Abstract

Scientific research of a stabilized flapping flight process (e.g. hovering) has been of great interest to a variety of fields including biology, aerodynamics, and bio-inspired robotics. Different from the current passive photogrammetry based methods, the digital fringe projection (DFP) technique has the capability of performing dense superfast (e.g. kHz) 3D topological reconstructions with the projection of defocused binary patterns, yet it is still a challenge to measure a flapping flight process with the presence of rapid flapping wings. This paper presents a novel absolute 3D reconstruction method for a stabilized flapping flight process. Essentially, the slow motion parts (e.g. body) and the fast-motion parts (e.g. wings) are segmented and separately reconstructed with phase shifting techniques and the Fourier transform, respectively. The topological relations between the wings and the body are utilized to ensure absolute 3D reconstruction. Experiments demonstrate the success of our computational framework by testing a flapping wing robot at different flapping speeds.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2017 (1)

2016 (4)

2015 (2)

B. Li, S. Ma, and Y. Zhai, “Fast temporal phase unwrapping method for the fringe reflection technique based on the orthogonal grid fringes,” Appl. Opt. 54, 6282–6290 (2015).
[Crossref] [PubMed]

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with Fourier-assisted phase shifting,” IEEE Journal of Selected Topics in Signal Processing 9, 396–408 (2015).
[Crossref]

2014 (3)

2013 (1)

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 (5)

2011 (2)

2010 (2)

2009 (2)

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

S. M. Walker, A. L. Thomas, and G. K. Taylor, “Photogrammetric reconstruction of high-resolution surface topographies and deformable wing kinematics of tethered locusts and free-flying hoverflies,” J. Royal Soc. Interface 6, 351–366 (2009).
[Crossref]

2007 (3)

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

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

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]

2006 (1)

U. M. L. Norberg and Y. Winter, “Wing beat kinematics of a nectar-feeding bat, glossophaga soricina, flying at different flight speeds and strouhal numbers,” J. Exp. Biol. 209, 3887–3897 (2006).
[Crossref]

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 (1)

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Laser Eng. 42, 245–261 (2004).
[Crossref]

1997 (1)

A. P. Willmott and C. P. Ellington, “The mechanics of flight in the hawkmoth manduca sexta. i. kinematics of hovering and forward flight,” J. Exp. Biol. 200, 2705–2722 (1997).

1993 (1)

R. J. Wootton, “Leading edge section and asymmetric twisting in the wings of flying butterflies (insecta, papilionoidea),” J. Exp. Biol. 180, 105 (1993).

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]

1985 (1)

1984 (1)

An, Y.

Biewener, A. A.

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

Bruning, J.-H.

H. Schreiber and J.-H. Bruning, “Phase Shifting Interferometry”, in Optical Shop Testing, 3 Edition, D. Malacaral,ed. (John Wiley & Sons, 2007).
[Crossref]

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] [PubMed]

Chen, W.

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Laser Eng. 42, 245–261 (2004).
[Crossref]

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]

Clark, C. J.

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

Combes, S. A.

T. L. Hedrick, S. A. Combes, and L. A. Miller, “Recent developments in the study of insect flight,” Canadian Journal of Zoology 93, 925–943 (2014).
[Crossref]

Cong, P.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with Fourier-assisted phase shifting,” IEEE Journal of Selected Topics in Signal Processing 9, 396–408 (2015).
[Crossref]

Deng, X.

Y. Ren, H. Dong, X. Deng, and B. Tobalske, “Turning on a dime: Asymmetric vortex formation in hummingbird maneuvering flight,” Physical Review Fluids 1, 050511 (2016).
[Crossref]

Dong, H.

Y. Ren, H. Dong, X. Deng, and B. Tobalske, “Turning on a dime: Asymmetric vortex formation in hummingbird maneuvering flight,” Physical Review Fluids 1, 050511 (2016).
[Crossref]

C. Koehler, Z. Liang, Z. Gaston, H. Wan, and H. Dong, “3d reconstruction and analysis of wing deformation in free-flying dragonflies,” J. Exp. Biol. 215, 3018–3027 (2012).
[Crossref] [PubMed]

Ellington, C. P.

A. P. Willmott and C. P. Ellington, “The mechanics of flight in the hawkmoth manduca sexta. i. kinematics of hovering and forward flight,” J. Exp. Biol. 200, 2705–2722 (1997).

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] [PubMed]

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] [PubMed]

Gaston, Z.

C. Koehler, Z. Liang, Z. Gaston, H. Wan, and H. Dong, “3d reconstruction and analysis of wing deformation in free-flying dragonflies,” J. Exp. Biol. 215, 3018–3027 (2012).
[Crossref] [PubMed]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, 1998).

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] [PubMed]

Guo, H.

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

Hedrick, T. L.

T. L. Hedrick, S. A. Combes, and L. A. Miller, “Recent developments in the study of insect flight,” Canadian Journal of Zoology 93, 925–943 (2014).
[Crossref]

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

Hsung, T.-C.

Huang, P. S.

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

Hyder, G. A.

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

Hyun, J.-S.

Karpinsky, N.

Koehler, C.

C. Koehler, Z. Liang, Z. Gaston, H. Wan, and H. Dong, “3d reconstruction and analysis of wing deformation in free-flying dragonflies,” J. Exp. Biol. 215, 3018–3027 (2012).
[Crossref] [PubMed]

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.

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

Liang, Z.

C. Koehler, Z. Liang, Z. Gaston, H. Wan, and H. Dong, “3d reconstruction and analysis of wing deformation in free-flying dragonflies,” J. Exp. Biol. 215, 3018–3027 (2012).
[Crossref] [PubMed]

Liu, Z.

Lohry, W.

W. Lohry and S. Zhang, “Fourier transform profilometry using a binary area modulation technique,” Opt. Eng. 51, 113602 (2012).
[Crossref]

Lun, P.

Ma, S.

Malacara, D.

D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).
[Crossref]

Miller, L. A.

T. L. Hedrick, S. A. Combes, and L. A. Miller, “Recent developments in the study of insect flight,” Canadian Journal of Zoology 93, 925–943 (2014).
[Crossref]

Norberg, U. M. L.

U. M. L. Norberg and Y. Winter, “Wing beat kinematics of a nectar-feeding bat, glossophaga soricina, flying at different flight speeds and strouhal numbers,” J. Exp. Biol. 209, 3887–3897 (2006).
[Crossref]

Oliver, J.

Oliver, J. H.

Padilla, J. M.

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (John Wiley & Sons, 2014).

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]

Powers, D. R.

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, 1998).

Quan, C.

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

Quiroga, J. A.

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (John Wiley & Sons, 2014).

Ren, Y.

Y. Ren, H. Dong, X. Deng, and B. Tobalske, “Turning on a dime: Asymmetric vortex formation in hummingbird maneuvering flight,” Physical Review Fluids 1, 050511 (2016).
[Crossref]

Schreiber, H.

H. Schreiber and J.-H. Bruning, “Phase Shifting Interferometry”, in Optical Shop Testing, 3 Edition, D. Malacaral,ed. (John Wiley & Sons, 2007).
[Crossref]

Servin, M.

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (John Wiley & Sons, 2014).

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]

Su, X.

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

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Laser Eng. 42, 245–261 (2004).
[Crossref]

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]

Tay, C.

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

Taylor, G. K.

S. M. Walker, A. L. Thomas, and G. K. Taylor, “Photogrammetric reconstruction of high-resolution surface topographies and deformable wing kinematics of tethered locusts and free-flying hoverflies,” J. Royal Soc. Interface 6, 351–366 (2009).
[Crossref]

Thomas, A. L.

S. M. Walker, A. L. Thomas, and G. K. Taylor, “Photogrammetric reconstruction of high-resolution surface topographies and deformable wing kinematics of tethered locusts and free-flying hoverflies,” J. Royal Soc. Interface 6, 351–366 (2009).
[Crossref]

Tobalske, B.

Y. Ren, H. Dong, X. Deng, and B. Tobalske, “Turning on a dime: Asymmetric vortex formation in hummingbird maneuvering flight,” Physical Review Fluids 1, 050511 (2016).
[Crossref]

Tobalske, B. W.

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

van der Weide, D.

Walker, S. M.

S. M. Walker, A. L. Thomas, and G. K. Taylor, “Photogrammetric reconstruction of high-resolution surface topographies and deformable wing kinematics of tethered locusts and free-flying hoverflies,” J. Royal Soc. Interface 6, 351–366 (2009).
[Crossref]

Wan, H.

C. Koehler, Z. Liang, Z. Gaston, H. Wan, and H. Dong, “3d reconstruction and analysis of wing deformation in free-flying dragonflies,” J. Exp. Biol. 215, 3018–3027 (2012).
[Crossref] [PubMed]

Wang, Y.

Warrick, D. R.

B. W. Tobalske, D. R. Warrick, C. J. Clark, D. R. Powers, T. L. Hedrick, G. A. Hyder, and A. A. Biewener, “Three-dimensional kinematics of hummingbird flight,” J. Exp. Biol. 210, 2368–2382 (2007).
[Crossref] [PubMed]

Willmott, A. P.

A. P. Willmott and C. P. Ellington, “The mechanics of flight in the hawkmoth manduca sexta. i. kinematics of hovering and forward flight,” J. Exp. Biol. 200, 2705–2722 (1997).

Winter, Y.

U. M. L. Norberg and Y. Winter, “Wing beat kinematics of a nectar-feeding bat, glossophaga soricina, flying at different flight speeds and strouhal numbers,” J. Exp. Biol. 209, 3887–3897 (2006).
[Crossref]

Wootton, R. J.

R. J. Wootton, “Leading edge section and asymmetric twisting in the wings of flying butterflies (insecta, papilionoidea),” J. Exp. Biol. 180, 105 (1993).

Wu, F.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with Fourier-assisted phase shifting,” IEEE Journal of Selected Topics in Signal Processing 9, 396–408 (2015).
[Crossref]

Wyant, J. C.

Xiao, Y.

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Supplementary Material (3)

NameDescription
» Visualization 1       Result of dynamic 3D reconstruction of a flapping flight process when the flapping speed is set at 7 cycles/second (slow).
» Visualization 2       Result of dynamic 3D reconstruction of a flapping flight process when the flapping speed is set at 12 cycles/second (moderate).
» Visualization 3       Result of dynamic 3D reconstruction of a flapping flight process when the flapping speed is set at 21 cycles/second (fast).

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Figures (9)

Fig. 1
Fig. 1 Proposed hybrid computational framework; the image with pure white fringe projection is used to separate the body and the wings; the relative phase maps of the wings were extracted using MFTP method [10] and spatial phase unwrapping [35]; the absolute phase of the body were extracted by phase shifting and enhanced two-wavelength phase unwrapping method [23]; the relative phase maps of the wings are shifted to absolute ones using geometric relations between the body and the wings; the final absolute phase map is produced by merging all absolute phase maps.
Fig. 2
Fig. 2 Segmentation of the body and the wings using template matching. (a) The template image manually extracted in the initial frame; (b) the matched body part on the next frame; (c) generated mask image for the body; (d) generated mask image for the wings by subtracting body from overall image.
Fig. 3
Fig. 3 Body absolute phase retrieval using enhanced two-wavelength phase-shifting. (a) a sample low frequency fringe image; (b) a sample high frequency fringe image; (c) resultant low frequency wrapped phase maps ϕ body L; (d) resultant high frequency wrapped phase maps ϕ body H; (e) an artificial phase plane Φmin; (f) unwrapped phase Φ body L of (c) by pixel-by-pixel referring to (e); (g) final absolute phase map Φ body H for the body.
Fig. 4
Fig. 4 Comparison of extracted 3D of the body part. (a) The texture image; (b) reconstructed 3D result using the absolute phases extracted from MFTP; (c) reconstructed 3D result using the absolute phases extracted from four-step phase-shifting.
Fig. 5
Fig. 5 Relative phase extraction for the wings. (a) fringe image I1 with left wing mask applied; (b) wrapped phase map for the left wing using MFTP; (c) unwrapped relative phase map of (b) using spatial phase unwrapping; (d) – (f) corresponding fringes and phase maps for the right wing.
Fig. 6
Fig. 6 Using geometric relations of the wings and the body to shift the relative phase maps of the wings to absolute ones. (a) – (c) The same cross-section picked for the body’s absolute phase and the wings’ relative phase; (d) the plotted cross-sections before shifting; (e) the plotted cross-sections after shifting; (f) the resultant wings’ absolute phase map Φ wings A.
Fig. 7
Fig. 7 Final absolute phase computation and 3D reconstruction. (a) Computed absolute phase map by merging Fig. 6(a) and 6(f); (b) comparing proposed method with conventional temporal phase unwrapping (enhanced two-wavelength method [23]) by plotting a cross-section; (c) the difference of the cross-section in (b) (mean difference: 0.27 rad; RMS difference: 1.34 rad); (d) – (e) reconstructed 3D geometry using proposed method and conventional temporal phase unwrapping (associated video Visualization 1).
Fig. 8
Fig. 8 Sample results of robotic bird measurement with a flapping speed of 12 cycles/second (associated video Visualization 2). (a) A sample downstroke 2D image; (b) reconstructed 3D geometry of (a) using proposed method; (b) reconstructed 3D geometry of (a) using conventional enhanced two-wavelength method [23]; (e) – (g) corresponding 2D image and 3D results of a sample upstroke frame.
Fig. 9
Fig. 9 Sample results of robotic bird measurement with a flapping speed of 21 cycles/second (associated video Visualization 3). (a) A sample downstroke 2D image; (b) reconstructed 3D geometry of (a) using proposed method; (b) reconstructed 3D geometry of (a) using conventional enhanced two-wavelength method [23]; (e) – (g) corresponding 2D image and 3D results of a sample upstroke frame.

Equations (19)

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I 1 = I ( x , y ) + I ( x , y ) cos [ φ ( x , y ) ] ,
I 2 = I ( x , y ) + I ( x , y ) cos [ φ ( x , y ) ] ,
I ( x , y ) = I 1 + I 2 2 .
I = ( I 1 I 2 ) / 2 = I ( x , y ) cos [ φ ( x , y ) ] .
I = 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 { Im [ I f ( x , y ) ] Re [ I f ( x , y ) ] } ,
I 3 = I ( x , y ) + I ( x , y ) sin [ φ ( x , y ) ] ,
I 4 = I ( x , y ) I ( x , y ) sin [ φ ( x , y ) ] ,
φ = tan 1 ( I 3 I 4 I 1 I 2 ) .
Φ ( x , y ) = φ ( x , y ) + k ( x , y ) × 2 π .
k L = ceil ( Φ min φ body L 2 π ) ,
Φ body L = φ body L + k L × 2 π .
k body = round ( Φ body L × T L T H φ body H 2 π ) ,
Φ body H = φ body H + k body × 2 π
y r e f = a × x + b .
k shift = mode [ round ( y ref y wings 2 π ) ] .
Φ wings A = Φ wings R + k shift × 2 π .
Φ final = Φ body H × M body + Φ wings A × M wings .

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