Abstract

Instead of various mathematical stitching algorithms, an aspheric subaperture stitching interferometric method relying on modern computer modeling technique is presented. Based on our previously reported non-null annular subaperture stitching interferometry (NASSI), a simultaneous reverse optimizing reconstruction (SROR) method based on system modeling is proposed for full aperture figure error reconstruction. All the subaperture measurements are simulated simultaneously with a multi-configuration model in a ray tracing program. With the multi-configuration model, full aperture figure error would be extracted in form of Zernike polynomials from subapertures wavefront data by the SROR method. This method concurrently accomplishes subaperture retrace error and misalignment correction, requiring neither complex mathematical algorithms nor subaperture overlaps. Experiment results showing the validity of SROR method are presented.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Non-null annular subaperture stitching interferometry for steep aspheric measurement

Lei Zhang, Chao Tian, Dong Liu, Tu Shi, Yongying Yang, Hanshuo Wu, and Yibing Shen
Appl. Opt. 53(25) 5755-5762 (2014)

Modified stitching algorithm for annular subaperture stitching interferometry for aspheric surfaces

Yongfu Wen, Haobo Cheng, Hon-Yuen Tam, and Dongmei Zhou
Appl. Opt. 52(23) 5686-5694 (2013)

Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method

Xi Hou, Fan Wu, Li Yang, and Qiang Chen
Opt. Express 15(20) 12890-12899 (2007)

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, Optical Shop Testing, 3rd ed (Wiley, 2007).
    [Crossref]
  2. M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
    [Crossref]
  3. W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8(9), 468–470 (1983).
    [Crossref] [PubMed]
  4. J. E. Negro, “Subaperture optical system testing,” Appl. Opt. 23(12), 1921–1930 (1984).
    [Crossref] [PubMed]
  5. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
    [Crossref]
  6. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
    [Crossref]
  7. Y. Wen, H. Cheng, H. Y. Tam, and D. Zhou, “Modified stitching algorithm for annular subaperture stitching interferometry for aspheric surfaces,” Appl. Opt. 52(23), 5686–5694 (2013).
    [Crossref] [PubMed]
  8. Y. M. Liu, G. N. Lawrence, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27(21), 4504–4513 (1988).
    [Crossref] [PubMed]
  9. M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
    [Crossref]
  10. F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11(2), 82–86 (2004).
    [Crossref]
  11. X. Hou, F. Wu, L. Yang, S. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces,” Appl. Opt. 45(15), 3442–3455 (2006).
    [Crossref] [PubMed]
  12. S. Chen, S. Li, Y. Dai, L. Ding, and S. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express 16(7), 4760–4765 (2008).
    [Crossref] [PubMed]
  13. L. Zhang, C. Tian, D. Liu, T. Shi, Y. Yang, H. Wu, and Y. Shen, “Non-null annular subaperture stitching interferometry for steep aspheric measurement,” Appl. Opt. 53(25), 5755–5762 (2014).
    [Crossref] [PubMed]
  14. D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 74260, 74260M (2009).
    [Crossref]
  15. J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
    [Crossref]
  16. D. Liu, T. Shi, L. Zhang, Y. Yang, S. Chong, and Y. Shen, “Reverse optimization reconstruction of aspheric figure error in a non-null interferometer,” Appl. Opt. 53(24), 5538–5546 (2014).
    [Crossref] [PubMed]
  17. C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50(20), 3559–3569 (2011).
    [Crossref] [PubMed]
  18. TRIOPTICS, SpectroMaster, http://www.trioptics.com/ spectromaster/description.php .
  19. TRIOPTICS, Spectrometers, http://www.trioptics.com/ spectromaster/description.php .
  20. Y. Luo, Y. Yang, and C. Tian, “Error analysis and preprocess of partial compensatory aspheric testing system,” J. Zhejiang Univ. 46(4), 636–733 (2012).
  21. Zhejiang University, “Aligning device and method for partial compensating lens in aspheric non-null interference testing,” China patent 201210054455.2 (2012).
  22. L. L. Deck, “Model-based phase shifting interferometry,” Appl. Opt. 53(21), 4628–4636 (2014).
    [Crossref] [PubMed]
  23. L. Zhang, D. Liu, T. Shi, Y. Yang, and Y. Shen, “Practical and accurate method for aspheric misalignment aberrations calibration in non-null interferometric testing,” Appl. Opt. 52(35), 8501–8511 (2013).
    [Crossref] [PubMed]
  24. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009).
    [Crossref] [PubMed]
  25. R. O. Gappinger and J. E. Greivenkamp, “Iterative reverse optimization procedure for calibration of aspheric wave-front measurements on a nonnull interferometer,” Appl. Opt. 43(27), 5152–5161 (2004).
    [Crossref] [PubMed]

2014 (3)

2013 (2)

2012 (1)

Y. Luo, Y. Yang, and C. Tian, “Error analysis and preprocess of partial compensatory aspheric testing system,” J. Zhejiang Univ. 46(4), 636–733 (2012).

2011 (1)

2009 (3)

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 74260, 74260M (2009).
[Crossref]

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
[Crossref]

2006 (1)

2004 (2)

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11(2), 82–86 (2004).
[Crossref]

R. O. Gappinger and J. E. Greivenkamp, “Iterative reverse optimization procedure for calibration of aspheric wave-front measurements on a nonnull interferometer,” Appl. Opt. 43(27), 5152–5161 (2004).
[Crossref] [PubMed]

2003 (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

1994 (1)

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[Crossref]

1993 (1)

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[Crossref]

1988 (1)

1984 (1)

1983 (1)

Chen, Q.

Chen, S.

Cheng, H.

Chong, S.

Chow, W. W.

Cornejo Rodríguez, A.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11(2), 82–86 (2004).
[Crossref]

Dai, Y.

Deck, L. L.

Ding, L.

Dumas, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Escobar-Romero, J. F.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11(2), 82–86 (2004).
[Crossref]

Fleig, J.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Forbes, G.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Gappinger, R. O.

Granados-Agustín, F.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11(2), 82–86 (2004).
[Crossref]

Greivenkamp, J. E.

Hou, X.

Koliopoulos, C. L.

Küchel, M. F.

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

Lawrence, G. N.

Li, S.

Liu, D.

Liu, Y. M.

Luo, Y.

Y. Luo, Y. Yang, and C. Tian, “Error analysis and preprocess of partial compensatory aspheric testing system,” J. Zhejiang Univ. 46(4), 636–733 (2012).

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 74260, 74260M (2009).
[Crossref]

D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009).
[Crossref] [PubMed]

Mazzoni, A.

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[Crossref]

Melozzi, M.

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[Crossref]

Murphy, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Negro, J. E.

Okada, K.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[Crossref]

Otsubo, M.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[Crossref]

Pezzati, L.

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[Crossref]

Shen, Y.

Shi, T.

Sullivan, J. J.

J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
[Crossref]

Tam, H. Y.

Tian, C.

Tricard, M.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Tsujiuchi, J.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[Crossref]

Wang, L.

Wei, T.

Wen, Y.

Wu, F.

Wu, H.

Wu, S.

Yang, L.

Yang, Y.

Zeng, S.

Zhang, L.

Zhou, D.

Zhuo, Y.

C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50(20), 3559–3569 (2011).
[Crossref] [PubMed]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 74260, 74260M (2009).
[Crossref]

Appl. Opt. (10)

J. E. Negro, “Subaperture optical system testing,” Appl. Opt. 23(12), 1921–1930 (1984).
[Crossref] [PubMed]

Y. M. Liu, G. N. Lawrence, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27(21), 4504–4513 (1988).
[Crossref] [PubMed]

R. O. Gappinger and J. E. Greivenkamp, “Iterative reverse optimization procedure for calibration of aspheric wave-front measurements on a nonnull interferometer,” Appl. Opt. 43(27), 5152–5161 (2004).
[Crossref] [PubMed]

X. Hou, F. Wu, L. Yang, S. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces,” Appl. Opt. 45(15), 3442–3455 (2006).
[Crossref] [PubMed]

C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50(20), 3559–3569 (2011).
[Crossref] [PubMed]

Y. Wen, H. Cheng, H. Y. Tam, and D. Zhou, “Modified stitching algorithm for annular subaperture stitching interferometry for aspheric surfaces,” Appl. Opt. 52(23), 5686–5694 (2013).
[Crossref] [PubMed]

L. Zhang, D. Liu, T. Shi, Y. Yang, and Y. Shen, “Practical and accurate method for aspheric misalignment aberrations calibration in non-null interferometric testing,” Appl. Opt. 52(35), 8501–8511 (2013).
[Crossref] [PubMed]

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

D. Liu, T. Shi, L. Zhang, Y. Yang, S. Chong, and Y. Shen, “Reverse optimization reconstruction of aspheric figure error in a non-null interferometer,” Appl. Opt. 53(24), 5538–5546 (2014).
[Crossref] [PubMed]

L. Zhang, C. Tian, D. Liu, T. Shi, Y. Yang, H. Wu, and Y. Shen, “Non-null annular subaperture stitching interferometry for steep aspheric measurement,” Appl. Opt. 53(25), 5755–5762 (2014).
[Crossref] [PubMed]

J. Zhejiang Univ. (1)

Y. Luo, Y. Yang, and C. Tian, “Error analysis and preprocess of partial compensatory aspheric testing system,” J. Zhejiang Univ. 46(4), 636–733 (2012).

Opt. Eng. (2)

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[Crossref]

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Opt. Photonics News (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Opt. Rev. (1)

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11(2), 82–86 (2004).
[Crossref]

Proc. SPIE (3)

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 74260, 74260M (2009).
[Crossref]

J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
[Crossref]

Other (4)

TRIOPTICS, SpectroMaster, http://www.trioptics.com/ spectromaster/description.php .

TRIOPTICS, Spectrometers, http://www.trioptics.com/ spectromaster/description.php .

D. Malacara, Optical Shop Testing, 3rd ed (Wiley, 2007).
[Crossref]

Zhejiang University, “Aligning device and method for partial compensating lens in aspheric non-null interference testing,” China patent 201210054455.2 (2012).

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 (9)

Fig. 1
Fig. 1 The sketch of NASSI.
Fig. 2
Fig. 2 The multi-configuration model for ray tracing.
Fig. 3
Fig. 3 The Zernike fitting coordinate transformation between subaperture and full aperture.
Fig. 4
Fig. 4 The experiment layout (a) NASSI system, (b) The DMI system.
Fig. 5
Fig. 5 The three subaperture interferograms.
Fig. 6
Fig. 6 The calculation of full aperture misalignments in different subaperture measurements, (a) Subaperture misalignment coefficients,(b) Full aperture misalignment coefficients, (c)) Full aperture misalignments.
Fig. 7
Fig. 7 The SROR process, (a) Zernike annular coefficients of three subaperture wavefronts in experiment, (b) Standard Zernike coefficient of full aperture figure error.
Fig. 8
Fig. 8 Testing results by the SROR method, the ROR-stitching method and Zygo Verifire Asphere interferometer, (a) and (d) are the reconstructed figure error of the two aspherics by the SROR method, respectively, (b) and (e) are the reconstructed figure error of the two aspherics by the ROR-stitching method, respectively, (c) and (f) are the two surface figure errors tested by Zygo Verifire Asphere interferometer, respectively.
Fig. 9
Fig. 9 Testing results of other two independent tests with the first paraboloidal mirror by the SROR method.

Tables (4)

Tables Icon

Table 1 Parameters of subapertures in experiment

Tables Icon

Table 2 Aspheric misalignment coefficients

Tables Icon

Table 3 Aspheric misalignments at each subaperture measurement

Tables Icon

Table 4 Parameters of figure error by SROR method, ROR-stitching method and Zygo interferometer

Equations (18)

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

{ ρ = S r ρ ' θ = θ ' ,
W d e f o c u s ( ρ , θ ) + W tilt ( ρ , θ ) + W c o m a ( ρ , θ ) = W defocus ( ρ ' , θ ' , ε ) + W tilt ( ρ ' , θ ' , ε ) + W c o m a ( ρ ' , θ ' , ε ) ,
C 4 3 ( 2 ρ 2 1 ) + 2 ρ ( C 2 cos θ + C 3 sin θ ) + 8 ( 3 ρ 3 2 ρ ) ( C 7 sin θ + C 8 cos θ ) = 3 C ' 4 R 2 0 ( ρ ' , ε ) + 2 R 1 1 ( ρ ' , ε ) ( C ' 2 cos θ ' + C ' 3 sin θ ' ) + 8 R 3 1 ( ρ ' , ε ) ( C ' 7 sin θ ' + C ' 8 cos θ ' ) ,
{ R 2 0 ( ρ ' , ε ) = ( 2 ρ ' 2 1 ε 2 ) / ( 1 ε 2 ) R 1 1 ( ρ ' , ε ) = ρ ' / ( 1 + ε 2 ) 1 / 2 R 3 1 ( ρ ' , ε ) = [ 3 ( 1 + ε 2 ) ρ ' 3 2 ( 1 + ε 2 + ε 4 ) ρ ' ] / { ( 1 ε 2 ) [ ( 1 + ε 2 ) ( 1 + 4 ε 2 + ε 4 ) ] 1 / 2 } .
{ C 2 = C ' 2 1 / [ S r ( 1 + ε 2 ) 1 / 2 ] + C ' 8 K C 3 = C ' 3 1 / [ S r ( 1 + ε 2 ) 1 / 2 ] + C ' 7 K C 4 = C ' 4 1 / [ S r 2 ( 1 ε 2 ) ] C 7 = C ' 7 ( T / S r 3 ) C 8 = C ' 8 ( T / S r 3 )
{ K = 8 ( 1 + ε 2 S r 3 1 + ε 2 + ε 4 S r ) / { ( 1 ε 2 ) [ ( 1 + ε 2 ) ( 1 + 4 ε 2 + ε 4 ) ] 1 / 2 } T = ( 1 + ε 2 ) / { ( 1 ε 2 ) [ ( 1 + ε 2 ) ( 1 + 4 ε 2 + ε 4 ) ] 1 / 2 } .
E asp _ m = { W m [ f s ( R m ) ] cos θ δ m } / 2 , m = 1 , , M ,
W m f ( E asp _ m + E mis _ m + E ret _ m ) ,
E asp _ m f - 1 ( W m ) E m i s _ m E ret _ m .
E ˜ asp _ m f - 1 ( W ˜ m ) E ˜ m i s _ m E ˜ ret _ m ,
U = ( E asp _ m E ˜ asp _ m ) 2 = [ ( f - 1 ( W m ) E m i s _ m E ret _ m ) ( f - 1 ( W ˜ m ) E ˜ m i s _ m E ˜ ret _ m ) ] 2 = min .
{ E ret _ m = E ˜ ret _ m E m i s _ m = E ˜ m i s _ m
U = ( E asp _ m E ˜ asp _ m ) 2 = [ f - 1 ( W m ) f - 1 ( W ˜ m ) ] 2 = min .
U = ( E ASP E ˜ ASP ) 2 = [ ( E asp_1 E asp_2 E asp_M ) ( E ˜ asp_1 E ˜ asp_2 E ˜ asp_M ) ] 2 = m = 1 M ( E asp _ m E ˜ asp _ m ) 2 = m = 1 M [ f - 1 ( W m ) f - 1 ( W ˜ m ) ] 2 = min ,
E ASP = E ˜ ASP .
{ E ASP i = 5 N C i Z i ( ρ , θ ) ; E ˜ ASP i = 5 N C ˜ i Z i ( ρ , θ ) W m i = 5 N C i _ m Z i ( ρ , θ , ε m ) ; W ˜ m i = 5 N C ˜ i _ m Z i ( ρ , θ , ε m ) ,
U = i = 5 N ( C i C ˜ i ) 2 +cons = i = 5 N m = 1 M ( C i _ m C ˜ i _ m ) 2 + cons = min ,
E ASP = i = 5 N C i Z i ( ρ , θ ) = i = 5 N C ˜ i Z i ( ρ , θ ) .

Metrics