Abstract

This paper proposes a robust noise and phase jump detection scheme for noisy phase maps containing height discontinuities. The detection scheme has two primary functions, namely to detect the positions of noise and to locate the positions of the phase jumps. Generally speaking, the removal of noise from a wrapped phase map causes a smearing of the phase jumps and therefore leads to a loss of definition in the unwrapped phase map. However, in the proposed scheme, the boundaries of the phase jump regions are preserved during the noise detection process. The validity of the proposed approach is demonstrated using the simulated and experimental wrapped phase maps of a 3D object containing height discontinuities, respectively. It is shown that the noise and phase jump detection scheme enables the precise and efficient detection of three different types of noise, namely speckle noise, residual noise, and noise at the lateral surfaces of the height discontinuities. Therefore, the proposed scheme represents an ideal solution for the pre-processing of noisy wrapped phase maps prior to their treatment using a filtering algorithm and phase unwrapping algorithm.

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2010 (1)

2009 (2)

2008 (3)

2007 (4)

2006 (1)

2005 (2)

2001 (1)

1999 (1)

H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

1998 (1)

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

1997 (2)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997).
[CrossRef] [PubMed]

1995 (1)

1993 (1)

1991 (1)

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

1989 (1)

1987 (2)

1985 (1)

1983 (1)

Aebischery, H. A.

H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

Bertani, D.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Brady, D.

Capanni, A.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Cetica, M.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Chang, H. Y.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Chen, C. W.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Creath, K.

Dalmau-Cedeño, O. S.

Dubey, S. K.

Eiju, T.

Fetterman, M.

Francini, F.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Ghiglia, D. C.

Hahn, J.

Hao, Q.

Hariharan, P.

Hassebrook, L. G.

Hirose, A.

R. Yamaki and A. Hirose, “Singularity-Spreading Phase Unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007).
[CrossRef]

Hooper, A.

Hossain, M. M.

Hu, C. P.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Huang, M. J.

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008).
[CrossRef]

Huntley, J. M.

Ishii, Y.

Javidi, B.

Kato, M.

Kim, E. H.

Kim, H.

Krishnaswamy, S.

Lau, D. L.

Lee, B.

Lee, C. K.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Legarda-Saenz, R.

Li, X. L.

Liou, J. K.

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008).
[CrossRef]

Liu, K.

Liu, Z.

Macy, W. W.

Marendic, B.

Mastin, G.

Mehta, D. S.

Moon, I.

Oreb, B. F.

Pezzati, L.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Potuluri, P.

Pouet, B. F.

Reichard, K.

Rivera, M.

Robinson, D. W.

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

Romero, L. A.

Saldner, H.

Saldner, H. O.

Shakher, C.

Shi, K.

Spik, A.

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

Stark, H.

Su, W. H.

Wada, A.

Waldner, S.

H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

Wang, B.

Wang, Y. C.

Yamaki, R.

R. Yamaki and A. Hirose, “Singularity-Spreading Phase Unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007).
[CrossRef]

Yang, Y.

Yau, S. T.

Yin, S.

Yuqing, S.

Zebker, H. A.

Zhang, S.

Appl. Opt. (8)

IEEE Trans. Geosci. Rem. Sens. (1)

R. Yamaki and A. Hirose, “Singularity-Spreading Phase Unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007).
[CrossRef]

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

Opt. Commun. (1)

H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

Opt. Eng. (1)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Opt. Express (5)

Opt. Lasers Eng. (2)

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Opt. Lett. (2)

Strain (1)

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008).
[CrossRef]

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Figures (21)

Fig. 1
Fig. 1

(a) Top view of four neighboring pixels. (b) 3D phase map corresponding to Fig. 1(a).

Fig. 2
Fig. 2

(a) Raw wrapped phase map of 3D object containing height discontinuities. Note that the phase map contains Noise A- speckle noise (within rectangular region), Noise B- residual noise (within square and circular regions), and Noise C- noise at the lateral surfaces of the height discontinuities (along red line). Note also that the phase map contains a significant phase jump (within ellipse). (b) Cross-sectional view of wrapped phase map at position corresponding to the 125th column pixel in Fig. 2(a).

Fig. 3
Fig. 3

Noise maps with threshold parameter value of σ A = 2.4 for (a) S1, (b) S2, (c) S3, and (d) S4.

Fig. 4
Fig. 4

Noise maps for case in which (a) S2 detects the noise but S1 does not, and (b) S4 detects the noise but S3 does not.

Fig. 5
Fig. 5

Noise map in superimposed noise detection results (S1, S2, S3, and S4) obtained using threshold parameter values of (a) σ A = 2.4, and (b) σ A = 1.4. Note that in both cases, the pixels within the phase jump region (i.e. within the ellipse) are considered to be “good” pixels by the detection scheme.

Fig. 6
Fig. 6

Phase jump map for threshold parameter values of (a) σ A = 2.4 and (b) σ A = 1.4.

Fig. 7
Fig. 7

Solutions of S(i, j) obtained from Eq. (12). Note that pixel positions returning a value of + 1 are marked as “O” while pixel positions returning a value of −1 are marked as “X”.

Fig. 8
Fig. 8

Raw wrapped phase map of the step height standard sample with (a) a white light source and (b) a He-Ne laser.

Fig. 9
Fig. 9

White-light source results of the noise and phase jump detection scheme with σ Α = 2.4. (a) Noise map and (b) phase jump map.

Fig. 10
Fig. 10

White-light source results of the detection scheme with σ Α = 2.75. (a) Noise map and (b) phase jump map.

Fig. 11
Fig. 11

He-Ne laser results of the detection scheme with σ Α = 2.4. (a) Noise map and (b) phase jump map.

Fig. 12
Fig. 12

He-Ne laser results of the detection scheme with σ Α = 2.75. (a) Noise map and (b) phase jump map.

Fig. 13
Fig. 13

(a) Unwrapping results of Fig. 8(a) with the detection scheme ( σ Α = 2.75). (a) Unwrapping results. (b) Column and row cross-sections.

Fig. 14
Fig. 14

Unwrapping results of Fig. 8(b) with the detection scheme ( σ Α = 2.75). (a) Unwrapping results. (b) Column and row cross-sections.

Fig. 15
Fig. 15

Raw wrapped phase map of the TaSiN sample with (a) a white light source and (b) a He-Ne laser.

Fig. 16
Fig. 16

White-light source results of the detection scheme with σ Α = 2.8. (a) Noise map and (b) phase jump map.

Fig. 17
Fig. 17

He-Ne laser results of the detection scheme with σ Α = 2.8. (a) Noise map and (b) phase jump map.

Fig. 18
Fig. 18

(a) Unwrapping results of Fig. 15(a) with the detection scheme ( σ Α = 2.8). (a) Unwrapping results. (b) Column and row cross-sections.

Fig. 19
Fig. 19

(a) Unwrapping results of Fig. 15(b) with the detection scheme ( σ Α = 2.8). (a) Unwrapping results. (b) Column and row cross-sections.

Fig. 20
Fig. 20

(a) Unwrapping result of Fig. 15(a) only with MACY algorithm. (b) Unwrapping results of Fig. 15(b) only with MACY algorithm.

Fig. 21
Fig. 21

White-light source case. (a) Raw wrapped phase map of the step height standard sample (b) Noise map with σ Α = 2.6 and (c) phase jump map with σ Α = 2.6.

Equations (12)

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S 1 ( i , j ) = [ ϕ ( i + 1 , j ) ϕ ( i , j ) σ Α 2 π ] + [ ϕ ( i + 1 , j + 1 ) ϕ ( i + 1 , j ) + σ Α 2 π ] + [ ϕ ( i , j + 1 ) ϕ ( i + 1 , j + 1 ) σ Α 2 π ] + [ ϕ ( i , j ) ϕ ( i , j + 1 ) + σ Α 2 π ] S 2 ( i , j ) = [ ϕ ( i + 1 , j ) ϕ ( i , j ) + σ Α 2 π ] + [ ϕ ( i + 1 , j + 1 ) ϕ ( i + 1 , j ) σ Α 2 π ] + [ ϕ ( i , j + 1 ) ϕ ( i + 1 , j + 1 ) + σ Α 2 π ] + [ ϕ ( i , j ) ϕ ( i , j + 1 ) σ Α 2 π ] S 3 ( i , j ) = [ ϕ ( i + 1 , j ) ϕ ( i , j ) + σ Α 2 π ] + [ ϕ ( i + 1 , j + 1 ) ϕ ( i + 1 , j ) + σ Α 2 π ] + [ ϕ ( i , j + 1 ) ϕ ( i + 1 , j + 1 ) σ Α 2 π ] + [ ϕ ( i , j ) ϕ ( i , j + 1 ) σ Α 2 π ] S 4 ( i , j ) = [ ϕ ( i + 1 , j ) ϕ ( i , j ) σ Α 2 π ] + [ ϕ ( i + 1 , j + 1 ) ϕ ( i + 1 , j ) σ Α 2 π ] + [ ϕ ( i , j + 1 ) ϕ ( i + 1 , j + 1 ) + σ Α 2 π ] + [ ϕ ( i , j ) ϕ ( i , j + 1 ) + σ Α 2 π ]
S 1 ( i , j ) = [ 0 2 π σ Α 2 π ] + [ 0 0 + σ Α 2 π ] + [ 2 π 0 σ Α 2 π ] + [ 2 π 2 π + σ Α 2 π ]            =-1+[ σ Α 2 π ]+[ σ Α 2 π ]+1+[ σ Α 2 π ]+[ σ Α 2 π ]            =0 
S 1 ( i , j ) = [ ϕ ( i + 1 , j ) ϕ ( i , j ) σ Α 2 π ] = [ ϕ ( i + 1 , j + 1 ) ϕ ( i + 1 , j ) + σ Α 2 π ]            = [ ϕ ( i , j + 1 ) ϕ ( i + 1 , j + 1 ) σ Α 2 π ] = [ ϕ ( i , j ) ϕ ( i , j + 1 ) + σ Α 2 π ]            = 0   ( = S 2 ( i , j ) = S 3 ( i , j ) = S 4 ( i , j ) )
[ P D σ Α 2 π ] = [ P D + σ Α 2 π ] = 0
P D σ Α 2 π < ± 0.5
P D + σ Α 2 π < ± 0.5
  | P D | < π σ Α
[ ϕ ( i + 1 , j ) ϕ ( i , j ) σ Α 2 π ] = [ ϕ ( 251 , 25 ) ϕ ( 250 , 25 ) σ Α 2 π ] = [ 4 .3114 4 .3568 2 . 4 2 π ] = 0
[ ϕ ( i + 1 , j + 1 ) ϕ ( i + 1 , j ) + σ Α 2 π ] = [ ϕ ( 251 , 26 ) ϕ ( 251 , 25 ) + σ Α 2 π ] = [ 4 .3552 4 .3114 + 2 . 4 2 π ] = 0
[ ϕ ( i , j + 1 ) ϕ ( i + 1 , j + 1 ) σ Α 2 π ] = [ ϕ ( 250 , 26 ) ϕ ( 251 , 26 ) σ Α 2 π ] = [ 4 .4075 4 .3552 2 . 4 2 π ] = 0
[ ϕ ( i , j ) ϕ ( i , j + 1 ) + σ Α 2 π ] = [ ϕ ( 250 , 26 ) ϕ ( 251 , 26 ) + σ Α 2 π ] = [ 4 .4075 4 .3552 + 2 . 4 2 π ] = 0
S ( i , j ) = [ ϕ ( i + 1 , j ) ϕ ( i , j ) 2 π ] + [ ϕ ( i + 1 , j + 1 ) ϕ ( i + 1 , j ) 2 π ] + [ ϕ ( i , j + 1 ) ϕ ( i + 1 , j + 1 ) 2 π ] + [ ϕ ( i , j ) ϕ ( i , j + 1 ) 2 π ]

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