Abstract

In order to quantify the effect of phase contrast on X-ray image formation, the theory of statistical decision making has been applied to a binary classification task between two signals known exactly, namely, a phase-contrast image (that combines both the absorption and phase contrast) and the corresponding hypothetical pure absorption image that would be obtained under the same imaging conditions but without diffraction/refraction effects. The signal-to-noise ratio (SNR) for two widely used observers, including the ideal observer (also known as the prewhitening matched filter) and a non-ideal observer (the non-prewhitening matched filter) has been estimated in the case of in-line phase-contrast imaging, thus providing a figure-of-merit for the optimisation of the imaging conditions. A broad class of edge objects has been investigated and simple analytical expressions for the corresponding SNRs have been obtained and discussed.

© 2014 Optical Society of America

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References

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  1. A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
    [CrossRef]
  2. Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
    [CrossRef]
  3. E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
    [CrossRef] [PubMed]
  4. T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
    [CrossRef] [PubMed]
  5. C. Y. Chou, M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express 17(17), 14466–14480 (2009).
    [CrossRef] [PubMed]
  6. M. A. Anastasio, C. Y. Chou, A. M. Zysk, J. G. Brankov, “Analysis of ideal observer signal detectability in phase-contrast imaging employing linear shift-invariant optical systems,” J. Opt. Soc. Am. A 27(12), 2648–2659 (2010).
    [CrossRef] [PubMed]
  7. P. C. Diemoz, A. Bravin, M. Langer, P. Coan, “Analytical and experimental determination of signal-to-noise ratio and figure of merit in three phase-contrast imaging techniques,” Opt. Express 20(25), 27670–27690 (2012).
    [CrossRef] [PubMed]
  8. H. H. Barrett and K. J. Myers, Foundations of Image Science (John Wiley & Sons, Inc., 2004).
  9. A. Bravin, P. Coan, P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013).
    [CrossRef] [PubMed]
  10. J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttg.) 49, 121–125 (1977).

2013

A. Bravin, P. Coan, P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013).
[CrossRef] [PubMed]

2012

2010

2009

2008

2005

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[CrossRef]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

1997

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[CrossRef]

1977

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttg.) 49, 121–125 (1977).

Anastasio, M. A.

Baruchel, J.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Brankov, J. G.

Bravin, A.

A. Bravin, P. Coan, P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013).
[CrossRef] [PubMed]

P. C. Diemoz, A. Bravin, M. Langer, P. Coan, “Analytical and experimental determination of signal-to-noise ratio and figure of merit in three phase-contrast imaging techniques,” Opt. Express 20(25), 27670–27690 (2012).
[CrossRef] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Chou, C. Y.

Cloetens, P.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Coan, P.

A. Bravin, P. Coan, P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013).
[CrossRef] [PubMed]

P. C. Diemoz, A. Bravin, M. Langer, P. Coan, “Analytical and experimental determination of signal-to-noise ratio and figure of merit in three phase-contrast imaging techniques,” Opt. Express 20(25), 27670–27690 (2012).
[CrossRef] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Diemoz, P. C.

Fezzaa, K.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Fiedler, S.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Gao, D.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[CrossRef]

Guigay, J. P.

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttg.) 49, 121–125 (1977).

Gureyev, T. E.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[CrossRef] [PubMed]

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[CrossRef]

Härtwig, J.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Langer, M.

Miller, P. R.

Nesterets, Y. I.

Nesterets, Ya. I.

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[CrossRef]

Pagot, E.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Pogany, A.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[CrossRef] [PubMed]

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[CrossRef]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[CrossRef]

Stevenson, A. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[CrossRef] [PubMed]

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[CrossRef]

Suortti, P.

A. Bravin, P. Coan, P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013).
[CrossRef] [PubMed]

Wilkins, S. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[CrossRef] [PubMed]

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[CrossRef]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[CrossRef]

Zysk, A. M.

J. Opt. Soc. Am. A

Opt. Express

Optik (Stuttg.)

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttg.) 49, 121–125 (1977).

Phys. Med. Biol.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

A. Bravin, P. Coan, P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[CrossRef]

Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[CrossRef]

Other

H. H. Barrett and K. J. Myers, Foundations of Image Science (John Wiley & Sons, Inc., 2004).

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic representation of several “edge” objects from the family described by Eqs. (13) and (14) and the corresponding phase contrast calculated using Eq. (12) with γ −1 = 0, for large Fresnel number (b) and small Fresnel number (c), N F = σ o b j 2 / ( λ z ) .

Fig. 2
Fig. 2

Dependencies of the integral F α on the Fresnel number, given by Eq. (28).

Fig. 3
Fig. 3

Dependencies of various SNRs (a-c) and of the corresponding Fresnel numbers (d) on the magnification of the imaging system, for different types of the edge (α = 0, ½ and 1). Gaussian distributions are assumed for the source, the edge smearing (σobj = 1μm) and the detector PSF (σdet = 10μm). Solid and dashed lines correspond to two different source sizes, σsrc = 5μm and 50μm, respectively.

Tables (1)

Tables Icon

Table 1 Edge length L = 1mm, image fluence I ¯ 0 = 10 ph/μm2, wavelength λ = 0.05nm, total distance R = 2m, linear phase shift ϕ = −5.5 rad/mm and γ = −1061 (Nylon in water), sheet thickness t = 100μm, cylinder radius r = 50μm, wedge slope tanβ = 1. Gaussian PSFs.

Equations (48)

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S N R I 2 = d u | I ^ ( u ) I ^ a b s ( u ) | 2 | P ^ s y s ( u ) | 2 W n ( u ) ,
S N R n p w 2 = [ d u | I ^ ( u ) I ^ a b s ( u ) | 2 | P ^ s y s ( u ) | 2 ] 2 d u | I ^ ( u ) I ^ a b s ( u ) | 2 | P ^ s y s ( u ) | 2 W n ( u ) .
S N R w h t 2 = W n 1 ( 0 ) d u | I ^ ( u ) I ^ a b s ( u ) | 2 | P ^ s y s ( u ) | 2 ,
I z I ¯ 0 { 1 2 B G a m 2 φ G p h } ,
G a m ( x ) = Re [ G ( x ) ] , G p h ( x ) = Im [ G ( x ) ] , G ( x ) = ( λ z ) 1 / 2 exp [ i π ( x 2 λ z 1 4 ) ] ,
I z , a b s I ¯ 0 { 1 2 B } .
I ^ z I ¯ 0 { δ D 2 B ^ G ^ a m 2 φ ^ G ^ p h } ,
G ^ a m ( u ) = Re [ G ^ ( u ) ] , G ^ p h ( u ) = Im [ G ^ ( u ) ] , G ^ ( u ) = exp [ i π λ z u 2 ] .
I d i f f I z I z , a b s = I ¯ 0 { 2 B 2 B G a m 2 φ G p h } ,
I ^ d i f f = I ¯ 0 { 2 B ^ [ 1 G ^ a m ] 2 φ ^ G ^ p h } = I ¯ 0 { 2 B ^ [ 1 cos ( π λ z u 2 ) ] + 2 φ ^ sin ( π λ z u 2 ) } .
B = γ 1 φ ,
I ^ d i f f = I ¯ 0 2 φ ^ { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } .
φ ( x ; σ o b j ) = ( φ i d P o b j ) ( x ; σ o b j ) .
φ i d ( x ) = { A x α , x > 0 , 0 , x 0 ,
φ ^ i d ( u ) = A Γ ( 1 + α ) ( 2 π i u ) 1 α .
S N R I 2 = L I ¯ 0 2 A 2 C α + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ t o t ( u ) | 2 W n ( u ) ,
S N R n p w 2 = L I ¯ 0 2 A 2 C α [ + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ t o t ( u ) | 2 ] 2 + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ t o t ( u ) | 2 W n ( u ) ,
S N R w h t 2 = L I ¯ 0 2 A 2 C α W n ( 0 ) + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ t o t ( u ) | 2 ,
W n ( u ) = W n , w h t | P ^ d e t ( u ) | 2 ,
S N R I 2 = L I ¯ 0 A 2 C α + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ s r c ( u ) P ^ o b j ( u ) | 2 ,
S N R n p w 2 = L I ¯ 0 A 2 C α [ + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ t o t ( u ) | 2 ] 2 + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ t o t ( u ) P ^ d e t ( u ) | 2 ,
S N R w h t 2 = L I ¯ 0 A 2 C α + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ t o t ( u ) | 2 .
P 1 ( x ; σ ) = ( 2 π σ ) 1 exp [ 1 / 2 ( x / σ ) 2 ] , P ^ 1 ( u ; σ ) = P ^ 1 ( σ u ) = exp [ 2 π 2 ( σ u ) 2 ] ,
P 2 ( x ; σ ) = ( π σ ) 1 1 + ( x / σ ) 2 , P ^ 2 ( u ; σ ) = P ^ 2 ( σ u ) = exp ( 2 π | σ u | ) .
F α ( λ z , γ , σ ) + d u | u 1 α { 2 γ 1 sin 2 ( π λ z u 2 / 2 ) + sin ( π λ z u 2 ) } | 2 | P ^ ( σ u ) | 2 .
F α ( λ z , γ , σ ) = ( λ z ) 1 / 2 + α F α ( γ , N F ) ,
F α ( γ , N F ) + d v | v 1 α { 2 γ 1 sin 2 ( π v 2 / 2 ) + sin ( π v 2 ) } | 2 | P ^ ( N F 1 / 2 v ) | 2
F 1 / 2 ( N F ) 4 1 ln [ 1 + ( 2 π N F ) 2 ] ,
F 0 ( N F ) π ( 2 π N F ) 1 / 2 [ 1 + 1 + ( 2 π N F ) 2 2 ] ,
F 1 / 2 ( N F ) π arc tan [ ( 2 π N F ) 1 ] ( π / 2 ) ( 2 π N F ) ln [ 1 + ( 2 π N F ) 2 ] ,
F 1 ( N F ) 3 1 ( 2 π ) 2 ( 2 π N F ) 3 / 2 [ 1 + ( 2 π N F ) 2 1 + 1 + ( 2 π N F ) 2 + 2 ] .
F α ( λ z , γ , σ ) | N F > > 1 + d u | u 1 α { 2 γ 1 ( π λ z u 2 / 2 ) 2 + π λ z u 2 } | 2 | P ^ ( σ u ) | 2 = b 0 ( α ) ( λ z ) 2 σ 3 2 α b 1 ( α ) γ 1 ( λ z ) 3 σ 5 2 α + b 2 ( α ) γ 2 ( λ z ) 4 σ 7 2 α = σ 1 + 2 α { b 0 ( α ) N F 2 b 1 ( α ) γ 1 N F 3 + b 2 ( α ) γ 2 N F 4 } ,
b 0 ( α ) π 2 + d v | P ^ ( v ) | 2 ( v 2 ) 1 α , b 1 ( α ) π 3 + d v | P ^ ( v ) | 2 ( v 2 ) 2 α , b 2 ( α ) π 4 4 + d v | P ^ ( v ) | 2 ( v 2 ) 3 α .
b 0 ( α ) = π 2 Γ ( 3 / 2 α ) ( 2 π ) 3 2 α , b 1 ( α ) = π 3 Γ ( 5 / 2 α ) ( 2 π ) 5 2 α , b 2 ( α ) = π 4 4 Γ ( 7 / 2 α ) ( 2 π ) 7 2 α .
b 0 ( α ) = 2 π 2 Γ ( 3 2 α ) ( 4 π ) 3 2 α , b 1 ( α ) = 2 π 3 Γ ( 5 2 α ) ( 4 π ) 5 2 α , b 2 ( α ) = π 4 2 Γ ( 7 2 α ) ( 4 π ) 7 2 α .
S N R I 2 | N F > > 1 L I ¯ 0 A 2 C α b 0 ( α ) ( λ z ) 2 σ s r c , o b j 3 2 α ,
S N R n p w 2 | N F > > 1 L I ¯ 0 A 2 C α b 0 ( α ) ( λ z ) 2 σ t o t , d e t 3 2 α [ σ t o t 3 2 α ] 2 .
S N R w h t 2 | N F > > 1 L I ¯ 0 A 2 C α b 0 ( α ) ( λ z ) 2 σ t o t 3 2 α .
z R 1 R 2 / ( R 1 + R 2 ) = R M 2 ( M 1 ) .
I ^ d i f f ( u ; σ t o t ) | N F > > 1 2 π I ¯ 0 λ z u 2 φ ^ i d ( u ) P ^ t o t ( σ u ) = σ t o t α 1 I ^ d i f f ( σ t o t u ; 1 ) | N F > > 1 .
I d i f f ( x ; σ t o t ) | N F > > 1 = σ t o t α 2 I d i f f ( x / σ t o t ; 1 ) | N F > > 1 .
F α ( γ , N F < < 1 ) + d v | v 1 α { 2 γ 1 sin 2 ( π v 2 / 2 ) + sin ( π v 2 ) } | 2 = c 0 ( α ) c 1 ( α ) γ 1 + c 2 ( α ) γ 2 ,
c 0 ( α ) + d u sin 2 ( π u 2 ) ( u 2 ) 1 + α = π 1 + α Γ ( 3 / 4 α / 2 ) ( 1 + 2 α ) Γ ( 3 / 4 + α / 2 ) , 1 / 2 α < 3 / 2 , c 0 ( 1 / 2 ) = , c 0 ( 0 ) = π , c 0 ( 1 / 2 ) = π 2 / 2 , c 0 ( 1 ) = 4 π 2 / 3 ,
c 1 ( α ) 4 + d u sin ( π u 2 ) sin 2 ( π u 2 / 2 ) ( u 2 ) 1 + α , c 1 ( 1 / 2 ) = π / 2 , c 1 ( 0 ) = 2 π ( 2 1 / 2 1 ) , c 1 ( 1 / 2 ) = 2 π ln 2 , c 1 ( 1 ) = 8 π 2 3 ( 1 2 1 / 2 ) ,
c 2 ( α ) 4 + d u sin 4 ( π u 2 / 2 ) ( u 2 ) 1 + α , c 2 ( 1 / 2 ) = , c 2 ( 0 ) = π ( 2 3 / 2 1 ) , c 2 ( 1 / 2 ) = π 2 / 2 , c 2 ( 1 ) = 4 π 2 3 ( 2 1 / 2 1 ) .
F α ( λ z , γ , σ ) | N F < < 1 = ( λ z ) 1 / 2 + α [ c 0 ( α ) c 1 ( α ) γ 1 + c 2 ( α ) γ 2 ] .
S N R i 2 | N F < < 1 L I ¯ 0 A 2 C α ( λ z ) 1 / 2 + α [ c 0 ( α ) c 1 ( α ) γ 1 + c 2 ( α ) γ 2 ] , i = I , n p w , w h t .
I d i f f ( x ; λ z ) | N F < < 1 = ( λ z ) α / 2 I d i f f ( x / λ z ; 1 ) | N F < < 1 .

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