Abstract

A new algorithm is proposed for unwrapping interferometric phase maps. Existing algorithms search the two-dimensional spatial domain for 2π discontinuities: only one phase map is required, but phase errors can propagate outward from regions of high noise, corrupting the rest of the image. An alternative approach based on one-dimensional unwrapping along the time axis is proposed. It is applicable to an important subclass of interferometry applications, in which a sequence of incremental phase maps can be obtained leading up to the final phase-difference map of interest. A particular example is quasi-static deformation analysis. The main advantages are (i) it is inherently simple, (ii) phase errors are constrained within the high-noise regions, and (iii) phase maps containing global discontinuities are unwrapped correctly, provided the positions of the discontinuities remain fixed with time. The possibility of real-time phase unwrapping is also discussed.

© 1993 Optical Society of America

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References

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  1. G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
    [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. W. Osten, R. Höfling, “The inverse modulo process in automatic fringe analysis: problems and approaches,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.
  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. A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng 14, 25–37 (1991).
    [CrossRef]
  6. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  7. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  8. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  9. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng818, 2–9 (1987).
  10. R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng1010, 36–43 (1989).
  11. T. R. Judge, “Quantitative digital image processing in fringe analysis and particle image velocimetry,” Ph.D. dissertation (University of Warwick, Warwick, UK, 1992).
  12. K. Creath, “Phase-measurement techniques for nondestructive testing,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 473–479.
  13. K. A. Stetson, “Theory and applications of electronic holography,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 294–300.

1991 (2)

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng 14, 25–37 (1991).
[CrossRef]

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

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)

1986 (1)

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[CrossRef]

1982 (1)

Bone, D. J.

Creath, K.

K. Creath, “Phase-measurement techniques for nondestructive testing,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 473–479.

Ghiglia, D. C.

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng818, 2–9 (1987).

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]

Green, R. J.

R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng1010, 36–43 (1989).

Höfling, R.

W. Osten, R. Höfling, “The inverse modulo process in automatic fringe analysis: problems and approaches,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.

Huntley, J. M.

Ina, H.

Judge, T. R.

T. R. Judge, “Quantitative digital image processing in fringe analysis and particle image velocimetry,” Ph.D. dissertation (University of Warwick, Warwick, UK, 1992).

Kobayashi, S.

Mastin, G. A.

Osten, W.

W. Osten, R. Höfling, “The inverse modulo process in automatic fringe analysis: problems and approaches,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.

Reid, G. T.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[CrossRef]

Robinson, D. W.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng 14, 25–37 (1991).
[CrossRef]

Romero, L. A.

Spik, A.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng 14, 25–37 (1991).
[CrossRef]

Stetson, K. A.

K. A. Stetson, “Theory and applications of electronic holography,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 294–300.

Takeda, M.

Walker, J. G.

R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng1010, 36–43 (1989).

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]

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. (2)

J. Opt. Soc. Am. (1)

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

Opt. Lasers Eng (1)

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng 14, 25–37 (1991).
[CrossRef]

Opt. Lasers Eng. (1)

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[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 (6)

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng818, 2–9 (1987).

R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng1010, 36–43 (1989).

T. R. Judge, “Quantitative digital image processing in fringe analysis and particle image velocimetry,” Ph.D. dissertation (University of Warwick, Warwick, UK, 1992).

K. Creath, “Phase-measurement techniques for nondestructive testing,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 473–479.

K. A. Stetson, “Theory and applications of electronic holography,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 294–300.

W. Osten, R. Höfling, “The inverse modulo process in automatic fringe analysis: problems and approaches,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.

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Figures (6)

Fig. 1
Fig. 1

Conventional (spatial) phase-unwrapping method. The phase at Q relative to P is obtained from the number of 2π discontinuities crossed by any path (e.g., A or B): (a) noise-free case, (b) noise-induced discontinuity sources 1 and 2, which cause unwrapping errors dependent on the path taken.

Fig. 2
Fig. 2

Temporal phase-unwrapping method. A stack of 2-D phase maps is assembled to form a 3-D phase distribution Φ(m, n, t) (t = 0, 1, 2, …, s). Unwrapping along the t axis (e.g., P to Q) avoids phase boundaries and noise encountered by spatial unwrapping (e.g., Q to R).

Fig. 3
Fig. 3

Wrapped phase map for a disk undergoing compression between two anvils. The disk diameter is 10 mm; black represents −π, white +π.

Fig. 4
Fig. 4

Distribution of discontinuity sources for the phase map shown in Fig. 3. Positive and negative sources are represented as white and black, respectively, on a gray background.

Fig. 5
Fig. 5

Unwrapped phase map corresponding to Fig. 3, formed by summation of intermediate phase differences.

Fig. 6
Fig. 6

Phase map from Fig. 5 after further localized spatial unwrapping (see text). Gray levels from black to white cover a total phase range of 31.5 rad.

Equations (10)

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d ( i ) = [ { Φ ( i ) - Φ ( i - 1 ) ] / 2 π } ] ,
ν = i = 1 N d ( i ) ,
d ( m , n , t ) = [ Δ Φ ( m , n , t ) / 2 π ] ,
Δ Φ ( m , n , t ) = Φ ( m , n , t ) - Φ ( m , n , t - 1 ) .
ν ( m , n , s ) = t = 1 s d ( m , n , t ) ,
I n ( t ) = I r + I o + 2 A r A o cos [ Φ ( t ) + ϕ n ] , [ ϕ n = ( n - 1 ) π / 2 , n = 1 , 2 , 3 , 4 ] ,
Φ ( t ) = tan - 1 ( Δ I 42 ( t ) Δ I 13 ( t ) ) ,
Δ I i j ( t ) = I i ( t ) - I j ( t )
Δ Φ ( t ) = tan - 1 [ Δ I 42 ( t ) Δ I 13 ( t - 1 ) - Δ I 13 ( t ) Δ I 42 ( t - 1 ) Δ I 13 ( t ) Δ I 13 ( t - 1 ) + Δ I 42 ( t ) Δ I 42 ( t - 1 ) ] .
Φ ( m , n , s ) = t = 1 s Δ Φ ( m , n , t ) .

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