Abstract

Research into two-dimensional phase unwrapping has uncovered interesting and troublesome inconsistencies that cause path-dependent results. Cellular automata, which are simple, discrete mathematical systems, offered promise of computation in a nondirectional, parallel manner. A cellular automaton was discovered that can unwrap consistent phase data in n dimensions in a path-independent manner and can automatically accommodate noise-induced (pointlike) inconsistencies and arbitrary boundary conditions (region partitioning). For data with regional (nonpointlike) inconsistencies, no phase-unwrapping algorithm will converge, including the cellular-automata approach. However, the automata method permits more simple visualization of the regional inconsistencies. Examples of its behavior on one- and two-dimensional data are presented.

© 1987 Optical Society of America

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References

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  1. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.
  2. H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris, “Extraction of phonon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
    [CrossRef]
  3. S. Nakadate, H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
    [CrossRef] [PubMed]
  4. See the special issue of J. Opt. Soc. Am. [67(3) (1977)] for several phase-related examples.
  5. J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
    [CrossRef]
  6. K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24, 3101–3105 (1985).
    [CrossRef] [PubMed]
  7. B. Breuckmann, W. Thieme, “Computer-aided analysis of holographic interferograms using the phase-shift method,” Appl. Opt. 24, 2145–2149 (1985).
    [CrossRef] [PubMed]
  8. K. A. Stetson, W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef] [PubMed]
  9. J. C. Dainty, ed., Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1984), p. 301.
  10. A. Odlyzko, D. J. Randall, AT&T Bell Laboratories, Murray Hill, New Jersey 07974, “On the periods of some graph transformations” ( personal communication).
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. N. G. Cooper, “From Turing and Von Neumann to the present,” (Los Alamos Science No. 9, Los Alamos National Laboratory, Los Alamos, N.M., Fall1983), pp. 22–27.
  14. S. Wolfram, “Cellular automata,” (Los Alamos Science No. 9, Los Alamos National Laboratory, Los Alamos, N.M., Fall1983), pp. 2–21.
  15. S. Wolfram, “Statistical mechanics of cellular automata,” Rev. Mod. Phys. 55, 601–644 (1983).
    [CrossRef]
  16. D. Farmer, T. Toffoli, S. Wolfram, eds., Cellular Automata: Proceedings of an Interdisciplinary Workshop (North-Holland, Amsterdam, 1984) [Physica 10D(1), (2) (1984)].
  17. S. Wolfram, “Twenty problems in the theory of cellular automata,” Phys. Scr. T9, 170–183 (1985).
    [CrossRef]
  18. M. Gardner, Wheels, Life, and other Mathematical Amusements (Freeman, San Francisco, 1983).

1985 (6)

1983 (2)

1982 (1)

1977 (1)

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

Baranova, N. B.

Breuckmann, B.

Brohinsky, W. R.

Cooper, N. G.

N. G. Cooper, “From Turing and Von Neumann to the present,” (Los Alamos Science No. 9, Los Alamos National Laboratory, Los Alamos, N.M., Fall1983), pp. 22–27.

Gardner, M.

M. Gardner, Wheels, Life, and other Mathematical Amusements (Freeman, San Francisco, 1983).

Ghiglia, D. C.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris, “Extraction of phonon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Hjalmarson, H. P.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris, “Extraction of phonon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Itoh, K.

Jones, E. D.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris, “Extraction of phonon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Mamaev, A. V.

Nakadate, S.

Norris, C. B.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris, “Extraction of phonon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Nugent, K. A.

Odlyzko, A.

A. Odlyzko, D. J. Randall, AT&T Bell Laboratories, Murray Hill, New Jersey 07974, “On the periods of some graph transformations” ( personal communication).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

Pilipetsky, N. F.

Randall, D. J.

A. Odlyzko, D. J. Randall, AT&T Bell Laboratories, Murray Hill, New Jersey 07974, “On the periods of some graph transformations” ( personal communication).

Romero, L. A.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris, “Extraction of phonon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Saito, H.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

Shkunov, V. V.

Stetson, K. A.

Thieme, W.

Tribolet, J. M.

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

Wolfram, S.

S. Wolfram, “Twenty problems in the theory of cellular automata,” Phys. Scr. T9, 170–183 (1985).
[CrossRef]

S. Wolfram, “Statistical mechanics of cellular automata,” Rev. Mod. Phys. 55, 601–644 (1983).
[CrossRef]

S. Wolfram, “Cellular automata,” (Los Alamos Science No. 9, Los Alamos National Laboratory, Los Alamos, N.M., Fall1983), pp. 2–21.

Zel’dovich, B. Ya.

Appl. Opt. (5)

IEEE Trans. Acoust. Speech Signal Process. (1)

J. M. Tribolet, “A new phase unwrapping algorithm,”IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. B (1)

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris, “Extraction of phonon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Phys. Scr. (1)

S. Wolfram, “Twenty problems in the theory of cellular automata,” Phys. Scr. T9, 170–183 (1985).
[CrossRef]

Rev. Mod. Phys. (1)

S. Wolfram, “Statistical mechanics of cellular automata,” Rev. Mod. Phys. 55, 601–644 (1983).
[CrossRef]

Other (8)

D. Farmer, T. Toffoli, S. Wolfram, eds., Cellular Automata: Proceedings of an Interdisciplinary Workshop (North-Holland, Amsterdam, 1984) [Physica 10D(1), (2) (1984)].

See the special issue of J. Opt. Soc. Am. [67(3) (1977)] for several phase-related examples.

J. C. Dainty, ed., Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1984), p. 301.

A. Odlyzko, D. J. Randall, AT&T Bell Laboratories, Murray Hill, New Jersey 07974, “On the periods of some graph transformations” ( personal communication).

N. G. Cooper, “From Turing and Von Neumann to the present,” (Los Alamos Science No. 9, Los Alamos National Laboratory, Los Alamos, N.M., Fall1983), pp. 22–27.

S. Wolfram, “Cellular automata,” (Los Alamos Science No. 9, Los Alamos National Laboratory, Los Alamos, N.M., Fall1983), pp. 2–21.

M. Gardner, Wheels, Life, and other Mathematical Amusements (Freeman, San Francisco, 1983).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

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Figures (13)

Fig. 1
Fig. 1

Graphical representation of the method used to check for all 2 × 2 sample path inconsistencies. The phase is unwrapped along the closed path indicated. All 2 × 2 sample regions in the entire principal-value array are tested.

Fig. 2
Fig. 2

A 512 × 512 sample principal-value array scaled to positive intensity for display purposes. Simulated noisy (signal-to-noise ratio = 2.0) interferometric data was used to obtain the phase principal values. Detected inconsistent points are highlighted in white.

Fig. 3
Fig. 3

The result of performing a conventional unwrapping operation (i.e., leftmost column followed by all rows) on the phase principal values shown in Fig. 2. Inconsistent points can cause phase discontinuities to propagate and generate streaks along the direction of unwrapping.

Fig. 4
Fig. 4

A 1-D phase function: a, 512 principal-value samples; b, after 512 iterations; c, 1000 iterations; d, complete unwrapping at iteration 3468.

Fig. 5
Fig. 5

Space–time diagram of the first 512 iterations of the automaton unwrapping process using the phase principal values shown in Fig. 4a. The image has been linearly scaled to positive intensity for display purposes.

Fig. 6
Fig. 6

The upper-left 128 × 128 pixel region of the space–time diagram shown in Fig. 5 magnified four times by pixel replication. Note how some of the phase discontinuities merge and disappear while others are created in locations where none existed at previous updates.

Fig. 7
Fig. 7

A 128 × 128 sample 2-D principal-value array scaled to intensity for display purposes.

Fig. 8
Fig. 8

A series of iterations of the automaton-unwrapping algorithm: a, after the first local iteration; b, after five local iterations; c, after ten local iterations; d, the first global iteration (averaged state). The dynamic range has increased, and several discontinuity contours have been removed at this point (compare with Fig. 7).

Fig. 9
Fig. 9

Complete phase unwrapping after 710 total local iterations and 13 total global iterations.

Fig. 10
Fig. 10

2-D phase unwrapping of data containing noise-induced inconsistencies. a, The same data as in Fig. 7 but recomputed from noisy data (with detected inconsistencies highlighted in white). b, Conventional unwrapping method generates streaks along the path of unwrapping. c, Complete unwrapping by the automaton method.

Fig. 11
Fig. 11

2-D phase unwrapping with aliasing-induced path inconsistencies. a, A 128 × 128 sample principal-value array with an aliasing-induced dislocation. Detected inconsistent points (highlighted in white) hint of a regional inconsistency. b, c, d, Three successive global iterations (global iterations 23, 24, and 25, respectively) showing almost-periodic, self-similar behavior. The automaton phase-unwrapping process does not terminate.

Fig. 12
Fig. 12

Successful unwrapping after manual intervention for region partitioning. White points highlight the originally detected inconsistencies (shown in Fig. 11), while additional white points were manually inserted to prevent phase unwrapping around other inconsistent paths. The unwrapped phase is continuous along any path that does not cross the chosen region boundary.

Fig. 13
Fig. 13

Successful unwrapping with arbitrarily chosen region partitioning. The unwrapped phase is continuous along any path that does not cross a selected boundary. Other region partitionings yield qualitatively similar results.

Equations (24)

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I ( x , y ) = B ( x , y ) + V ( x , y ) cos [ 2 π ( ω x + ω y ) + ϕ ( x , y ) + θ i ] ,
W l [ ϕ ( n ) ] = ϕ p v ( n )             ( n = 0 , 1 , 2 , , N ) ,
W l [ ϕ ( n ) ] = ϕ ( n ) + 2 π k l ( n )             ( n = 0 , 1 , 2 , , N ) ,
- π W l [ ϕ ( n ) ] π .
Δ ϕ ( n ) = ϕ ( n ) - ϕ ( n - 1 )             ( n = 1 , 2 , 3 , , N ) .
Δ W 1 [ ϕ ( n ) ] = Δ ϕ ( n ) + 2 π Δ k 1 ( n ) .
W 2 { Δ W 1 [ ϕ ( n ) ] } = Δ ϕ ( n ) + 2 π [ Δ k 1 ( n ) + k 2 ( n ) ] ,
- π Δ ϕ ( n ) π ,
Δ ϕ ( n ) = W 2 { Δ W 1 [ ϕ ( n ) ] } ,
ϕ ( m ) = ϕ ( 0 ) + n = 1 m W 2 { Δ W 1 [ ϕ ( n ) ] } .
W l [ ϕ ( m , n ) ] = ϕ ( m , n ) + 2 π k l ( m , n ) ,
Δ m ϕ ( m , n ) = ϕ ( m , n ) - ϕ ( m - 1 , n ) ,
Δ n ϕ ( m , n ) = ϕ ( m , n ) - ϕ ( m , n - 1 ) .
W 2 { Δ n W 1 [ ϕ ( m , n ) ] } = Δ n ϕ ( m , n ) + 2 π [ Δ n k l ( m , n ) + k 2 ( m , n ) ] ,
W 3 { Δ m W 1 [ ϕ ( m , n ) ] } = Δ m ϕ ( m , n ) + 2 π [ Δ m k l ( m , n ) + k 3 ( m , n ) ] ,
- π Δ m ϕ ( m , n ) π
- π Δ n ϕ ( m , n ) π ,
Δ n ϕ ( m , n ) = W 2 { Δ n W 1 [ ϕ ( m , n ) ] } ,
Δ m ϕ ( m , n ) = W 3 { Δ m W 1 [ ϕ ( m , n ) ] } .
ϕ ( m , 0 ) = ϕ ( 0 , 0 ) + j = 1 m W 3 { Δ m W 1 [ ϕ ( j , 0 ) ] } ,
ϕ ( m , n ) = ϕ ( m , 0 ) + l = 1 n W 2 { Δ n W 1 [ ϕ ( m , l ) ] } .
ϕ ( 0 , n ) = ϕ ( 0 , 0 ) + l = 1 n W 2 { Δ n W 1 [ ϕ ( 0 , l ) ] } ,
ϕ ( m , n ) = ϕ ( 0 , n ) + j = 1 m W 3 { Δ m W 1 [ ϕ ( j , n ) ] } .
n ( 1 ) , n ( 2 ) , n ( 3 ) , , n ( N )

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