Abstract

Our paper mainly separates the specific aberration contributions of third-order astigmatism and third-order coma from the total aberration fields, on the framework of the modified nodal aberration theory (NAT), for the perturbed off-axis telescope. Based on the derived aberration functions, two alignment models for the same off-axis two-mirror telescope are established and compared. Among them, one is based on third-order NAT, the other is based on fifth-order NAT. By comparison, it is found that the calculated perturbations based on fifth-order NAT are more accurate. It illustrates that third-order astigmatism and third-order coma contributed from fifth-order aberrations can’t be neglected in the alignment process. Then the fifth-order NAT is used for the alignment of off-axis three-mirror telescopes. After simulation, it is found that the perturbed off-axis three-mirror telescope can be perfectly aligned as well. To further demonstrate the application of the alignment method based on fifth-order NAT (simplified as NAT method), Monte-Carlo simulations for both off-axis two-mirror telescope and off-axis three-mirror telescope are conducted in the end. Meantime, a comparison between NAT method and sensitivity table method is also conducted. It is proven that the computation accuracy of NAT method is much higher, especially in poor conditions.

© 2016 Optical Society of America

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References

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  1. H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
    [Crossref]
  2. J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
    [Crossref]
  3. R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
    [Crossref]
  4. M. Liang, V. Krabbendam, C. F. Claver, S. Chandrasekharan, and B. Xin, “Active Optics in Large Synoptic Survey Telescope,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 84444Q–84444Q–13 (2012).
    [Crossref]
  5. R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
    [Crossref]
  6. M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
    [Crossref]
  7. R. Tessieres, “Analysis for alignment of optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  18. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).
  19. T. Schmid, “Misalignment Induced Nodal Aberration Fields and Their Use in the Alignment of Astronomical Telescopes,” Ph.D. dissertation (University of Central Florida Orlando, Florida, 2010).
  20. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
    [Crossref] [PubMed]
  21. M. L. Lampton, M. J. Sholl, and M. E. Levi, “Off-axis telescopes for dark energy investigations,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 77311G–77311G (2010).
    [Crossref]
  22. L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
    [Crossref]

2015 (1)

2013 (1)

2012 (1)

2011 (1)

2010 (2)

2009 (1)

2008 (1)

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

2006 (1)

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

2005 (1)

1999 (1)

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

1998 (1)

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

1991 (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

1987 (1)

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

1980 (1)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

1979 (1)

L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
[Crossref]

Cook, L. G.

L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
[Crossref]

Dempewolf, G.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Fan, Z.

Franza, F.

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Gu, Z.

Guo, B.

Harnisch, B.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Hawley, S. L.

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Hu, H.

Hubbard, R.

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

Juranek, H. J.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Kuhn, J. R.

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Kunkel, B.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Litzelmann, A.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Liu, J.

Lu, Z.

Lundgren, M. A.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Noethe, L.

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Rakich, A.

Rimmele, T.

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

Rolland, J. P.

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

Sand, R.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schillke, F.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schmid, T.

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

Schmidt, E.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schweizer, J.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Sun, Q.

Thompson, K.

Thompson, K. P.

Upton, R.

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

Wang, J.

Wang, Y.

Wilson, R. N.

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Wolfe, W. L.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Yan, C.

J. Mod. Opt. (1)

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Opt. Express (4)

Proc. SPIE (5)

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
[Crossref]

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Publ. Astron. Soc. Pac. (1)

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Other (6)

M. Liang, V. Krabbendam, C. F. Claver, S. Chandrasekharan, and B. Xin, “Active Optics in Large Synoptic Survey Telescope,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 84444Q–84444Q–13 (2012).
[Crossref]

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).

T. Schmid, “Misalignment Induced Nodal Aberration Fields and Their Use in the Alignment of Astronomical Telescopes,” Ph.D. dissertation (University of Central Florida Orlando, Florida, 2010).

M. L. Lampton, M. J. Sholl, and M. E. Levi, “Off-axis telescopes for dark energy investigations,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 77311G–77311G (2010).
[Crossref]

R. Tessieres, “Analysis for alignment of optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

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Figures (6)

Fig. 1
Fig. 1 Pupil vector transformation between decentered pupil and parent pupil. The black circle represents the pupil of an on-axis system. The red circle represents the pupil of an off-axis section of the on-axis system.
Fig. 2
Fig. 2 Layout of the main optical system of New Solar Telescope (NST). It can be seen that the offset of aperture stop is 1840mm in the positive direction of Y axis. Then the defined parameter A in Eq. (25) and Eq. (26) can be determined. A = 1840 / 800 .
Fig. 3
Fig. 3 Average RMS WFE before and after alignment for different cases based on NAT method. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after first alignment. The red spot represents the RMS WFE after second alignment. It is the same as the RMA WFE of the nominal design (0.062 waves).
Fig. 4
Fig. 4 Layout of an off-axis Cook-TMA telescope. It can be seen that the offset of aperture stop is −460mm in the negative direction of Y axis. Then the defined parameter A in Eq. (56) and Eq. (57) can be determined. A = 460 / 300 .
Fig. 5
Fig. 5 Average RMS WFE before and after alignment for different cases based on NAT method. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after alignment. The red spot represents the RMS WFE in nominal design (0.072 waves).
Fig. 6
Fig. 6 Average RMS WFE before and after alignment for different cases based on sensitivity table method (SMT). (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after alignment. The red spot represents the RMS WFE in nominal design (0.072 waves).

Tables (15)

Tables Icon

Table 1 Optical Parameters of NST

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Table 2 Wave Aberration Coefficients of SM for NST

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Table 3 Introduced Misalignments of SM for NST

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Table 4 Calculated Misalignments of SM for NST Based on Third-order NAT and Relative Errors between Calculated and Introduced Misalignments

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Table 5 Calculated Misalignments of SM for NST Based on Fifth-order NAT and Relative Errors between Calculated and Introduced Misalignments

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Table 6 Introduced Astigmatic Figure Errors on PM

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Table 7 Calculated Perturbations Based on Fifth-order NAT and Relative Errors between Calculated and Introduced Perturbations

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Table 8 Four Different Cases Considered in Monte-Carlo Simulations

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Table 9 Root Mean Square Deviations (RMSDs) between Introduced and Computed Perturbations

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Table 10 Wave Aberration Coefficients of SM and TM for Three-mirror Parent Telescope and Values of C A & C B & C C

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Table 11 Introduced Misalignments of SM and TM

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Table 12 Calculated Misalignments of SM and TM and Their Relative Errors

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Table 13 Four Different Cases Considered in Monte-Carlo Simulations

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Table 14 RMSDs between Introduced and Calculated Misalignments Based on NAT and STM for Different Cases

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Table 15 Parameter Definitions and Vector Identities

Equations (63)

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W = j p n m ( W k l m ) j ( H H ) p ( ρ ρ ) n ( H ρ ) m , k = 2 p + m , l = 2 n + m ,
ρ = B ρ + h ,
h = h x i + h y j , B = r o r O , h x = o x r O , h y = o y r O ,
W = j p n m ( W k l m ) j ( H H ) p [ ( B ρ + h ) ( B ρ + h ) ] n [ H ( B ρ + h ) ] m .
H A j = H σ j ,
W = j p n m ( W k l m ) j ( H A j H A j ) p [ ( B ρ + h ) ( B ρ + h ) ] n [ H A j ( B ρ + h ) ] m .
W = 1 2 j W 222 j [ H A j 2 ( B ρ + h ) 2 ] + j W 131 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] + j W 040 j [ ( B ρ + h ) ( B ρ + h ) ] 2 + j W 240 M j ( H A j H A j ) [ ( B ρ + h ) ( B ρ + h ) ] 2 + j W 331 M j ( H A j H A j ) [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] + 1 2 j W 422 j ( H A j H A j ) [ H A j 2 ( B ρ + h ) 2 ] + 1 4 j W 333 j [ H A j 3 ( B ρ + h ) 3 ] + 1 2 j W 242 j [ H A j 2 ( B ρ + h ) 2 ] [ ( B ρ + h ) ( B ρ + h ) ] + j W 151 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] 2 + j W 060 j [ ( B ρ + h ) ( B ρ + h ) ] 3 .
W = i C i ( H x , H y ) Z i ( ρ , φ ) .
W = 1 2 j W 222 j H A j 2 ( B ρ + h ) 2 ,
W = 1 2 j W 222 j [ B 2 ( H A j 2 ρ 2 ) + Δ ] ,
B 2 [ H x 2 H y 2 H x H y 1 0 2 H x H y H y H x 0 1 ] [ W 222 2 A 222 , x 2 A 222 , y B 222 , x 2 B 222 , y 2 ] = 2 [ C 5 w 222 C 6 w 222 ] .
W = j W 131 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] ,
W = j W 131 j [ B 3 ( H A j ρ ) ( ρ ρ ) + B 2 ( H A j h ρ 2 ) + Δ ] ,
B 3 [ H x 1 0 H y 0 1 ] [ W 131 A 131 , x A 131 , y ] = 3 [ C 7 w 131 C 8 w 131 ] ,
B 2 [ h y 0 0 h y ] [ H y 0 1 H x 1 0 ] [ W 131 A 131 , x A 131 , y ] = [ C 5 w 131 C 6 w 131 ] .
C 5 w 131 = 3 h y B C 8 w 131 C 6 w 131 = 3 h y B C 7 w 131 .
W = j W 040 j [ ( B ρ + h ) ( B ρ + h ) ] 2 ,
W = j W 040 j [ B 4 ( ρ ρ ) 2 + 4 B 3 ( h ρ ) ( ρ ρ ) + 2 B 2 ( h 2 ρ 2 ) + Δ ] ,
B 4 W 040 = 6 C 9 w 040 ,
4 B 3 h y W 040 = 3 C 8 040 ,
2 B 2 h y 2 W 040 = C 5 w 040 .
C 8 w 040 = 8 h y B C 9 w 040 C 5 w 040 = 12 h y B C 9 w 040 .
{ C 5 w 222 = C 5 C 5 w 131 C 5 w 040 C 6 w 222 = C 6 C 6 w 131 .
{ C 7 w 131 = C 7 C 8 w 131 = C 8 C 8 w 040 .
{ C 5 w 222 = C 5 + 3 A C 8 + 12 A C 9 24 A 2 C 9 C 6 w 222 = C 6 3 A C 7 ,
{ C 7 w 131 = C 7 C 8 w 131 = C 8 8 A C 9 .
B 2 [ H x 2 H y 2 H x H y 1 0 2 H x H y H y H x 0 1 ] [ W 222 2 A 222 , x 2 A 222 , y B 222 , x 2 B 222 , y 2 ] = 2 [ C 5 + 3 A C 8 + 12 A C 9 24 A 2 C 9 C 6 3 A C 7 ] ,
B 3 [ H x 1 0 H y 0 1 ] [ W 131 A 131 , x A 131 , y ] = 3 [ C 7 C 8 8 A C 9 ] .
{ A 222 x = W 222 , S M s p h σ S M , x s p h + W 222 , S M a s p h σ S M , x a s p h A 222 y = W 222 , S M s p h σ S M , y s p h + W 222 , S M a s p h σ S M , y a s p h A 131 x = W 131 , S M s p h σ S M , x s p h + W 131 , S M a s p h σ S M , x a s p h A 131 y = W 131 , S M s p h σ S M , y s p h + W 131 , S M a s p h σ S M , y a s p h ,
{ X D E S M = u ¯ P M d 1 σ S M , x a s p h Y D E S M = u ¯ P M d 1 σ S M , y a s p h A D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , y s p h c S M Y D E S M B D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , x s p h + c S M Y D E S M ,
W = 1 2 j W 422 j ( H A j H A j ) [ H A j 2 ( B ρ + h ) 2 ] ,
W = 1 2 j W 422 j [ B 2 ( H A j H A j ) ( H A j 2 ρ 2 ) + Δ ] ,
B 2 [ H x 4 H y 4 4 H x 3 4 H y 3 2 H x H y ( H x 2 + H y 2 ) 6 H x 2 H y 2 H y 3 2 H x 3 6 H x H y 2 ] [ W 422 A 422 , x A 422 , y ] = 2 [ C 5 w 422 C 6 w 422 ] ,
W = j W 331 M j ( H A j H A j ) [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] ,
W = j W 331 M j [ ( H A j H A j ) ( H A j ρ ) ( ρ ρ ) + ( H A j H A j ) ( H A j h ρ 2 ) + Δ ] ,
B 3 [ H x 3 + H x H y 2 3 H x 2 H y 2 2 H x H y H x 2 H y + H y 3 2 H x H y H x 2 3 H y 2 ] [ W 331 M A 331 M , x A 331 M , y ] = 3 [ C 7 w 331 M C 8 w 331 M ] ,
B 2 [ h y 0 0 h y ] [ H x 2 H y + H y 3 2 H x H y H x 2 3 H y 2 H x 3 + H x H y 2 3 H x 2 H y 2 2 H x H y ] [ W 331 M A 331 M , x A 331 M , y ] = [ C 5 w 331 M C 6 w 331 M ] ,
C 5 w 331 M = 3 h y B C 8 w 331 M C 6 w 331 M = 3 h y B C 7 w 331 M .
W = 1 4 j W 333 j H A j 3 ( B ρ + h ) 3 ,
W = 1 4 j W 333 j [ B 3 ( H A j 3 ρ 3 ) + 3 B 2 ( H A j 3 h ρ 2 ) + Δ ] ,
C 5 w 333 = 3 h y B C 11 w 333 C 6 w 333 = 3 h y B C 10 w 333 .
W = j W 240 M j ( H A j H A j ) [ ( B ρ + h ) ( B ρ + h ) ] 2 ,
W = j W 240 M j [ B 4 ( H A j H A j ) ( ρ ρ ) 2 + 4 B 3 ( H A j H A j ) ( h ρ ) ( ρ ρ ) + 2 B 2 ( H A j H A j ) ( h 2 ρ 2 ) + Δ ] ,
C 8 w 240 M = 8 h y B C 9 w 240 M C 5 w 240 M = 12 h y B C 9 w 240 M .
W = 1 2 j W 242 j [ H A j 2 ( B ρ + h ) 2 ] [ ( B ρ + h ) ( B ρ + h ) ] ,
W = 1 2 j W 242 j [ B 4 ( H A j 2 ρ 2 ) ( ρ ρ ) + B 3 ( h H A j 2 ρ 3 ) + 3 B 3 ( H A j 2 h ρ ) ( ρ ρ ) + 3 B 2 ( h h ) ( H A j 2 ρ 2 ) + Δ ] ,
{ C 10 w 242 = 4 h y B C 13 w 242 C 11 w 242 = 4 h y B C 12 w 242 C 7 w 242 = 4 h y B C 13 w 242 C 8 w 242 = 4 h y B C 12 w 242 C 5 w 242 = 12 h y 2 B 2 C 12 w 242 C 6 w 242 = 4 h y 2 B 2 C 13 w 242 .
W = j W 151 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] 2 ,
W = j W 151 j [ B 5 ( H A j ρ ) ( ρ ρ ) 2 + 2 B 4 ( H A j h ρ 2 ) ( ρ ρ ) + B 3 ( H A j h 2 ρ 3 ) + 3 B 4 ( H A j h ) ( ρ ρ ) ( ρ ρ ) +6 B 3 ( h h ) ( H A j ρ ) ( ρ ρ ) + 3 B 3 ( h 2 H A j ρ ) ( ρ ρ ) + 2 B 2 ( h h ) ( H A j h ρ 2 ) + 2 B 2 ( H A j h ) ( h 2 ρ 2 ) + Δ ] ,
{ C 12 w 151 = 5 h y B C 15 w 151 C 13 w 151 = 5 h y B C 14 w 151 C 10 w 151 = 10 h y 2 B 2 C 14 w 151 C 11 w 151 = 10 h y 2 B 2 C 15 w 151 C 9 w 151 = 5 h y B C 15 w 151 C 7 w 151 = 10 h y 2 B 2 C 14 w 151 C 8 w 151 = 30 h y 2 B 2 C 15 w 151 C 5 w 151 = 40 h y 2 B 2 C 15 w 151 C 6 w 151 = 20 h y 2 B 2 C 14 w 151 .
W = j W 060 j [ ( B ρ + h ) ( B ρ + h ) ] 3 ,
W = j W 060 j [ B 6 ( ρ ρ ) 3 + 6 B 5 ( ρ ρ ) 2 ( ρ h ) + 6 B 4 ( ρ ρ ) ( ρ 2 h 2 ) + 2 B 3 ( ρ 3 h 3 ) + 9 B 4 ( ρ ρ ) 2 ( h h ) + 18 B 3 ( ρ ρ ) ( ρ h ) ( h h ) + 6 B 2 ( ρ 2 h 2 ) ( h h ) + Δ ] ,
{ C 5 w 060 = 20 h y 4 B 4 C 16 w 060 C 8 w 060 = 120 h y 3 B 3 C 16 w 060 C 9 w 060 = 30 h y 2 B 2 C 16 w 060 C 11 w 060 = 40 h y 3 B 3 C 16 w 060 C 12 w 060 = 30 h y 2 B 2 C 16 w 060 C 15 w 060 = 12 h y B C 16 w 060 .
{ C 5 w 222 + C 5 w 422 = C 5 s u m C 5 w 060 C 5 w 151 C 5 w 242 C 5 w 040 + w 240 M C 5 w 131 + w 331 M C 5 w 333 C 6 w 222 + C 6 w 422 = C 6 s u m C 6 w 151 C 6 w 242 C 6 w 131 + w 331 M C 6 w 333 .
{ C 7 w 131 + C 7 w 331 M = C 7 s u m C 7 w 151 C 7 w 242   C 8 w 131 + C 8 w 331 M = C 8 s u m C 8 w 060 C 8 w 151 C 8 w 242 C 8 w 040 + w 240 M   .
{ C 5 w 222 + C 5 w 422 = C 5 + 40 A 2 C 15 + 3 A C 8 12 A 2 C 9 3 A C 11 + 12 A 2 C 12 + 360 A 4 C 16 480 A 3 C 16 C 6 w 222 + C 6 w 422 = C 6 3 A C 7 20 A 2 C 14 + 3 A C 10 + 12 A 2 C 13
{ C 7 w 131 + C 7 w 331 M = C 7 4 A C 13 + 10 A 2 C 14 C 8 w 131 + C 8 w 331 M = C 8 8 A C 9 + 4 A C 12 + 30 A 2 C 15 + 120 A 2 C 16 240 A 3 C 16
B 2 [ H x 2 H y 2 2 H x H y H x H y H y H x 1 0 0 1 H x 4 H y 4 2 H x H y ( H x 2 + H y 2 ) 4 H x 3 6 H x 2 H y 2 H y 3 4 H y 3 2 H x 3 6 H x H y 2 ] T [ W 222 2 A 222 , x 2 A 222 , y B 222 , x 2 B 222 , y 2 W 422 A 422 , x A 422 , y ] = 2 [ C 5 w 222 + C 5 w 422 C 6 w 222 + C 6 w 422 ] ,
B 3 [ H x H y 1 0 0 1 H x 3 + H x H y 2 H x 2 H y + H y 3 3 H x 2 H y 2 2 H x H y 2 H x H y H x 2 3 H y 2 ] T [ W 131 A 131 , x A 131 , y W 331 M A 331 M , x A 331 M , y ] = 3 [ C 7 w 131 + C 7 w 331 M C 8 w 131 + C 8 w 331 M ] ,
B 222 , F i g 2 = B 2 ( B 222 2 B 222 , M i s 2 )
R M S D i = 1 150 n = 1 150 [ X i ( n ) x i ( n ) ] 2
{ H T M , x = C A σ S M , x s p h H T M , y = C A σ S M , y s p h H I M A G E , x = C B σ S M , x s p h + C C σ T M , x s p h H I M A G E , y = C B σ S M , y s p h + C C σ T M , y s p h A 222 x = W 222 , S M s p h σ S M , x s p h + W 222 , S M a s p h σ S M , x a s p h + W 222 , T M s p h σ T M , x s p h + W 222 , T M a s p h σ T M , x a s p h A 222 y = W 222 , S M s p h σ S M , y s p h + W 222 , S M a s p h σ S M , y a s p h + W 222 , T M s p h σ T M , y s p h + W 222 , T M a s p h σ T M , y a s p h A 131 x = W 131 , S M s p h σ S M , x s p h + W 131 , S M a s p h σ S M , x a s p h + W 131 , T M s p h σ T M , x s p h + W 131 , T M a s p h σ T M , x a s p h A 131 y = W 131 , S M s p h σ S M , y s p h + W 131 , S M a s p h σ S M , y a s p h + W 131 , T M s p h σ T M , y s p h + W 131 , T M a s p h σ T M , y a s p h
{ X D E S M = u ¯ P M d 1 σ S M , x a s p h Y D E S M = u ¯ P M d 1 σ S M , y a s p h A D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , y s p h c S M Y D E S M B D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , x s p h + c S M Y D E S M X D E T M = [ d 2 + d 1 ( 2 c S M d 2 1 ) ] u ¯ P M σ T M , x a s p h + 2 d 2 ( B D E S M + c S M X D E S M ) Y D E T M = [ d 2 + d 1 ( 2 c S M d 2 1 ) ] u ¯ P M σ T M , y a s p h + 2 d 2 ( A D E S M + c S M X D E S M ) A D E T M = [ c T M ( d 2 d 1 ) + 2 c S M ( c T M d 1 d 2 + d 1 ) + 1 ] u ¯ P M σ T M , y s p h + 2 ( 1 + c T M d 2 ) ( c S M Y D E S M + A D E S M ) c T M Y D E T M B D E T M = [ c T M ( d 2 d 1 ) + 2 c S M ( c T M d 1 d 2 + d 1 ) + 1 ] u ¯ P M σ T M , x s p h 2 ( 1 + c T M d 2 ) ( c S M X D E S M B D E S M ) + c T M X D E T M

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