Abstract

Phase unwrapping recovers the actual surface topography from a calculated phase map, which is often degraded by a combination of surface defects and measurement noise. Prevalent unwrapping techniques are unable to differentiate between the surface and noise-created discontinuities, resulting in an improper surface recovery. In our approach the stochastic nature of the phase map is used by the data-dependent systems (DDS) methodology to separate the surface from noisy measurements. The DDS methodology identifies an adequate autoregressive moving-average model representing the measured phase values. An adaptive thresholding scheme carries out the phase discontinuity removal from the model residuals, employing statistical outlier detection. We then recover the surface by using the model and clean residuals. The proposed technique is found useful for unwrapping especially noisy and degraded phase maps.

© 1994 Optical Society of America

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References

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  1. S. M. Pandit, N. Jordache, G. A. Joshi, “Computer rigid disk surface assessment with laser interferometry using DDS,” in Symposium on Manufacturing Science and Engineering, K. K. Ehman, ed., Vol. PED 64 (American Society of Mechanical Engineers, New York, 1993), pp. 13–21.
  2. D. M. Perry, P. J. Morgan, G. M. Robinson, “Three-dimensional surface metrology of magnetic recording materials through direct-phase-detecting microscopic interferometry,” J. Inst. Electron. Radio Eng. 55, 145–150 (1985).
    [CrossRef]
  3. K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 350–391.
  4. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  5. G. Gierloff, “Phase unwrapping by regions,” in Industrial Applications for Optical Data Processing, E. Conley, J. Robillard, eds. (CRC Press, Boca Raton, Fla., 1992), pp. 195–203.
  6. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Philadelphia, Pa., 1993), pp. 194–230.
  7. 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]
  8. S. M. Pandit, G. A. Joshi, “Image enhancement: a data dependent systems approach,” J. Eng. Ind. 116, 247–252 (1994).
    [CrossRef]
  9. S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted by Krieger, Malabar, Fla., 1993).
  10. S. M. Pandit, Modal and Spectrum Analysis: Data Dependent Systems in State Space (Wiley, New York, 1991).
  11. S. M. Pandit, C. R. Weber, G. A. Joshi, “Deterministic and stochastic separation of digital images,” J. Visual Commun. Image Representation 5, 52–61 (1994).
    [CrossRef]

1994 (2)

S. M. Pandit, G. A. Joshi, “Image enhancement: a data dependent systems approach,” J. Eng. Ind. 116, 247–252 (1994).
[CrossRef]

S. M. Pandit, C. R. Weber, G. A. Joshi, “Deterministic and stochastic separation of digital images,” J. Visual Commun. Image Representation 5, 52–61 (1994).
[CrossRef]

1987 (1)

1985 (1)

D. M. Perry, P. J. Morgan, G. M. Robinson, “Three-dimensional surface metrology of magnetic recording materials through direct-phase-detecting microscopic interferometry,” J. Inst. Electron. Radio Eng. 55, 145–150 (1985).
[CrossRef]

1982 (1)

Creath, K.

K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 350–391.

Ghiglia, D. C.

Gierloff, G.

G. Gierloff, “Phase unwrapping by regions,” in Industrial Applications for Optical Data Processing, E. Conley, J. Robillard, eds. (CRC Press, Boca Raton, Fla., 1992), pp. 195–203.

Itoh, K.

Jordache, N.

S. M. Pandit, N. Jordache, G. A. Joshi, “Computer rigid disk surface assessment with laser interferometry using DDS,” in Symposium on Manufacturing Science and Engineering, K. K. Ehman, ed., Vol. PED 64 (American Society of Mechanical Engineers, New York, 1993), pp. 13–21.

Joshi, G. A.

S. M. Pandit, G. A. Joshi, “Image enhancement: a data dependent systems approach,” J. Eng. Ind. 116, 247–252 (1994).
[CrossRef]

S. M. Pandit, C. R. Weber, G. A. Joshi, “Deterministic and stochastic separation of digital images,” J. Visual Commun. Image Representation 5, 52–61 (1994).
[CrossRef]

S. M. Pandit, N. Jordache, G. A. Joshi, “Computer rigid disk surface assessment with laser interferometry using DDS,” in Symposium on Manufacturing Science and Engineering, K. K. Ehman, ed., Vol. PED 64 (American Society of Mechanical Engineers, New York, 1993), pp. 13–21.

Mastin, G. A.

Morgan, P. J.

D. M. Perry, P. J. Morgan, G. M. Robinson, “Three-dimensional surface metrology of magnetic recording materials through direct-phase-detecting microscopic interferometry,” J. Inst. Electron. Radio Eng. 55, 145–150 (1985).
[CrossRef]

Pandit, S. M.

S. M. Pandit, G. A. Joshi, “Image enhancement: a data dependent systems approach,” J. Eng. Ind. 116, 247–252 (1994).
[CrossRef]

S. M. Pandit, C. R. Weber, G. A. Joshi, “Deterministic and stochastic separation of digital images,” J. Visual Commun. Image Representation 5, 52–61 (1994).
[CrossRef]

S. M. Pandit, N. Jordache, G. A. Joshi, “Computer rigid disk surface assessment with laser interferometry using DDS,” in Symposium on Manufacturing Science and Engineering, K. K. Ehman, ed., Vol. PED 64 (American Society of Mechanical Engineers, New York, 1993), pp. 13–21.

S. M. Pandit, Modal and Spectrum Analysis: Data Dependent Systems in State Space (Wiley, New York, 1991).

S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted by Krieger, Malabar, Fla., 1993).

Perry, D. M.

D. M. Perry, P. J. Morgan, G. M. Robinson, “Three-dimensional surface metrology of magnetic recording materials through direct-phase-detecting microscopic interferometry,” J. Inst. Electron. Radio Eng. 55, 145–150 (1985).
[CrossRef]

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Philadelphia, Pa., 1993), pp. 194–230.

Robinson, G. M.

D. M. Perry, P. J. Morgan, G. M. Robinson, “Three-dimensional surface metrology of magnetic recording materials through direct-phase-detecting microscopic interferometry,” J. Inst. Electron. Radio Eng. 55, 145–150 (1985).
[CrossRef]

Romero, L. A.

Weber, C. R.

S. M. Pandit, C. R. Weber, G. A. Joshi, “Deterministic and stochastic separation of digital images,” J. Visual Commun. Image Representation 5, 52–61 (1994).
[CrossRef]

Wu, S. M.

S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted by Krieger, Malabar, Fla., 1993).

Appl. Opt. (1)

J. Eng. Ind. (1)

S. M. Pandit, G. A. Joshi, “Image enhancement: a data dependent systems approach,” J. Eng. Ind. 116, 247–252 (1994).
[CrossRef]

J. Inst. Electron. Radio Eng. (1)

D. M. Perry, P. J. Morgan, G. M. Robinson, “Three-dimensional surface metrology of magnetic recording materials through direct-phase-detecting microscopic interferometry,” J. Inst. Electron. Radio Eng. 55, 145–150 (1985).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Visual Commun. Image Representation (1)

S. M. Pandit, C. R. Weber, G. A. Joshi, “Deterministic and stochastic separation of digital images,” J. Visual Commun. Image Representation 5, 52–61 (1994).
[CrossRef]

Other (6)

S. M. Pandit, N. Jordache, G. A. Joshi, “Computer rigid disk surface assessment with laser interferometry using DDS,” in Symposium on Manufacturing Science and Engineering, K. K. Ehman, ed., Vol. PED 64 (American Society of Mechanical Engineers, New York, 1993), pp. 13–21.

S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted by Krieger, Malabar, Fla., 1993).

S. M. Pandit, Modal and Spectrum Analysis: Data Dependent Systems in State Space (Wiley, New York, 1991).

K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 350–391.

G. Gierloff, “Phase unwrapping by regions,” in Industrial Applications for Optical Data Processing, E. Conley, J. Robillard, eds. (CRC Press, Boca Raton, Fla., 1992), pp. 195–203.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Philadelphia, Pa., 1993), pp. 194–230.

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Figures (6)

Fig. 1
Fig. 1

(a) Wrapped phase in an expanded version of the zone bet 300 and 320 from Fig. 2. (b) Unwrapped phase in an expanded version of the zone between pixels 300 and 320 from Fig. 3.

Fig. 2
Fig. 2

Wrapped phase in a scan line obtained from a computer hard-disk surface.

Fig. 3
Fig. 3

Unwrapped phase with the discontinuity introduced by the sequential phase-unwrapping algorithm applied to Fig. 2.

Fig. 4
Fig. 4

Residuals from modeling the scan line from Fig. 2.

Fig. 5
Fig. 5

Residuals of Fig. 4 after the outliers are replaced by adaptive thresholding.

Fig. 6
Fig. 6

Phase reconstructed with outlier-free residuals from Fig. 5.

Tables (2)

Tables Icon

Table 1 Autoregressive Parameters (Weights for Preceding Pixel Intensities)

Tables Icon

Table 2 Moving-Average Parameters (Weights for Preceding Residuals)

Equations (27)

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I i ( x , y ) = I 0 ( x , y ) { 1 + γ ( x , y ) cos [ θ ( x , y ) + ϕ i ] } ,
I 1 ( x , y ) = I 0 ( x , y ) { 1 + γ ( x , y ) cos [ θ ( x , y ) + ¼ π ] } ,
I 2 ( x , y ) = I 0 ( x , y ) { 1 + γ ( x , y ) cos [ θ ( x , y ) + ³ / π ] } ,
I 3 ( x , y ) = I 0 ( x , y ) { 1 + γ ( x , y ) cos [ θ ( x , y ) + / π ] } .
θ ( x , y ) = tan - 1 [ I 3 ( x , y ) - I 2 ( x , y ) I 1 ( x , y ) - I 2 ( x , y ) ] .
W l [ θ n ] = θ n pv             ( n = 0 , 1 , 2 , , N ) ,
W l [ θ n ] = θ n + 2 π k l n             ( n = 1 , 2 , , N ) ,
- π W l [ θ n ] π .
Δ θ n = θ n - θ n - 1             ( n = 1 , 2 , , N ) .
Δ W 1 [ θ n ] = Δ θ n + 2 π Δ k 1 n .
W 2 { Δ W 1 [ θ n ] } = Δ θ n + 2 π ( Δ k 1 n + k 2 n ) ,
- π Δ θ n π ,
Δ θ n = W 2 { Δ W 1 [ θ n ] } ,
θ n = θ 0 + n = 1 n W 2 { Δ W 1 [ θ n ] } .
X t = ϕ 1 X t - 1 + ϕ 2 X t - 2 + + ϕ n X t - n + a t - θ 1 a t - 1 - - θ m a t - m .
X t = Φ t X 0 + j = 0 t G j a t - j ,
G j = g 1 λ 1 j + g 2 λ 2 j + g 3 λ 3 j + + g n - 1 λ n - 1 j + g n λ n j ,
λ n - ϕ 1 λ n - 1 - - ϕ n = 0 ,
g i = λ i n - 1 - θ 1 λ i n - 2 - - θ m λ i n - m - 1 ( λ i - λ 1 ) ( λ i - λ i - 1 ) ( λ i - λ i + 1 ) ( λ i - λ n ) .
γ 0 = d 1 + d 2 + + d n ,
d i = ( g i g 1 1 - λ i λ n + g i g 2 1 - λ i λ n + + g i g n 1 - λ i λ n ) σ a 2 .
g i λ i j + g i + 1 λ i + 1 j = r j A cos ( ω j + β ) ,
r = λ i = λ i + 1 ,             ω = arccos [ λ i + λ i + 1 2 ( λ i λ i + 1 ) 1 / 2 ] ,
A = 2 g i = 2 g i + 1 ,             β = arctan [ I ( g ) R ( g ) ] .
ζ = - [ ln ( λ i λ i + 1 ) 2 ω ] .
X t - X t - 1 = a t ,
r i = a t / d t ,

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