Abstract

A general framework is presented for the design of parallel algorithms for two-dimensional, path-independent phase unwrapping of locally inconsistent, noisy principal-value phase fields that may contain regions of invalid information. This framework is based in Bayesian estimation theory with the use of Markov random field models to construct the prior distribution, so that the solution to the unwrapping problem is characterized as the minimizer of a piecewise-quadratic functional. This method allows one to design a variety of parallel algorithms with different computational properties, which simultaneously perform the desired path-independent unwrapping, interpolate over regions with invalid data, and reduce the noise. It is also shown how this approach may be extended to the case of discontinuous phase fields, incorporating information from fringe patterns of different frequencies.

© 1995 Optical Society of America

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References

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  1. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
  2. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  3. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  4. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993), Chap. 6, pp. 194–229.
  5. 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]
  6. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  7. Th. Kreis, “Computer aided evaluation of fringe patterns,” Proceedings of the Fifth International Conference Fringe Analysis ’92 (Institute of Physics, Bristol, UK, 1992), pp. 15–30.
  8. H. A. Vrooman, A. A. M. Maas, “Image processing in digital speckle interferometry,” in Proceedings of the Fourth International Conference Fringe Analysis ’89 (Institute of Physics, Bristol, UK, 1989).
  9. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  10. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  11. R. J. Green, J. G. Walker, “Phase unwrapping using a prioriknowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Intrum. Eng.1010, 36–43 (1988).
    [CrossRef]
  12. D. C. Ghiglia, G. A. Masting, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  13. A. Spik, D. W. Robinson “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
    [CrossRef]
  14. M. Takeda, K. Nagatome, Y. Watanabe, “Phase unwrapping by neural network,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns Fringe ’93, W. Jupner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 136–141.
  15. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  16. J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Asoc. 82, 76–89 (1987).
    [CrossRef]
  17. J. M. Ortega, W. C. Rheinholdt, Iterative Solutions of Nonlinear Equations in Several Variables (Academic, New York, 1970).
  18. J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
    [CrossRef]
  19. D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2946 (1994).
    [CrossRef] [PubMed]
  20. M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
    [CrossRef]
  21. A. Lumsdaine, J. Wyatt, I. Elfadel, “Parallel distributed networks for image smoothing and segmentation in analog VLSI,” in Proceedings of the 28th IEEE Conference on Decision and Control (IEEE Computer Society Press, Washington, DC, 1989).
    [CrossRef]

1994 (3)

1993 (1)

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
[CrossRef]

1991 (2)

A. Spik, D. W. Robinson “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

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

1989 (1)

1987 (2)

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Asoc. 82, 76–89 (1987).
[CrossRef]

D. C. Ghiglia, G. A. Masting, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
[CrossRef]

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1982 (2)

1979 (1)

Bone, D. J.

Burton, D. R.

Creath, K.

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

Elfadel, I.

A. Lumsdaine, J. Wyatt, I. Elfadel, “Parallel distributed networks for image smoothing and segmentation in analog VLSI,” in Proceedings of the 28th IEEE Conference on Decision and Control (IEEE Computer Society Press, Washington, DC, 1989).
[CrossRef]

Ghiglia, D. C.

Green, R. J.

R. J. Green, J. G. Walker, “Phase unwrapping using a prioriknowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Intrum. Eng.1010, 36–43 (1988).
[CrossRef]

Hunt, B. R.

Huntley, J. M.

Ina, H.

Itoh, K.

Kobayashi, S.

Kreis, Th.

Th. Kreis, “Computer aided evaluation of fringe patterns,” Proceedings of the Fifth International Conference Fringe Analysis ’92 (Institute of Physics, Bristol, UK, 1992), pp. 15–30.

Lalor, M. J.

Lumsdaine, A.

A. Lumsdaine, J. Wyatt, I. Elfadel, “Parallel distributed networks for image smoothing and segmentation in analog VLSI,” in Proceedings of the 28th IEEE Conference on Decision and Control (IEEE Computer Society Press, Washington, DC, 1989).
[CrossRef]

Maas, A. A. M.

H. A. Vrooman, A. A. M. Maas, “Image processing in digital speckle interferometry,” in Proceedings of the Fourth International Conference Fringe Analysis ’89 (Institute of Physics, Bristol, UK, 1989).

Marroquin, J. L.

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
[CrossRef]

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Asoc. 82, 76–89 (1987).
[CrossRef]

Masting, G. A.

Mitter, S.

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Asoc. 82, 76–89 (1987).
[CrossRef]

Moore, A. J.

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

Nagatome, K.

M. Takeda, K. Nagatome, Y. Watanabe, “Phase unwrapping by neural network,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns Fringe ’93, W. Jupner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 136–141.

Ortega, J. M.

J. M. Ortega, W. C. Rheinholdt, Iterative Solutions of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Poggio, T.

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Asoc. 82, 76–89 (1987).
[CrossRef]

Rheinholdt, W. C.

J. M. Ortega, W. C. Rheinholdt, Iterative Solutions of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Robinson, D. W.

A. Spik, D. W. Robinson “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993), Chap. 6, pp. 194–229.

Rodriguez-Vera, R.

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

Romero, L. A.

Servin, M.

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

Spik, A.

A. Spik, D. W. Robinson “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

Takeda, M.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

M. Takeda, K. Nagatome, Y. Watanabe, “Phase unwrapping by neural network,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns Fringe ’93, W. Jupner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 136–141.

Vrooman, H. A.

H. A. Vrooman, A. A. M. Maas, “Image processing in digital speckle interferometry,” in Proceedings of the Fourth International Conference Fringe Analysis ’89 (Institute of Physics, Bristol, UK, 1989).

Walker, J. G.

R. J. Green, J. G. Walker, “Phase unwrapping using a prioriknowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Intrum. Eng.1010, 36–43 (1988).
[CrossRef]

Watanabe, Y.

M. Takeda, K. Nagatome, Y. Watanabe, “Phase unwrapping by neural network,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns Fringe ’93, W. Jupner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 136–141.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Wyatt, J.

A. Lumsdaine, J. Wyatt, I. Elfadel, “Parallel distributed networks for image smoothing and segmentation in analog VLSI,” in Proceedings of the 28th IEEE Conference on Decision and Control (IEEE Computer Society Press, Washington, DC, 1989).
[CrossRef]

Appl. Opt. (4)

CVGIP Graph. Models Image Process. (1)

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
[CrossRef]

J. Am. Stat. Asoc. (1)

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Asoc. 82, 76–89 (1987).
[CrossRef]

J. Mod. Opt. (1)

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Opt. Laser Eng. (1)

A. Spik, D. W. Robinson “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

Other (8)

M. Takeda, K. Nagatome, Y. Watanabe, “Phase unwrapping by neural network,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns Fringe ’93, W. Jupner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 136–141.

J. M. Ortega, W. C. Rheinholdt, Iterative Solutions of Nonlinear Equations in Several Variables (Academic, New York, 1970).

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993), Chap. 6, pp. 194–229.

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

Th. Kreis, “Computer aided evaluation of fringe patterns,” Proceedings of the Fifth International Conference Fringe Analysis ’92 (Institute of Physics, Bristol, UK, 1992), pp. 15–30.

H. A. Vrooman, A. A. M. Maas, “Image processing in digital speckle interferometry,” in Proceedings of the Fourth International Conference Fringe Analysis ’89 (Institute of Physics, Bristol, UK, 1989).

R. J. Green, J. G. Walker, “Phase unwrapping using a prioriknowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Intrum. Eng.1010, 36–43 (1988).
[CrossRef]

A. Lumsdaine, J. Wyatt, I. Elfadel, “Parallel distributed networks for image smoothing and segmentation in analog VLSI,” in Proceedings of the 28th IEEE Conference on Decision and Control (IEEE Computer Society Press, Washington, DC, 1989).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Principal-value (wrapped) phase field of an elliptical object. (b) Location of local inconsistencies (marked by black pixels) and regions with unreliable information (shown as two black circles). (c) Final correction field obtained with the Newtonian descent algorithm, second-order prior potentials, and λ = 9. (d) Reconstructed phase. (e) Same as (c), but for first-order potentials and λ = 50. (f) Reconstructed phase corresponding to (e).

Fig. 2
Fig. 2

(a) Image of an object with surface shape discontinuities. (b) Low-frequency fringe pattern projected over the object shown in (a). (c) High-frequency pattern projected over the same object.

Fig. 3
Fig. 3

(a) Phase recovered from the fringe pattern of Fig. 2(b) with the use of the Fourier method. (b) Same as (a), but for the pattern of Fig. 2(c). (c) Corrected low-frequency phase obtained with Algorithm A1. (d) Final reconstructed phase obtained with the method of Section 4.

Equations (38)

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ϕ x ( x , y ) = Δ x g ( x , y ) + 2 π k x ( x , y ) , ϕ y ( x , y ) = Δ y g ( x , y ) + 2 π k y ( x , y ) ,
Δ x ϕ y ( x , y ) = Δ y ϕ x ( x , y )
Δ y ϕ x ( x 0 , y 0 ) = 2 π 3 2 π 3 = 4 π 3 , Δ x ϕ y ( x 0 , y 0 ) = 2 π 3 0 = 2 π 3 .
U ( ϕ ) = x , y [ Δ x ϕ ( x , y ) ϕ x ( x , y ) ] 2 + [ Δ y ϕ ( x , y ) ϕ y ( x , y ) ] 2 .
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ u x + υ y + ϕ ( x , y ) ] .
g ( x , y ) = ϕ ( x , y ) + 2 π k ( x , y ) ,
g i = ϕ i 2 π k i + n i + q , i S .
f i = k i + q 2 π .
f i f j = r ( f i f j ) ,
P g , f | ϕ ( ϕ , f ) = 1 K exp [ α i S ( g i ϕ i + 2 π f i ) 2 ] ,
i N i , i N j , if and only if j N i for all i , j L .
V C ( ϕ ) = V C ( ϕ i , i C ) .
P ϕ ( ϕ ) = 1 Z exp [ c V C ( ϕ ) ] ,
P ϕ , f | g ( ϕ , f ) = P g , f | ϕ ( ϕ , f ) P ϕ ( ϕ ) P g ( g ) = 1 Z P exp [ U ( ϕ , f ) ] ,
U ( ϕ , f ) = C V C ( ϕ ) + γ i S ( g i ϕ i + 2 π f i ) 2 .
ϕ i = g i + 2 π f i , i S = 2 π f i , i S
U ( f ) = C V C ( g + 2 π f ) + λ i , j [ f i f j r ( f i f j ) ] 2 s i s j ,
f i ( 0 ) = g i 2 π ,
F γ ( f ) = C V C ( g + 2 π f ) + γ λ i , j [ f i f j r ( f i f j ) ] 2 s i s j .
f i ( t + 1 ) = f i ( t ) h U [ f ( t ) ] f i ,
U [ f ( t ) ] f i = C : i C V C [ g + 2 π f ( t ) ] f i + 2 λ j : i j = 1 { f i ( t ) f j ( t ) r [ f i ( t ) f j ( t ) ] } s i s j
V i j ( z i , z j ) = ( z i z j ) 2 .
f i ( t + 1 ) = f i ( t ) 2 h j N i { 2 π ( g i g j ) + 2 π [ f i ( t ) f j ( t ) ] + λ [ f i f j r ( f i f j ) ] s i s j } ,
f i ( t + 1 ) = 2 π | N i | g i + j N i { ( 4 π 2 + λ s i s j ) f j ( t ) + 2 π g j λ s i s j r [ f i ( t ) f j ( t ) ] } 4 π 2 | N i | + λ s i j N i s j ,
V 3 ( z i , z j , z k ) = ( z i + 2 z j z k ) 2 , V 4 ( z i , z j , z k , z l ) = ( z i + z j + z k z l ) 2 ,
2 f i t 2 U ( f ) f i + η f i t = 0 ,
f i ( t + h ) = A f i ( t ) + B f i ( t h ) + C U [ f ( t ) ] f i ,
A = 2 η h + 1 , B = η h 1 η h + 1 , C = h 2 η h + 1
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ u h x + υ h y + u h ϕ ( x , y ) ] .
u l < 2 π θ .
c ( x , y ) = cos [ u l x + υ l y + u l ϕ ( x , y ) ]
V i j ( z i , z j ) = { ( z i z j ) 2 | z i z j | < a a 2 otherwise ,
i S [ g l i B ( g h 2 π f ) i ] 2 ,
U ( f ) = i , j { V i j ( g h + 2 π f ) + λ [ f i f j r ( f i f j ) ] 2 s i s j } + ν i S ( ĝ l i g h i 2 π f i ) 2 .
U ( ϕ ) = x , y S [ Δ x ϕ ( x , y ) ϕ x ( x , y ) ] 2 + [ Δ y ϕ ( x , y ) ϕ y ( x , y ) ] 2
| g l ( x , y ) x g h ( x , y ) x | | g l x ( x , y ) g h x ( x , y ) | ,
| g l x ( x ̂ j , y ) g h x ( x ̂ j , y ) | > | g l x ( x , y ) g h x ( x , y ) |
ĝ l ( x , y ) = { g l [ x 1 ( j ) , y ] x [ x 1 ( j ) , x j ) , j = 1 , . . . , J g l [ x 2 ( j ) , y ] x [ x j , x j ( j ) ) , j = 1 , . . . , J g l ( x , y ) elsewhere .

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