Abstract

Near-field coded-aperture data from a single view contain information useful for three-dimensional (3D) reconstruction. A common approach is to reconstruct the 3D image one plane at a time. An analytic expression is derived for the 3D point-spread function of coded-aperture laminography. Comparison with computer simulations and experiments for apertures with different size, pattern, and pattern family shows good agreement in all cases considered. The expression is discussed in the context of the completeness conditions for projection data and is applied to explain an example of nonlinear behavior inherent in 3D laminographic imaging.

© 2005 Optical Society of America

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    [CrossRef]
  2. E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. R. Accorsi, F. Gasparini, R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273–284 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  15. H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM (Soc. Ind. Appl. Math.)J. Appl. Math. 43, 546–552 (1983).
    [CrossRef]
  16. S. D. Metzler, J. E. Bowsher, R. J. Jaszczak, “Geometrical similarities of the Orlov and Tuy sampling criteria and a numerical algorithm for assessing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1550–1555 (2003).
    [CrossRef]
  17. S. D. Metzler, K. L. Greer, R. J. Jaszczak, “Helical pinhole SPECT for small-animal imaging: a method for addressing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1575–1583 (2003).
    [CrossRef]

2003 (2)

S. D. Metzler, J. E. Bowsher, R. J. Jaszczak, “Geometrical similarities of the Orlov and Tuy sampling criteria and a numerical algorithm for assessing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1550–1555 (2003).
[CrossRef]

S. D. Metzler, K. L. Greer, R. J. Jaszczak, “Helical pinhole SPECT for small-animal imaging: a method for addressing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1575–1583 (2003).
[CrossRef]

2001 (3)

R. Accorsi, F. Gasparini, R. C. Lanza, “A coded aperture for high-resolution nuclear medicine planar imaging with a conventional Anger camera: experimental results,” IEEE Trans. Nucl. Sci. 48, 2411–2417 (2001).
[CrossRef]

R. Accorsi, F. Gasparini, R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273–284 (2001).
[CrossRef]

R. Accorsi, R. C. Lanza, “Near-field artifact reduction in coded aperture imaging,” Appl. Opt. 40, 4697–4705 (2001).
[CrossRef]

1989 (1)

1987 (1)

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

1984 (1)

G. K. Skinner, “Imaging with coded aperture masks,” Nucl. Instrum. Methods Phys. Res. 221, 33–40 (1984).
[CrossRef]

1983 (1)

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM (Soc. Ind. Appl. Math.)J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

1981 (1)

1980 (1)

1979 (1)

1978 (2)

1968 (2)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

J. G. Ables, “Fourier transformphotography: a new method for x-ray astronomy,” Proc. Astron. Soc. Australia 1, 172–173 (1968).

Ables, J. G.

J. G. Ables, “Fourier transformphotography: a new method for x-ray astronomy,” Proc. Astron. Soc. Australia 1, 172–173 (1968).

Accorsi, R.

R. Accorsi, R. C. Lanza, “Near-field artifact reduction in coded aperture imaging,” Appl. Opt. 40, 4697–4705 (2001).
[CrossRef]

R. Accorsi, F. Gasparini, R. C. Lanza, “A coded aperture for high-resolution nuclear medicine planar imaging with a conventional Anger camera: experimental results,” IEEE Trans. Nucl. Sci. 48, 2411–2417 (2001).
[CrossRef]

R. Accorsi, F. Gasparini, R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273–284 (2001).
[CrossRef]

J. D. Idoine, R. Accorsi, J. A. Parker, R. C. Lanza, J. V. Frangioni, “An optical/radioscintigraphic dual-modality intra-operative imaging system for sentinel lymph node mapping and cancer resection,” in Proceedings of the Third Annual Meeting of the Society for Molecular Imaging Meeting, St. Louis, Mo, September 2004, Mol. Imaging3, 258 (2004).

Bowsher, J. E.

S. D. Metzler, J. E. Bowsher, R. J. Jaszczak, “Geometrical similarities of the Orlov and Tuy sampling criteria and a numerical algorithm for assessing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1550–1555 (2003).
[CrossRef]

Cannon, T. M.

Caroli, E.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Di Cocco, G.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

Fenimore, E. E.

Frangioni, J. V.

J. D. Idoine, R. Accorsi, J. A. Parker, R. C. Lanza, J. V. Frangioni, “An optical/radioscintigraphic dual-modality intra-operative imaging system for sentinel lymph node mapping and cancer resection,” in Proceedings of the Third Annual Meeting of the Society for Molecular Imaging Meeting, St. Louis, Mo, September 2004, Mol. Imaging3, 258 (2004).

Gasparini, F.

R. Accorsi, F. Gasparini, R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273–284 (2001).
[CrossRef]

R. Accorsi, F. Gasparini, R. C. Lanza, “A coded aperture for high-resolution nuclear medicine planar imaging with a conventional Anger camera: experimental results,” IEEE Trans. Nucl. Sci. 48, 2411–2417 (2001).
[CrossRef]

Gottesman, S. R.

Greer, K. L.

S. D. Metzler, K. L. Greer, R. J. Jaszczak, “Helical pinhole SPECT for small-animal imaging: a method for addressing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1575–1583 (2003).
[CrossRef]

Idoine, J. D.

J. D. Idoine, R. Accorsi, J. A. Parker, R. C. Lanza, J. V. Frangioni, “An optical/radioscintigraphic dual-modality intra-operative imaging system for sentinel lymph node mapping and cancer resection,” in Proceedings of the Third Annual Meeting of the Society for Molecular Imaging Meeting, St. Louis, Mo, September 2004, Mol. Imaging3, 258 (2004).

Jaszczak, R. J.

S. D. Metzler, K. L. Greer, R. J. Jaszczak, “Helical pinhole SPECT for small-animal imaging: a method for addressing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1575–1583 (2003).
[CrossRef]

S. D. Metzler, J. E. Bowsher, R. J. Jaszczak, “Geometrical similarities of the Orlov and Tuy sampling criteria and a numerical algorithm for assessing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1550–1555 (2003).
[CrossRef]

Lanza, R. C.

R. Accorsi, F. Gasparini, R. C. Lanza, “A coded aperture for high-resolution nuclear medicine planar imaging with a conventional Anger camera: experimental results,” IEEE Trans. Nucl. Sci. 48, 2411–2417 (2001).
[CrossRef]

R. Accorsi, R. C. Lanza, “Near-field artifact reduction in coded aperture imaging,” Appl. Opt. 40, 4697–4705 (2001).
[CrossRef]

R. Accorsi, F. Gasparini, R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273–284 (2001).
[CrossRef]

J. D. Idoine, R. Accorsi, J. A. Parker, R. C. Lanza, J. V. Frangioni, “An optical/radioscintigraphic dual-modality intra-operative imaging system for sentinel lymph node mapping and cancer resection,” in Proceedings of the Third Annual Meeting of the Society for Molecular Imaging Meeting, St. Louis, Mo, September 2004, Mol. Imaging3, 258 (2004).

Metzler, S. D.

S. D. Metzler, K. L. Greer, R. J. Jaszczak, “Helical pinhole SPECT for small-animal imaging: a method for addressing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1575–1583 (2003).
[CrossRef]

S. D. Metzler, J. E. Bowsher, R. J. Jaszczak, “Geometrical similarities of the Orlov and Tuy sampling criteria and a numerical algorithm for assessing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1550–1555 (2003).
[CrossRef]

Natalucci, L.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Parker, J. A.

J. D. Idoine, R. Accorsi, J. A. Parker, R. C. Lanza, J. V. Frangioni, “An optical/radioscintigraphic dual-modality intra-operative imaging system for sentinel lymph node mapping and cancer resection,” in Proceedings of the Third Annual Meeting of the Society for Molecular Imaging Meeting, St. Louis, Mo, September 2004, Mol. Imaging3, 258 (2004).

Skinner, G. K.

G. K. Skinner, “Imaging with coded aperture masks,” Nucl. Instrum. Methods Phys. Res. 221, 33–40 (1984).
[CrossRef]

Spizzichino, A.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Stephen, J. B.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Tuy, H. K.

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM (Soc. Ind. Appl. Math.)J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

Appl. Opt. (7)

Astrophys. J. (1)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

IEEE Trans. Nucl. Sci. (3)

S. D. Metzler, J. E. Bowsher, R. J. Jaszczak, “Geometrical similarities of the Orlov and Tuy sampling criteria and a numerical algorithm for assessing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1550–1555 (2003).
[CrossRef]

S. D. Metzler, K. L. Greer, R. J. Jaszczak, “Helical pinhole SPECT for small-animal imaging: a method for addressing sampling completeness,” IEEE Trans. Nucl. Sci. 50, 1575–1583 (2003).
[CrossRef]

R. Accorsi, F. Gasparini, R. C. Lanza, “A coded aperture for high-resolution nuclear medicine planar imaging with a conventional Anger camera: experimental results,” IEEE Trans. Nucl. Sci. 48, 2411–2417 (2001).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. (1)

G. K. Skinner, “Imaging with coded aperture masks,” Nucl. Instrum. Methods Phys. Res. 221, 33–40 (1984).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

R. Accorsi, F. Gasparini, R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273–284 (2001).
[CrossRef]

Proc. Astron. Soc. Australia (1)

J. G. Ables, “Fourier transformphotography: a new method for x-ray astronomy,” Proc. Astron. Soc. Australia 1, 172–173 (1968).

SIAM (Soc. Ind. Appl. Math.) (1)

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM (Soc. Ind. Appl. Math.)J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

Space Sci. Rev. (1)

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Other (1)

J. D. Idoine, R. Accorsi, J. A. Parker, R. C. Lanza, J. V. Frangioni, “An optical/radioscintigraphic dual-modality intra-operative imaging system for sentinel lymph node mapping and cancer resection,” in Proceedings of the Third Annual Meeting of the Society for Molecular Imaging Meeting, St. Louis, Mo, September 2004, Mol. Imaging3, 258 (2004).

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Figures (7)

Fig. 1
Fig. 1

Coded-aperture geometry and definition of variables. All vectors lie inside transverse planes (i.e., normal to the z axis): continuous irregular curve, 3D object; dashed irregular curve, its cross section on the transverse plane at a distance a from the aperture. The longitudinal direction is that parallel to the z axis.

Fig. 2
Fig. 2

Pictorial representation of the two grids of Eq. (24). Each irregular shape represents the projection of a hole of the aperture cast by an ideal point source and blurred by the detector, i.e., PSFsp. On one of these is shown the sampling grid. Each of its dots represents a square in the array of Fig. 3.

Fig. 3
Fig. 3

Values of the coefficients of the sampling grid for a 62 × 62 NTHT MURA aperture: black, −1; gray, 0; white, 1. (a) Aperture array A; (b) decoding array G; (c) coefficients of the sampling grid for a point source not at the center of the field of view for a copy of PSFsp for which there is no relative shift of A and G; (d) coefficients of the sampling grid for a point source not at the center of the field of view for a copy of PSFsp for which there is a relative shift of A and G.

Fig. 4
Fig. 4

Longitudinal PSF for a point source imaged with the 62 × 62 NTHT MURA described in the text and shown in Fig. 3(a). Experimental data (solid curve) versus theoretical prediction, with (long-dashed curve) and without (short-dashed curve) compensation for the finite size of the point source used in the experiment. The FWHM is 6.4 mm.

Fig. 5
Fig. 5

Longitudinal PSF for a point source imaged with three different patterns. Experimental data, solid curve, versus the theoretical prediction for, long-dashed curve, square and, short-dashed curve, round holes. All patterns have round holes: (a) 62 × 62 NTHT MURA. The FWHM is 5.7 mm. (b) 38 × 38 NTHT MURA. The FWHM is 5.9 mm. (c) 118 × 118 NTHT MURA. Data are acquired in a 1024 matrix (instead of 512) to ensure proper sampling with smaller holes. The FWHM is 3.1 mm. The distance of the source from the aperture is not the same in all cases.

Fig. 6
Fig. 6

Longitudinal PSF for a point source imaged with four different patterns. Computer simulation of different arrays with square holes and of approximately (exactly in three cases) the same size.

Fig. 7
Fig. 7

Longitudinal PSF for a point source imaged in two different conditions: alone in the field of view (thin curves) and with a Gaussian distribution of activity also present around it (σs = 1 mm, thick curves). Comparison of theory [Eq. (45), dashed curves] to computer-simulated profiles (solid curves) from the 62 × 62 NTHT MURA of previous examples. In both cases σi = 1.3 mm.

Equations (57)

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A ( r a ) = u , v J A u * , v * S [ r a - p a ( u i ^ + v j ^ ) ] ,
{ - N - 1 2 , - N - 3 2 , , N - 1 2 } ;
R ( r d ) = B ( r d ) * z r o O ( r o , z ) A [ r a ( r d , r o ) ] × cos 3 ( θ ) z 2 d 2 r o d z ,
r a ( r d , r o ) = r o + a z ( r d - r o ) = b z r o + a z r d ,
R ( r d ) = z u , v J A u * , v * r o O ( r o , z ) B ( r d ) * S [ b z r o + a z r d - p a ( u i ^ + v j ^ ) ] d 2 r o d z .
R ( r d ) = z r o δ ( r o , z - z 0 ) B ( r d ) * S ( b z r o + a z r d ) d 2 r o d z = B ( r d ) * S ( a 0 z 0 r d ) PSF s p ( r d ) ,
R ( r d ) = z u , v J A u * , v * r o O ( r o , z ) r B ( r d - r ) × S [ b z r o + a z r - p a ( u i ^ + v j ^ ) ] d 2 r d 2 r o d z .
R ( r d ) = z u , v J A u * , v * r o O ( r o , z ) η B ( r d + b a r o - z a p a ( u i ^ + v j ^ ) - b a η ) S ( b z η ) d 2 η d 2 r o d z .
R ( r d ) = z u , v J A u * , v * r o O ( r o , z ) PSF s p [ r d + b a r o - z a p a ( u i ^ + v j ^ ) ] d 2 r o d z .
O ( r o , z ) = O ( r o ) δ ( z - z 0 ) ,
R ( r d ) = r o O ( r o ) δ ( r d + b a 0 r o ) d 2 r o = O ( - a 0 b r d ) ,
G ( r d , m ) = r , s G r * , s * δ [ r d - m p a ( r i ^ + s j ^ ) ] ,
O ^ ( r , m ) = R G .
O ^ ( r , m ) = u , v , r , s J A u * , v * G r * , s * ×     r o z r d O ( r o , z ) PSF s p [ r d + b a r o - z a p a ( u i ^ + v j ^ ) ] δ [ r d + r - m p a ] × ( r i ^ + s j ^ ) ] d 2 r d d z d 2 r o = u , v , r , s J A u * , v * G r * , s * r o z O ( r o , z ) PSF s p × [ b a r o - r + m p a ( r i ^ + s j ^ ) - z a p a × ( u i ^ + v j ^ ) ] d z d 2 r o .
O ^ ( r , z 0 a 0 ) = u , v , r , s J A u * , v * G r * , s * r o O ( r o ) × PSF s p { b a o r o - r - z 0 a 0 p a [ ( r - u ) i ^ + ( s - v ) j ^ ] } d 2 r o .
O ^ ( r , z 0 a 0 ) = k , l J r o O ( r o ) PSF s p [ b a 0 r o - r + z 0 a 0 × p a ( k i ^ + l j ^ ) ] r , s J A ( k + r ) * , ( l + s ) * G r * , s * d 2 r o .
A G = r , s J A ( k + r ) * , ( l + s ) * G r * , s * = δ k , l .
O ^ ( r , z 0 a 0 ) = r o O ( r o ) PSF s p ( b a 0 r o - r ) d 2 r o
O ^ ( r , z 0 a 0 ) = r o δ ( r o - r δ ) PSF s p ( b a 0 r o - r ) d 2 r o = PSF s p ( b a 0 r δ - r o ) ,
m = z a = a + b a = 1 + b a .
O ^ ( r , z 0 a 0 ) = r o O ( r o ) δ ( b a 0 r o - r ) d 2 r o = O ( a 0 b r ) .
O ( r o ) = δ ( r o - r o δ , z - z δ ) .
O ^ ( r , m ) = u , v , r , s J A u * , v * G r * , s * PSF s p [ b a s r o δ - r + m p a ( r i ^ + s j ^ ) - z δ a δ p a ( u i ^ + v j ^ ) ] ,
z = b m m - 1 .
O ^ ( r , m ) = k , l , r , s J A ( k + r ) * , ( l + s ) * G r * , s * PSF s p [ b a δ r o δ - r + Δ m p a ( r i ^ + s j ^ ) - m δ p a ( k i ^ + l j ^ ) ] .
O ^ ( r , m ) = k , l J ξ PSF s p [ b a δ r 0 δ - r - m δ p a ( k i ^ + l j ^ ) + ξ ] × r , s J A ( k + r ) * , ( l + s ) * G r * , s * [ ξ - Δ m p a × ( r i ^ × s j ^ ) ] d 2 ξ .
r , s J A ( k + r ) * , ( l + s ) * G r * , s * δ [ ξ - Δ m p a ( r i ^ + s j ^ ) ] H x c ( ξ ) 4 x c 2 r , s J A ( k + r ) * , ( l + s ) * G r * , s * = H x c ( ξ ) 4 x c 2 δ k , l ,
x c Δ m p a N - 1 2 ,
H x c ( ξ ) { 1 ξ i ^ x c ,     ξ j ^ x c 0 otherwise .
O ^ ( r , m ) = 1 4 x c 2 k , l J - x c x c - x c x c PSF s p [ b a δ r 0 δ - r - m δ p a ( k i ^ + l j ^ ) + ξ ] δ k , l d 2 ξ .
O ^ ( r , m ) = 1 4 x c 2 - x c x c - x c x c PSF s p ( b a δ r 0 δ - r + ξ ) d 2 ξ PSF s p ( b a δ r 0 δ - r + ξ ) ξ ,
O ^ ( r , m δ ) = PSF s p ( b a δ r 0 δ - r + ξ ) ξ = PSF s p ( b a δ r 0 δ - r ) .
O ^ ( r δ , m ) = PSF s p ( ξ ) ξ ,
S ( r a - r c ) = { 1 { ( r a - r c ) i ^ p a 2 ( r a - r c ) j ^ p a 2 0 otherwise .
B ( r d ) = 1 2 π σ i 2 exp ( - r d 2 2 σ i 2 ) .
PSF s p ( r d ) = 1 2 π σ i 2 r S ( a 0 z 0 r ) exp ( - r d - r 2 2 σ i 2 ) d 2 r = 1 2 π σ i 2 - m δ p a / 2 m δ p a / 2 - m δ p a / 2 m δ p a / 2 exp ( - r d - r 2 2 σ i 2 ) d 2 r .
erf ( x ) = 2 π 0 x exp ( - t 2 ) d t
PSF s p ( r d ) = 1 4 [ erf ( ξ 0 + r d i ^ 2 σ i ) + erf ( ξ 0 - r d i ^ 2 σ i ) ] × [ erf ( ξ 0 + r d j ^ 2 σ i ) + erf ( ξ 0 - r d j ^ 2 σ i ) ] ,
ξ 0 m δ p a 2 2 σ i .
O ^ ( r δ , m ) = PSF s p ( ξ ) ξ = 1 4 x c 2 - x c x c - x c x c PSF s p ( ξ ) d 2 ξ = 1 16 x c 2 [ - x c x c erf ( ξ 0 + x 2 σ i ) + erf ( ξ 0 - x 2 σ i ) d x ] 2 .
O ^ ( r δ , m ) = σ i 2 2 x c 2 { ξ + erf ( ξ + ) - ξ - erf ( ξ - ) + 1 π [ exp ( - ξ + 2 ) - exp ( - ξ - 2 ) ] } 2 ,
lim x c 0 σ i 2 2 x c 2 { ξ + erf ( ξ + ) - ξ - erf ( ξ - ) + 1 π [ exp ( - ξ + 2 ) - exp ( - ξ - 2 ) ] } 2 = erf 2 ξ 0 ,
σ i 2 2 x c 2 { ξ + erf ( ξ + ) - ξ - erf ( ξ - ) + 1 π [ exp ( - ξ + 2 ) - exp ( - ξ - 2 ) ] } 2 = k erf 2 ξ 0 ,
a ± = a δ 1 ± 2 a δ x c ( k ) ( N - 1 ) b p a .
Δ a = a - - a + = 4 ( N - 1 ) a δ a δ b x c ( k ) p a ( N - 1 ) 2 - 4 [ a δ b x c ( k ) p a ] 2 .
σ = [ σ i 2 + 1 8 ln 2 ( b a 0 FWHM s ) 2 ] 1 / 2 ,
lim N Δ a = 0.
PSF s p ξ 1 4 x c 2 - x c x c - x c x c S ( a 0 z 0 ξ ) d 2 ξ = { 1 x c m δ p a 2 m δ 2 p a 2 4 x c 2 x c > m δ p a 2 .
m δ 2 p a 2 4 x c 2 = k .
x c ( k ) = m δ p a 2 k .
PSF s p ( r d ) B ( r d ) * S ( a 0 z 0 r d ) = S ( a 0 z 0 r d ) ,
Δ a = 4 ( N - 1 ) a δ a δ b m δ 2 ( N - 1 ) 2 - 4 [ a δ b m δ 2 ] 2 m δ p a a δ b .
( a δ b ) 2 + [ 2 b p a ( N - 1 ) + 2 ] a δ b + 1 - ( N - 1 ) 2 2 ( a δ b ) 2 + 2 b p a N a δ b - N 2 2 0 ,
a δ b N 2 b p a { [ 1 + ( p a b ) 2 ] 1 / 2 - 1 } .
a δ p a N 2 2 ,
2 2 a δ d .
θ max = arctan d 2 a δ .

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