Abstract

The literature deals with the modulation transfer function (MTF) only for object brightness distribution functions (OBDFs) oriented along the meridional and sagittal directions. This paper addresses computation of the geometrical MTF for an off-axis source point when the OBDF is oriented along any arbitrarily defined direction. This study finds that the MTF is not a monotonic increasing or decreasing function when the direction of the OBDF is changing. Consequently, the extreme MTF values may occur when the OBDF is aligned at any direction between the meridional and sagittal directions. Four theorems are provided for the MTF and the phase shift variations that take place when the OBDF is translated or rotated. It is found that the MTF and the phase shift are symmetrical or antisymmetrical about certain directions. Thus, to observe all possible changes in the MTF and the phase shift, it is sufficient to rotate the OBDF through a range of just 90°. The presented method is based on a recent irradiance method for MTF computation that does not rely on counting the number of ray hits on a mesh, making the method immune to effects of grid size and thus improving traditional accuracy.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2011

P. D. Lin and C. S. Liu, “Geometrical MTF computation method based on the irradiance model,” Appl. Phys. B 102, 243–249 (2011).
[CrossRef]

2010

2009

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

1998

1997

S. Inoue, N. Tsumura, and Y. Miyake, “Measuring MTF of paper by sinusoidal test pattern projection,” J. Imaging Sci. Technol. 41, 657–661 (1997).

1995

1989

1984

1976

1965

Barakat, R.

Boreman, G. D.

Burkhard, D. G.

David, S.

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

Ferrell, R. K.

Goddard, J. S.

Inoue, S.

S. Inoue, N. Tsumura, and Y. Miyake, “Measuring MTF of paper by sinusoidal test pattern projection,” J. Imaging Sci. Technol. 41, 657–661 (1997).

Kaczynski, M.

Kalyvas, N.

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

Kandarakis, I.

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

Kassim, A. M.

Lin, P. D.

P. D. Lin and C. S. Liu, “Geometrical MTF computation method based on the irradiance model,” Appl. Phys. B 102, 243–249 (2011).
[CrossRef]

C. S. Liu and P. D. Lin, “Computational method for deriving the geometrical point spread function of an optical system,” Appl. Opt. 49, 126–136 (2010).
[CrossRef] [PubMed]

Liu, C. S.

P. D. Lin and C. S. Liu, “Geometrical MTF computation method based on the irradiance model,” Appl. Phys. B 102, 243–249 (2011).
[CrossRef]

C. S. Liu and P. D. Lin, “Computational method for deriving the geometrical point spread function of an optical system,” Appl. Opt. 49, 126–136 (2010).
[CrossRef] [PubMed]

Michail, C.

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

Miyake, Y.

S. Inoue, N. Tsumura, and Y. Miyake, “Measuring MTF of paper by sinusoidal test pattern projection,” J. Imaging Sci. Technol. 41, 657–661 (1997).

Panayiotakis, G. S.

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

Park, S. K.

Rogers, G. L.

Schowengerdt, R.

Shealy, D. L.

Sitter, D. N.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 372–375.

Toutountzis, A.

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

Tsumura, N.

S. Inoue, N. Tsumura, and Y. Miyake, “Measuring MTF of paper by sinusoidal test pattern projection,” J. Imaging Sci. Technol. 41, 657–661 (1997).

Valais, I.

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

Yang, S.

Appl. Opt.

Appl. Phys. B

P. D. Lin and C. S. Liu, “Geometrical MTF computation method based on the irradiance model,” Appl. Phys. B 102, 243–249 (2011).
[CrossRef]

C. Michail, A. Toutountzis, S. David, N. Kalyvas, I. Valais, I. Kandarakis, and G. S. Panayiotakis, “Imaging performance and light emission efficiency of Lu2 SiO5:Ce (LSO:Ce) powder scintillator under X-ray mammographic conditions,” Appl. Phys. B 95, 131–139 (2009).
[CrossRef]

J. Imaging Sci. Technol.

S. Inoue, N. Tsumura, and Y. Miyake, “Measuring MTF of paper by sinusoidal test pattern projection,” J. Imaging Sci. Technol. 41, 657–661 (1997).

J. Opt. Soc. Am.

Other

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 372–375.

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Figures (12)

Fig. 1
Fig. 1

Radiation of light rays from an on-axis source point P 0 0 onto the image plane in an axis-symmetrical optical system. Note that the MTF and LSF are unchanged when the direction of the OBDF is changed.

Fig. 2
Fig. 2

Radiation of light rays from off-axis source point P 0 0 on the image plane in an axis-symmetrical optical system. Note that the imaged spot is not symmetrical due to coma and astigmatism aberrations.

Fig. 3
Fig. 3

Ray density of the source point P 0 0 = [ 0 507 150 1 ] T on the image plane from the ray-counting method with grid size 0.1 × 0.1 and 56,529 rays.

Fig. 4
Fig. 4

Ray-counting method in which the ray density (i.e., the number of rays intercepted by each square grid on the image plane) is taken as a measure of the point spread function and LSF.

Fig. 5
Fig. 5

Representation of unit directional vector 0 0 originating from source point P 0 0 .

Fig. 6
Fig. 6

Infinitesimally small area d π n = d x n d z n centered at the incident point on the image plane receives the energy flux from the source point.

Fig. 7
Fig. 7

Schematic illustration of Ω 0 , i.e., the solid angle subtended by a ray cone with its apex at the source point.

Fig. 8
Fig. 8

LSF of P 0 0 = [ 0 507 150 1 ] T as determined using the ray-counting method. It is seen that the LSF depends on both the number of rays traced and the grid size on the image plane.

Fig. 9
Fig. 9

Variations of the MTF and phase shift of a source point P 0 0 = [ 0 507 150 1 ] T when tracing 2294 rays and using ν = 2 . It is shown that the MTF computed using the ray-counting method (B, μ = 0 ° ; D, μ = 45 ° ; F, μ = 90 ° ) is sensitive to the grid size meshed on the image plane. However, the MTF computed using the irradiance method (A, μ = 0 ° ; C, μ = 45 ° ; E, μ = 90 ° ) is insensitive to the grid size.

Fig. 10
Fig. 10

Effects of coma and astigmatism aberrations in changing the shape of the LSF when the OBDF is rotated through different angles ( P 0 0 = [ 0 507 150 1 ] T ; 25,781 rays; A, μ = 0 ° ; B, μ = 30 ° ; C, μ = 60 ° ; D, μ = 90 ° ).

Fig. 11
Fig. 11

Nonmonotonic variation of MTF in the domain 0 ° μ 90 ° ( P 0 0 = [ 0 507 150 1 ] T , 2294 rays, ν = 2 ). It is noted that the extreme MTF values occur at intermediate directions between the sagittal and meridional directions.

Fig. 12
Fig. 12

Variation of MTF with the number of traced rays for ν = 2 and μ = 45 ° . It is noted that, for both the ray-counting and irradiance methods, the MTF values oscillate significantly unless a sufficient number of rays are traced.

Equations (47)

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I ( x 0 ) = b 0 + b 1 cos ( 2 π ν x 0 ) ,
M 0 = I max I min I max + I min = ( b 0 + b 1 ) ( b 0 b 1 ) ( b 0 + b 1 ) + ( b 0 b 1 ) = b 1 b 0 .
I ( x n ) = L ( δ ) I ( x n δ ) d δ = b 0 L ( δ ) d δ + b 1 cos ( 2 π ν x n ) L ( δ ) cos ( 2 π ν δ ) d δ + b 1 sin ( 2 π ν x n ) L ( δ ) sin ( 2 π ν δ ) d δ = b 0 + b 1 [ L c ( ν ) cos ( 2 π ν x n ) + L s ( ν ) sin ( 2 π ν x n ) ] = b 0 + b 1 G ( ν ) cos ( 2 π ν x n ϖ ( ν ) ) .
L c ( ν ) = L ( δ ) cos ( 2 π ν δ ) d δ = L ( x n ) cos ( 2 π ν x n ) d x n ,
L s ( ν ) = L ( δ ) sin ( 2 π ν δ ) d δ = L ( x n ) sin ( 2 π ν x n ) d x n ,
G ( ν ) = L c 2 ( ν ) + L s 2 ( ν ) ,
ϖ ( ν ) = atan 2 ( L s ( ν ) , L c ( ν ) ) ,
MTF ( ν ) = M n / M 0 = G ( ν ) = L c 2 ( ν ) + L s 2 ( ν ) .
[ x n 0 z n 1 ] = A n n [ x n 0 z n 1 ] = [ cos μ 0 sin μ x n c cos μ z n c sin μ 0 1 0 0 sin μ 0 cos μ x n c sin μ z n c cos μ 0 0 0 1 ] [ x n 0 z n 1 ] = [ ( x n x n c ) cos μ + ( z n z n c ) sin μ 0 ( x n x n c ) sin μ + ( z n z n c ) cos μ 1 ] .
L c ( ν , μ , x n c , z n c ) = L ( x n , μ ) cos ( 2 π ν x n ) d x n ,
L s ( ν , μ , x n c , z n c ) = L ( x n , μ ) sin ( 2 π ν x n ) d x n ,
I * ( x n ) = I ( x n ) L ( x n ) Δ x n = b 0 + b 1 [ L c * ( ν ) cos ( 2 π ν x n ) + L s * ( ν ) sin ( 2 π ν x n ) ] = b 0 + b 1 G * ( ν ) cos ( 2 π ν x n ϖ * ) ,
L c ( ν , μ , x n c , z n c ) = L ( x n , μ ) cos ( 2 π ν x n ) Δ x n L ( x n , μ ) Δ x n ,
L s ( ν , μ , x n c , z n c ) = L ( x n , μ ) sin ( 2 π ν x n ) Δ x n L ( x n , μ ) Δ x n .
d F 0 = I 0 d Ω 0 = I 0 cos β 0 d α 0 d β 0 .
F 0 = I 0 cos β 0 d α 0 d β 0 = I 0 Ω 0 .
d F 0 = I 0 d Ω 0 = cos β 0 Ω 0 d α 0 d β 0 = d F n = B ( x n , z n ) d x n d z n .
d π n = d x n d z n = | ( x n , z n ) ( α 0 , β 0 ) | d α 0 d β 0 .
B ( x n , z n ) = cos β 0 Ω 0 | ( x n , z n ) / ( α 0 , β 0 ) | .
L ( x n , μ ) = B ( x n , z n ) d z n = 1 Ω 0 cos β 0 | ( x n , z n ) / ( α 0 , β 0 ) | d z n .
L c ( ν , μ , x n c , z n c ) = 1 Ω 0 cos β 0 cos ( 2 π ν x n ) | ( x n , z n ) / ( α 0 , β 0 ) | d z n d x n = 1 Ω 0 cos ( 2 π ν x n ) cos β 0 d α 0 d β 0 = 1 Ω 0 cos ( 2 π ν [ ( x n x n c ) cos μ + ( z n z n c ) sin μ ] ) C β 0 d α 0 d β 0 ,
L s ( ν , μ , x n c , z n c ) = 1 Ω 0 cos β 0 sin ( 2 π ν x n ) | ( x n , z n ) / ( α 0 , β 0 ) | d z n d x n = 1 Ω 0 sin ( 2 π ν x n ) cos β 0 d α 0 d β 0 = 1 Ω 0 sin ( 2 π ν [ ( x n x n c ) cos μ + ( z n z n c ) sin μ ) ] ) C β 0 d α 0 d β 0 .
yrot ( ω i y ) = [ cos ω i y 0 sin ω i y 0 0 1 0 0 sin ω i y 0 cos ω i y 0 0 0 0 1 ] ,
tran ( t i x , t i y , t i z ) = [ 1 0 0 t i x 0 1 0 t i y 0 0 1 t i z 0 0 0 1 ] .
MTF 2 ( ν , 0 , x n c , z n c ) x n c = 2 MTF ( ν , 0 , x n c , z n c ) MTF ( ν , 0 , x n c , z n c ) x n c = MTF 2 ( ν , 0 , x n c , z n c ) x n x n x n c + MTF 2 ( ν , 0 , x n c , z n c ) z n z n x n c = 2 ( L c ( ν , 0 , x n c , z n c ) L c ( ν , 0 , x n c , z n c ) x n + L s ( ν , 0 , x n c , z n c ) L s ( ν , 0 , x n c , z n c ) x n ) x n x n c + 2 ( L c ( ν , 0 , x n c , z n c ) L c ( ν , 0 , x n c , z n c ) z n + L s ( ν , 0 ) L s ( ν , 0 , x n c , z n c ) z n ) z n x n c = 0 .
MTF 2 ( ν , 0 , x n c , z n c ) z n c = 2 MTF ( ν , 0 , x n c , z n c ) MTF ( ν , 0 , x n c , z n c ) z n c = MTF 2 ( ν , μ , x n c , z n c ) x n x n z n c + MTF 2 ( ν , μ , x n c , z n c ) z n z n z n c = 0.
ϖ ( ν , 0 , x n c , z n c ) x n c = ϖ ( ν , 0 , x n c , z n c ) x n x n x n c + ϖ ( ν , 0 , x n c , z n c ) z n z n x n c = L c ( ν , 0 , x n c , z n c ) L c ( ν , 0 , x n c , z n c ) 2 + ( L s ( ν , 0 , x n c , z n c ) 2 L s ( ν , 0 , x n c , z n c ) x n x n x n c L s ( ν , 0 , x n c , z n c ) L c ( ν , 0 , x n c , z n c ) 2 + ( L s ( ν , 0 , x n c , z n c ) 2 L c ( ν , 0 , x n c , z n c ) x n x n x n c = x n x n c = 1 ,
ϖ ( ν , 0 , x n c , z n c ) z n c = ϖ ( ν , 0 , x n c , z n c ) x n x z n c + ϖ ( ν , 0 , x n c , z n c ) z n z n z n c = L c ( ν , 0 , x n c , z n c ) L c ( ν , 0 , x n c , z n c ) 2 + ( L s ( ν , 0 , x n c , z n c ) 2 L s ( ν , 0 , x n c , z n c ) x n x n z n c L s ( ν , 0 , x n c , z n c ) L c ( ν , 0 , x n c , z n c ) 2 + ( L s ( ν , 0 , x n c , z n c ) 2 L c ( ν , 0 , x n c , z n c ) x n x n z n c = x n z n c = 0 .
L c ( ν , μ ^ + 180 ° , x n c , z n c ) = 1 Ω 0 cos [ 2 π ν ( x n x n c ) cos ( μ ^ + 180 ° ) + 2 π ν ( z n z n c ) sin ( μ ^ + 180 ° ) ] C β 0 d α 0 d β 0 = 1 Ω 0 cos [ 2 π ν ( x n x n c ) cos μ ^ + 2 π ν ( z n z n c ) sin μ ^ ] cos β 0 d α 0 d β 0 = L c ( ν , μ ^ , x n c , z n c ) ,
L s ( ν , μ ^ + 180 ° , x n c , z n c ) = 1 Ω 0 sin [ 2 π ν ( x n x n c ) cos ( μ ^ + 180 ° ) + 2 π ν ( z n z n c ) sin ( μ ^ + 180 ° ) ] cos β 0 d α 0 d β 0 = 1 Ω 0 sin [ 2 π ν ( x n x n c ) cos ( μ ^ ) + 2 π ν ( z n z n c ) sin ( μ ^ ) ] cos β 0 d α 0 d β 0 = L s ( ν , μ ^ , x n c , z n c ) .
MTF ( ν , μ ^ + 180 ° , x n c , z n c ) = [ L c ( ν , μ ^ + 180 ° , x n c , z n c ) ] 2 + [ L s ( ν , μ ^ + 180 ° , x n c , z n c ) ] 2 = [ L c ( ν , μ ^ , x n c , z n c ) ] 2 + [ L s ( ν , μ ^ , x n c , z n c ) ] 2 = MTF ( ν , μ ^ , x n c , z n c ) , ϖ ( ν , μ ^ + 180 ° , x n c , z n c ) = atan 2 ( L s ( ν , μ ^ + 180 ° , x n c , z n c ) , L c ( ν , μ ^ + 180 ° , x n c , z n c ) ) = atan 2 ( L s ( ν , μ ^ , x n c , z n c ) , L c ( ν , μ ^ , x n c , z n c ) ) = ϖ ( ν , μ ^ , x n c , z n c ) .
x n c = y n c = 0.
0 , 1 0 = [ cos β 0 cos ( 90 ° α 0 ) cos β 0 sin ( 90 ° α 0 ) sin β 0 0 ] T ,
0 , 2 0 = [ cos β 0 cos ( 90 ° + α 0 ) cos β 0 sin ( 90 ° + α 0 ) sin β 0 0 ] T ,
x n ( 90 ° α 0 , β 0 ) = x n ( 90 ° + α 0 , β 0 ) ,
z n ( 90 ° α 0 , β 0 ) = z n ( 90 ° + α 0 , β 0 ) .
L c ( ν , 90 ° + μ ^ , x n c , z n c ) = 1 Ω 0 cos ( 2 π ν [ x n cos ( 90 ° + μ ^ ) + ( z n z n c ) sin ( 90 ° + μ ^ ) ] ) cos β 0 d α 0 d β 0 = cos ( 2 π ν z n c C μ ^ ) Ω 0 cos ( 2 π ν ( x n sin μ ^ z n cos μ ^ ) ) cos β 0 d α 0 d β 0 sin ( 2 π ν z n c cos μ ^ ) Ω 0 sin ( 2 π ν ( x n sin μ ^ z n cos μ ^ ) ) cos β 0 d α 0 d β 0 = cos ( 2 π ν z n c cos μ ^ ) Ω 0 [ cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 + sin ( 2 π ν x n sin μ ^ ) sin ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 ] sin ( 2 π ν z n c cos μ ^ ) Ω 0 [ sin ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 cos ( 2 π ν x n sin μ ^ ) sin ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 ] .
cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 = [ 90 ° α ^ 0   limit 90 ° + α ^ 0   limit cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) d ( 90 ° + α ^ 0 ) ] cos β 0 d β 0 = [ α ^ 0   limit α ^ 0   limit cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) d α ^ 0 ] cos β 0 d β 0 = [ α ^ 0   limit 0 cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) d α ^ 0 + 0 α ^ 0   limit cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) d α ^ 0 ] cos β 0 d β 0 = 2 [ 0 α ^ 0   limit cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) d α ^ 0 ] cos β 0 d β 0 .
sin ( 2 π ν x n sin μ ^ ) sin ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 = 0 ,
sin ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 = 0 ,
cos ( 2 π ν x n sin μ ^ ) sin ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 = 2 [ 0 α ^ 0   limit cos ( 2 π ν x n sin μ ^ ) sin ( 2 π ν z n cos μ ^ ) d α ^ 0 ] cos β 0 d β 0 .
L c ( ν , 90 ° + μ ^ , x n c , z n c ) = 2 cos ( 2 π ν z n c cos μ ^ ) Ω 0 [ 0 α ^ 0   limit cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) d α ^ 0 ] cos β 0 d β 0 + 2 sin ( 2 π ν z n c cos μ ^ ) Ω 0 [ 0 α ^ 0   limit cos ( 2 π ν x n sin μ ^ ) sin ( 2 π ν z n cos μ ^ ) d α ^ 0 ] cos β 0 d β 0 .
L c ( ν , 90 ° + μ ^ , x n c , z n c ) = L c ( ν , 90 ° μ ^ , x n c , z n c ) .
L s ( ν , 90 ° + μ ^ , x n c , z n c ) = 1 Ω 0 sin ( 2 π ν [ x n cos ( 90 ° + μ ^ ) + ( z n z n c ) sin ( 90 ° + μ ^ ) ] ) cos β 0 d α 0 d β 0 = sin ( 2 π ν cos μ ^ z n c ) Ω 0 cos ( 2 π ν ( x n sin μ ^ z n cos μ ^ ) ) cos β 0 d α 0 d β 0 cos ( 2 π ν z n c cos μ ^ ) Ω 0 sin ( 2 π ν ( x n sin μ ^ z n cos μ ^ ) ) cos β 0 d α 0 d β 0 = sin ( 2 π ν z n c cos μ ^ ) Ω 0 [ cos ( 2 π ν x n sin μ ^ ) cos ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 + sin ( 2 π ν x n sin μ ^ ) sin ( 2 π ν z n cos μ ^ ) cos β 0 d α 0 d β 0 ] C ( 2 π ν z n c C μ ^ ) Ω 0 [ S ( 2 π ν x n S μ ^ ) C ( 2 π ν z n C μ ^ ) C β 0 d α 0 d β 0 C ( 2 π ν x n S μ ^ ) S ( 2 π ν z n C μ ^ ) C β 0 d α 0 d β 0 ] = 2 sin ( 2 π ν z n c cos μ ) Ω 0 [ 0 α ^ 0   limit cos ( 2 π ν x n sin μ ) cos ( 2 π ν z n cos μ ) d α ^ 0 ] cos β 0 d β 0 + 2 cos ( 2 π ν z n c cos μ ) Ω 0 [ 0 α ^ 0   limit cos ( 2 π ν x n sin μ ) sin ( 2 π ν z n cos μ ) d α ^ 0 ] cos β 0 d β 0 .
L s ( ν , 90 ° + μ ^ , x n c , z n c ) = L s ( ν , 90 ° μ ^ , x n c , z n c ) .
MTF ( ν , 90 ° + μ ^ , x n c , z n c ) = [ L c ( ν , 90 ° + μ ^ , x n c , z n c ) ] 2 + [ L s ( ν , 90 ° + μ ^ , x n c , z n c ) ] 2 = [ L c ( ν , 90 ° μ ^ , x n c , z n c ) ] 2 + [ L s ( ν , 90 ° μ ^ , x n c , z n c ) ] 2 = MTF ( ν , 90 ° μ ^ , x n c , z n c ) ,
ϖ ( ν , 90 ° + μ ^ , x n c , z n c ) = atan 2 ( L s ( ν , 90 ° + μ ^ , x n c , z n c ) , L c ( ν , 90 ° + μ ^ , x n c , z n c ) ) = atan 2 ( L s ( ν , 90 ° μ ^ , x n c , z n c ) , L c ( ν , 90 ° μ ^ , x n c , z n c ) ) = ϖ ( ν , 90 ° μ ^ , x n c , z n c ) .

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