1428–1436 (1993).
[CrossRef]

1991 (1)

P. Strobach, “Quadtree-structured recursive plane decomposition coding of images,” IEEE Trans. Signal Process. 39, 1380–1397 (1991).
[CrossRef]

1989 (1)

Bryanston-Cross, P. J.

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

Burton, D.

Burton, D. R.

P. R. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Clegg, D. B.

Francos, J. M.

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3006 (1996).
[CrossRef]

Friedlander, B.

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3006 (1996).
[CrossRef]

Geirloff, J.

J. Geirloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

Ghiglia, D. C.

Glover, G. H.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of mr phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–675 (1995).
[CrossRef]

Harraez, M. A.

Huntly, J. M.

Judge, T. R.

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

Lalor, M. J.

M. A. Harraez, D. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

P. R. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Liang, Z.-P.

Z.-P. Liang, “A model-based method for phase unwrapping of 2-D signals,” IEEE Trans. Med. Imag. 15, 893–897 (1996).
[CrossRef]

Napel, S.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of mr phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–675 (1995).
[CrossRef]

Pauw, W. D.

W. D. Pauw, “Multitrees with internal storage,” IEEE Trans. Comput. Aided Design Integr. Circuits Sys. 12, 1428–1436 (1993).
[CrossRef]

Pelc, N. J.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of mr phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–675 (1995).
[CrossRef]

Pritt, M. D.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
[CrossRef]

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase-unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sensing 32, 706–708 (1994).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry Principles and Methods (Springer-Verlag, Berlin, 1994).
[CrossRef]

Romero, L. A.

Shipman, J. S.

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase-unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sensing 32, 706–708 (1994).
[CrossRef]

Shum, H.-Y.

R. Szeliski, H.-Y. Shum, “Motion estimation with quadtree lines,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 1199–1210 (1996).
[CrossRef]

Song, S. M.-H.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of mr phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–675 (1995).
[CrossRef]

Spagnolini, U.

U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
[CrossRef]

Stephenson, P. R.

P. R. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Strobach, P.

P. Strobach, “Quadtree-structured recursive plane decomposition coding of images,” IEEE Trans. Signal Process. 39, 1380–1397 (1991).
[CrossRef]

Szeliski, R.

R. Szeliski, H.-Y. Shum, “Motion estimation with quadtree lines,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 1199–1210 (1996).
[CrossRef]

Therrien, C. W.

C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).

Appl. Opt. (2)

IEEE Trans. Comput. Aided Design Integr. Circuits Sys. (1)

W. D. Pauw, “Multitrees with internal storage,” IEEE Trans. Comput. Aided Design Integr. Circuits Sys. 12, 1428–1436 (1993).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (3)

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
[CrossRef]

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase-unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sensing 32, 706–708 (1994).
[CrossRef]

U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
[CrossRef]

IEEE Trans. Image Process. (1)

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of mr phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–675 (1995).
[CrossRef]

IEEE Trans. Med. Imag. (1)

Z.-P. Liang, “A model-based method for phase unwrapping of 2-D signals,” IEEE Trans. Med. Imag. 15, 893–897 (1996).
[CrossRef]

IEEE Trans. Patt. Anal. Mach. Intell. (1)

R. Szeliski, H.-Y. Shum, “Motion estimation with quadtree lines,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 1199–1210 (1996).
[CrossRef]

IEEE Trans. Signal Process. (2)

P. Strobach, “Quadtree-structured recursive plane decomposition coding of images,” IEEE Trans. Signal Process. 39, 1380–1397 (1991).
[CrossRef]

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3006 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

P. R. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Opt. Lasers Eng. (1)

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

Other (3)

P. K. Rastogi, Holographic Interferometry Principles and Methods (Springer-Verlag, Berlin, 1994).
[CrossRef]

J. Geirloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).

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Figures (9)

Fig. 1
Fig. 1

Illustration of two adjacent image blocks with one choice of subblock shaded.

Fig. 2
Fig. 2

Illustration of the MLBT phase-unwrapping procedure at (a) level 1, (b) level 2, (c) level 3, and (d) level 4.

Fig. 3
Fig. 3

Illustration of the parallel architecture for the MLBT phase-unwrapping algorithm.

Fig. 4
Fig. 4

Phase-unwrapping results for a ramp with additive Gaussian noise with a standard deviation of 1. (a) Raw noisy wrapped-phase data with σ = 1.0. (b) Line unwrapped data with σ = 1.0. (c) Path-independent phase-unwrapping algorithm output with σ = 1.0. (d) MLBT algorithm output for L = 3 and σ = 1.0.

Fig. 5
Fig. 5

Intermediate MLBT phase-unwrapping results for a 32 × 32 portion of the ramp data.

Fig. 6
Fig. 6

Quantitative error analysis for the ramp data with various unwrapping methods. The MAE per pixel versus the additive Gaussian noise standard deviation is shown.

Fig. 7
Fig. 7

MAE per pixel versus L for noise with a standard deviation of σ = 1.3.

Fig. 8
Fig. 8

Phase-unwrapping results for a contoured phase surface with additive Gaussian noise with a standard deviation of 1. (a) Ideal contoured phase surface. (b) Raw noisy wrapped-phase data with σ = 1.0. (c) Path-independent phase-unwrapping algorithm output with σ = 1.0. (d) MLBT algorithm output for L = 3 and σ = 1.0.

Fig. 9
Fig. 9

Quantitative performance analysis for the contoured surface phase data with various unwrapping methods.

Tables (1)

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Table 1 Proposed MLBT Phase-Unwrapping Algorithm

Equations (13)

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b 1 = b 1 , 1 ,   b 1 , 2 , ,   b 1 , M T ,
b 2 = b 2 , 1 ,   b 2 , 2 , ,   b 2 , M T .
b ˜ 1 = b ˜ 1 , 1 ,   b ˜ 1 , 2 , ,   b ˜ 1 , M ˜ T ,
b ˜ 2 = b ˜ 2 , 1 ,   b ˜ 2 , 2 , ,   b ˜ 2 , M ˜ T ,
b ˜ 2 = b ˜ 1 + k 2 π 1 + n ,
Pr n = 1 2 π M ˜ / 2 σ η M ˜ exp - 1 2 σ η 2 n T n ,
k ˆ = k arg   max   Pr b 2 | k ,   b 1 ,
k ˆ = k arg   max 1 2 π M ˜ / 2 σ η M ˜ exp - 1 2 σ η 2 ( b ˜ 2 - b ˜ 1 + k 2 π 1 ) T ( b ˜ 2 - b ˜ 1 + k 2 π 1 ) .
k ˆ = k arg   min { ( b ˜ 2 - b ˜ 1 + k 2 π 1 ) T ( b ˜ 2 - b ˜ 1 + k 2 π 1 ) } = k arg   min b ˜ 2 - b ˜ 1 + k 2 π 1 2 .
b ˆ 2 = b 2 - k ˆ 2 π 1 ,
b ˆ 1 = b 1 + k ˆ 2 π 1 .
= 2   log N log 2 .
N 2 l = 1 1 2 l = N 2 1 - 1 2 .

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