Abstract

Most computer-aided alignment methods for optical systems are based on numerical algorithms at present, which omit aberration theory. This paper presents a novel alignment algorithm for three-mirror anastigmatic (TMA) telescopes using Nodal Aberration Theory (NAT). The aberration field decenter vectors and boresight error of misaligned TMA telescopes are derived. Two alignment models based on 3rd and 5th order NAT are established successively and compared in the same alignment example. It is found that the average and the maximum RMS wavefront errors in the whole field of view of 0.3° × 0.15° are 0.063 λ (λ = 1 μm) and 0.068 λ respectively after the 4th alignment action with the 3rd order model, and 0.011 λ and 0.025 λ (nominal values) respectively after the 3rd alignment action with the 5th order model. Monte-Carlo alignment simulations are carried out with the 5th order model. It shows that the 5th order model still has good performance even when the misalignment variables are large (−1 mm≤linear misalignment≤1 mm, −0.1°≤angular misalignment≤0.1°), and multiple iterative alignments are needed when the misalignment variables increase.

© 2015 Optical Society of America

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References

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  1. J. W. Figoski, T. E. Shrode, and G. F. Moore, “Computer-aided alignment of a wide-field, three-mirror, unobscured, high-resolution sensor,” Proc. SPIE 1049, 166–177 (1989).
    [Crossref]
  2. S. Kim, H.-S. Yang, Y.-W. Lee, and S.-W. Kim, “Merit function regression method for efficient alignment control of two-mirror optical systems,” Opt. Express 15(8), 5059–5068 (2007).
    [Crossref] [PubMed]
  3. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II :Optical system alignment using differential wavefront sampling,” Opt. Express 15(23), 15424–15437 (2007).
    [Crossref] [PubMed]
  4. K. P. Thompson and J. P. Rolland, “A page from “the drawer”: how Roland Shack opened the door to the aberration theory of freeform optics,” Proc. SPIE 9186, 91860A (2014).
    [Crossref]
  5. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).
  6. T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
    [Crossref]
  7. K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
    [Crossref] [PubMed]
  8. K. P. Thompson, T. Schmid, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 6834, 68340B (2007).
    [Crossref]
  9. K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
    [Crossref]
  10. The Fringe Zernike polynomial was developed by John Loomis at the University of Arizona, Optical Sciences Center in the 1970s, and is described on page C-8 of the CODE V® Version 10.4 Reference Manual (Synopsys, Inc.) (2012).
  11. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
    [Crossref] [PubMed]
  12. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).
  13. R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 7(6), 262–264 (1982).
    [Crossref] [PubMed]
  14. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
    [Crossref] [PubMed]
  15. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
    [Crossref] [PubMed]
  16. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
    [Crossref] [PubMed]
  17. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
    [Crossref] [PubMed]
  18. J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt. 49(16), D69–D95 (2010).
    [Crossref] [PubMed]

2014 (1)

K. P. Thompson and J. P. Rolland, “A page from “the drawer”: how Roland Shack opened the door to the aberration theory of freeform optics,” Proc. SPIE 9186, 91860A (2014).
[Crossref]

2011 (1)

2010 (2)

2009 (3)

2008 (2)

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[Crossref]

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
[Crossref] [PubMed]

2007 (3)

2005 (1)

1989 (1)

J. W. Figoski, T. E. Shrode, and G. F. Moore, “Computer-aided alignment of a wide-field, three-mirror, unobscured, high-resolution sensor,” Proc. SPIE 1049, 166–177 (1989).
[Crossref]

1982 (1)

Cakmakci, O.

Dalton, G. B.

Figoski, J. W.

J. W. Figoski, T. E. Shrode, and G. F. Moore, “Computer-aided alignment of a wide-field, three-mirror, unobscured, high-resolution sensor,” Proc. SPIE 1049, 166–177 (1989).
[Crossref]

Fuerschbach, K.

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

Kim, S.

Kim, S.-W.

Lee, H.

Lee, Y.-W.

Moore, G. F.

J. W. Figoski, T. E. Shrode, and G. F. Moore, “Computer-aided alignment of a wide-field, three-mirror, unobscured, high-resolution sensor,” Proc. SPIE 1049, 166–177 (1989).
[Crossref]

Rolland, J. P.

K. P. Thompson and J. P. Rolland, “A page from “the drawer”: how Roland Shack opened the door to the aberration theory of freeform optics,” Proc. SPIE 9186, 91860A (2014).
[Crossref]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
[Crossref] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[Crossref]

K. P. Thompson, T. Schmid, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 6834, 68340B (2007).
[Crossref]

Sasián, J.

Schmid, T.

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
[Crossref] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[Crossref]

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 6834, 68340B (2007).
[Crossref]

Shrode, T. E.

J. W. Figoski, T. E. Shrode, and G. F. Moore, “Computer-aided alignment of a wide-field, three-mirror, unobscured, high-resolution sensor,” Proc. SPIE 1049, 166–177 (1989).
[Crossref]

Thompson, K.

Thompson, K. P.

K. P. Thompson and J. P. Rolland, “A page from “the drawer”: how Roland Shack opened the door to the aberration theory of freeform optics,” Proc. SPIE 9186, 91860A (2014).
[Crossref]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
[Crossref] [PubMed]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[Crossref]

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 6834, 68340B (2007).
[Crossref]

Tosh, I. A. J.

Tyson, R. K.

Yang, H.-S.

Appl. Opt. (1)

J. Opt. Soc. Am. A (5)

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (5)

K. P. Thompson, T. Schmid, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 6834, 68340B (2007).
[Crossref]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

K. P. Thompson and J. P. Rolland, “A page from “the drawer”: how Roland Shack opened the door to the aberration theory of freeform optics,” Proc. SPIE 9186, 91860A (2014).
[Crossref]

J. W. Figoski, T. E. Shrode, and G. F. Moore, “Computer-aided alignment of a wide-field, three-mirror, unobscured, high-resolution sensor,” Proc. SPIE 1049, 166–177 (1989).
[Crossref]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[Crossref]

Other (3)

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

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

The Fringe Zernike polynomial was developed by John Loomis at the University of Arizona, Optical Sciences Center in the 1970s, and is described on page C-8 of the CODE V® Version 10.4 Reference Manual (Synopsys, Inc.) (2012).

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Figures (9)

Fig. 1
Fig. 1 The optical layout of the TMA telescope.

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Fig. 2
Fig. 2 FFDs for Fringe Zernike coefficients (a) C5/6, (b) C7/8, and (c) RMS wavefront error for the nominal TMA telescope.

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Fig. 3
Fig. 3 Concepts of effective field height, aberration field decenter vectors and boresight error.

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Fig. 4
Fig. 4 FFDs for Fringe Zernike coefficients (a) C5/6 and (b) C7/8 for the misaligned TMA telescope.

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Fig. 5
Fig. 5 The results after each alignment action. (a) The residual linear misalignments (b) The residual angular misalignments (c) The residual RMS wavefront error.

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Fig. 6
Fig. 6 FFDs for Fringe Zernike coefficients (a) C5/6 and (b) C7/8 for the TMA telescope after the 4th alignment action.

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Fig. 7
Fig. 7 The results after each alignment action. (a) The residual linear misalignments (b) The residual angular misalignments (c) The residual RMS wavefront error.

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Fig. 8
Fig. 8 The results of Monte-Carlo alignment simulations in the (a) case 1 (b) case 2 (c) case 3.

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Fig. 9
Fig. 9 The alignment process with 3rd/5th order NAT model.

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Tables (7)

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Table 1 Optical Prescription of the Example System

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Table 2 Third Order Aberration Coefficients of the TMA Telescope

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Table 3 Misalignments of the TMA Telescope

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Table 4 Calculated Aberration Coefficients

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Table 5 Alignment Eesults of the 3rd Order Model after the 4th Alignment Action and the 5th Order Model after the 3rd Alignment Action

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Table 6 Ranges of Misalignment Variables Used for the Simulations

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Table 7 Acronyms and Parameter Definitions

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Equations (45)

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W = ∑ j ∑ p ∞ ∑ n ∞ ∑ m ∞ W k l m , j ( s p h , a s p h ) ( H → A j ( s p h , a s p h ) ⋅ H → A j ( s p h , a s p h ) ) p ( ρ → ⋅ ρ → ) n ( H → A j ( s p h , a s p h ) ⋅ ρ → ) m , k = 2 p + m , l = 2 n + m ,
H → A j ( s p h , a s p h ) = H → − σ → j ( s p h , a s p h ) .
σ → j ( s p h ) = − i ¯ j * → i ¯ j = − u ¯ # O A R → j − β 0 # → j + y ¯ O A R # → j ⋅ c j u ¯ j + y ¯ j c j ,
β 0 # → j = [ − B D E j + X D E j c j A D E j + Y D E j c j ] .
u ¯ S M = − u ¯ P M ,
u ¯ T M = u ¯ P M   ( 2   d 1 + r S M ) r S M ,
y ¯ P M = 0 ,
y ¯ S M = − d 1   u ¯ P M ,
y ¯ T M = 2   d 1   d 2   u ¯ P M r S M − u ¯ P M   ( d 1 − d 2 ) ,
σ → j ( a s p h ) = δ v j * → y ¯ j = 1 y ¯ j ( [ X D E j Y D E j ] − y ¯ O A R # → j ) ,
u ¯ # O A R → P M = [ 0 0 ] ,
u ¯ # O A R → S M = [ − 2   B D E P M 2   A D E P M ] ,
u ¯ # O A R → T M = [ − 2   B D E S M 2   A D E S M ] − 2 r S M [ X D E P M Y D E P M ] + 2 r S M [ X D E S M Y D E S M ] + ( 4   d 1 r S M + 2 ) [ B D E P M − A D E P M ] ,
y ¯ O A R # → P M = [ X D E P M Y D E P M ] ,
y ¯ O A R # → S M = [ X D E P M − 2   B D E P M   d 1 Y D E P M + 2   A D E P M   d 1 ] ,
y ¯ O A R # → T M = d 2 [ − 2   B D E S M 2   A D E S M ] + ( 2   d 2 − 2   d 1 + 4   d 1   d 2 r S M ) [ B D E P M − A D E P M ] .                                                                           − ( 2   d 2 r S M − 1 ) [ X D E P M Y D E P M ] + 2     d 2 r S M [ X D E S M Y D E S M ]
σ → P M s p h = 1 u ¯ P M [ − B D E P M A D E P M ] ,
σ → S M s p h = 1 u ¯ P M [ − 2   B D E P M 2   A D E P M ] + 1 u ¯ P M   ( d 1 + r S M ) [ X D E P M − X D E S M + B D E S M   r S M − Y D E S M + Y D E P M − A D E S M   r S M ] ,
σ → T M s p h = 1 u ¯ P M [ − 2 B D E P M 2 A D E P M ] + [ B D E S M   ( 2   d 2   r S M + 2   r S M   r T M ) + X D E P M   ( 2   d 2 − r S M + 2   r T M ) − X D E S M   ( 2   d 2 + 2   r T M ) − A D E S M   ( 2   d 2   r S M + 2   r S M   r T M ) + Y D E P M   ( 2   d 2 − r S M + 2   r T M ) − Y D E S M   ( 2   d 2 + 2   r T M ) ] u ¯ P M   ( 2   d 1   d 2 − d 1   r S M + d 2   r S M + 2   d 1   r T M + r S M   r T M ) ,
σ → P M a s p h = [ 0 0 ] ,
σ → S M a s p h = 1 d 1   u ¯ P M [ X D E P M − X D E S M − 2   B D E P M   d 1 Y D E P M − Y D E S M + 2   A D E P M   d 1 ] ,
σ → T M a s p h = 1 u ¯ P M [ − 2 B D E P M 2 A D E P M ] + [ − 2   X D E S M   d 2 + X D E P M   ( 2   d 2 − r S M ) + 2   B D E S M   d 2   r S M − 2   Y D E S M   d 2 + Y D E P M   ( 2   d 2 − r S M ) − 2   A D E S M   d 2   r S M ] u ¯ P M   ( 2   d 1   d 2 − d 1   r S M + d 2   r S M ) .
Δ H → I M G = ( ( 4   d 2   d 3 − 4   d 1   d 3 + 2   d 1   r T M − 2   d 2   r T M + 2   d 3   r T M ) r T M + 4   d 1   ( 2   d 2   d 3 − d 2   r T M + d 3   r T M ) r S M   r T M ) [ − B D E P M A D E P M ]                                                                     −   ( ( 2   d 3 − r T M ) r T M − ( 4   d 2   d 3 − 2   d 2   r T M + 2   d 3   r T M ) r S M   r T M ) [ X D E P M Y D E P M ]                                                                   +   ( 4   d 2   d 3 − 2   d 2   r T M + 2   d 3   r T M ) r T M [ B D E S M − A D E S M ] −   ( 4   d 2   d 3 − 2   d 2   r T M + 2   d 3   r T M ) r S M r T M [ X D E S M Y D E S M ]
W C O M A 3 = [ ( W 131 H → − A → 131 ) ⋅ ρ → ] ( ρ → ⋅ ρ → ) ,
W C O M A 3       = [ W 131 H → x − A → 131 , x W 131 H → y − A → 131 , y ] ⋅ [ | ρ → | 3 cos φ | ρ → | 3 sin φ ] ,
[ A → 131 , x A → 131 , y ] = [ W 131 H → x − 3 C C O M A , x W 131 H → y − 3 C C O M A , y ] ,
W A S T 3 = 1 2 [ ∑ j W 222 j H → 2 − 2 H → A 222 + B → 222 2 ] ⋅ ρ → 2 ,
W A S T 3   = [ W 222   ( H → x 2 − H → y 2 ) 2 − H → x   A → 222 , x + H → y   A → 222 , y + B → 222 , x 2 2 W 222 H → x H → y − H → x   A → 222 , y − H → y A → 222 , x + B → 222 , y 2 2 ] ⋅ [ | ρ → | 2 cos ( 2 φ ) | ρ → | 2 sin ( 2 φ ) ] .
[ − H → x H → y 1 2 0 − H → y − H → x 0 1 2 ] [ A → 222 , x A → 222 , y B → 222 , x 2 B → 222 , y 2 ] = [ C A S T , x − W 222 2 ( H → x 2 − H → y 2 ) C A S T , y − W 222 H → x H → y ] ,
W C O M A 5 = [ W 331 M ( H → ⋅ H → ) H → − 2 ( H → ⋅ A → 331 M ) H → + 2 B 331 M H → − ( H → ⋅ H → ) A → 331 M + B → 331 M 2 H → * − C → 331 M ] ⋅ ρ → ( ρ → ⋅ ρ → ) ,
[ W 331 M ( H → ⋅ H → ) H → − 2 ( H → ⋅ A → 331 M ) H → + ( W 131 + 2 B 331 M ) H → − ( H → ⋅ H → ) A → 331 M + B → 331 M 2 H → * − ( A → 131 + C → 331 M ) ] = [ 3 C c o m a , x 3 C c o m a , y ] .
H C O M A P C O M A = Z C O M A ,
H C O M A = [ − 3 H → x 2 − H → y 2 − 2   H → x   H → y − 2   H → x   H → y − H → x 2 − 3 H → y 2 − 1 0 0 − 1 H → x − H → y H → y H → x H → x 3 + H → x   H → y 2 H → x 2   H → y + H → y 3 H → x H → y ] T , P C O M A = [ A → 331 M , x A → 331 M , y A → 131 , x + C → 331 M , x A → 131 , y + C → 331 M , y B → 331 M , x 2 B → 331 M , y 2 W 331 M W 131 + 2 B 331 M ] and Z C O M A = [ 3 C c o m a , x 3 C c o m a , y ] .
W A S T 5 = [ 1 2 W 422 ( H → ⋅ H → ) H → 2 − ( H → ⋅ H → ) ( H → A → 422 ) + 3 2 ( H → ⋅ H → ) B → 422 2 − ( H → ⋅ A → 422 ) H → 2 − 1 2 C → 422 3 H → * + 3 2 B 422 H → 2 − 3 2 ( H → C → 422 ) + 1 2 D → 422 2 ] ⋅ ρ → 2 ,
[ 1 2 W 422 ( H → ⋅ H → ) H → 2 − ( H → ⋅ H → ) ( H → A → 422 ) + 3 2 ( H → ⋅ H → ) B → 422 2 − ( H → ⋅ A → 422 ) H → 2 − 1 2 C → 422 3 H → * + ( 1 2 W 222 + 3 2 B 422 ) H → 2 − ( A → 222 + 3 2 C → 422 ) H → + ( 1 2 D → 422 2 + 1 2 B → 222 2 ) ] = [ C A S T , x C A S T , y ] .
H A S T P A S T = Z A S T ,
H A S T = [ H → x 4 − H → y 4 2 H → x   H → y   ( H → x 2 + H → y 2 ) 3   H → x 2 + 3   H → y 2 0 0 3   H → x 2 + 3   H → y 2 − H → x H → y − H → y − H → x H → x 2 − H → y 2 2 H → x   H → y − H → x − H → y H → y − H → x 1 0 0 1 − 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 , P A S T = [ W 422 B → 422 , x 2 B → 422 , y 2 C → 422 , x 3 C → 422 , y 3 W 222 + 3 B 422 2 A → 222 , x + 3 C → 422 , x 2 A → 222 , y + 3 C → 422 , y D → 422 , x 2 + B → 2 22 , x 2 D → 422 , y 2 + B → 2 22 , y 2 A → 422 , x A → 422 , y ] , Z A S T = [ 2 C A S T , x 2 C A S T , y ] .
P C O M A = S C O M A W C O M A ,
P A S T = S A S T W A S T .
( ( H 1 C O M A ⋮ H m C O M A ) ⋱ ( H 1 C O M A ⋮ H m C O M A ) ) 2 m n × 8 n ( S C O M A ( 1 ) ⋮ S C O M A ( n ) ) 8 n × 12 ( W C O M A ) 12 × 1 = ( ( Z 1 C O M A ( 1 ) ⋮ Z m C O M A ( 1 ) ) ⋮ ( Z 1 C O M A ( n ) ⋮ Z m C O M A ( n ) ) ) 2 m n × 1 ,
( ( H 1 A S T ⋮ H m A S T ) ⋱ ( H 1 A S T ⋮ H m A S T ) ) 2 m n × 12 n ( S A S T ( 1 ) ⋮ S A S T ( n ) ) 12 n × 12 ( W A S T ) 12 × 1 = ( ( Z 1 A S T ( 1 ) ⋮ Z m A S T ( 1 ) ) ⋮ ( Z 1 A S T ( n ) ⋮ Z m A S T ( n ) ) ) 2 m n × 1 ,
H ′ C O M A S ′ C O M A W C O M A = Z ′ C O M A ,
H ′ A S T S ′ A S T W A S T = Z ′ A S T .
W C O M A = p i n v ( H ′ C O M A S ′ C O M A ) Z ′ C O M A ,
W A S T = p i n v ( H ′ A S T S ′ A S T ) Z ′ A S T ,

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