Abstract

In this paper we derive an approximate analytical representation for the modulation transfer function (MTF) of an imaging system possessing a defocused cubic-phase pupil function. This expression is based on an approximation using the Arctan function and significantly reduces the computational time required to calculate the resulting MTF. We derive rigorous bounds on the minimum and average accuracy of our approximation. Using this approximate representation of the MTF, the analytical solution of the problem of calculating the extension of the depth of field for a circular aperture with a cubic phase mask becomes possible. We also comment on how one can modify our method to construct a lower-bound or an upper-bound approximate analytical expression for the MTF.

© 2008 Optical Society of America

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  1. W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” J. Opt. A, Pure Appl. Opt. 41, 6080-6092 (2002).
  2. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  3. E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
    [CrossRef]
  4. R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” J. Opt. A, Pure Appl. Opt. 44, 701-712 (2005).
  5. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signal and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).
    [CrossRef]
  6. R. Piestun and A. B. Miller, “Electromegnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892-902 (2000).
    [CrossRef]
  7. G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
    [CrossRef]
  8. S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “Reduced-complexity representation of the coherent point-spread function in the presence of aberrations and arbitrarily large defocus,” J. Opt. Soc. Am. A 23, 2476-2493 (2006).
    [CrossRef]
  9. S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “On the computation of the coherent point-spread function using a low-complexity representation,” Proc. SPIE 6311, 631108 (2006).
    [CrossRef]
  10. M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45, 2911-2923 (2006).
    [CrossRef] [PubMed]
  11. G. Muya and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. 30, 2715-2717 (2005).
    [CrossRef]
  12. S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. Pucci de Farias, “Analytical optimal solution of the extension of the depth of field using cubic phase wavefront coding. Part II. Design and optimization of the cubic phase,” J. Opt. Soc. Am. A 25, 1064-1074 (2007).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  14. S. Bagheri, P. X. Silveira, and D. Pucci de Farias, “The maximum extension of the depth of field of SNR-limited wavefront coded imaging systems,” in Technical Digest, OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings (Optical Society of America, 2007), paper CMD4, on CD.
  15. K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the otf,” J. Opt. Commun. 44, 323-326 (1983).
    [CrossRef]
  16. A. Mattuck, Introduction to Analysis (Prentice Hall, 1999).

2007 (1)

2006 (3)

2005 (3)

G. Muya and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. 30, 2715-2717 (2005).
[CrossRef]

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” J. Opt. A, Pure Appl. Opt. 44, 701-712 (2005).

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

2002 (1)

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” J. Opt. A, Pure Appl. Opt. 41, 6080-6092 (2002).

2000 (1)

1999 (1)

E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

1996 (1)

1995 (1)

1983 (1)

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the otf,” J. Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Bagheri, S.

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. Pucci de Farias, “Analytical optimal solution of the extension of the depth of field using cubic phase wavefront coding. Part II. Design and optimization of the cubic phase,” J. Opt. Soc. Am. A 25, 1064-1074 (2007).
[CrossRef]

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “Reduced-complexity representation of the coherent point-spread function in the presence of aberrations and arbitrarily large defocus,” J. Opt. Soc. Am. A 23, 2476-2493 (2006).
[CrossRef]

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “On the computation of the coherent point-spread function using a low-complexity representation,” Proc. SPIE 6311, 631108 (2006).
[CrossRef]

S. Bagheri, P. X. Silveira, and D. Pucci de Farias, “The maximum extension of the depth of field of SNR-limited wavefront coded imaging systems,” in Technical Digest, OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings (Optical Society of America, 2007), paper CMD4, on CD.

Barbastathis, G.

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “On the computation of the coherent point-spread function using a low-complexity representation,” Proc. SPIE 6311, 631108 (2006).
[CrossRef]

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “Reduced-complexity representation of the coherent point-spread function in the presence of aberrations and arbitrarily large defocus,” J. Opt. Soc. Am. A 23, 2476-2493 (2006).
[CrossRef]

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the otf,” J. Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Cathey, W. T.

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” J. Opt. A, Pure Appl. Opt. 41, 6080-6092 (2002).

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1866 (1995).
[CrossRef] [PubMed]

Christensen, M. P.

de Farias, D. Pucci

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. Pucci de Farias, “Analytical optimal solution of the extension of the depth of field using cubic phase wavefront coding. Part II. Design and optimization of the cubic phase,” J. Opt. Soc. Am. A 25, 1064-1074 (2007).
[CrossRef]

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “Reduced-complexity representation of the coherent point-spread function in the presence of aberrations and arbitrarily large defocus,” J. Opt. Soc. Am. A 23, 2476-2493 (2006).
[CrossRef]

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “On the computation of the coherent point-spread function using a low-complexity representation,” Proc. SPIE 6311, 631108 (2006).
[CrossRef]

S. Bagheri, P. X. Silveira, and D. Pucci de Farias, “The maximum extension of the depth of field of SNR-limited wavefront coded imaging systems,” in Technical Digest, OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings (Optical Society of America, 2007), paper CMD4, on CD.

Dorsch, R. G.

Dowski, E. R.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” J. Opt. A, Pure Appl. Opt. 41, 6080-6092 (2002).

E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1866 (1995).
[CrossRef] [PubMed]

Ferreira, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Harvey, A. R.

Johnson, G. E.

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” J. Opt. A, Pure Appl. Opt. 44, 701-712 (2005).

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signal and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).
[CrossRef]

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the otf,” J. Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Mattuck, A.

A. Mattuck, Introduction to Analysis (Prentice Hall, 1999).

Mendlovic, D.

Miller, A. B.

Muya, G.

Narayanswamy, R.

Neifeld, M. A.

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “On the computation of the coherent point-spread function using a low-complexity representation,” Proc. SPIE 6311, 631108 (2006).
[CrossRef]

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “Reduced-complexity representation of the coherent point-spread function in the presence of aberrations and arbitrarily large defocus,” J. Opt. Soc. Am. A 23, 2476-2493 (2006).
[CrossRef]

Ojeda-Castaneda, J.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the otf,” J. Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Piestun, R.

Silveira, P. E. X.

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. Pucci de Farias, “Analytical optimal solution of the extension of the depth of field using cubic phase wavefront coding. Part II. Design and optimization of the cubic phase,” J. Opt. Soc. Am. A 25, 1064-1074 (2007).
[CrossRef]

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” J. Opt. A, Pure Appl. Opt. 44, 701-712 (2005).

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

Silveira, P. X.

S. Bagheri, P. X. Silveira, and D. Pucci de Farias, “The maximum extension of the depth of field of SNR-limited wavefront coded imaging systems,” in Technical Digest, OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings (Optical Society of America, 2007), paper CMD4, on CD.

Somayaji, M.

Wach, H. B.

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” J. Opt. A, Pure Appl. Opt. 44, 701-712 (2005).

Zalevsky, Z.

Appl. Opt. (2)

J. Opt. A, Pure Appl. Opt. (2)

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” J. Opt. A, Pure Appl. Opt. 44, 701-712 (2005).

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” J. Opt. A, Pure Appl. Opt. 41, 6080-6092 (2002).

J. Opt. Commun. (1)

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the otf,” J. Opt. Commun. 44, 323-326 (1983).
[CrossRef]

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

Opt. Lett. (1)

Proc. SPIE (3)

E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

S. Bagheri, D. Pucci de Farias, G. Barbastathis, and M. A. Neifeld, “On the computation of the coherent point-spread function using a low-complexity representation,” Proc. SPIE 6311, 631108 (2006).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

S. Bagheri, P. X. Silveira, and D. Pucci de Farias, “The maximum extension of the depth of field of SNR-limited wavefront coded imaging systems,” in Technical Digest, OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings (Optical Society of America, 2007), paper CMD4, on CD.

A. Mattuck, Introduction to Analysis (Prentice Hall, 1999).

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of the imaging system.

Fig. 2
Fig. 2

Comparison between the original integral in Eq. (4) (solid curve) and its novel approximation (dashed curve). The original integral is equal to the left-hand side of Eq. (4), and its approximation is equal to the right-hand side of Eq. (4).

Fig. 3
Fig. 3

Plot of the exact ( MTF e ) and the first approximation of MTF ( u , 0 ) ( MTF a 1 ) for different values of α and W 20 . Note how the exact MTF has many oscillations, whereas the approximated MTF is a smooth function. Also note that as α λ gets larger, the accuracy of the approximation gets better. (a) α λ = 5 , W 20 λ = 1 ; (b) α λ = 1 , W 20 λ = 1 ; (c) α λ = 5 , W 20 λ = 5 ; (d) α λ = 1 , W 20 λ = 5 .

Fig. 4
Fig. 4

Plot of the exact ( MTF e ) and the second approximation of MTF ( u , 0 ) ( MTF a 2 ) for different values of α and W 20 . Note how the exact MTF has many oscillations, whereas the approximated MTF is a smooth function. Also note that as α λ gets larger, the accuracy of the approximation gets better. (a) α λ = 5 , W 20 λ = 1 ; (b) α λ = 1 , W 20 λ = 1 ; (c) α λ = 5 , W 20 λ = 5 ; (d) α λ = 1 , W 20 λ = 5 .

Fig. 5
Fig. 5

Plot of ϵ ( α ̂ , 4 , u ) as a function of α ̂ and u (note that all variables are dimensionless).

Fig. 6
Fig. 6

Plot of MTF a 2 MTF a 1 as a function of α ̂ and u when W ̂ 20 = 4 (note that all variables are dimensionless).

Equations (83)

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MTF e ( f x , f y ) = P ( x + λ d i f x 2 , y + λ d i f y 2 ) P * ( x λ d i f x 2 , y λ d i f y 2 ) d x d y P ( x , y ) P * ( x , y ) d x d y ,
MTF e ( u , 0 ) = 1 π y m y m x m x m exp { k i [ ( 4 W 20 u ) x ̂ + ( 6 α u ) x ̂ 2 ] } d x ̂ d y ̂ ,
x m = 1 y ̂ 2 u ,
y m = 1 u 2 .
0 X exp ( i t 2 ) d t i + 1 2 π Arctan ( π X ) ,
MTF a 1 ( u , 0 ) = 2 b π 3 π k u α 2 Arcsin ( 1 u 2 ) i ( a 1 i ) 2 b 2 Arctan [ b 1 u 2 ( a 1 i ) 2 b 2 ] + i ( a 1 + i ) 2 b 2 Arctan [ b 1 u 2 ( a 1 + i ) 2 b 2 ] + i ( a 2 i ) 2 b 2 Arctan [ b 1 u 2 ( a 2 i ) 2 b 2 ] i ( a 2 + i ) 2 b 2 Arctan [ b 1 u 2 ( a 2 + i ) 2 b 2 ] + i ( a 1 i ) 2 b 2 Arctan [ ( a 1 i ) 1 u 2 [ ( a 1 i ) 2 b 2 ] u 2 ] i ( a 1 + i ) 2 b 2 Arctan [ ( a 1 + i ) 1 u 2 [ ( a 1 + i ) 2 b 2 ] u 2 ] + i ( a 2 i ) 2 b 2 Arctan [ ( a 2 i ) 1 u 2 [ ( a 2 i ) 2 b 2 ] u 2 ] i ( a 2 + i ) 2 b 2 Arctan [ ( a 2 + i ) 1 u 2 [ ( a 2 + i ) 2 b 2 ] u 2 ] ,
a 1 = 2 π k u 3 α ( W 20 3 u α ) ,
a 2 = 2 π k u 3 α ( W 20 + 3 u α ) ,
b = 6 π k u α .
MTF a 2 ( u , 0 ) = 2 3 π 2 k u α [ Arcsin ( u ) + 3 π k u α 2 1 u 2 × ( Arctan { 2 π k u 3 α [ W 20 + 3 α ( 1 u ) ] } Arctan { 2 π k u 3 α [ W 20 + 3 α ( 1 + u ) ] } ) ] .
MTF e ( u , 0 ) MTF a 1 ( u , 0 ) ϵ ( α ̂ , W ̂ 20 , u ) ,
C { 0.2 < u < 1 } × { 2 < α ̂ < 10 } × { 8 < W ̂ 20 < 8 } .
max C { ϵ ( α ̂ , W ̂ 20 , u ) } 0.1 ,
1 C C ϵ ( α ̂ , W ̂ 20 , u ) d α ̂ d W ̂ 20 d u 0.03 ,
MTF a 2 MTF a 1 0.1 .
MTF e ( u , v ) = P ( x ̂ + u , y ̂ + v ) P * ( x ̂ u , y ̂ v ) d x ̂ d y ̂ P ( x ̂ , y ̂ ) P * ( x ̂ , y ̂ ) d x ̂ d y ̂ ,
P ( x ̂ , y ̂ ) = exp { k i [ Φ ( x ̂ , y ̂ ) + W 20 ( x ̂ 2 + y ̂ 2 ) ] } circ ( x ̂ , y ̂ ) ,
circ ( x ̂ , y ̂ ) = { 1 if x ̂ 2 + y ̂ 2 1 0 otherwise } .
MTF e ( u , 0 ) = P ( x ̂ + u , y ̂ ) P * ( x ̂ u , y ̂ ) d x ̂ d y ̂ P ( x ̂ , y ̂ ) P * ( x ̂ , y ̂ ) d x ̂ d y ̂ , = 1 circ ( x ̂ , y ̂ ) circ * ( x ̂ , y ̂ ) d x ̂ d y ̂ exp ( k i { Φ ( x ̂ + u , y ̂ ) + W 20 [ ( x ̂ + u ) 2 + y ̂ 2 ] } ) circ ( x ̂ + u , y ̂ ) exp ( k i { Φ ( x ̂ u , y ̂ ) + W 20 [ ( x ̂ u ) 2 + y ̂ 2 ] } ) circ * ( x ̂ u , y ̂ ) d x ̂ d y ̂ , = 1 π exp [ k i ( 6 α δ 2 u + 2 α u 3 ) ] exp { k i [ ( 12 α δ u + 4 u W 20 ) x ̂ + ( 6 α u ) x ̂ 2 ] } circ ( x ̂ + u , y ̂ ) circ ( x ̂ u , y ̂ ) d x ̂ d y ̂ , = 1 π exp { k i [ ( 12 α δ u + 4 u W 20 ) x ̂ + ( 6 α u ) x ̂ 2 ] } circ ( x ̂ + u , y ̂ ) circ ( x ̂ u , y ̂ ) d x ̂ d y ̂ .
MTF e ( u , 0 ) = 1 π y m y m x m x m exp { k i [ ( 12 α δ u + 4 u W 20 ) x ̂ + ( 6 α u ) x ̂ 2 ] } d x ̂ d y ̂ ,
Φ ( x ̂ , y ̂ ) = α [ ( x ̂ 3 + y ̂ 3 ) + 3 δ ( x ̂ 2 + y ̂ 2 ) + 3 δ 2 ( x ̂ + y ̂ ) + 6 δ 3 ] .
MTF e ( u , 0 ) = 1 π y m y m x m x m exp ( i X 2 ) d x ̂ d y ̂ ,
= 1 π 6 k u α y m y m [ 0 X 2 exp ( i X 2 ) d X ]
[ 0 X 1 exp ( i X 2 ) d X ] d y ̂ ,
X 6 k u α x ̂ + 2 k u 3 α W 20 ,
X 1 6 k u α x m + 2 k u 3 α W 20 ,
X 2 6 k u α x m + 2 k u 3 α W 20 .
0 X exp ( i t 2 ) d t = π 2 i + 1 2 Erf [ ( 1 i ) X 2 ] .
Erf [ ( 1 i ) X 2 ] 2 π arctan ( π X ) ,
0 X exp ( i t 2 ) d t i + 1 2 π arctan ( π X ) .
MTF a 1 ( u , 0 ) = 1 π 6 k u α y m y m i + 1 2 π [ Arctan ( π X 2 ) Arctan ( π X 1 ) ] d y ̂ = 1 π 6 k u α π y m y m [ Arctan ( π X 2 ) Arctan ( π X 1 ) ] d y ̂ ,
MTF a 1 ( u , 0 ) = 1 π 6 π k u α y m y m { Arctan [ 2 π k u 3 α ( W 20 3 u α + 3 α 1 y ̂ 2 ) ] Arctan [ 2 π k u 3 α ( W 20 + 3 u α 3 α 1 y ̂ 2 ) ] } d y ̂ .
I = Arctan ( a + b 1 y ̂ 2 ) d y ̂ ,
I = y ̂ Arctan ( a + b 1 y ̂ 2 ) 1 b Arcsin ( y ̂ ) + i 2 b { ( a + i ) 2 b 2 [ Arctan ( b y ̂ ( a + i ) 2 b 2 ) Arctan ( ( a + i ) y ̂ ( ( a + i ) 2 b 2 ) ( 1 y ̂ 2 ) ) ] ( a i ) 2 b 2 [ Arctan ( b y ̂ ( a i ) 2 b 2 ) Arctan ( ( a i ) y ̂ ( ( a i ) 2 b 2 ) ( 1 y ̂ 2 ) ) ] } .
lim X Erf [ ( 1 i ) X 2 ] = lim X 2 π Arctan ( π X ) = 1 ,
d d X { Erf [ ( 1 i ) X 2 ] } X = 0 = d d X [ 2 π Arctan ( π X ) ] X = 0 = 2 π .
MTF e ( u , 0 ) = 1 π y m y m x m x m exp { k i [ ( 4 W 20 u ) x ̂ + ( 6 α u ) x ̂ 2 ] } d x ̂ d y ̂ ,
MTF e ( u , 0 ) = 1 4 y m y m x m x m exp { k i [ ( 4 W 20 u ) x ̂ + ( 6 α u ) x ̂ 2 ] } d x ̂ d y ̂ ,
x m = 1 u , y m = 1 .
MTF e ( u , 0 ) = 1 4 6 k u α y m y m X 1 X 2 exp ( i X 2 ) d X d y ̂ ,
= 1 4 6 k u α y m y m { 0 X 2 exp ( i X 2 ) d X 0 X 1 exp ( i X 2 ) d X } d y ̂ ,
X 6 k u α x ̂ + 2 k u 3 α W 20 ,
X 1 6 k u α x m + 2 k u 3 α W 20 ,
X 2 6 k u α x m + 2 k u 3 α W 20 .
MTF e ( u , 0 ) = 1 2 6 k u α 0 X 2 exp ( i X 2 ) d X 0 X 1 exp ( i X 2 ) d X = 1 2 6 k u α X 1 X 2 n = 0 ( i X 2 ) n n ! d X = 1 2 6 k u α n = 0 X 1 X 2 ( i X 2 ) n n ! d X = 1 2 6 k u α n = 0 [ i n X 2 2 n + 1 ( 2 n + 1 ) n ! i n X 1 2 n + 1 ( 2 n + 1 ) n ! ] .
MTF a 1 ( u , 0 ) = 1 4 6 k u α y m y m i + 1 2 π { Arctan ( π X 2 ) Arctan ( π X 1 ) } d y ̂ = 1 2 6 k u α i + 1 2 π { Arctan ( π X 2 ) Arctan ( π X 1 ) } ,
Δ MTF e ( u , 0 ) MTF a 1 ( u , 0 ) .
Δ = 1 2 6 k u α n = 0 [ i n X 2 2 n + 1 ( 2 n + 1 ) n ! i n X 1 2 n + 1 ( 2 n + 1 ) n ! ] i + 1 2 π { Arctan ( π X 2 ) Arctan ( π X 1 ) } 1 2 6 k u α n = 0 [ i n X 2 2 n + 1 ( 2 n + 1 ) n ! i n X 1 2 n + 1 ( 2 n + 1 ) n ! ] i + 1 2 π { Arctan ( π X 2 ) Arctan ( π X 1 ) } , 1 2 6 k u α { n = 0 [ i n X 2 2 n + 1 ( 2 n + 1 ) n ! ] i + 1 2 π Arctan ( π X 2 ) + n = 0 [ i n X 1 2 n + 1 ( 2 n + 1 ) n ! ] i + 1 2 π Arctan ( π X 1 ) } 1 2 6 k u α [ E r ( X 2 ) + E r ( X 1 ) ] ,
E r ( X ) min ( 11 20 , 3 2 π X , π X 2 ) .
MTF e ( u , 0 ) MTF a 1 ( u , 0 ) 1 2 6 k u α [ E r ( X 2 ) + E r ( X 1 ) ] .
C { 0.2 < u < 1 } × { 2 < α ̂ < 10 } × { 8 < W ̂ 20 < 8 } .
ϵ ( α ̂ , W ̂ 20 , u ) 1 2 12 π u α ̂ [ E r ( 12 π u α ̂ x m + 4 π u 3 α ̂ W ̂ 20 ) + E r ( 12 π u α ̂ x m + 4 π u 3 α ̂ W ̂ 20 ) ] ,
max C { ϵ ( α ̂ , W ̂ 20 , u ) } 0.1 ,
1 C C ϵ ( α ̂ , W ̂ 20 , u ) d α ̂ d W ̂ 20 d u 0.03 ,
MTF a 2 ( u , 0 ) = 2 3 π 2 k u α Arcsin ( u ) + 3 π k u α 2 1 u 2 × ( Arctan { 2 π k u 3 α [ W 20 + 3 α ( + 1 u ) ] } Arctan { 2 π k u 3 α [ W 20 + 3 α ( 1 + u ) ] } ) .
max C MTF a 2 MTF a 1 0.1 ,
MTF a 2 ( u , 0 ) = 2 3 π 2 k u α [ Arcsin ( u ) + 3 π k u α 2 1 u 2 Φ ] ,
d Φ d χ 1 = 1 1 + ( χ 1 + χ 2 ) 2 1 1 + ( χ 1 χ 2 ) 2 .
S = d MTF a 2 ( u , 0 ) d u 0 ,
MTF a 2 ( u , 0 ) = 2 Arcsin ( u ) 3 π 2 k u α + 2 ( 1 u 2 ) 3 π 3 k u α { Arctan [ c 1 u + c 2 u ( 1 u ) ] Arctan [ c 1 u c 2 u ( 1 u ) ] } ,
S ( u , c 1 , c 2 ) = 2 3 α π 2 k [ 1 u 1 u 2 + Arcsin ( u ) u 2 ] + 2 3 π 3 k α d d u ( 1 u 2 u { Arctan [ c 1 u + c 2 u ( 1 u ) ] Arctan [ c 1 u c 2 u ( 1 u ) ] } ) .
S 2 ( u , c 1 , c 2 ) = d d u ( 1 u 2 u × { Arctan [ c 1 u + c 2 u ( 1 u ) ] Arctan [ c 1 u c 2 u ( 1 u ) ] } ) .
d d u 1 u 2 = u 1 u 2 0 ,
1 u { Arctan [ c 1 u + c 2 u ( 1 u ) ] Arctan [ c 1 u c 2 u ( 1 u ) ] } 0 ,
S 3 ( u , c 1 , c 2 ) = d d u ( 1 u × { Arctan [ c 1 u + c 2 u ( 1 u ) ] Arctan [ c 1 u c 2 u ( 1 u ) ] } ) .
S 3 ( u , c 1 , c 2 ) = 1 2 u 3 2 { 2 c 2 u { 1 + u [ 3 c 1 2 ( 1 + u ) c 2 2 ( 1 u ) 2 ( 3 u 1 ) ] } ( 1 + c 1 2 u ) 2 2 c 2 2 u ( u 1 ) 2 ( c 1 2 u 1 ) + c 2 4 u 2 ( u 1 ) 4 Arctan [ c 1 u + c 2 u ( 1 u ) ] + Arctan [ c 1 u c 2 u ( 1 u ) ] } .
sup u , c 1 , c 2 { S 4 ( u , c 1 , c 2 ) } 0 , u ( 0 , 1 ) , c 1 0 , c 2 0 ,
S 4 ( u , c 1 , c 2 ) = 2 c 2 u { 1 + u [ 3 c 1 2 ( 1 + u ) c 2 2 ( 1 u ) 2 ( 3 u 1 ) ] } ( 1 + c 1 2 u ) 2 2 c 2 2 u ( u 1 ) 2 ( c 1 2 u 1 ) + c 2 4 u 2 ( u 1 ) 4 Arctan [ c 1 u + c 2 u ( 1 u ) ] + Arctan [ c 1 u c 2 u ( 1 u ) ] .
c 1 1 * = 0 ,
A 1 ( u , c 2 ) S 4 ( u , c 1 1 * , c 2 ) = 2 Arctan [ c 2 u ( u 1 ) ] 2 c 2 u ( 3 u 1 ) 1 + u ( 1 u ) 2 c 2 2 ,
c 1 2 * = ( 1 u ) [ c 2 2 ( u 1 ) ( 2 u 1 ) + u 1 + 4 c 2 2 u 3 ( 1 + c 2 2 u ( 1 u ) 2 ) ] 1 ,
A 2 ( u , c 2 ) S 3 ( u , c 1 2 * , c 2 ) ,
c 1 3 * = ,
A 3 ( u , c 2 ) lim c 1 S 4 ( u , c 1 , c 2 ) = π .
sup u , c 2 { A 1 ( u , c 2 ) } 0 , u ( 0 , 1 ) , c 2 0 ,
sup u , c 2 { A 2 ( u , c 2 ) } 0 , u ( 0 , 1 ) , c 2 0 .
c 2 1 * = 0 ,
B 1 ( u ) A 1 ( u , c 2 1 * ) = 0 ,
c 2 2 * = 1 ( 1 u ) 2 u 1 , for u ( 1 2 , 1 ) ,
B 2 ( u ) A 1 ( u , c 2 2 * )
= 2 Arctan ( u 2 u 1 ) 2 u ( 2 u 1 ) 1 u ,
c 2 3 * = ,
B 3 ( u ) lim c 2 A 1 ( u , c 2 ) = π .

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