Abstract

We discuss the limit of the depth-of-field (DOF) extension for an imaging system using aspheric surfaces. In particular we consider an imaging system with an arbitrary pupil function and present the rigorous tradeoff between the DOF of the system and the spectral signal-to-noise ratio (SNR) over an extended DOF, to our knowledge for the first time. In doing so we use the relation between the conservation of ambiguity and modulation-transfer function (MTF) on one hand and the relation between the spectral SNR and MTF on the other. Using this, we rigorously derive the expression for an upper bound for the minimum spectral SNR, i.e., the limit of spectral SNR improvement. This leads to the introduction of our spectral SNR conservation principle. We also draw the relation between our result and the conservation of brightness theorem and establish that our result is the spectral version of the brightness conservation theorem.

© 2009 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
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    [CrossRef] [PubMed]
  3. P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. 45, 2924-2934 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
  5. S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A 25, 1051-1063 (2008).
    [CrossRef]
  6. S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. de Farias, “Design and optimization of the cubic phase pupil for the extension of the depth of field of task-based imaging systems,” Proc. SPIE 6311, 63,110R (2006).
  7. R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
    [CrossRef]
  8. S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “A novel approximation for the defocused modulation transfer function of a cubic-phase pupil,” in OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings, Technical Digest (OSA, 2007), paper CMB5.
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. 44, 701-712 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  23. J. Ojeda-Castaneda and E. E. Sicre, “Bilinear optical systems Wigner distribution function and ambiguity representation,” Opt. Acta 31, 255-260 (1984).
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2008 (3)

2006 (2)

P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. 45, 2924-2934 (2006).
[CrossRef] [PubMed]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. de Farias, “Design and optimization of the cubic phase pupil for the extension of the depth of field of task-based imaging systems,” Proc. SPIE 6311, 63,110R (2006).

2005 (2)

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
[CrossRef]

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

2004 (1)

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

2003 (2)

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

K. Kubala, E. R. Dowski, and W. T. Cathey, “Reducing complexity in computational imaging systems,” Opt. Express 11, 2102-2108 (2003).
[CrossRef] [PubMed]

2002 (1)

1999 (1)

1995 (2)

1984 (1)

J. Ojeda-Castaneda and E. E. Sicre, “Bilinear optical systems Wigner distribution function and ambiguity representation,” Opt. Acta 31, 255-260 (1984).

1983 (1)

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

1974 (1)

1960 (1)

Bagheri, S.

Baron, A. E.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Sec. 4.8.3.

Brenner, K. H.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

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

Cathey, W. T.

Chumachenko, V.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

de Farias, D. P.

S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A 25, 1051-1063 (2008).
[CrossRef]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. 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 (2008).
[CrossRef]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. de Farias, “Design and optimization of the cubic phase pupil for the extension of the depth of field of task-based imaging systems,” Proc. SPIE 6311, 63,110R (2006).

S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “A novel approximation for the defocused modulation transfer function of a cubic-phase pupil,” in OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings, Technical Digest (OSA, 2007), paper CMB5.

Diniz, P. S. R.

P. S. R. Diniz, Adaptive Filtering, Algorithms and Practical Implementation (Kluwer Academic, 1997).

Dowski, E. R.

Goodman, J. W.

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

Greengard, A.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Hall, J.

J. Hall, “F-number, numerical aperture, and depth of focus,” Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 556-559.

Javidi, B.

Johnson, G. E.

Kailath, T.

Kubala, K.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Liu, X.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

Lohmann, A. W.

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

Mattuck, A.

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

Narayanswamy, R.

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. 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 (2008).
[CrossRef]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. de Farias, “Design and optimization of the cubic phase pupil for the extension of the depth of field of task-based imaging systems,” Proc. SPIE 6311, 63,110R (2006).

P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. 45, 2924-2934 (2006).
[CrossRef] [PubMed]

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
[CrossRef]

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

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Ojeda-Castaneda, J.

J. Ojeda-Castaneda and E. E. Sicre, “Bilinear optical systems Wigner distribution function and ambiguity representation,” Opt. Acta 31, 255-260 (1984).

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

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Papoulis, A.

Pati, Y. C.

Pauca, V. P.

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
[CrossRef]

Setty, H.

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
[CrossRef]

Sheppard, C. J. R.

Sicre, E. E.

J. Ojeda-Castaneda and E. E. Sicre, “Bilinear optical systems Wigner distribution function and ambiguity representation,” Opt. Acta 31, 255-260 (1984).

Silveira, P. E. X.

S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A 25, 1051-1063 (2008).
[CrossRef]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. 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 (2008).
[CrossRef]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. de Farias, “Design and optimization of the cubic phase pupil for the extension of the depth of field of task-based imaging systems,” Proc. SPIE 6311, 63,110R (2006).

P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. 45, 2924-2934 (2006).
[CrossRef] [PubMed]

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
[CrossRef]

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

S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “A novel approximation for the defocused modulation transfer function of a cubic-phase pupil,” in OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings, Technical Digest (OSA, 2007), paper CMB5.

van der Gracht, J.

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
[CrossRef]

Wach, H. B.

Wang, Y.-T.

Welford, W. T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Sec. 4.8.3.

Woodward, P. M.

P. M. Woodward, Probability and Information Theory, with Applications to Radar (Artech House, 1980).

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

J. Ojeda-Castaneda and E. E. Sicre, “Bilinear optical systems Wigner distribution function and ambiguity representation,” Opt. Acta 31, 255-260 (1984).

Opt. Commun. (2)

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (3)

Proc. SPIE (3)

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D. P. de Farias, “Design and optimization of the cubic phase pupil for the extension of the depth of field of task-based imaging systems,” Proc. SPIE 6311, 63,110R (2006).

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, “Extended depth-of-field iris recognition system for workstation environment,” Proc. SPIE 5779, 41-50 (2005).
[CrossRef]

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Other (10)

J. Hall, “F-number, numerical aperture, and depth of focus,” Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 556-559.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Sec. 4.8.3.

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).

P. M. Woodward, Probability and Information Theory, with Applications to Radar (Artech House, 1980).

S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “A novel approximation for the defocused modulation transfer function of a cubic-phase pupil,” in OSA Computational Optical Sensing and Imaging (COSI) Topical Meetings, Technical Digest (OSA, 2007), paper CMB5.

P. S. R. Diniz, Adaptive Filtering, Algorithms and Practical Implementation (Kluwer Academic, 1997).

http://domino.research.ibm.com/comm/research_projects.nsf/pages/pupil_function_engr.index.html.

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

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

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Figures (13)

Fig. 1
Fig. 1

Schematic view of the imaging system under consideration. O is the center of the aperture, O 0 is the center of the object plane and O 1 is the center of the image plane. I obj is the power reflected (and/or emitted) by the object and I in is the power incident on the image plane. D is the width of the aperture.

Fig. 2
Fig. 2

(a) Plot of the integral of spectral SNR over all DOF (i.e., the available spectral SNR) for the first examplary imaging system (the solid curve) as well as the bound predicted by Theorem 4.1 for this plot (the dashed curve) with arbitrary unit. (b) Relative difference between the two plots shown in (a).

Fig. 3
Fig. 3

(a) Plot of the integral of spectral SNR over all DOF (i.e., the available spectral SNR) for the second examplary imaging system (the solid curve) as well as the bound predicted by Theorem 4.1 for this plot (the dashed curve) with arbitrary unit. (b) Relative difference between the two plots shown in (a).

Fig. 4
Fig. 4

Plot of the spectral SNR at a given spatial frequency u 0 as a function of s i ( Δ s i = { s i } 2 { s i } 1 ) for an imaging system that is optimized to operate uniformly in DOF = { s i : { s i } 1 s i { s i } 2 } .

Fig. 5
Fig. 5

Plot of the upper bound for the integral of the 1D spectral SNR as a function of normalized spatial frequency u. Here, UB represents upper bound.

Fig. 6
Fig. 6

Plot of the upper bound for the integral of the 2D spectral SNR as a function of normalized spatial frequency u. Here, UB represents upper bound and we have considered a rectangular aperture.

Fig. 7
Fig. 7

Plot of the regions U 1 and U 2 in the result of Theorem 4.2. The light region represents U 1 and the dark region U 2 .

Fig. 8
Fig. 8

Plot of the upper bound of the minimum 1D spectral SNR as a function of normalized spatial frequency u and DOF [nominal DOF divided by λ ( 2 N A ) ]. Here, UB represents upper bound.

Fig. 9
Fig. 9

Plot of the upper bound of the minimum 2D spectral SNR as a function of normalized spatial frequency u 1 = u 2 = u and DOF [nominal DOF divided by λ ( 2 N A 2 ) ]. Here, UB represents upper bound and we have considered a rectangular aperture.

Fig. 10
Fig. 10

Plot of the upper bound for the integral of the 2D spectral SNR as a function of normalized spatial frequency u. Here, UB represents upper bound and we have considered a circular aperture.

Fig. 11
Fig. 11

Plot of the upper bound of the minimum 2D spectral SNR as a function of normalized spatial frequency u 1 = u 2 = u and DOF [nominal DOF divided by λ ( 2 N A 2 ) ]. Here, UB represents upper bound and we have considered a circular aperture.

Fig. 12
Fig. 12

Plot of bound c 1 in Lemma A.4.

Fig. 13
Fig. 13

Plot of bound c 2 in Lemma A.5.

Tables (2)

Tables Icon

Table 1 Imaging System Parameters for Example 1 in Fig. 2

Tables Icon

Table 2 Imaging System Parameters for Example 2 in Fig. 3

Equations (49)

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

P ( x ̂ ; s i ) = rect ( x ̂ 1 ) rect ( x ̂ 2 ) exp { i k [ ψ ( x ̂ ) + w ( x ̂ ; s i ) ] }
SNR n ( u ; s i ) = SNR c , n I ̂ obj 2 ( u ) MTF 2 ( u ; s i ) ,
A f 1 ( ξ , ζ ) = f 1 ( t + ξ 2 ) f 1 * ( t ξ 2 ) exp ( 2 π i ζ t ) d t .
A f 2 ( ξ , ζ ) = f 2 ( t + ξ 2 ) f 2 * ( t ξ 2 ) exp ( 2 π i ζ t ) d t 1 d t 2 .
P ¯ ( x ̂ ; s i ) = rect ( x ̂ ) exp { i k [ ψ ( x ̂ ) + w ( x ̂ ; s i ) W ̂ 20 x ̂ 2 ] } .
MTF ( u ; W ̂ 20 ) = 1 A A F ( P ¯ ) ( 4 W ̂ 20 u , 2 u ) ,
MTF ( u ; W ̂ 20 ) = 1 A A F ( P ¯ ) ( 4 W ̂ 20 u 1 , 2 u 1 , 4 W ̂ 20 u 2 , 2 u 2 ) .
A f 1 ( ξ , ζ ) 2 d ξ d ζ = [ E ( f 1 ) ] 2 ,
E ( f 1 ) = f 1 ( t ) f 1 * ( t ) d t .
E ( f 1 ) = F t τ { f 1 ( t ) } [ F t τ { f 1 ( t ) } ] * d τ .
A f 2 ( ξ , ζ ) 2 d ξ 1 d ξ 2 d ζ 1 d ζ 2 = E ( f 2 ) 2 ,
E ( f 2 ) = f 2 ( t ) f 2 * ( t ) d t 1 d t 2 .
E ( f 2 ) = F t τ { f 2 ( t ) } [ F t τ { f 2 ( t ) } ] * d τ 1 d τ 2 .
S 1 ( u ) = 0 SNR 1 ( u ; s i ) d s i .
S 1 ( u ) S total , 1 I ̂ obj 2 ( u ) c 1 ( F D ) 1 u 8 u rect ( u 2 ) ,
S 2 ( u ) = 0 SNR 2 ( u ; s i ) d s i .
S 2 ( u ) { S total , 2 I ̂ obj 2 ( u ) c 2 ( F D ) ( u 1 1 ) 2 ( u 1 2 u 1 3 u 2 2 + 3 u 2 ) 96 u 2 2 rect ( u 1 2 ) rect ( u 2 2 ) , u U 1 S total , 2 I ̂ obj 2 ( u ) c 2 ( F D ) ( u 2 1 ) 2 ( u 2 2 u 2 3 u 1 2 + 3 u 1 ) 96 u 1 2 rect ( u 1 2 ) rect ( u 2 2 ) , u U 2 } ,
SNR ¯ 1 ( u , Δ s i ) E 0 2 T 2 c 1 ( F D ) ( η h ν ) 2 e t 2 f f 2 p 2 m N 2 I ̂ obj 2 ( u ) 1 u 4 u rect ( u 2 ) λ Δ s i .
S 1 ( u ) λ S total , 1 λ c 1 ( F D ) I ̂ obj 2 ( u ) 1 u 8 u rect ( u 2 ) = E 0 2 T 2 c 1 ( F D ) ( η h ν ) 2 e t 2 f f 2 p 2 m N 2 I ̂ obj 2 ( u ) 1 u 4 u rect ( u 2 ) .
SNR ¯ 1 ( u , Δ s i ) = S 1 ( u ) Δ s i .
SNR ¯ 1 ( u , Δ s i ) E 0 2 T 2 c 1 ( F D ) ( η h ν ) 2 e t 2 f f 2 p 2 m N 2 I ̂ obj 2 ( u ) 1 u 4 u rect ( u 2 ) λ Δ s i ,
SNR ¯ 2 ( u , Δ s i ) = S 2 ( u ) Δ s i ,
SNR ¯ c ( u , Δ s i ) = S c ( u ) Δ s i .
T S N ̃ R , n = 0 S N ̃ R n ( s i ) d s i = α n 0 P n ( s i ) A n ( s i ) N A ( s i ) n d s i ,
T S N ̃ R , n = α n P n 0 A n ( s i ) sin [ θ ( s i ) ] n d s i = ,
S 1 ( u ) S total , 1 1 u 8 u rect ( u 2 ) .
S total , 1 1 u 8 u rect ( u 2 ) d u = S total 1 1 1 u 8 u d u = S total , 1 4 0 1 ( 1 u 1 ) d u = .
P ¯ ( x ̂ ; s i ) = rect ( x ̂ ) A ( x ̂ ) exp { i k [ ϕ ( x ̂ ) + w ( x ̂ ; s i ) W ̂ 20 x 2 ] } .
G ( B ) = g ( T 1 , T 2 ) exp [ 2 π i ( B T 1 + α B T 2 ) ] d T 1 d T 2 .
F B b { G ( B ) } = g ( b α T 2 , T 2 ) d T 2 .
F B b { G ( B ) } = lim R g R ( b ) ,
g R ( b ) = R R G ( B ) exp ( 2 π i b B ) .
g R ( b ) = R R { g ( T 1 , T 2 ) exp [ 2 π i ( B T 1 + α B T 2 ) ] d T 1 d T 2 } exp ( 2 π i b B ) d B = g ( T 1 , T 2 ) d T 1 d T 2 R R exp [ 2 π i ( B T 1 + α B T 2 ) ] d B = g ( T 1 , T 2 ) sin [ 2 π R ( T 1 + α T 2 b ) ] π ( T 1 + α T 2 b ) b B d T 1 d T 2 .
lim R g R ( b ) = g ( T 1 , T 2 ) { 2 lim R sin [ 2 π R ( T 1 + α T 2 b ) ] 2 π ( T 1 + α T 2 b ) } d T 1 d T 2 = 2 g ( T 1 , T 2 ) δ [ 2 ( T 1 + α T 2 b ) ] d T 1 d T 2 = g ( T 1 , T 2 ) δ ( T 1 + α T 2 b ) d T 1 d T 2 = g ( b α T 2 , T 2 ) d T 2 ,
MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 = 1 u 8 u rect ( u 2 ) .
Φ ( u ) = MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 = 1 A 2 A F ( P ¯ ) ( 4 u W ̂ 20 , 2 u ) 2 d W ̂ 20 = 1 4 u A 2 A F ( P ¯ ) ( ξ , ζ ) 2 d ξ = 1 4 u A 2 F ξ a { A F ( P ¯ ) ( ξ , ζ ) } 2 d a = 1 4 u A 2 P ¯ ( a + ζ 2 ; s i ) P ¯ * ( a ζ 2 ; s i ) 2 d a = 1 4 u A 2 rect ( a 4 + ζ 4 ) rect ( a 2 ζ 4 ) d a = 1 2 u A 2 ( 1 u ) rect ( u 2 ) = 1 u 8 u rect ( u 2 ) .
Φ ( u ) = 1 + u 8 u rect ( u 2 ) .
Φ ( u ) = 1 u 8 u rect ( u 2 )
MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 { 2 ( u 1 1 ) 2 ( u 1 2 u 1 3 u 2 2 + 3 u 2 ) 24 u 2 2 rect ( u 1 2 ) rect ( u 2 2 ) , u U 1 2 ( u 2 1 ) 2 ( u 2 2 u 2 3 u 1 2 + 3 u 1 ) 24 u 1 2 rect ( u 1 2 ) rect ( u 2 2 ) , u U 2 } ,
Φ ( u ) = MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 = 1 A 2 A F ( P ¯ ) ( 4 u 1 W ̂ 20 , 2 u 1 , 4 u 2 W ̂ 20 , 2 u 2 ) 2 d W ̂ 20 = 1 4 u 1 A 2 A F ( P ¯ ) ( ξ 1 , ζ 1 , u 2 u 1 ξ 2 , ζ 2 ) 2 d ξ 1 = 1 4 u 1 A 2 F ξ 1 a { A F ( P ¯ ) ( ξ 1 , ζ 1 , u 2 u 1 ξ 2 , ζ 2 ) } 2 d a = 1 4 u 1 A 2 { P ¯ ( a u 2 u 1 τ 2 + ζ 1 2 , τ 2 + ζ 2 2 ; s i ) P ¯ * ( a u 2 u 1 τ 2 ζ 1 2 , τ 2 ζ 2 2 ; s i ) d τ 2 } 2 d a 1 4 u 1 A 2 [ P ¯ ( a u 2 u 1 τ 2 + u 1 , τ 2 + u 2 ; s i ) P ¯ ( a u 2 u 1 τ 2 u 1 , τ 2 u 2 ; s i ) d τ 2 ] 2 d a = 1 4 u 1 A 2 [ rect [ 1 2 ( a u 2 u 1 τ 2 + u 1 ) ] rect [ 1 2 ( τ 2 + u 2 ) ] rect [ 1 2 ( a u 2 u 1 τ 2 u 1 ) ] rect [ 1 2 ( τ 2 u 2 ) ] d τ 2 ] 2 d a = 1 4 u 1 A 2 [ u 2 1 1 u 2 rect [ 1 2 ( a u 2 u 1 τ 2 + u 1 ) ] rect [ 1 2 ( a u 2 u 1 τ 2 u 1 ) ] d τ 2 ] 2 d a = 1 4 u 1 A 2 ( 1 u 2 max { min [ u 2 u 2 2 , u 1 u 1 2 + a u 1 ] max [ u 2 + u 2 2 , u 1 + u 1 2 + a u 1 ] , 0 } ) 2 d a = { ( u 1 1 ) 2 ( u 1 2 u 1 3 u 2 2 + 3 u 2 ) 24 u 2 2 rect ( u 1 2 ) rect ( u 2 2 ) , u U 1 ( u 2 1 ) 2 ( u 2 2 u 2 3 u 1 2 + 3 u 1 ) 24 u 1 2 rect ( u 1 2 ) rect ( u 2 2 ) , u U 2 } .
Φ ( u ; c ) = { 2 ( u 1 1 ) 2 ( u 1 2 u 1 3 u 2 2 + 3 u 2 ) 24 u 2 2 rect ( u 1 2 ) rect ( u 2 2 ) u U 1 2 ( u 2 1 ) 2 ( u 2 2 u 2 3 u 1 2 + 3 u 1 ) 24 u 1 2 rect ( u 1 2 ) rect ( u 2 2 ) u U 2 }
g ( x , D , λ , F ) = 4 ( 8 λ x D 2 + 1 F ) 2 D 2 + 4 ( 8 λ x D 2 + 1 F ) 2 .
a g ( x , D , λ , F ) h ( x ) d x c 1 ( F D ) h ( x ) d x ,
c 1 ( x ) = { 1 + 1 + 4 x 2 4 , x 3 2 8 x 2 1 + 4 x 2 , x > 3 2 } .
g ( x , D , λ , F ) = 16 D 2 ( 8 λ x D 2 + 1 F ) 2 [ D 2 + 4 ( 8 λ x D 2 + 1 F ) 2 ] 2 .
a g ( x , D , λ , F ) h ( x ) d x c 2 ( F D ) h ( x ) d x ,
c 2 ( x ) = { 10 + 47 x 2 + 52 x 3 20 , x 8 9 16 x 2 ( 1 + 4 x 2 ) 2 , 8 9 < x 1 + 2 2 . 1 2 , x > 1 + 2 2 . }
S 1 ( u ) = 0 SNR 1 ( u ; s i ) d s i = 0 E 0 2 T 2 D 2 D 2 + 4 s i 2 ( η h ν ) 2 e t 2 f f 2 p 2 m N 2 I ̂ obj 2 ( u ) MTF 2 ( u ; s i ) d s i = E 0 2 T 2 ( η h ν ) 2 e t 2 f f 2 p 2 m N 2 I ̂ obj 2 ( u ) 0 D 2 D 2 + 4 s i 2 MTF 2 ( u ; s i ) d s i = E 0 2 T 2 ( η h ν ) 2 e t 2 f f 2 p 2 m N 2 I ̂ obj 2 ( u ) W ̂ 20 0 D 2 D 2 + 4 s i 2 MTF 2 ( u ; W ̂ 20 ) 8 λ s i 2 D 2 d W ̂ 20 = S total , 1 I ̂ obj 2 ( u ) W ̂ 20 0 4 s i 2 D 2 + 4 s i 2 MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 S total , 1 I ̂ obj 2 ( u ) c 1 ( F D ) MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 S total , 1 I ̂ obj 2 ( u ) c 1 ( F D ) 1 u 8 u rect ( u 2 ) .
S 2 ( u ) = 0 SNR 2 ( u ; s i ) d s i = 0 E 0 2 T 2 D 4 ( D 2 + 4 s i 2 ) 2 ( η h ν ) 2 e t 2 f f 4 p 4 m N 2 I ̂ obj 2 ( u ) MTF 2 ( u ; s i ) d s i = E 0 2 T 2 ( η h ν ) 2 e t 2 f f 4 p 4 m N 2 I ̂ obj 2 ( u ) 0 D 4 ( D 2 + 4 s i 2 ) 2 MTF 2 ( u ; s i ) d s i = E 0 2 T 2 ( η h ν ) 2 e t 2 f f 4 p 4 m N 2 I ̂ obj 2 ( u ) W ̂ 20 0 D 4 ( D 2 + 4 s i 2 ) 2 MTF 2 ( u ; W ̂ 20 ) 8 λ s i 2 D 2 d W ̂ 20 = 1 4 S total , 2 I ̂ obj 2 ( u ) W ̂ 20 0 16 s i 2 D 2 ( D 2 + 4 s i 2 ) 2 MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 1 4 S total , 2 I ̂ obj 2 ( u ) c 2 ( F D ) MTF 2 ( u ; W ̂ 20 ) d W ̂ 20 { S total , 2 I ̂ obj 2 ( u ) c 2 ( F D ) ( u 1 1 ) 2 ( u 1 2 u 1 3 u 2 2 + 3 u 2 ) 96 u 2 2 rect ( u 1 2 ) rect ( u 2 2 ) u U 1 S total , 2 I ̂ obj 2 ( u ) c 2 ( F D ) ( u 2 1 ) 2 ( u 2 2 u 2 3 u 1 2 + 3 u 1 ) 96 u 1 2 rect ( u 1 2 ) rect ( u 2 2 ) u U 2 . }

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