Abstract

We propose a hybrid computational framework to reduce motion-induced measurement error by combining the Fourier transform profilometry (FTP) and phase-shifting profilometry (PSP). The proposed method is composed of three major steps: Step 1 is to extract continuous relative phase maps for each isolated object with single-shot FTP method and spatial phase unwrapping; Step 2 is to obtain an absolute phase map of the entire scene using PSP method, albeit motion-induced errors exist on the extracted absolute phase map; and Step 3 is to shift the continuous relative phase maps from Step 1 to generate final absolute phase maps for each isolated object by referring to the absolute phase map with error from Step 2. Experiments demonstrate the success of the proposed computational framework for measuring multiple isolated rapidly moving objects.

© 2016 Optical Society of America

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References

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2016 (5)

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87, 97–102 (2016).
[Crossref]

J.-S. Hyun and S. Zhang, “Enhanced two-frequency phase-shifting method,” Appl. Opt. 55, 4395–4401 (2016).
[Crossref]

Y. An, J.-S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24, 18445–18459 (2016).
[Crossref]

B. Li, Y. An, and S. Zhang, “Single-shot absolute 3d shape measurement with fourier transform profilometry,” Appl. Opt. 55, 5219–5225 (2016).
[Crossref]

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

2015 (2)

M. Servin, J. M. Padilla, A. Gonzalez, and G. Garnica, “Temporal phase-unwrapping of static surfaces with 2-sensitivity fringe-patterns,” Opt. Express 23, 15806–15815 (2015).
[Crossref]

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process 9, 396–408 (2015).
[Crossref]

2014 (4)

2013 (2)

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51, 1213–1222 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

2012 (3)

2011 (3)

2010 (1)

2009 (2)

H. Guo and P. S. Huang, “Absolute phase technique for the fourier transform method,” Opt. Eng. 48, 043609 (2009).
[Crossref]

S. Lei and S. Zhang, “Flexible 3-d shape measurement using projector defocusing,” Opt. Lett. 34, 3080–3082 (2009).
[Crossref]

2008 (1)

H. Guo and P. Huang, “3-d shape measurement by use of a modified fourier transform method,” Proc. SPIE 7066, 70660E (2008).
[Crossref]

2007 (4)

Y. Xiao, X. Su, Q. Zhang, and Z. Li, “3-d profilometry for the impact process with marked fringes tracking,” Optoelectron. Eng. 34, 46–52 (2007).

S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46, 113603 (2007).
[Crossref]

S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
[Crossref]

Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[Crossref]

2006 (1)

2005 (1)

J. Pan, P. S. Huang, and F.-P. Chiang, “Color-coded binary fringe projection technique for 3-d shape measurement,” Opt. Eng. 44, 023606 (2005).
[Crossref]

2004 (2)

2003 (1)

D. P. Towers, J. D. C. Jones, and C. E. Towers, “Optimum frequency selection in multi-frequency interferometry,” Opt. Lett. 28, 1–3 (2003).
[Crossref]

1997 (1)

1990 (1)

L. Guo, X. Su, and J. Li, “Improved fourier transform profilometry for the automatic measurement of 3d object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[Crossref]

1988 (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1985 (1)

1984 (1)

1983 (1)

An, Y.

Budianto, B.

Chen, Q.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Chen, V.

Cheng, Y.-Y.

Chiang, F.-P.

J. Pan, P. S. Huang, and F.-P. Chiang, “Color-coded binary fringe projection technique for 3-d shape measurement,” Opt. Eng. 44, 023606 (2005).
[Crossref]

Cong, P.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process 9, 396–408 (2015).
[Crossref]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Dietrich, P.

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

Feng, F.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Feng, S.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Garnica, G.

Geng, J.

J. Geng, “Structured-light 3d surface imaging: a tutorial,” Adv. Opt. Photonics 3, 128–160 (2011).
[Crossref]

Gonzalez, A.

Gu, G.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Guo, H.

H. Guo and P. S. Huang, “Absolute phase technique for the fourier transform method,” Opt. Eng. 48, 043609 (2009).
[Crossref]

H. Guo and P. Huang, “3-d shape measurement by use of a modified fourier transform method,” Proc. SPIE 7066, 70660E (2008).
[Crossref]

Guo, L.

L. Guo, X. Su, and J. Li, “Improved fourier transform profilometry for the automatic measurement of 3d object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[Crossref]

Heist, S.

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

Hsung, T.-C.

Huang, P.

H. Guo and P. Huang, “3-d shape measurement by use of a modified fourier transform method,” Proc. SPIE 7066, 70660E (2008).
[Crossref]

Huang, P. S.

H. Guo and P. S. Huang, “Absolute phase technique for the fourier transform method,” Opt. Eng. 48, 043609 (2009).
[Crossref]

P. S. Huang and S. Zhang, “Fast three-step phase-shifting algorithm,” Appl. Opt. 45, 5086–5091 (2006).
[Crossref]

J. Pan, P. S. Huang, and F.-P. Chiang, “Color-coded binary fringe projection technique for 3-d shape measurement,” Opt. Eng. 44, 023606 (2005).
[Crossref]

Hyun, J.-S.

Jones, J. D. C.

D. P. Towers, J. D. C. Jones, and C. E. Towers, “Optimum frequency selection in multi-frequency interferometry,” Opt. Lett. 28, 1–3 (2003).
[Crossref]

Karpinsky, N.

Kemao, Q.

Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[Crossref]

Q. Kemao, “Windowed fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[Crossref]

Kühmstedt, P.

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

Lei, S.

Lei, Y.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51, 1213–1222 (2013).
[Crossref]

Li, B.

Li, J.

L. Guo, X. Su, and J. Li, “Improved fourier transform profilometry for the automatic measurement of 3d object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[Crossref]

Li, R.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

Li, X.

Li, Z.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51, 1213–1222 (2013).
[Crossref]

Y. Xiao, X. Su, Q. Zhang, and Z. Li, “3-d profilometry for the impact process with marked fringes tracking,” Optoelectron. Eng. 34, 46–52 (2007).

Lohry, W.

Lun, P.

Lutzke, P.

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

Mutoh, K.

Notni, G.

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

Oliver, J. H.

Padilla, J. M.

Pan, J.

J. Pan, P. S. Huang, and F.-P. Chiang, “Color-coded binary fringe projection technique for 3-d shape measurement,” Opt. Eng. 44, 023606 (2005).
[Crossref]

Quan, C.

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87, 97–102 (2016).
[Crossref]

Sandoz, P.

Schmidt, I.

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

Servin, M.

Shen, G.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

Shi, Y.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51, 1213–1222 (2013).
[Crossref]

Su, X.

Y. Xiao, X. Su, Q. Zhang, and Z. Li, “3-d profilometry for the impact process with marked fringes tracking,” Optoelectron. Eng. 34, 46–52 (2007).

L. Guo, X. Su, and J. Li, “Improved fourier transform profilometry for the automatic measurement of 3d object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[Crossref]

Takeda, M.

Tay, C.

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87, 97–102 (2016).
[Crossref]

Towers, C. E.

D. P. Towers, J. D. C. Jones, and C. E. Towers, “Optimum frequency selection in multi-frequency interferometry,” Opt. Lett. 28, 1–3 (2003).
[Crossref]

Towers, D. P.

D. P. Towers, J. D. C. Jones, and C. E. Towers, “Optimum frequency selection in multi-frequency interferometry,” Opt. Lett. 28, 1–3 (2003).
[Crossref]

Tünnermann, A.

S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, “High-speed three-dimensional shape measurement using gobo projection,” Opt. Lasers Eng. 87, 90–96 (2016).
[Crossref]

Wang, C.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51, 1213–1222 (2013).
[Crossref]

Wang, Y.

Weng, J.

Wu, F.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process 9, 396–408 (2015).
[Crossref]

Wyant, J. C.

Xiao, Y.

Y. Xiao, X. Su, Q. Zhang, and Z. Li, “3-d profilometry for the impact process with marked fringes tracking,” Optoelectron. Eng. 34, 46–52 (2007).

Xing, Y.

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87, 97–102 (2016).
[Crossref]

Xiong, Z.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process 9, 396–408 (2015).
[Crossref]

Yau, S.-T.

S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46, 113603 (2007).
[Crossref]

S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
[Crossref]

Zhang, Q.

Y. Xiao, X. Su, Q. Zhang, and Z. Li, “3-d profilometry for the impact process with marked fringes tracking,” Optoelectron. Eng. 34, 46–52 (2007).

Zhang, S.

J.-S. Hyun and S. Zhang, “Enhanced two-frequency phase-shifting method,” Appl. Opt. 55, 4395–4401 (2016).
[Crossref]

Y. An, J.-S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24, 18445–18459 (2016).
[Crossref]

B. Li, Y. An, and S. Zhang, “Single-shot absolute 3d shape measurement with fourier transform profilometry,” Appl. Opt. 55, 5219–5225 (2016).
[Crossref]

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22, 1287–1301 (2014).
[Crossref]

W. Lohry and S. Zhang, “High-speed absolute three-dimensional shape measurement using three binary dithered patterns,” Opt. Express 22, 26752–26762 (2014).
[Crossref]

B. Li, N. Karpinsky, and S. Zhang, “Novel calibration method for structured light system with an out-of-focus projector,” Appl. Opt. 53, 3415–3426 (2014).
[Crossref]

Y. Wang and S. Zhang, “Three-dimensional shape measurement with binary dithered patterns,” Appl. Opt. 51, 6631–6636 (2012).
[Crossref]

Y. Wang and S. Zhang, “Novel phase coding method for absolute phase retrieval,” Opt. Lett. 37, 2067–2069 (2012).
[Crossref]

Y. Wang, S. Zhang, and J. H. Oliver, “3-d shape measurement technique for multiple rapidly moving objects,” Opt. Express 19, 5149–5155 (2011).
[Crossref]

Y. Wang and S. Zhang, “Superfast multifrequency phase-shifting technique with optimal pulse width modulation,” Opt. Express 19, 5143–5148 (2011).

S. Zhang, “Flexible 3d shape measurement using projector defocusing: Extended measurement range,” Opt. Lett. 35, 931–933 (2010).

S. Lei and S. Zhang, “Flexible 3-d shape measurement using projector defocusing,” Opt. Lett. 34, 3080–3082 (2009).
[Crossref]

S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
[Crossref]

S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46, 113603 (2007).
[Crossref]

P. S. Huang and S. Zhang, “Fast three-step phase-shifting algorithm,” Appl. Opt. 45, 5086–5091 (2006).
[Crossref]

Zhang, Y.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process 9, 396–408 (2015).
[Crossref]

Zhao, S.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process 9, 396–408 (2015).
[Crossref]

Zhong, J.

Zhong, K.

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Supplementary Material (2)

NameDescription
» Visualization 1: MP4 (346 KB)      Visualization 1
» Visualization 2: MP4 (1120 KB)      Visualization 2

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Figures (14)

Fig. 1
Fig. 1 Illustration of the geometric mapping between the camera image region and the corresponding region on the projector sensor if a virtual plane is positioned as zmin [25].
Fig. 2
Fig. 2 Concept of removing 2π discontinuities using the minimum phase map determined from geometric constraints [25]. (a) Regions acquired by the camera at different depth z plane: red dashed windowed region where z = zmin and solid blue windowed region where z > zmin; (b) Φmin and Φ defined on the projector; (c) cross sections of the wrapped phase maps, ϕ1 and ϕ, and their correctly unwrapped phase map Φmin and Φ; (d) case for using fringe patterns with four periods.
Fig. 3
Fig. 3 Simulation of motion-induced measurement error. (a) – (c) high frequency phase shifted patterns; (d) – (f) low frequency phase shifted patterns; (g) reconstructed 3D shape; (h) a cross section of (g) and the ideal sphere; (i) difference between the reconstructed sphere and the ideal sphere.
Fig. 4
Fig. 4 The pipeline of proposed hybrid absolute phase computational framework. The first step is to generate continuous relative phase map Φr using single shot FTP and spatial phase unwrapping; the second step is to generate absolute phase map with error Φe through PSP and geometric constraint; the final step is to retrieve absolute phase map by finding rigid fringe order shift ks.
Fig. 5
Fig. 5 Photograph of our experimental system setup.
Fig. 6
Fig. 6 A cycle of 6 continuously captured fringe images. (a) I 1 h; (b) I 2 h; (c) I 3 h; (d) I 1 l; (e) I 2 l; (f) I 3 l; (g) – (l) close-up views of the left ball in (a) – (f).
Fig. 7
Fig. 7 A sample frame of result using enhanced two-frequency PSP method [24]. (a) Retrieved absolute phase map; (b) reconstructed 3D geometries ( Visualization 1).
Fig. 8
Fig. 8 Continuous relative phase map Φr extraction from single-shot FTP. (a) Captured fringe image I 3 h; (b) wrapped phase map obtained from (a) using FTP; (c) separately unwrapped phase map of each ball in (b) with spatial phase unwrapping.
Fig. 9
Fig. 9 Extraction of absolute phase map with motion-induced error Φe from PSP. (a) One of the three phase shifted fringe pattern; (b) extracted wrapped phase map from I 1 l - I 3 l with motion-induced error before applying geometric constraints; (c) unwrapped phase map Φe using geometric constraints; (d) difference fringe order map ke obtained from using low frequency phase map shown in Fig. 8(c) and phase map shown in (c) using Eq. (13).
Fig. 10
Fig. 10 Histogram method for fringe order determination. (a) Histogram of Fig. 9(d) for the left ball; (b) histogram of Fig. 9(d) for the right ball.
Fig. 11
Fig. 11 Absolute phase Φa retrieval and 3D reconstruction. (a) Retrieved final absolute phase map Φa; (b) reconstructed 3D geometries ( Visualization 1).
Fig. 12
Fig. 12 Comparison between proposed computational framework and PSP based approach. (a) 3D result from PSP approach; (b) residual error of (a) after sphere fitting (RMS error: 6.92 mm); (c) a cross section of sphere fitting; (d) a cross section of residual error; (e) – (f) corresponding plots results from our proposed approach (RMS error: 0.26 mm).
Fig. 13
Fig. 13 3D shape measurement of multiple free-falling ping-pong balls ( Visualization 2). (a) A sample frame image; (b) 3D reconstructed geometry.
Fig. 14
Fig. 14 Illustration of artifacts induced by texture variation. (a) Zoom-in view of the ball inside of the red bounding box of Fig. 13(a); (b) corresponding zoom-in view of the 3D result inside of the red bounding box of Fig. 13(b).

Equations (15)

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I ( x , y ) = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) ] ,
I ( x , y ) = I ( x , y ) + I ( x , y ) 2 [ e j ϕ ( x , y ) + e j ϕ ( x , y ) ] .
I f ( x , y ) = I ( x , y ) 2 e j ϕ ( x , y ) .
ϕ ( x , y ) = tan 1 { I m [ I f ( x , y ) ] R e [ I f ( x , y ) ] } ,
Φ ( x , y ) = ϕ ( x , y ) + 2 π × k ( x , y ) .
I 1 ( x , y ) = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) 2 π / 3 ] ,
I 2 ( x , y ) = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) ] ,
I 3 ( x , y ) = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) + 2 π / 3 ] .
ϕ ( x , y ) = tan 1 3 ( I 1 I 3 ) 2 I 2 I 1 I 3 .
2 π × ( k 1 ) < Φ m i n ϕ < 2 π × k ,
k = c e i l [ Φ m i n ϕ 2 π ] ,
k s = r o u n d { [ Φ a ( u , v ) Φ r ( u , v ) ] / ( 2 π ) } .
k e ( u , v ) = r o u n d { [ Φ e ( u + Δ u , v + Δ v ) × T l T Φ r ( u , v ) ] / ( 2 π ) } ,
k s = m o d e [ k e ( u , v ) ] ,
Φ a ( u , v ) = Φ r ( u , v ) + 2 π × k s .

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