Abstract

We implement a proposed method for noise-immune phase unwrapping through the calculation of what may be called wrap regions. The theory of this method is reformulated and discussed with regard to random wrapping, a phenomenon that occurs with diffusely reflecting objects if the optical phase change caused by a deformation is calculated by subtraction of the phases calculated before and after the deformation. Two methods of eliminating random wrapping are presented. The algorithm for phase unwrapping is described, and experimental results are presented.

© 1997 Optical Society of America

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References

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  1. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  2. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  3. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  4. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  5. K. A. Stetson, “Phase-step interferometry of irregular shapes by using an edge-following algorithm,” Appl. Opt. 31, 5320–5325 (1992).
    [CrossRef] [PubMed]
  6. 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]
  7. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1994).
    [CrossRef]
  8. X. Tian, M. Itoh, T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral shearing interferometry,” Appl. Opt. 34, 7213–7219 (1995).
    [CrossRef] [PubMed]
  9. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  10. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  11. H. A. Vrooman, A. A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [CrossRef] [PubMed]
  12. K. A. Stetson, “Noise-immune method of locating wrap regions in phase-step interferometry,” Opt. Lett. 21, 1268–1270 (1996).
    [CrossRef] [PubMed]
  13. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).
  14. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3059 (1985).
    [CrossRef] [PubMed]
  15. K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef]
  16. K. Creath, “Phase-shifting holographic interferometry” in Holographic Interferometry, P. K. Rostogi, ed. (Springer-Verlag, Heidelberg, 1994), Chap. 5, p. 143.
  17. J. Wang, I. Grant, “Electronic speckle interferometry, phase-mapping, and nondestructive testing techniques applied to real-time, thermal loading,” Appl. Opt. 34, 3620–3627 (1995).
    [CrossRef] [PubMed]
  18. P. Ettl, K. Creath, “Comparison of phase-unwrapping algorithms by using gradient of first failure,” Appl. Opt. 35, 5108–5114, (1996).
    [CrossRef] [PubMed]
  19. J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [CrossRef] [PubMed]

1996 (2)

1995 (3)

1994 (2)

1992 (1)

1991 (2)

1989 (1)

1985 (2)

1979 (1)

1978 (1)

1977 (2)

Bone, D. J.

Brohinsky, W. R.

Buckland, J. R.

Creath, K.

Ettl, P.

Fried, D. L.

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. SPIE818, 2–9 (1987).

Grant, I.

Hudgin, R. H.

Hunt, B. R.

Huntley, J. M.

Itoh, M.

Maas, A. A. M.

Marroquin, J. L.

Noll, R. J.

Rivera, M.

Romero, L. A.

Stetson, K. A.

Tian, X.

Turner, S. R. E.

Vrooman, H. A.

Wang, J.

Yatagai, T.

Appl. Opt. (10)

X. Tian, M. Itoh, T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral shearing interferometry,” Appl. Opt. 34, 7213–7219 (1995).
[CrossRef] [PubMed]

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

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

H. A. Vrooman, A. A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[CrossRef] [PubMed]

K. A. Stetson, “Phase-step interferometry of irregular shapes by using an edge-following algorithm,” Appl. Opt. 31, 5320–5325 (1992).
[CrossRef] [PubMed]

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3059 (1985).
[CrossRef] [PubMed]

K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
[CrossRef]

J. Wang, I. Grant, “Electronic speckle interferometry, phase-mapping, and nondestructive testing techniques applied to real-time, thermal loading,” Appl. Opt. 34, 3620–3627 (1995).
[CrossRef] [PubMed]

P. Ettl, K. Creath, “Comparison of phase-unwrapping algorithms by using gradient of first failure,” Appl. Opt. 35, 5108–5114, (1996).
[CrossRef] [PubMed]

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (4)

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

Opt. Lett. (1)

Other (2)

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

K. Creath, “Phase-shifting holographic interferometry” in Holographic Interferometry, P. K. Rostogi, ed. (Springer-Verlag, Heidelberg, 1994), Chap. 5, p. 143.

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Figures (6)

Fig. 1
Fig. 1

Phase unwrap of an interferogram of the deformation of a brandy goblet caused by the weight of a leaning rod: (a) The wrapped phase from 0 to 2π without random wrapping. (b) The wrapped phase from π/2 to 3π/2. (c) The wrap regions for (b) on the basis of its construction from (a). (d) The image of the object used for a mask in the unwrapping process.

Fig. 2
Fig. 2

Unwrapped phase of the interferogram shown in Fig. 1(a). The image is compressed in the horizontal direction as a result of display in a square pixel format rather than in the aspect ratio of the TV camera. The pixel levels have been shifted and scaled so that they range from 0 to 255.

Fig. 3
Fig. 3

Wrap regions for two different severe deformations of the goblet caused by warm water: (a) Introduction of warm water. (b) Introduction of additional warm water. (c) Unwrap for (a). (d) Unwrap for (b). The discontinuities indicate the failure of the program.

Fig. 4
Fig. 4

Enlargements of the high-gradient regions showing bridging of the fringes: (a) Bridging in Fig. 3(a). (b) Bridging in Fig. 3(b).

Fig. 5
Fig. 5

Unwraps with bridging suppressed: (a) Unwrap for Fig. 3(a). (b) Unwrap for Fig. 3(b). Two anomalous regions remain in (a), as indicated by the arrows.

Fig. 6
Fig. 6

Data arrays with differing data-related >π transitions but identical residue patterns. The arrows indicate a transition greater than π and point in the direction of an increasing phase step. (a) Three upward-pointing >π steps. (b) Two downward-pointing >π steps.

Equations (9)

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ΔPh+1-2>π,
ΔΦ=arctand1n2-n1d2/n1n2+d1d2,
ϕ0x, y=arctann/d,
ϕ90x, y=arctand/-n,
ϕ180x, y=arctan-n/-d,
ϕ270x, y=arctan-d/n,
ϕ90x, y=wrap±πϕ0x, y+π/2,
ϕ180x, y=wrap±πϕ0x, y+π,
ϕ270x, y=wrap±πϕ0x, y+3π/2,

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