Abstract

The comparison of phase-unwrapping algorithms has been an enigma because there has been no quantitative means of comparison. Noting that unwrapping routines are sensitive to noise and the local gradient of the phase array as well as fringe modulation, we have linked unwrapping performance to the gradient of first failure of the algorithm. When the gradient of first failure is plotted versus the signal-to-noise ratio, this can be used as an indicator of which algorithm to use in a given situation without the need for user intervention during the measurement and calculation. This study introduces three algorithms developed for their trade-offs in speed versus pixel noise (pointwise) sensitivity. The algorithms developed have been applied to live, noisy measurement data and have been proven to be robust.

© 1996 Optical Society of America

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References

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  1. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), pp. 349–393.
    [CrossRef]
  2. M. Takeda, H. Ina, S. Kobayashi, “Fourier transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  3. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993) pp. 194–229.
  4. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  5. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  6. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  7. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
    [CrossRef]
  8. K. A. Stetson, “Phase-step interferometry of irregular shapes by using an edge following algorithm,” Appl. Opt. 31, 5320–5325 (1992).
    [CrossRef] [PubMed]
  9. J. A. Quiroga, E. Bernabeu, “Phase unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
    [CrossRef] [PubMed]
  10. H. A. Vrooman, A. M. Maas, “Image processing algorithms for the analysis of phase shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [CrossRef] [PubMed]
  11. J. M. Huntley, “New Methods for unwrapping noisy phase maps,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 110–123 (1994).
  12. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).
  13. P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
    [CrossRef]
  14. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  15. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, ed. D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993) pp. 141–193.

1994 (2)

J. A. Quiroga, E. Bernabeu, “Phase unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
[CrossRef] [PubMed]

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

1992 (1)

1991 (3)

1989 (1)

1987 (2)

1982 (1)

Bernabeu, E.

Bone, D. J.

Bryanston-Cross, P. J.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).

Burton, D. R.

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

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), pp. 349–393.
[CrossRef]

Eiju, T.

Ghiglia, D. C.

Hariharan, P.

Huntley, J. M.

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

J. M. Huntley, “New Methods for unwrapping noisy phase maps,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 110–123 (1994).

Ina, H.

Judge, T. R.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).

Kobayashi, S.

Kujawinska, M.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, ed. D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993) pp. 141–193.

Lalor, M. I.

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

Maas, A. M.

Mastin, G. A.

Oreb, B. F.

Quiroga, J. A.

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993) pp. 194–229.

Romero, L. A.

Stephenson, P.

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

Stetson, K. A.

Takeda, M.

Towers, D. P.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).

Vrooman, H. A.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

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

Opt. Lasers Eng. (1)

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

Other (5)

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, ed. D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993) pp. 141–193.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), pp. 349–393.
[CrossRef]

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993) pp. 194–229.

J. M. Huntley, “New Methods for unwrapping noisy phase maps,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 110–123 (1994).

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).

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Figures (6)

Fig. 1
Fig. 1

Height distribution of phase map Φ(x, y).

Fig. 2
Fig. 2

Single line of 8-bit intensity array I 0(x, y).

Fig. 3
Fig. 3

Overlay of all 128 lines of unwrapped phase (at σ = 80) in the phase array. The first propagated 2π phase jump error at pixel 79 is where the first propagating 2π phase jump occurs, indicating the GOFF.

Fig. 4
Fig. 4

GOFF for the three algorithms as a function of signal-to-noise level.

Fig. 5
Fig. 5

Interferogram of 6.5-m-diameter test mirror. The fringe contrast is low because of the long path and the air turbulence.

Fig. 6
Fig. 6

Unwrapped phase map of data in Fig. 5.

Tables (1)

Tables Icon

Table 1 Mean GOFF Values at all Simulated Noise Levels for the Three Algorithmsa

Equations (9)

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Φ ( x , y ) = x 2 ( π 255 ) ,
G x ( x , y ) = x ( 2 π 255 ) ,
G y ( x , y ) = 0.
I i ( x , y ) = I 0 { 1 + γ cos [ Φ ( x , y ) + α i ] } α i = ( i - 3 ) ( π 2 ) ;             i = 1 , 2 , 3 , 4 , 5.
W ( x , y ) = arctan [ 2 ( I 2 - I 4 ) 2 I 3 - I 1 - I 5 ] ,
M ( x , y ) = 3 [ 4 ( I 4 - I 2 ) 2 + ( I 1 + I 5 - 2 I 3 ) 2 ] 1 / 2 2 ( I 1 + I 2 + 2 I 3 + I 4 + I 5 ) .
D i ( x , y ) = I i ( x , y ) + N i ( x , y ) .
P ( z ) = 1 2 π σ exp [ - 1 2 ( z σ ) 2 ] .
SNR = s i g n a l σ ,

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