Abstract

The classical problem of phase unwrapping in two dimensions, that of how to create a path-independent unwrapped map, is extended to the case of a three-dimensional phase distribution. Whereas in two dimensions the path dependence problem arises from isolated phase singularity points, in three dimensions the phase singularities are shown to form closed loops in space. A closed path that links one such loop will cross a nonzero number of phase discontinuities. In two dimensions, path independence is achieved when branch-cut lines are placed between singular points of opposite sign; an equivalent path-independent algorithm for three dimensions is developed that places branch-cut surfaces so as to prevent unwrapping through the phase singularity loops. The placing of the cuts is determined uniquely by the phase data, which contrasts with the two-dimensional case for which there are many possible ways in which to pair up the singular points. The performance of the new algorithm is demonstrated on three-dimensional phase data from a high-speed phase-shifting speckle pattern interferometer.

© 2001 Optical Society of America

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References

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2000 (2)

1999 (1)

1998 (1)

M. Constantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

1997 (1)

1996 (2)

1995 (2)

1994 (1)

1993 (1)

1991 (2)

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

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

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)

1982 (1)

Barr, L. D.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

Bone, D. J.

Buckland, J. R.

Chen, C. W.

Constantini, M.

M. Constantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

Coudé du Foresto, V.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

Cuevas, F. J.

Cusack, R.

R. Cusack, “Unwrapping 3d maps of magnetic field deviations for use in undistorting fMRI images of the brain,” paper presented at the Institute of Physics Applied Optics Divisional Conference, Loughborough, UK, September 2000.

Flynn, T. J.

Fox, J.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

Ghiglia, D. C.

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]

Greivenkamp, J. E.

Haible, P.

Huntley, J. M.

Itoh, K.

Kaufmann, G. H.

Kerr, D.

Kothiyal, M. P.

Malacara, D.

Poczulp, G. A.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Richardson, J.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

Roddier, C.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

Roddier, F.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

Romero, L. A.

Saldner, H.

Servin, M.

Tiziani, H. J.

Turner, S. R. E.

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.

C. W. Chen, H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
[CrossRef]

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. (9)

IEEE Trans. Geosci. Remote Sens. (1)

M. Constantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the national-optical-astronomy observatories,” Opt. Eng. 30, 1405–1414 (1991).

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 (2)

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

R. Cusack, “Unwrapping 3d maps of magnetic field deviations for use in undistorting fMRI images of the brain,” paper presented at the Institute of Physics Applied Optics Divisional Conference, Loughborough, UK, September 2000.

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Figures (8)

Fig. 1
Fig. 1

(a) Wrapped phase map (black and white representing, respectively, -π and +π) containing phase singularities. (b) Corresponding graph with vertices (filled circles) representing the pixels at which the phase is sampled and edges (lines) representing potential unwrapping paths between the vertices.

Fig. 2
Fig. 2

Two FCSs containing residues.

Fig. 3
Fig. 3

Three-dimensional representation of parts of two FCSs.

Fig. 4
Fig. 4

Two-dimensional representations of the surfaces from Fig. 3.

Fig. 5
Fig. 5

(a) Example of the formation of a pair of residues within a 2-D phase map through the intersection of the plane (shaded) with a PSL. (b) Closed path A linking a PSL encircles a single residue when represented on a FCS.

Fig. 6
Fig. 6

(a) Example of two PSLs passing through a single cube volume element. Phase values are specified as fractions of 2π. (b) Resulting knot point.

Fig. 7
Fig. 7

Phase singularity loops and cuts for a 3-D phase distribution, measured with a high-speed speckle pattern interferometer.

Fig. 8
Fig. 8

Result of unwrapping the phase distribution from Fig. 7 with (a) a spatial method, (b) a temporal method, and (c) the proposed new method.

Equations (5)

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dA(i)=NINT{[ϕA(i)-ϕA(i-1)]/2π},
νA=i=1NA dA(i).
s(x, y)=NINT{[ϕ(x+1, y)-ϕ(x, y)]/2π}+NINT{[ϕ(x+1, y+1)-ϕ(x+1,y)]/2π}+NINT{[ϕ(x, y+1)-ϕ(x+1,y+1)]/2π}+NINT{[ϕ(x, y)-ϕ(x, y+1)]/2π}.
S=x,y s(x, y),
S=i s*(xi, yi, zi),

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