Abstract

To achieve high measurement accuracy with less computational time-in-phase shifting interferometry, a random phase-shifting algorithm based on principal component analysis and Lissajous ellipse fitting (PCA&LEF) is proposed. It doesn’t need pre-filtering and can obtain relatively accurate phase distribution with only two phase shifted interferograms and less computational time and is suitable for different background intensity, modulation amplitude distributions and noises. Moreover, it can obtain absolutely accurate result when the background intensity and modulation amplitude are perfect and can partly suppress the effect of imperfect background intensity and modulation amplitude. Last but not least, it removes the restriction that PCA needs more than three interferograms with well-distributed phase shifts to subtract relatively accurate mean. The simulations and experiments verify the correctness and feasibility of PCA&LEF.

© 2019 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 (3)

2016 (4)

2015 (1)

2014 (2)

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (3)

2010 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

2008 (2)

2007 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

2006 (2)

2005 (1)

2004 (2)

2003 (2)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

1992 (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

1974 (1)

Belenguer, T.

Brangaccio, D. J.

Brock, N.

Bruning, J. H.

Cai, L. Z.

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[Crossref] [PubMed]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

Carazo, J. M.

Chai, L.

Chen, Q.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Chen, W.

Chen, Y. C.

Cheng, X. C.

Deng, J.

Dong, G. Y.

Estrada, J. C.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Gallagher, J. E.

Han, B.

Hao, J.

Hayes, J.

Herriott, D. R.

Hou, X.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Ishikawa, K.

Jin, W.

Lee, C. M.

Li, C.

Liang, C. W.

Liang, R.

Lin, P. C.

Liu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Liu, Q.

Liu, S.

Lu, X.

Luo, C.

Lv, X.

Ma, S.

Meng, X. F.

Millerd, J.

North-Morris, M.

Novak, M.

Oikawa, Y.

Patorski, K.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Qin, J.

Quiroga, J. A.

Rosenfeld, D. P.

Shen, X. X.

Sorzano, C. O.

Sun, W. J.

Tian, C.

Tian, X.

Trusiak, M.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Vargas, J.

Wan, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wang, D.

Wang, H.

Wang, J.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wang, K.

Wang, Y. R.

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Wang, Z.

White, A. D.

Wu, D.

Wu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wu, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wyant, J.

Xu, J.

Xu, Q.

Xu, X. F.

Yan, R. S.

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

Yang, X. L.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

Yatabe, K.

Zhang, H.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

Zhang, Y.

Zhong, L.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

J. Opt. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

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

J. Optics-UK (1)

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Meas. Sci. Technol. (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Opt. Commun. (1)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Opt. Express (7)

C. Tian and S. Liu, “Two-frame phase-shifting interferometry for testing optical surfaces,” Opt. Express 24(16), 18695–18708 (2016).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[Crossref] [PubMed]

J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Hyper ellipse fitting in subspace method for phase-shifting interferometry: Practical implementation with automatic pixel selection,” Opt. Express 25(23), 29401–29416 (2017).
[Crossref]

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

Y. Zhang, X. Tian, and R. Liang, “Fringe-print-through error analysis and correction in snapshot phase-shifting interference microscope,” Opt. Express 25(22), 26554–26566 (2017).
[Crossref] [PubMed]

Opt. Lett. (10)

J. Vargas, J. A. Quiroga, C. O. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the Gram-Schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012).
[Crossref] [PubMed]

D. Wang and R. Liang, “Simultaneous polarization Mirau interferometer based on pixelated polarization camera,” Opt. Lett. 41(1), 41–44 (2016).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

Optik (Stuttg.) (1)

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

Other (3)

https://www.4dtechnology.com/products/polarimeters/polarcam/ .

https://doi.org/10.6084/m9.figshare.8195426 .

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007).

Supplementary Material (1)

NameDescription
» Code 1       Matlab code of PCA and LEF

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Figures (9)

Fig. 1
Fig. 1 Flow chart of PCA&LEF.
Fig. 2
Fig. 2 Simulated phase distribution and two phase shifted interferograms. (a) The theoretical phase distribution (PV = 31.4159 rad, RMS = 6.6561 rad), (b) and (c) the first interferogram and the second interferogram.
Fig. 3
Fig. 3 The phase distributions and phase error distributions calculated by PCA&LEF and GS in different situations.
Fig. 4
Fig. 4 The ellipses before and after using LEF for PCA&LEF in different situations.
Fig. 5
Fig. 5 RMS phase errors of PCA&LEF and GS with different phase shifts in different situations.
Fig. 6
Fig. 6 The compared results of PCA&LEF and GS in different situations.
Fig. 7
Fig. 7 Experimental results of the circular fringes . (a) One of the phase shifted interferograms, (b) the reference phase distribution extracted by 4-step PSA (PV = 24.9105rad, RMS = 5.0283 rad), (c) and (d) the phase distributions extracted by PCA&LEF (PV = 25.1416 rad, RMS = 5.0314 rad) and GS (PV = 25.1539 rad, RMS = 4.9495 rad), (e) and (f) the differences between the reference and phase distributions extracted by PCA&LEF and GS, (g) and (h) the ellipses before and after using LEF.
Fig. 8
Fig. 8 Experimental results of the straight fringes. (a) One of the phase shifted interferograms, (b) the reference phase distribution extracted by 4-step PSA (PV = 22.2217rad, RMS = 6.0449 rad), (c) and (d) the phase distributions extracted by PCA&LEF (PV = 22.1837 rad, RMS = 6.0488 rad) and GS (PV = 24.0806 rad, RMS = 6.0492 rad), (e) and (f) the differences between the reference and phase distributions extracted by PCA&LEF and GS, (g) and (h) the ellipses before and after using LEF.
Fig. 9
Fig. 9 Experimental results of the complex fringes . (a) One of the phase shifted interferograms, (b) the reference phase distribution extracted by 4-step PSA (PV = 40.4129rad, RMS = 8.4209 rad), (c) and (d) the phase distributions extracted by PCA&LEF (PV = 40.7467 rad, RMS = 8.4380 rad) and GS (PV = 24.0806 rad, RMS = 6.0492 rad), (e) and (f) the differences between the reference and phase distributions extracted by PCA&LEF and GS, (g) and (h) the ellipses before and after using LEF.

Tables (3)

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Table 1 The RMS phase errors and computational time of PCA&LEF and GS in different situations

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Table 2 The computational time of every step for PCA&LEF in different situations

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Table 3 The computational time of every step for PCA&LEF with the different fringes

Equations (33)

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I n (x,y)= a n (x,y)+b (x,y) n cos( φ(x,y)+ δ n )+ ξ n (x,y).
I n = α n I c + β n I s + κ n
I=QΓ+.
C= I T I= ( QΓ+ ) T ( QΓ+ )= Γ T Q T QΓ+ Γ T Q T + T QΓ+ T Γ T Q T QΓ+ T
Γ Γ T =[ α β ][ α β ]=( α 2 α,β α,β β 2 ).
Γ Γ T = P Γ T D Γ P Γ =( P Γ T D Γ 1/2 Γ ^ ) ( Γ ^ T ( D Γ 1/2 ) T P Γ ) T .
Γ= P Γ T D Γ 1/2 Γ ^ .
Γ ^ = D Γ 1/2 P Γ Γ.
D Γ =( λ 1 0 0 λ 2 ).
λ 1,2 = ( α 2 + β 2 )± ( α 2 β 2 ) 2 +4 αβ 2 2 .
P Γ =( ( λ 1 α 2 ) αβ 2 + ( λ 1 α 2 ) 2 αβ αβ 2 + ( λ 2 β 2 ) 2 αβ αβ 2 + ( λ 1 α 2 ) 2 ( λ 2 β 2 ) αβ 2 + ( λ 2 β 2 ) 2 ).
Q T Q=( p q )( p q )=( p 2 pq pq q 2 )= N x × N y ( b 2 cos 2 ( φ ) b 2 cos( φ )sin( φ ) b 2 cos( φ )sin( φ ) b 2 sin 2 ( φ ) ) .
N x × N y b 2 cos( φ )sin( φ ) 0.
N x × N y b 2 cos 2 ( φ ) N x × N y b 2 sin 2 ( φ ) σ.
Q T Q= D Q σ( 1 0 0 1 ).
C Γ ^ T D Γ 1/2 P Γ D Q P Γ T D Γ 1/2 Γ ^ + T .
C Γ ^ T ( D Γ D Q + D ) Γ ^ .
T = Γ ^ T ( Γ ^ T ) T ( Γ ^ T ) Γ ^ = Γ ^ T D Γ ^ .
IQ( P Γ T D Γ 1/2 Γ ^ )+=( Q P Γ T ) D Γ 1/2 Γ ^ +.
C= U T DU.
Z=I U T .
φ=arctan( I s I c )=arctan( z 2 z 1 ).
z i =I Γ ^ T = ( ( Q P Γ T ) D Γ 1/2 + Γ ^ ) i = ( Q ^ D Γ 1/2 + Γ ^ ) i ,i=1,2.
P Γ T =( cos( θ ) sin( θ ) sin( θ ) cos( θ ) ).
Q ^ =Q P Γ T =( bcos( φ ) bsin( φ ) )( cos( θ ) sin( θ ) sin( θ ) cos( θ ) ). =( bcos( φ )cos( θ )bsin( φ )sin( θ ) bcos( φ )sin( θ )+bsin( φ )cos( θ ) ) =( bcos( φ+θ ) bsin( φ+θ ) )=( p ^ q ^ )
z 1 = p ^ λ 1 1/2 + ( Γ ^ ) 1 =bcos( φ+θ ) λ 1 1/2 + ( Γ ^ ) 1 . z 2 = q ^ λ 2 1/2 + ( Γ ^ ) 2 =bsin( φ+θ ) λ 2 1/2 + ( Γ ^ ) 2
cos( φ+θ )= z 1 ( Γ ^ ) 1 b λ 1 1/2 . sin( φ+θ )= z 2 ( Γ ^ ) 2 b λ 2 1/2
( X x 0 a x ) 2 + ( Y y 0 a y ) 2 =1.
a x =b λ 2 1/2 , a y =b λ 2 1/2 , x 0 = ( Γ ^ ) 2 , y 0 = ( Γ ^ ) 1 .
1 a x 2 X 2 + 1 a y 2 Y 2 2 x 0 a x 2 X2 y 0 a y 2 Y+ x 0 2 a x 2 + y 0 2 a y 2 1=0.
F=c x 2 +dxy+e y 2 +fx+gy+h.
a x = 2 c g 2 +e f 2 +h d 2 dfg4ceh ( d 2 4ce )( ( ce ) 2 + d 2 ( c+e ) ) , a y = 2 c g 2 +e f 2 +h d 2 dfg4ceh ( d 2 4ce )( ( ce ) 2 + d 2 ( c+e ) ) . x 0 = 2efdg d 2 4ce , y 0 = 2cgdf d 2 4ce
Φ=φ+θ= tan 1 ( X x 0 Y y 0 a y a x ).

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