Abstract

For 3D objects with height discontinuities, the image reconstruction performance of interferometric systems is adversely affected by the presence of noise in the wrapped phase map. Various schemes have been proposed for detecting residual noise, speckle noise and noise at the lateral surfaces of the discontinuities. However, in most schemes, some noisy pixels are missed and noise detection errors occur. Accordingly, this paper proposes two robust filters (designated as Filters A and B, respectively) for improving the performance of the phase unwrapping process for objects with height discontinuities. Filter A comprises a noise and phase jump detection scheme and an adaptive median filter, while Filter B replaces the detected noise with the median phase value of an N × N mask centered on the noisy pixel. Filter A enables most of the noise and detection errors in the wrapped phase map to be removed. Filter B then detects and corrects any remaining noise or detection errors during the phase unwrapping process. Three reconstruction paths are proposed, Path I, Path II and Path III. Path I combines the path-dependent MACY algorithm with Filters A and B, while Paths II and III combine the path-independent cellular automata (CA) algorithm with Filters A and B. In Path II, the CA algorithm operates on the whole wrapped phase map, while in Path III, the CA algorithm operates on multiple sub-maps of the wrapped phase map. The simulation and experimental results confirm that the three reconstruction paths provide a robust and precise reconstruction performance given appropriate values of the parameters used in the detection scheme and filters, respectively. However, the CA algorithm used in Paths II and III is relatively inefficient in identifying the most suitable unwrapping paths. Thus, of the three paths, Path I yields the lowest runtime.

© 2012 OSA

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2012 (2)

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

2011 (4)

J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express 19(4), 3086–3105 (2011).
[CrossRef] [PubMed]

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011).
[CrossRef]

L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011).
[CrossRef]

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011).
[CrossRef]

2010 (1)

2009 (1)

2008 (2)

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

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]

2007 (2)

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45(10), 3240–3251 (2007).
[CrossRef]

S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007).
[CrossRef] [PubMed]

2005 (1)

2002 (1)

A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. 40(3), 699–709 (2002).
[CrossRef]

1999 (2)

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

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999).
[CrossRef] [PubMed]

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 (3)

1995 (2)

B. F. Pouet and S. Krishnaswamy, “Technique for the removal of speckle phase in electronic speckle interferometry,” Opt. Lett. 20(3), 318–320 (1995).
[CrossRef] [PubMed]

R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process. 3(5), 325–333 (1995).
[CrossRef]

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]

1988 (1)

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

1987 (1)

1985 (1)

T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[CrossRef]

1983 (1)

Aebischer, H. A.

H. A. Aebischer 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]

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]

Cheng, X.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011).
[CrossRef]

Chu, T. C.

T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[CrossRef]

Cuche, E.

Cui, H.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011).
[CrossRef]

Dai, N.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011).
[CrossRef]

Depeursinge, C.

Dubey, S. K.

Estrada, J. C.

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

Flynn, T. J.

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.

Goldstein, R.

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Hao, Q.

Hassebrook, L. G.

Hirose, A.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45(10), 3240–3251 (2007).
[CrossRef]

A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. 40(3), 699–709 (2002).
[CrossRef]

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.

Javidi, B.

Krishnaswamy, S.

Lau, D. L.

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]

Liao, W.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011).
[CrossRef]

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, Y.

L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011).
[CrossRef]

L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011).
[CrossRef]

Lo, Y. L.

Lopez-Sanchez, J. M.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

Macy, W. W.

Marquet, P.

Martinez-Espla, J. J.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

Martinez-Marin, T.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

Mastin, G.

Mehta, D. S.

Moon, I.

Navarro, M. A.

Peters, W. H.

T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[CrossRef]

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]

Pouet, B. F.

Quiroga, J. A.

Ranson, W. F.

T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[CrossRef]

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.

Servin, M.

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

Shakher, C.

Smits, R.

R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process. 3(5), 325–333 (1995).
[CrossRef]

Song, L.

L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011).
[CrossRef]

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]

Suksmono, A. B.

A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. 40(3), 699–709 (2002).
[CrossRef]

Sutton, M. A.

T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[CrossRef]

Vargas, J.

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).

Waldner, S.

H. A. Aebischer 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, Y. C.

Weng, J. F.

Werner, C.

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Xianming, X.

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011).
[CrossRef]

Yamaki, R.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45(10), 3240–3251 (2007).
[CrossRef]

Yegnanarayana, B.

R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process. 3(5), 325–333 (1995).
[CrossRef]

Yiming, P.

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011).
[CrossRef]

Yue, H.

L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011).
[CrossRef]

Yuqing, S.

Zebker, H.

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Appl. Opt. (5)

Exp. Mech. (1)

T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[CrossRef]

IEEE Trans. Geosci Remote Sens. (1)

A. B. Suksmono and A. Hirose, “Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its application to phase unwrapping problem,” IEEE Trans. Geosci Remote Sens. 40(3), 699–709 (2002).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45(10), 3240–3251 (2007).
[CrossRef]

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a grid-based filter to solve InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

IEEE Trans. Speech Audio Process. (1)

R. Smits and B. Yegnanarayana, “Determination of instants of significant excitation in speech using group delay function,” IEEE Trans. Speech Audio Process. 3(5), 325–333 (1995).
[CrossRef]

IET Radar Sonar Navig. (1)

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011).
[CrossRef]

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

Opt. Commun. (1)

H. A. Aebischer 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. (3)

L. Song, H. Yue, Y. Liu, and Y. Liu, “Phase unwrapping method based on reliability and digital point array,” Opt. Eng. 50(4), 043605–043612 (2011).
[CrossRef]

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602–063608 (2011).
[CrossRef]

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 (4)

Opt. Laser. Eng. (1)

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Laser. Eng. available online (2012).

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)

Radio Sci. (1)

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

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]

Other (3)

I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Springer, 1990).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, 1984).

R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge Univ. Press, 1989).

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Figures (30)

Fig. 1
Fig. 1

N × N mask used in Filter B during phase unwrapping process.

Fig. 2
Fig. 2

Extension of wrapped phase sub-maps by 3 pixels in row and column directions.

Fig. 3
Fig. 3

Flowchart of three image reconstruction paths.

Fig. 4
Fig. 4

(a) Noisy wrapped phase map. (b) Phase difference values of pixels in pixel column 125 in noisy wrapped phase map.

Fig. 5
Fig. 5

Detection results obtained from Filter A for noisy wrapped phase map in Fig. 4(a) given threshold parameter of σ Α = 2.4: (a) noise map and (b) phase jump map.

Fig. 6
Fig. 6

(a) Noise-reduced wrapped phase map after repeated filtering by Filter A with detection threshold setting of = 2.4. (b) Phase difference values of pixels in pixel column 125 of noise-reduced wrapped phase map in Fig. 6(a).

Fig. 7
Fig. 7

Detection results obtained when applying Filter A twice: (a) noise map and (b) phase jump map.

Fig. 8
Fig. 8

(a) Unwrapping results obtained using Path I (G = 20). (b) Phase difference values of pixels in pixel column 125 (upper) and pixel row 203 (lower).

Fig. 9
Fig. 9

(a) Unwrapping results obtained using Path I without Filter B (G = 20). (b) Unwrapping results obtained using Path I with Filter B (G = 6).

Fig. 10
Fig. 10

(a) Unwrapping results obtained using Path II (G = 20). (b) Phase difference values of pixels in pixel column 125 (upper) and pixel row 203 (lower).

Fig. 11
Fig. 11

(a) Unwrapping results obtained using Path II without Filter B and with G = 20. (b) Unwrapping results obtained using Path II with Filter B and G = 6.

Fig. 12
Fig. 12

Noise-reduced wrapped phase sub-map extending from pixel rows 99~147 and pixel columns 83~123. Note that the green line indicates the stitching line.

Fig. 13
Fig. 13

(a) Unwrapping results obtained using Path III (G = 20). (b) Phase difference values of pixels in pixel column 125 (upper) and pixel row 203 (lower).

Fig. 14
Fig. 14

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

Fig. 15
Fig. 15

Detection results obtained for standard step height sample given white-light source and σ Α,Β = 2.75. (a) Noise map and (b) phase jump map.

Fig. 16
Fig. 16

Detection results obtained for standard step height sample given laser source and = 2.75. (a) Noise map and (b) phase jump map.

Fig. 17
Fig. 17

(a) Path I reconstruction results for standard step height sample given white-light source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

Fig. 18
Fig. 18

(a) Path II reconstruction results for standard step height sample given white-light source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

Fig. 19
Fig. 19

(a) Path III reconstruction results for standard step height sample given white-light source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

Fig. 20
Fig. 20

(a) Path I reconstruction results for standard step height sample given laser source. (b) Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower).

Fig. 21
Fig. 21

Pixel height distributions in pixel column 125 (upper) and pixel row 70 (lower) given laser source and: (a) Path II reconstruction route and (b) Path III reconstruction route

Fig. 22
Fig. 22

Detection results obtained for rough TaSiN sample given white-light source and = 2.75. (a) Noise map and (b) phase jump map.

Fig. 23
Fig. 23

(a) Raw wrapped phase map of TaSiN sample obtained using white-light source. (b) Noise-reduced wrapped phase map of TaSiN sample. Note that the arrows indicate the position of noise at the lateral surface.

Fig. 24
Fig. 24

(a) Path I reconstruction results obtained for TaSiN sample given white-light source. (b) Phase difference values of pixels in pixel column 120 (upper) and pixel row 170 (lower).

Fig. 25
Fig. 25

Path I detection results and reconstruction results given laser source and threshold parameter setting of σ Α,Β = 2.75 in detection scheme. (a) Noise map, (b) phase jump map, and (c) reconstructed image.

Fig. 26
Fig. 26

Path III detection results and reconstruction results given white-light source and threshold parameter setting of = 2.75 in detection scheme (a) Noise map, (b) phase jump map, and (c) reconstructed image.

Fig. 27
Fig. 27

Comparisons in the Path II reconstruction results. (a) Corresponding Path I result of Fig. 24(a), (b) corresponding Path I result of Fig. 25(c), and (c) corresponding Path III result of Fig. 26(c).

Fig. 28
Fig. 28

(a) Path I reconstruction results for sample with two different height discontinuities given white-light source. (b) Phase difference values of pixels in pixel column 190 (upper) and pixel row 120 (lower).

Fig. 29
Fig. 29

(a) Path III reconstruction results for sample with two different height discontinuities given white-light source. (b) Phase difference values of pixels in pixel column 190 (upper) and pixel row 120 (lower).

Fig. 30
Fig. 30

(a) Path I reconstruction results for sample with two different height discontinuities given laser source. (b) Phase difference values of pixels in pixel column 190 (upper) and pixel row 120 (lower).

Tables (4)

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Table 1 Standard Deviation of Phase Values of Pixels along Stitching Lines in Wrapped and Unwrapped Phase Sub-Maps

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Table 2 Summary of Simulation Results for Paths I, II and III

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Table 3 Standard Deviation of Phase Values of Pixels in Stitching Lines in Wrapped and Unwrapped Phase Sub-maps (without Filter B)

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Table 4 Summary of Experimental Results Obtained for Standard Step Height Sample

Equations (1)

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S1(i,j)=[ ϕ(i+1,j)ϕ(i,j) σ Α,Β 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j)+ σ Α,Β 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1) σ Α,Β 2π ]+[ ϕ(i,j)ϕ(i,j+1)+ σ Α,Β 2π ] S2(i,j)=[ ϕ(i+1,j)ϕ(i,j)+ σ Α,Β 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j) σ Α,Β 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1)+ σ Α,Β 2π ]+[ ϕ(i,j)ϕ(i,j+1) σ Α,Β 2π ] S3(i,j)=[ ϕ(i+1,j)ϕ(i,j)+ σ Α,Β 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j)+ σ Α,Β 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1) σ Α,Β 2π ]+[ ϕ(i,j)ϕ(i,j+1) σ Α,Β 2π ] S4(i,j)=[ ϕ(i+1,j)ϕ(i,j) σ Α,Β 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j) σ Α,Β 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1)+ σ Α,Β 2π ]+[ ϕ(i,j)ϕ(i,j+1)+ σ Α,Β 2π ]

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