Abstract

A new phase-unwrapping algorithm is described that uses two phase images with different precision in the unwrapping; this technique can produce an approximately correct unwrapping in the presence of discontinuities. We introduce it into the measurement of a three-dimensional object shape and also present the experimental results.

© 1994 Optical Society of America

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References

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  1. M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
    [CrossRef]
  2. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  3. 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]
  4. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  5. J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  6. J. H. Bruning, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, D. R. Herriott, “Digital wave-front measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  7. C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

1993 (1)

1991 (1)

1989 (1)

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

1987 (1)

1983 (1)

1974 (1)

Bone, D. J.

Brangaccio, D. J.

Bruning, J. H.

Gallagher, J. E.

Ghiglia, D. C.

Halioua, M.

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Herriott, D. R.

Huntley, J. M.

Koliopoulos, C. L.

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

Liu, H.-C.

M. Halioua, H.-C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Macy, W. W.

Mastin, G. A.

Romero, L. A.

Rosenfeld, D. P.

Saldner, H.

White, A. D.

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Figures (4)

Fig. 1
Fig. 1

General block diagram for a shape-measurement system. R, reference plane.

Fig. 2
Fig. 2

Geometry for projecting and imaging a grating pattern on the object.

Fig. 3
Fig. 3

Exemplary measurement: (a) object, (b) the wrapped phase with many-order fringes, (c) the wrapped phase with only the zero-order fringe, (d) the unwrapped phase obtained by the current algorithm.

Fig. 4
Fig. 4

Three-dimensional plot of the reconstructed surface of the object.

Equations (20)

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I ( x , y ) = r ( x , y ) [ a ( x , y ) + b ( x , y ) cos ϕ ( x , y ) ] ,
I 1 ( x , y ) = r ( x , y ) [ a ( x , y ) + b ( x , y ) cos ϕ ( x , y ) ] ,
I 2 ( x , y ) = r ( x , y ) [ a ( x , y ) b ( x , y ) sin ϕ ( x , y ) ] ,
I 3 ( x , y ) = r ( x , y ) [ a ( x , y ) b ( x , y ) cos ϕ ( x , y ) ] ,
I 4 ( x , y ) = r ( x , y ) [ a ( x , y ) + b ( x , y ) sin ϕ ( x , y ) ] .
ϕ ( x , y ) = arctan ( I 4 I 2 ) / ( I 1 I 3 ) .
ϕ 0 ( x , y ) = ϕ ( x , y ) ϕ r ( x , y )
Φ ( x , y ) = ϕ 0 ( x , y ) + 2 n ( x , y ) π ,
h ( x , y ) = k 1 Φ ( x , y ) ,
h ( x , y ) = k 2 θ 0 ( x , y ) ,
n ( x , y ) = INT [ Φ ( x , y ) ) 2 π ] = INT [ h ( x , y ) / k 1 2 π ] = INT [ k 2 θ 0 ( x , y ) k 1 2 π ] .
Δ ( m ) = k 1 [ ϕ 0 ( x , y ) + 2 m π ] k 2 θ 0 ( x , y ) [ m = n ( x , y ) , n ( x , y ) ± 1 ] .
( m ) = Δ ( m ) / k 1 = ϕ 0 ( x , y ) + 2 m π k 2 / k 1 θ 0 ( x , y ) [ m = n ( x , y ) , n ( x , y ) ± 1 ] .
Φ ( x , y ) = ϕ 0 ( x , y ) + 2 m 0 ( x , y ) π { m 0 ( x , y ) [ n ( x , y ) , n ( x , y ) ± 1 ] } .
AC = ϕ C D / ( 2 π f ) ,
h ( x , y ) = AC ( s / d ) ( 1 + AC / d ) 1 ,
h ( x , y ) = AC ( s / d ) .
h ( x , y ) = [ s / ( 2 π f d ) ] ϕ C D .
k 1 = s / ( 2 π f 1 d ) ,
k 2 = s / ( 2 π f 2 d ) ,

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