Abstract

A new, to our knowledge, two-dimensional phase-unwrapping algorithm is proposed. The algorithm, which is based on the global continuity of physical information (e.g. the three-dimensional surface profile of an object) being measured, uses the principle of least-phase difference to rectify errors caused by an erroneous 2π-phase jump in the initial unwrapped phase map obtained by the conventional phase-unwrapping method. Experimental results show that the proposed algorithm works well on a phase map that contains error sources, such as noises, phase discontinuity of more than π, and insufficient sampling. Moreover, the algorithm is most suitable for unwrapping a phase map generated during shape measurement with a step-change surface on the test object, which is usually a critical problem in shape measurement.

© 2002 Optical Society of America

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References

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2001 (2)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Optics Commun. 189, 21–29 (2001).
[CrossRef]

C. W. Chen, H. A. Zebker, “Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization,” J. Opt. Soc. Am. A 18, 338–351 (2001).
[CrossRef]

2000 (1)

M. Costantini, P. A. Rosen, C. L. Werner, “Preventing and masking out unreliable results for critical quantitative applications of phase unwrapping,” International Geoscience and Remote Sensing Symposium (IGARSS) 7, 3199–3201 (2000).

1999 (1)

M. Costantini, A. Farina, F. Zirilli, “A fast phase unwrapping algorithm for SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 37, 452–460 (1999).
[CrossRef]

1998 (5)

1997 (1)

1996 (3)

M. Takeda, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

1995 (4)

1994 (2)

M. D. Pritt, J. S. Shiman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transform and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
[CrossRef]

1992 (1)

N. H. Ching, D. Rosenfeld, M. Braum, “Two-dimensional phase unwrapping algorithm using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef]

1990 (1)

J. Li, X. Y. Su, L. R. Guo, “An improved Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

1989 (1)

1983 (1)

Bernabeu, E.

Braum, M.

N. H. Ching, D. Rosenfeld, M. Braum, “Two-dimensional phase unwrapping algorithm using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef]

Buckland, J. R.

Chen, C. W.

Ching, N. H.

N. H. Ching, D. Rosenfeld, M. Braum, “Two-dimensional phase unwrapping algorithm using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef]

Collaro, A.

Costantini, M.

M. Costantini, P. A. Rosen, C. L. Werner, “Preventing and masking out unreliable results for critical quantitative applications of phase unwrapping,” International Geoscience and Remote Sensing Symposium (IGARSS) 7, 3199–3201 (2000).

M. Costantini, A. Farina, F. Zirilli, “A fast phase unwrapping algorithm for SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 37, 452–460 (1999).
[CrossRef]

Cusack, R.

Farina, A.

M. Costantini, A. Farina, F. Zirilli, “A fast phase unwrapping algorithm for SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 37, 452–460 (1999).
[CrossRef]

Ferreiro, M. S.

Flynn, J. T.

J. T. Flynn, “Phase unwrapping using discontinuity optimization,” IGARSS 1, 80–82 (1998).

Fornaro, G.

Franceschetti, G.

Ghiglia, D. C.

Goldrein, H. T.

Gonzalez-Cano, A.

Guo, L. R.

J. Li, X. Y. Su, L. R. Guo, “An improved Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Guo, Y.

X. He, D. Zou, S. Liu, Y. Guo, “Phase shifting analysis in moiré interferometry and its applications in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

He, X.

X. He, D. Zou, S. Liu, Y. Guo, “Phase shifting analysis in moiré interferometry and its applications in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

He, X. Y.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Optics Commun. 189, 21–29 (2001).
[CrossRef]

Huntley, J. M.

Huntly, J. M.

Lanari, R.

Li, J.

J. Li, X. Y. Su, L. R. Guo, “An improved Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Liu, S.

X. He, D. Zou, S. Liu, Y. Guo, “Phase shifting analysis in moiré interferometry and its applications in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

Lu, Y. P.

Marklund, O.

Mutoh, K.

Palmieri, F.

Pritt, M. D.

M. D. Pritt, J. S. Shiman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

Quan, C.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Optics Commun. 189, 21–29 (2001).
[CrossRef]

Quiroga, J. A.

Romero, L. A.

Rosen, P. A.

M. Costantini, P. A. Rosen, C. L. Werner, “Preventing and masking out unreliable results for critical quantitative applications of phase unwrapping,” International Geoscience and Remote Sensing Symposium (IGARSS) 7, 3199–3201 (2000).

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, M. Braum, “Two-dimensional phase unwrapping algorithm using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef]

Sansosti, E.

Shang, H. M.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Optics Commun. 189, 21–29 (2001).
[CrossRef]

Shiman, J. S.

M. D. Pritt, J. S. Shiman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

Su, X. Y.

J. Li, X. Y. Su, L. R. Guo, “An improved Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Takeda, M.

M. Takeda, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[CrossRef] [PubMed]

Tay, C. J.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Optics Commun. 189, 21–29 (2001).
[CrossRef]

Tesauro, M.

Turner, S. R. E.

Wang, C. F.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Optics Commun. 189, 21–29 (2001).
[CrossRef]

Werner, C. L.

M. Costantini, P. A. Rosen, C. L. Werner, “Preventing and masking out unreliable results for critical quantitative applications of phase unwrapping,” International Geoscience and Remote Sensing Symposium (IGARSS) 7, 3199–3201 (2000).

Zebker, H. A.

Zirilli, F.

M. Costantini, A. Farina, F. Zirilli, “A fast phase unwrapping algorithm for SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 37, 452–460 (1999).
[CrossRef]

Zou, D.

X. He, D. Zou, S. Liu, Y. Guo, “Phase shifting analysis in moiré interferometry and its applications in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Geosci. Remote Sens. (3)

M. Costantini, A. Farina, F. Zirilli, “A fast phase unwrapping algorithm for SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 37, 452–460 (1999).
[CrossRef]

M. D. Pritt, J. S. Shiman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

IEEE Trans. Image Process. (1)

N. H. Ching, D. Rosenfeld, M. Braum, “Two-dimensional phase unwrapping algorithm using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef]

IGARSS (1)

J. T. Flynn, “Phase unwrapping using discontinuity optimization,” IGARSS 1, 80–82 (1998).

International Geoscience and Remote Sensing Symposium (IGARSS) (1)

M. Costantini, P. A. Rosen, C. L. Werner, “Preventing and masking out unreliable results for critical quantitative applications of phase unwrapping,” International Geoscience and Remote Sensing Symposium (IGARSS) 7, 3199–3201 (2000).

J. Opt. Soc. Am. A (8)

Opt. Eng. (3)

X. He, D. Zou, S. Liu, Y. Guo, “Phase shifting analysis in moiré interferometry and its applications in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[CrossRef]

J. Li, X. Y. Su, L. R. Guo, “An improved Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

M. Takeda, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

Optics Commun. (1)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Optics Commun. 189, 21–29 (2001).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Flow chart for proposed phase-unwrapping algorithm.

Fig. 2
Fig. 2

Plaster model of a human oral gum.

Fig. 3
Fig. 3

Fringe patterns on part of the plaster model with phase shifts (a) 0, (b) π/2, (c) π, (d) 3π/2.

Fig. 4
Fig. 4

(a) Wrapped phase map, (b) 2π-phase jump lines.

Fig. 5
Fig. 5

(a) Unwrapped phase map by use of the conventional method, (b) locations of erroneous 2π-phase jump.

Fig. 6
Fig. 6

(a) d2φ/dx 2 distribution along section x-x marked in Fig. 5(a). (b) corresponding ∂2(φ, φ′)/∂x∂y distribution along section x-x.

Fig. 7
Fig. 7

(a) Unwrapped phase map by use of the proposed algorithm. (b) corresponding height contour map. (c) a 3-D plot of the section profile.

Fig. 8
Fig. 8

(a) Wrapped phase map, (b) 2π-phase jump lines.

Fig. 9
Fig. 9

(a) Unwrapped phase map by use of the conventional method, (b) unwrapped phase map by use of the proposed algorithm.

Fig. 10
Fig. 10

Triangular prism test object.

Fig. 11
Fig. 11

Wrapped phase map by use of four-step phase-shifting algorithm.

Fig. 12
Fig. 12

(a) Unwrapped phase map with conventional method, (b) corresponding 3-D plot.

Fig. 13
Fig. 13

(a) Unwrapped phase map with the proposed algorithm, (b) corresponding 3-D plot.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

ki=φi-φi-12π,
d2φdx2i,j=φi,j+1+φi,j-1-2φi,j/2Δx,
2φ, φxyi,j=φi,j+1-φi-1,j+1Δy-φi,j-φi-1,jΔyΔx,
d2φdx2i,j>Txx 2φ, φxyi,j>Tx,y,
φi,j=l=0jdφdxi,l
ΔSi,j=l=jqrφi,l-φi-1,l,
ΔSi,j,k=l=jqrφi,l+k · 2π-φi-1,l k=0, ±1, ±2, K,
ΔSmin=min|ΔSi,j,k|k=0, ±1, ±2K.
φi,j=φi,j+p · 2π.

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