Abstract

We present a robust method of phase unwrapping that was designed for use on noisy phase images with arbitrary fringe patterns. The method proceeds by first identifying distinct regions between fringe boundaries in an image and then phase shifting the regions with respect to one another by multiples of 2π to unwrap the phase. Image pixels are segmented between interfringe and fringe boundary areas by fitting a plane model using least squares to overlapping domains centered on all pixels. The method is tolerant of fringe gradient degradation caused by noise, filtering artifacts, and finite instrumentation bandwidth.

© 1996 Optical Society of America

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References

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  1. P. G. Charette, “A method for full-field mechanical testing of biological membranes based on electronic speckle pattern interferometry (ESPI),” Ph.D. dissertation (McGill University, Montreal, 1994).
  2. G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Laser Eng. 7, 37–68 (1986/1987).
    [CrossRef]
  3. D. W. Robinson, Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, London, 1993), Chap. 6, p. 194.
  4. J. A. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-8, 679–698 (1986).
    [CrossRef]
  5. M. Petrou, J. Kittler, “Optimal edge detector for ramp edges,” IEEE Trans. Pattern Anal. Machine Intell. 13, 483–491 (1991).
    [CrossRef]
  6. B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
    [CrossRef]
  7. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989), Chap. 3, p. 153.
  8. S. Krishnaswamy, “Algorithm for computer tracing of interference fringes,” Appl. Opt. 30, 1624–1628 (1991).
    [CrossRef] [PubMed]
  9. J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review” Opt. Eng. 31, 304–314 (1992).
    [CrossRef]
  10. K. Ramesh, B. R. Pramod, “Digital image processing of fringe patterns in photomechanics,” Opt. Eng. 31, 1487–1497 (1992).
    [CrossRef]
  11. Q. Lin, J. F. Vesecky, H. A. Zebker, “Phase unwrapping through fringe-line detection in synthetic aperture radar interferometry,” Appl. Opt. 33, 201–208 (1994).
    [CrossRef] [PubMed]
  12. 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]
  13. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  14. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  15. R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef] [PubMed]
  16. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  17. 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]
  18. J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  19. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).
  20. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Laser Eng. 14, 239–282 (1991).
    [CrossRef]
  21. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, London, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
  22. E. R. Dougherty, An Introduction to Morphological Image Processing (SPIE Optical Engineering, Bellingham, Wash., 1992), Chap. 2, p. 17.

1995 (1)

1994 (1)

1993 (1)

1992 (2)

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review” Opt. Eng. 31, 304–314 (1992).
[CrossRef]

K. Ramesh, B. R. Pramod, “Digital image processing of fringe patterns in photomechanics,” Opt. Eng. 31, 1487–1497 (1992).
[CrossRef]

1991 (5)

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]

M. Petrou, J. Kittler, “Optimal edge detector for ramp edges,” IEEE Trans. Pattern Anal. Machine Intell. 13, 483–491 (1991).
[CrossRef]

S. Krishnaswamy, “Algorithm for computer tracing of interference fringes,” Appl. Opt. 30, 1624–1628 (1991).
[CrossRef] [PubMed]

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

J. A. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-8, 679–698 (1986).
[CrossRef]

1985 (1)

B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
[CrossRef]

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. Laser Eng. 14, 239–282 (1991).
[CrossRef]

Button, B. L.

B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
[CrossRef]

Canny, J. A.

J. A. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-8, 679–698 (1986).
[CrossRef]

Charette, P. G.

P. G. Charette, “A method for full-field mechanical testing of biological membranes based on electronic speckle pattern interferometry (ESPI),” Ph.D. dissertation (McGill University, Montreal, 1994).

Chen, Y. M.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review” Opt. Eng. 31, 304–314 (1992).
[CrossRef]

Cheng, A. Y.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review” Opt. Eng. 31, 304–314 (1992).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, London, 1988), Vol. 26, pp. 349–393.
[CrossRef]

Cusack, R.

Cutts, J.

B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
[CrossRef]

Dobbins, B. N.

B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
[CrossRef]

Dougherty, E. R.

E. R. Dougherty, An Introduction to Morphological Image Processing (SPIE Optical Engineering, Bellingham, Wash., 1992), Chap. 2, p. 17.

Ghiglia, D. C.

Gierloff, J. J.

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

Goldrein, H. T.

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]

Huntley, J. M.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989), Chap. 3, p. 153.

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. Laser Eng. 14, 239–282 (1991).
[CrossRef]

Kittler, J.

M. Petrou, J. Kittler, “Optimal edge detector for ramp edges,” IEEE Trans. Pattern Anal. Machine Intell. 13, 483–491 (1991).
[CrossRef]

Krishnaswamy, S.

Lin, Q.

Maas, A. A. M.

Mastin, G. A.

Moxon, C. J.

B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
[CrossRef]

Petrou, M.

M. Petrou, J. Kittler, “Optimal edge detector for ramp edges,” IEEE Trans. Pattern Anal. Machine Intell. 13, 483–491 (1991).
[CrossRef]

Pramod, B. R.

K. Ramesh, B. R. Pramod, “Digital image processing of fringe patterns in photomechanics,” Opt. Eng. 31, 1487–1497 (1992).
[CrossRef]

Ramesh, K.

K. Ramesh, B. R. Pramod, “Digital image processing of fringe patterns in photomechanics,” Opt. Eng. 31, 1487–1497 (1992).
[CrossRef]

Reid, G. T.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Laser Eng. 7, 37–68 (1986/1987).
[CrossRef]

Robinson, D. W.

D. W. Robinson, Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, London, 1993), Chap. 6, p. 194.

Romero, L. A.

Saldner, H.

Singh, H.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review” Opt. Eng. 31, 304–314 (1992).
[CrossRef]

Sirkis, J. S.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review” Opt. Eng. 31, 304–314 (1992).
[CrossRef]

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. Laser Eng. 14, 239–282 (1991).
[CrossRef]

Vesecky, J. F.

Vrooman, H. A.

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]

Wykes, C.

B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
[CrossRef]

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989), Chap. 3, p. 153.

Zebker, H. A.

Q. Lin, J. F. Vesecky, H. A. Zebker, “Phase unwrapping through fringe-line detection in synthetic aperture radar interferometry,” Appl. Opt. 33, 201–208 (1994).
[CrossRef] [PubMed]

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. (7)

IEEE Trans. Pattern Anal. Machine Intell. (2)

J. A. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-8, 679–698 (1986).
[CrossRef]

M. Petrou, J. Kittler, “Optimal edge detector for ramp edges,” IEEE Trans. Pattern Anal. Machine Intell. 13, 483–491 (1991).
[CrossRef]

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

Opt. Eng. (2)

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review” Opt. Eng. 31, 304–314 (1992).
[CrossRef]

K. Ramesh, B. R. Pramod, “Digital image processing of fringe patterns in photomechanics,” Opt. Eng. 31, 1487–1497 (1992).
[CrossRef]

Opt. Laser Eng. (2)

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Laser Eng. 7, 37–68 (1986/1987).
[CrossRef]

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

Opt. Laser Technol. (1)

B. L. Button, J. Cutts, B. N. Dobbins, C. J. Moxon, C. Wykes, “The identification of fringe positions in speckle patterns,” Opt. Laser Technol. 17, 189–192 (1985).
[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, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989), Chap. 3, p. 153.

D. W. Robinson, Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, London, 1993), Chap. 6, p. 194.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, London, 1988), Vol. 26, pp. 349–393.
[CrossRef]

E. R. Dougherty, An Introduction to Morphological Image Processing (SPIE Optical Engineering, Bellingham, Wash., 1992), Chap. 2, p. 17.

P. G. Charette, “A method for full-field mechanical testing of biological membranes based on electronic speckle pattern interferometry (ESPI),” Ph.D. dissertation (McGill University, Montreal, 1994).

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Figures (9)

Fig. 1
Fig. 1

Gray-level-encoded differential phase image resulting from the subtraction of two images obtained with phase-stepping speckle interferometry before and after in-plane deformation of a biological membrane (∼10 mm in diameter and 0.25 mm thick). The black pixels correspond to values of −πrad and the white pixels to πrad.

Fig. 2
Fig. 2

Image resulting from the application of an 11 × 11 median filter to the differential phase image of Fig. 1.

Fig. 3
Fig. 3

Gray-level-encoded image of the %VAF by the local fitting of plane models to the median filtered image shown in Fig. 2. Black pixels correspond to values of 0% VAF and white pixels to 100% VAF.

Fig. 4
Fig. 4

Normalized probability density function of the %VAF data of Fig. 3 within the object boundary.

Fig. 5
Fig. 5

Binary-valued image of the data shown in Fig. 3 using a threshold value calculated from the probability density function of the %VAF data shown in Fig. 4.

Fig. 6
Fig. 6

Diagram showing how relative phase offsets between two regions are calculated from the difference in the average phase values in 3 × 3 square areas at equal distances on either side of the common boundary.

Fig. 7
Fig. 7

Offset image obtained by first segmenting the image into distinct regions using the binary-valued data in Fig. 5 as a starting point and then expanding the regions until all pixels within the object boundary are covered. Fixed offsets between regions (in multiples of 2π) are then calculated with the procedure depicted in Fig. 6. The normals along the region boundaries used in the computation are shown as an overlay.

Fig. 8
Fig. 8

Final unwrapped-phase image obtained by adding the offset image of Fig. 7 to the raw phase image of Fig. 1 and median filtering the result.

Fig. 9
Fig. 9

Initial raw differential phase image of Fig. 1 with the fringe boundaries identified by the algorithm shown as an overlay.

Equations (7)

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θ x y = tan 1 [ I 4 ( x , y ) I 2 ( x , y ) I 1 ( x , y ) I 3 ( x , y ) ] ,
r x y = 1 / 2 { [ I 4 ( x , y ) I 2 ( x , y ) ] 2 + [ I 1 ( x , y ) I 3 ( x , y ) ] 2 } 1 / 2 ,
= u 1 + u 2 x + u 3 y ,
R i j u n = 0 , n = 1 3 ,
R i j = x = k 2 k 2 y = k 2 k 2 ( z i + x , j + y i + x , j + y ) 2 w i + x , j + y ,
( Σ w Σ x w Σ y w Σ x w Σ x 2 w Σ xyw Σ y w Σ xyw Σ y 2 w ) A ( u 1 u 2 u 3 ) u = ( Σ z w Σ xzw Σ yzw ) f ,
% VAF i j = 100 [ 1 R i j ( x = k 2 k 2 y = k 2 k 2 z i + x , j + y 2 w i + x , j + y ) ] .

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