Abstract

Current whole-field interferometric techniques yield a phase distribution in modulo 2π. Removal of the resulting cyclic discontinuities is a process known as unwrapping, which must be performed before the data can be interpreted. We investigate an iterative unwrapping technique recently published by Ghiglia and Romero [J. Opt. Soc. Am. A 11, 107 (1994)], which is based on least-squares minimization, obtained by the discrete cosine transform. We apply this technique to remove phase wraps from electronic speckle pattern interferometry data, using modest personal computer hardware. The algorithm is shown to be fast, easy to implement, robust in the presence of noise, and able to handle phase inconsistencies without propagating local errors.

© 1996 Optical Society of America

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References

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  1. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [Crossref] [PubMed]
  2. 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]
  3. T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform techniques with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
    [Crossref]
  4. D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2948 (1994).
    [Crossref] [PubMed]
  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. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

1994 (2)

1992 (1)

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform techniques with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[Crossref]

1989 (1)

1987 (1)

Bryanston-Cross, P. J.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform techniques with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[Crossref]

Burton, D. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Ghiglia, D. C.

Huntley, J. M.

Judge, T. R.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform techniques with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[Crossref]

Lalor, M. J.

Mastin, G. A.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Quan, C.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform techniques with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[Crossref]

Romero, L. A.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Appl. Opt. (2)

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

Opt. Eng. (1)

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform techniques with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[Crossref]

Other (1)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

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Figures (5)

Fig. 1
Fig. 1

Graphs of linear phase unwrapping with DCT algorithm I in the presence of increasing levels of additive random noise: (a) 10%, (b) 40%, and (c) 50% noise levels. Curve 1, normal unwrap; curve 2, DCT unwrap; curve 3, wrapped.

Fig. 2
Fig. 2

Unwrapping of computer-generated phase fringes with a localized artifically introduced inconsistency: (a) wrapped gray-level map with a line-segment inconsistency visible just to right of center; (b) mesh plot of a 32 × 32 pixel region centered on the inconsistency; (c) mesh plot of the same region after DCT algorithm II unwrapping, showing absolute phase error compared with ideal phase distribution without inconsistency.

Fig. 3
Fig. 3

DCT unwrapping of computer-generated phase fringes with a rectangular, artifically introduced block of random gray-level noise: (a) wrapped gray-level map with a 70 × 70 pixel noise region; (b) gray-level horizontal line profiles (line 142) across the entire image width with curve A showing the idealized unwrapped-phase distribution (without the noise region), curve B showing the unwrapped-phase distribution after complete masking and five iterations of algorithm II, and curve C showing the unwrapped phase after partial (boundary) masking and five iterations of algorithm II. The curve are offset in the vertical axis for greater clarity.

Fig. 4
Fig. 4

Unwrapping of computer-generated phase fringes with an artifically introduced inconsistency in the top left-hand quadrant: (a) wrapped gray-level map; (b) unwrapped-phase distribution after 500 iterations of algorithm II; (c) mesh plot of the centerl 64 × 64 pixel region of Fig. 4(b); (d) mesh plot of the central 64 × 64 regioon of the idealized unwrapped-phase distribution.

Fig. 5
Fig. 5

Unwrapping of real ESPI phase fringes with phase inconsistencies from noise and uneven illumination: (a) wrapped-phase distribution; (b) wrapped phase with a superimposed weighting mask used for the DCT unwrapper; (c) unwrapped image after five iterations of algorithm II.

Tables (1)

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Table 1 Unwrapping of Linear Phase Function with Additive Noise

Equations (11)

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i = 0 M 2 j = 0 N 1    ( ϕ i + 1 , j ϕ i , j Δ i , j x ) 2 + i = 0 M 1 j = 0 N 2 ( ϕ i , j + 1 ϕ i , j Δ i , j y ) 2 ,
Δ i , j x = W ( ψ i + 1 , j ψ i , j ) , i = 0 , ... , M 2 , j = 0 , ... , N 1
Δ i , j y = W ( ψ i , j + 1 ψ i , j ) , i = 0 , ... , M 1 , j = 0 , ... , N 2
( ϕ i + 1 , j 2 ϕ i , j + ϕ i 1 , j ) + ( ϕ i , j + 1 2 ϕ i , j + ϕ i , j 1 ) = ρ i , j ,
ρ i , j = ( Δ i , j x Δ i 1 , j x ) + ( Δ i , j y Δ i , j 1 y ) .
2 x 2 ϕ ( x , y ) + 2 y 2 ϕ ( x , y ) ( x , y )
P ϕ k + 1 = C D ϕ k ,
ϕ ^ m , n = ρ ^ m , n 2 ( cos π m M + cos π n N 2 ) .
C i , j = min ( Wt i + 1 , j 2 , Wt i , j 2 ) Δ i , j x min ( Wt i , j 2 , Wt i 1 , j 2 ) Δ i 1 , j x + min ( Wt i , j + 1 2 , Wt i , j 2 ) Δ i , j y min ( Wt i , j 2 , Wt i , j 1 2 ) Δ i , j 1 y .
ρ k = c [ ρϕ w k −ρ ϕ u w k ] = c D ϕ k ,
f = 1 ( x i , j y i , j ) x i , j 2 ,

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