Abstract

The accuracy of phase-shifting interferometers (PSI) is crippled by nonlinearity of the phase shifter and instability of the environment such as vibration and air turbulence. A general algorithm, utilizing Lissajous figures and ellipse fitting, of correcting the phase extraction error in the phase shifting interferometry is described in this paper. By plotting N against D, where N and D represent the numerator and denominator terms of the phase extraction function (i.e. an arctangent function) respectively, a Lissajous ellipse is created. Once the parameters of the ellipse are determined by ellipse fitting, one can transform the ellipse to a unit circle (ETC). Through this process the phase extraction error caused by random phase shift errors can be corrected successfully. Proposed method is non-iterated, adapts to all phase shifting algorithms (PSAs), and has high accuracy. Some factors that may affect the performance of proposed method are discussed in numerical simulations. Optical experiments are implemented to validate the effectiveness of proposed algorithm.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies

Yu Zhang, Xiaobo Tian, and Rongguang Liang
Opt. Express 26(12) 15059-15071 (2018)

Fringe-print-through error analysis and correction in snapshot phase-shifting interference microscope

Yu Zhang, Xiaobo Tian, and Rongguang Liang
Opt. Express 25(22) 26554-26566 (2017)

Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method

Jiancheng Xu, Weimin Jin, Liqun Chai, and Qiao Xu
Opt. Express 19(21) 20483-20492 (2011)

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing,2nd ed.(Taylor and Francis, 2005).
  2. Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985).
    [Crossref] [PubMed]
  3. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993).
    [Crossref] [PubMed]
  4. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
    [Crossref] [PubMed]
  5. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
    [Crossref]
  6. L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009).
    [Crossref] [PubMed]
  7. L. L. Deck, “Model-based phase shifting interferometry,” Appl. Opt. 53(21), 4628–4636 (2014).
    [Crossref] [PubMed]
  8. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23(4), 361–364 (1984).
    [Crossref]
  9. B. K. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for micro-surface profiling of flat surfaces,” Opt. Commun. 190(1-6), 109–116 (2001).
    [Crossref]
  10. J. Millerd, N. Brock, and J. Hayeset, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
    [Crossref]
  11. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
    [Crossref]
  12. B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
    [Crossref]
  13. 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]
  14. Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
    [Crossref]
  15. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007).
    [Crossref] [PubMed]
  16. K. Kinnstaetter, A. W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27(24), 5082–5089 (1988).
    [Crossref] [PubMed]
  17. 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]
  18. C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
    [Crossref]
  19. K. S. Moon and Y. D. Wang, “Accuracy improvement of phase shift interferometry,” Proc. SPIE 5264, 0277 (2003).
    [Crossref]
  20. A. Armando, V. F. Analucia, and F. M. Allison, “Use of Generalized N-dimensional Lissajous Figures for Phase Retrieval from Sequences of Interferometric Images with Unknown Phase Shifts,” W. Osten (ed.), Fringe 2013, 191–196 (2014).
  21. B. Kimbrough, “Correction of Errors in Polarization Based Dynamic Phase Shifting Interferometers,” International Journal of Optomechatronics 8(4), 304–312 (2014).
    [Crossref]
  22. G. Vladimir, I. Sergey, K. Roman, and H. Dmitry, “Generic algorithm of phase reconstruction in phase-shifting interferometry,” Opt. Eng. 52(3), 030501(2013).
  23. J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus 18(5), 65–67 (1982).
  24. C. Wei, M. Chen, H. Guo, and Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” Proc. SPIE 3478, 0277 (1998).
    [Crossref]
  25. C. J. Evans, R. E. Parks, P. J. Sullivan, and J. S. Taylor, “Visualization of surface figure by the use of Zernike polynomials,” Appl. Opt. 34(34), 7815–7819 (1995).
    [Crossref] [PubMed]
  26. J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23(4), 350–352 (1984).
    [Crossref]

2014 (2)

B. Kimbrough, “Correction of Errors in Polarization Based Dynamic Phase Shifting Interferometers,” International Journal of Optomechatronics 8(4), 304–312 (2014).
[Crossref]

L. L. Deck, “Model-based phase shifting interferometry,” Appl. Opt. 53(21), 4628–4636 (2014).
[Crossref] [PubMed]

2013 (1)

G. Vladimir, I. Sergey, K. Roman, and H. Dmitry, “Generic algorithm of phase reconstruction in phase-shifting interferometry,” Opt. Eng. 52(3), 030501(2013).

2009 (1)

2007 (2)

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007).
[Crossref] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

2004 (2)

J. Millerd, N. Brock, and J. Hayeset, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

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]

2003 (1)

K. S. Moon and Y. D. Wang, “Accuracy improvement of phase shift interferometry,” Proc. SPIE 5264, 0277 (2003).
[Crossref]

2001 (1)

B. K. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for micro-surface profiling of flat surfaces,” Opt. Commun. 190(1-6), 109–116 (2001).
[Crossref]

1998 (1)

C. Wei, M. Chen, H. Guo, and Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” Proc. SPIE 3478, 0277 (1998).
[Crossref]

1997 (1)

1995 (2)

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

C. J. Evans, R. E. Parks, P. J. Sullivan, and J. S. Taylor, “Visualization of surface figure by the use of Zernike polynomials,” Appl. Opt. 34(34), 7815–7819 (1995).
[Crossref] [PubMed]

1994 (1)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

1993 (1)

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]

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

1988 (1)

1987 (1)

1985 (1)

1984 (2)

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[Crossref]

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23(4), 361–364 (1984).
[Crossref]

1982 (1)

J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus 18(5), 65–67 (1982).

Bo, T.

B. K. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for micro-surface profiling of flat surfaces,” Opt. Commun. 190(1-6), 109–116 (2001).
[Crossref]

Brock, N.

J. Millerd, N. Brock, and J. Hayeset, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Chen, M.

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007).
[Crossref] [PubMed]

C. Wei, M. Chen, H. Guo, and Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” Proc. SPIE 3478, 0277 (1998).
[Crossref]

Cheng, Y.-Y.

Deck, L. L.

Dmitry, H.

G. Vladimir, I. Sergey, K. Roman, and H. Dmitry, “Generic algorithm of phase reconstruction in phase-shifting interferometry,” Opt. Eng. 52(3), 030501(2013).

Eiju, T.

Evans, C. J.

Farrant, D. I.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

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]

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[Crossref]

Guo, H.

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007).
[Crossref] [PubMed]

C. Wei, M. Chen, H. Guo, and Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” Proc. SPIE 3478, 0277 (1998).
[Crossref]

Han, B.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

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]

Hariharan, P.

Hayeset, J.

J. Millerd, N. Brock, and J. Hayeset, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Hibino, K.

Kim, S.-W.

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kimbrough, B.

B. Kimbrough, “Correction of Errors in Polarization Based Dynamic Phase Shifting Interferometers,” International Journal of Optomechatronics 8(4), 304–312 (2014).
[Crossref]

Kinnstaetter, K.

Kong, B.

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Larkin, K. G.

Lohmann, A. W.

Millerd, J.

J. Millerd, N. Brock, and J. Hayeset, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Moon, K. S.

K. S. Moon and Y. D. Wang, “Accuracy improvement of phase shift interferometry,” Proc. SPIE 5264, 0277 (2003).
[Crossref]

Moore, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23(4), 361–364 (1984).
[Crossref]

Ngoi, B. K.

B. K. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for micro-surface profiling of flat surfaces,” Opt. Commun. 190(1-6), 109–116 (2001).
[Crossref]

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

Oreb, B. F.

Parks, R. E.

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

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]

Roman, K.

G. Vladimir, I. Sergey, K. Roman, and H. Dmitry, “Generic algorithm of phase reconstruction in phase-shifting interferometry,” Opt. Eng. 52(3), 030501(2013).

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

Schwider, J.

Sergey, I.

G. Vladimir, I. Sergey, K. Roman, and H. Dmitry, “Generic algorithm of phase reconstruction in phase-shifting interferometry,” Opt. Eng. 52(3), 030501(2013).

Sivakumar, N. R.

B. K. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for micro-surface profiling of flat surfaces,” Opt. Commun. 190(1-6), 109–116 (2001).
[Crossref]

Smythe, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23(4), 361–364 (1984).
[Crossref]

Streibl, N.

Sullivan, P. J.

Surrel, Y.

Taylor, J. S.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

Venkatakrishnan, K.

B. K. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for micro-surface profiling of flat surfaces,” Opt. Commun. 190(1-6), 109–116 (2001).
[Crossref]

Vladimir, G.

G. Vladimir, I. Sergey, K. Roman, and H. Dmitry, “Generic algorithm of phase reconstruction in phase-shifting interferometry,” Opt. Eng. 52(3), 030501(2013).

Wang, Y. D.

K. S. Moon and Y. D. Wang, “Accuracy improvement of phase shift interferometry,” Proc. SPIE 5264, 0277 (2003).
[Crossref]

Wang, Z.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

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]

C. Wei, M. Chen, H. Guo, and Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” Proc. SPIE 3478, 0277 (1998).
[Crossref]

Wei, C.

C. Wei, M. Chen, H. Guo, and Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” Proc. SPIE 3478, 0277 (1998).
[Crossref]

Wyant, J. C.

Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985).
[Crossref] [PubMed]

J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus 18(5), 65–67 (1982).

Yu, Y.

Appl. Opt. (7)

International Journal of Optomechatronics (1)

B. Kimbrough, “Correction of Errors in Polarization Based Dynamic Phase Shifting Interferometers,” International Journal of Optomechatronics 8(4), 304–312 (2014).
[Crossref]

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

Laser Focus (1)

J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus 18(5), 65–67 (1982).

Meas. Sci. Technol. (2)

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]

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

Opt. Commun. (2)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

B. K. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for micro-surface profiling of flat surfaces,” Opt. Commun. 190(1-6), 109–116 (2001).
[Crossref]

Opt. Eng. (4)

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23(4), 361–364 (1984).
[Crossref]

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

G. Vladimir, I. Sergey, K. Roman, and H. Dmitry, “Generic algorithm of phase reconstruction in phase-shifting interferometry,” Opt. Eng. 52(3), 030501(2013).

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[Crossref]

Opt. Lasers Eng. (1)

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (3)

K. S. Moon and Y. D. Wang, “Accuracy improvement of phase shift interferometry,” Proc. SPIE 5264, 0277 (2003).
[Crossref]

J. Millerd, N. Brock, and J. Hayeset, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

C. Wei, M. Chen, H. Guo, and Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” Proc. SPIE 3478, 0277 (1998).
[Crossref]

Other (2)

A. Armando, V. F. Analucia, and F. M. Allison, “Use of Generalized N-dimensional Lissajous Figures for Phase Retrieval from Sequences of Interferometric Images with Unknown Phase Shifts,” W. Osten (ed.), Fringe 2013, 191–196 (2014).

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing,2nd ed.(Taylor and Francis, 2005).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Lissajous figures by plotting N against D: (a) an ellipse (with errors); (b) a unit circle (without errors).
Fig. 2
Fig. 2 Simulation results, (a) the fringe; (b) the given measured surface; (c) and (d) the extracted surface and the residual error of 4-buck PSA; (e) and (f) the compensated surface and the residual error of using proposed method in 4-buck PSA method.
Fig. 3
Fig. 3 Proposed circular mode: (a) Lissajous ellipse at circle1; (b) Lissajous ellipse at circle2; (c) Lissajous ellipse at circle6; (d) the calculated offset of the Lissajous ellipse at different circles; (e) the calculated bias of the Lissajous ellipse at different circles; (f) the less of quadrature of the Lissajous ellipse at different circles;
Fig. 4
Fig. 4 Discussions of the performance of proposed method and C.T. Farrell’s method with different influence factors: (a) the number of the frames; (b) the SNR of the interferogram; (c) the amplitude of the phase shift error; (d) the difference between proposed method and AIA at the different amplitudes of phase shift error.
Fig. 5
Fig. 5 Experimental results:(a) the fringe; (b) the measured surface Zygo gives; (c) and (d) the extracted surface and the residual error of LSM (with artificial phase shift errors); (e) and (f) the extracted surface and the residual error of AIA; (g) and (h) the extracted surface and the residual error of proposed method with the “circular mode”; (i) the residual error of proposed method with the “linear mode”.
Fig. 6
Fig. 6 Experimental results: (a) the fringe in vibration; (b) the measured surface with no vibration; (c) the surface measured by 13-frame PSA in vibration; (d) the compensated surface by proposed ETC method; (e) the residual surface error caused by vibration; (f) the residual surface error of compensated surface.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

ϕ= tan -1 ( N D ).
N(φ)= x 0 + a x sin(φ)
D ε (φ)= y 0 + a y cos(φ+ε)
x 0 = y 0 =0; a x = a y ;ε=0.
ϕ= tan -1 [ sin(φ) cos(φ) ]
D ε (φ)=y + 0 (D y 0 )cos(ε)( a y / a x )(N x 0 )sin(ε)
D=[( D ε y 0 )+( a y / a x )(Nx ) 0 sin(ε)]/cos(ε)+ y 0
N ¯ =(N x 0 )/ a x ; D ¯ =(D y 0 )/ a y .
φ=ϕ=arctan( N ¯ D ¯ )=arctan( 1 a x a y ( D ε y 0 N x 0 ) 1 cosε +tanε )
I k (x,y)=A(x,y)+B(x,y)cos[φ(x,y)+ β k ] k=1,2,3m
ϕ=arctan k=0 m1 [ ( I mod(k+1,m) I mod(m+k1,m) )cos( β k ) ] k=0 m1 [ ( I mod(k+1,m) I mod(m+k1,m) )sin( β k ) ]
ϕ=arctan I 4 I 2 I 1 I 3
ϕ=arctan N D =arctan I 4 I 2 I 1 I 3 =arctan B[ sin(φ+ δ 3 )+sin(φ+ δ 1 ) ] B[ cosφ+cos(φ+ δ 2 ) ] =arctan B[ sin(φ+ δ 3 + δ 1 2 )cos( δ 3 δ 1 2 ) ] B[ cos(φ+ δ 2 2 )cos( δ 2 2 ) ] =arctan b 1 sin(φ+ δ 3 + δ 1 2 ) b 2 cos(φ+ δ 2 2 ) =arctan b 1 sin(φ) b 2 cos(φ+ε)
A I 1 2 +2B I 1 I 2 +C I 2 2 +2D I 1 +2E I 2 +F=0
( I 1 x 0 ) 2 a x 2 + 2sinε( I 1 x ) 0 ( I 2 y ) 0 a x a y + ( I 2 y 0 ) 2 a y 2 = cos 2 ε
A= 1 a x 2 B= sinε a x a y C= 1 a y 2 D= x 0 a x 2 y 0 sinε a x a y E= y 0 a y 2 x 0 sinε a x a y F= x 0 2 a x 2 + y 0 2 a y 2 + 2 x 0 y 0 sinε a x a y cos 2 ε
ε=arcsin( B AC ), x 0 = BECD δ , y 0 = BDAE δ , a x = CΔ δ , a y = AΔ δ .
Δ ¯ =[ A B D B C E D E F ],Δ=det Δ ¯ ,δ=| A B B C |
ϕ(x,y)=π( 0.1+2x+0.5y+0.2( x 2 + y 2 1)+0.1( y 2 x 2 ) )(1x,y1)
β k =(π/2)(k1)+δ(k1);(1k4)
ε=0.5810.5809= 0.7278+0.1924+0.2417 2 = δ 2 δ 1 δ 3 2
δ 1 =0rad, δ 2 =1.5778rad, δ 3 =3.1838rad, δ 4 =4.7513rad

Metrics