Abstract

Recent advances in model observers that predict human perceptual performance now make it possible to optimize medical imaging systems for human task performance. We illustrate the procedure by considering the design of a lens for use in an optically coupled digital mammography system. The channelized Hotelling observer is used to model human performance, and the channels chosen are differences of Gaussians. The task performed by the model observer is detection of a lesion at a random but known location in a clustered lumpy background mimicking breast tissue. The entire system is simulated with a Monte Carlo application according to physics principles, and the main system component under study is the imaging lens that couples a fluorescent screen to a CCD detector. The signal-to-noise ratio (SNR) of the channelized Hotelling observer is used to quantify this detectability of the simulated lesion (signal) on the simulated mammographic background. Plots of channelized Hotelling SNR versus signal location for various lens apertures, various working distances, and various focusing places are presented. These plots thus illustrate the trade-off between coupling efficiency and blur in a task-based manner. In this way, the channelized Hotelling SNR is used as a merit function for lens design.

© 2005 Optical Society of America

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References

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  1. H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).
  2. M. Rabbani, R. Shaw, R. Van Metter, “Detective quantum efficiency of imaging systems with amplification and scattering mechanisms,” J. Opt. Soc. Am. A 4, 895–901 (1987).
    [CrossRef] [PubMed]
  3. I. A. Cunningham, M. S. Westmore, A. Fenster, “A spacial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems,” Med. Phys. 21, 417–427 (1994).
    [CrossRef] [PubMed]
  4. B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
  5. K. J. Myers, H. H. Barrett, “Addition of a channel mechanism to the ideal-observer model,” J. Opt. Soc. Am. A 4, 2447–2457 (1987).
    [CrossRef] [PubMed]
  6. C. K. Abbey, H. H. Barrett, “Human- and model-observer performance in ramp-spectrum noise: effects of regularization and object variability,” J. Opt. Soc. Am. A 18, 473–488 (2001).
    [CrossRef]
  7. N. Graham, “Complex channels, early nonlinearities, and normalization in texture segmentation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1990), pp. 273–290.
  8. L. J. Frishman, A. W. Freeman, J. B. Troy, D. E. Schweitzer-Tong, C. Enroth-Cugell, “Spatiotemporal frequency responses of cat retinal ganglion cells,” J. Gen. Physiol. 89, 599–628 (1987).
    [CrossRef] [PubMed]
  9. M. J. Hawken, A. J. Parker, “Spatial properties of neurons on the monkey striate cortex,” Proc. R. Soc. London, Ser. B 231, 251–288 (1987).
    [CrossRef]
  10. M. A. Webster, R. L. De Valois, “Relationship between spatial-frequency and orientation tuning of striate-cortex cells,” J. Opt. Soc. Am. A 2, 1124–1132 (1985).
    [CrossRef] [PubMed]
  11. D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
    [CrossRef]
  12. S. A. Klein, D. M. Levi, “Hyperacuity thresholds of 1 sec: theoretical predictions and empirical validation,” J. Opt. Soc. Am. A 2, 1170–1190 (1985).
    [CrossRef] [PubMed]
  13. R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
    [CrossRef] [PubMed]
  14. E. B. Cargill, “A mathematical liver model and its application to system optimization and texture analysis,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1989).
  15. M. P. Eckstein, J. S. Whiting, “Lesion detection in structured noise,” Acad. Radiol. 2, 249–253 (1995).
    [CrossRef] [PubMed]
  16. H. H. Barrett, R. F. Wagner, K. J. Myers, “Correlated point processes in radiological imaging,” in Physics of Medical Imaging, R. L. Van Metter, J. Beutel, eds., Proc. SPIE3032, 110–124 (1998).
    [CrossRef]
  17. R. K. Swank, “Absorption and noise in x-ray phosphors,” J. Appl. Phys. 44, 4199–4203 (1973).
    [CrossRef]
  18. F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Statistical texture synthesis of mammographic images with clustered lumpy backgrounds,” Opt. Exp. 4, 33–43 (1999), http://www.opticsexpress.org .
    [CrossRef]
  19. C. Berg, J. P. Reus Christensen, P. Russel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions (Springer-Verlag, New York, 1984).
  20. M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, Vol. 15 of the American Mathematical Society Colloquium (American Mathematical Society, New York, 1932).
  21. N. Aronszajn, “Theory of reproducing kernels,” Trans. Am. Math. Soc. 68, 337–404 (1950).
    [CrossRef]
  22. H. Roehrig, T. Yu, E. Krupinski, “Image quality control for digital mammographic systems: initial experience and outlook,” J. Digital Imaging 8, 52–66 (1995).
    [CrossRef]
  23. I. M. Blevis, D. C. Hunt, J. A. Rowlands, “X-ray imaging using amorphous selenium: detection of swank factor by pulse height spectroscopy,” Med. Phys. 25, 638–641 (1998).
    [CrossRef] [PubMed]
  24. D. P. Trauernicht, R. Van Metter, “The measurement of conversion noise in x-ray intensifying screens,” in Medical Imaging II: Image Formation, Detection, Processing, and Interpretation, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 100–116 (1988).
    [CrossRef]

2001 (1)

1999 (1)

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Statistical texture synthesis of mammographic images with clustered lumpy backgrounds,” Opt. Exp. 4, 33–43 (1999), http://www.opticsexpress.org .
[CrossRef]

1998 (1)

I. M. Blevis, D. C. Hunt, J. A. Rowlands, “X-ray imaging using amorphous selenium: detection of swank factor by pulse height spectroscopy,” Med. Phys. 25, 638–641 (1998).
[CrossRef] [PubMed]

1995 (2)

H. Roehrig, T. Yu, E. Krupinski, “Image quality control for digital mammographic systems: initial experience and outlook,” J. Digital Imaging 8, 52–66 (1995).
[CrossRef]

M. P. Eckstein, J. S. Whiting, “Lesion detection in structured noise,” Acad. Radiol. 2, 249–253 (1995).
[CrossRef] [PubMed]

1994 (1)

I. A. Cunningham, M. S. Westmore, A. Fenster, “A spacial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems,” Med. Phys. 21, 417–427 (1994).
[CrossRef] [PubMed]

1987 (5)

1985 (2)

1980 (1)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

1973 (1)

R. K. Swank, “Absorption and noise in x-ray phosphors,” J. Appl. Phys. 44, 4199–4203 (1973).
[CrossRef]

1950 (1)

N. Aronszajn, “Theory of reproducing kernels,” Trans. Am. Math. Soc. 68, 337–404 (1950).
[CrossRef]

Abbey, C. K.

C. K. Abbey, H. H. Barrett, “Human- and model-observer performance in ramp-spectrum noise: effects of regularization and object variability,” J. Opt. Soc. Am. A 18, 473–488 (2001).
[CrossRef]

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Statistical texture synthesis of mammographic images with clustered lumpy backgrounds,” Opt. Exp. 4, 33–43 (1999), http://www.opticsexpress.org .
[CrossRef]

Aronszajn, N.

N. Aronszajn, “Theory of reproducing kernels,” Trans. Am. Math. Soc. 68, 337–404 (1950).
[CrossRef]

Barrett, H. H.

Berg, C.

C. Berg, J. P. Reus Christensen, P. Russel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions (Springer-Verlag, New York, 1984).

Blevis, I. M.

I. M. Blevis, D. C. Hunt, J. A. Rowlands, “X-ray imaging using amorphous selenium: detection of swank factor by pulse height spectroscopy,” Med. Phys. 25, 638–641 (1998).
[CrossRef] [PubMed]

Bochud, F. O.

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Statistical texture synthesis of mammographic images with clustered lumpy backgrounds,” Opt. Exp. 4, 33–43 (1999), http://www.opticsexpress.org .
[CrossRef]

Cargill, E. B.

E. B. Cargill, “A mathematical liver model and its application to system optimization and texture analysis,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1989).

Cunningham, I. A.

I. A. Cunningham, M. S. Westmore, A. Fenster, “A spacial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems,” Med. Phys. 21, 417–427 (1994).
[CrossRef] [PubMed]

De Valois, R. L.

Eckstein, M. P.

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Statistical texture synthesis of mammographic images with clustered lumpy backgrounds,” Opt. Exp. 4, 33–43 (1999), http://www.opticsexpress.org .
[CrossRef]

M. P. Eckstein, J. S. Whiting, “Lesion detection in structured noise,” Acad. Radiol. 2, 249–253 (1995).
[CrossRef] [PubMed]

Enroth-Cugell, C.

L. J. Frishman, A. W. Freeman, J. B. Troy, D. E. Schweitzer-Tong, C. Enroth-Cugell, “Spatiotemporal frequency responses of cat retinal ganglion cells,” J. Gen. Physiol. 89, 599–628 (1987).
[CrossRef] [PubMed]

Fenster, A.

I. A. Cunningham, M. S. Westmore, A. Fenster, “A spacial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems,” Med. Phys. 21, 417–427 (1994).
[CrossRef] [PubMed]

Fiete, R. D.

Freeman, A. W.

L. J. Frishman, A. W. Freeman, J. B. Troy, D. E. Schweitzer-Tong, C. Enroth-Cugell, “Spatiotemporal frequency responses of cat retinal ganglion cells,” J. Gen. Physiol. 89, 599–628 (1987).
[CrossRef] [PubMed]

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).

Frishman, L. J.

L. J. Frishman, A. W. Freeman, J. B. Troy, D. E. Schweitzer-Tong, C. Enroth-Cugell, “Spatiotemporal frequency responses of cat retinal ganglion cells,” J. Gen. Physiol. 89, 599–628 (1987).
[CrossRef] [PubMed]

Graham, N.

N. Graham, “Complex channels, early nonlinearities, and normalization in texture segmentation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1990), pp. 273–290.

Hawken, M. J.

M. J. Hawken, A. J. Parker, “Spatial properties of neurons on the monkey striate cortex,” Proc. R. Soc. London, Ser. B 231, 251–288 (1987).
[CrossRef]

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

Hunt, D. C.

I. M. Blevis, D. C. Hunt, J. A. Rowlands, “X-ray imaging using amorphous selenium: detection of swank factor by pulse height spectroscopy,” Med. Phys. 25, 638–641 (1998).
[CrossRef] [PubMed]

Klein, S. A.

Krupinski, E.

H. Roehrig, T. Yu, E. Krupinski, “Image quality control for digital mammographic systems: initial experience and outlook,” J. Digital Imaging 8, 52–66 (1995).
[CrossRef]

Levi, D. M.

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

Myers, K. J.

K. J. Myers, H. H. Barrett, “Addition of a channel mechanism to the ideal-observer model,” J. Opt. Soc. Am. A 4, 2447–2457 (1987).
[CrossRef] [PubMed]

R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
[CrossRef] [PubMed]

H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

H. H. Barrett, R. F. Wagner, K. J. Myers, “Correlated point processes in radiological imaging,” in Physics of Medical Imaging, R. L. Van Metter, J. Beutel, eds., Proc. SPIE3032, 110–124 (1998).
[CrossRef]

Parker, A. J.

M. J. Hawken, A. J. Parker, “Spatial properties of neurons on the monkey striate cortex,” Proc. R. Soc. London, Ser. B 231, 251–288 (1987).
[CrossRef]

Rabbani, M.

Reus Christensen, J. P.

C. Berg, J. P. Reus Christensen, P. Russel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions (Springer-Verlag, New York, 1984).

Roehrig, H.

H. Roehrig, T. Yu, E. Krupinski, “Image quality control for digital mammographic systems: initial experience and outlook,” J. Digital Imaging 8, 52–66 (1995).
[CrossRef]

Rowlands, J. A.

I. M. Blevis, D. C. Hunt, J. A. Rowlands, “X-ray imaging using amorphous selenium: detection of swank factor by pulse height spectroscopy,” Med. Phys. 25, 638–641 (1998).
[CrossRef] [PubMed]

Russel, P.

C. Berg, J. P. Reus Christensen, P. Russel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions (Springer-Verlag, New York, 1984).

Schweitzer-Tong, D. E.

L. J. Frishman, A. W. Freeman, J. B. Troy, D. E. Schweitzer-Tong, C. Enroth-Cugell, “Spatiotemporal frequency responses of cat retinal ganglion cells,” J. Gen. Physiol. 89, 599–628 (1987).
[CrossRef] [PubMed]

Shaw, R.

Smith, W. E.

Stone, M. H.

M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, Vol. 15 of the American Mathematical Society Colloquium (American Mathematical Society, New York, 1932).

Swank, R. K.

R. K. Swank, “Absorption and noise in x-ray phosphors,” J. Appl. Phys. 44, 4199–4203 (1973).
[CrossRef]

Trauernicht, D. P.

D. P. Trauernicht, R. Van Metter, “The measurement of conversion noise in x-ray intensifying screens,” in Medical Imaging II: Image Formation, Detection, Processing, and Interpretation, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 100–116 (1988).
[CrossRef]

Troy, J. B.

L. J. Frishman, A. W. Freeman, J. B. Troy, D. E. Schweitzer-Tong, C. Enroth-Cugell, “Spatiotemporal frequency responses of cat retinal ganglion cells,” J. Gen. Physiol. 89, 599–628 (1987).
[CrossRef] [PubMed]

Van Metter, R.

M. Rabbani, R. Shaw, R. Van Metter, “Detective quantum efficiency of imaging systems with amplification and scattering mechanisms,” J. Opt. Soc. Am. A 4, 895–901 (1987).
[CrossRef] [PubMed]

D. P. Trauernicht, R. Van Metter, “The measurement of conversion noise in x-ray intensifying screens,” in Medical Imaging II: Image Formation, Detection, Processing, and Interpretation, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 100–116 (1988).
[CrossRef]

Wagner, R. F.

H. H. Barrett, R. F. Wagner, K. J. Myers, “Correlated point processes in radiological imaging,” in Physics of Medical Imaging, R. L. Van Metter, J. Beutel, eds., Proc. SPIE3032, 110–124 (1998).
[CrossRef]

Webster, M. A.

Westmore, M. S.

I. A. Cunningham, M. S. Westmore, A. Fenster, “A spacial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems,” Med. Phys. 21, 417–427 (1994).
[CrossRef] [PubMed]

Whiting, J. S.

M. P. Eckstein, J. S. Whiting, “Lesion detection in structured noise,” Acad. Radiol. 2, 249–253 (1995).
[CrossRef] [PubMed]

Yu, T.

H. Roehrig, T. Yu, E. Krupinski, “Image quality control for digital mammographic systems: initial experience and outlook,” J. Digital Imaging 8, 52–66 (1995).
[CrossRef]

Acad. Radiol. (1)

M. P. Eckstein, J. S. Whiting, “Lesion detection in structured noise,” Acad. Radiol. 2, 249–253 (1995).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

R. K. Swank, “Absorption and noise in x-ray phosphors,” J. Appl. Phys. 44, 4199–4203 (1973).
[CrossRef]

J. Digital Imaging (1)

H. Roehrig, T. Yu, E. Krupinski, “Image quality control for digital mammographic systems: initial experience and outlook,” J. Digital Imaging 8, 52–66 (1995).
[CrossRef]

J. Gen. Physiol. (1)

L. J. Frishman, A. W. Freeman, J. B. Troy, D. E. Schweitzer-Tong, C. Enroth-Cugell, “Spatiotemporal frequency responses of cat retinal ganglion cells,” J. Gen. Physiol. 89, 599–628 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (6)

Med. Phys. (2)

I. A. Cunningham, M. S. Westmore, A. Fenster, “A spacial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems,” Med. Phys. 21, 417–427 (1994).
[CrossRef] [PubMed]

I. M. Blevis, D. C. Hunt, J. A. Rowlands, “X-ray imaging using amorphous selenium: detection of swank factor by pulse height spectroscopy,” Med. Phys. 25, 638–641 (1998).
[CrossRef] [PubMed]

Opt. Exp. (1)

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Statistical texture synthesis of mammographic images with clustered lumpy backgrounds,” Opt. Exp. 4, 33–43 (1999), http://www.opticsexpress.org .
[CrossRef]

Proc. R. Soc. London, Ser. B (2)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

M. J. Hawken, A. J. Parker, “Spatial properties of neurons on the monkey striate cortex,” Proc. R. Soc. London, Ser. B 231, 251–288 (1987).
[CrossRef]

Trans. Am. Math. Soc. (1)

N. Aronszajn, “Theory of reproducing kernels,” Trans. Am. Math. Soc. 68, 337–404 (1950).
[CrossRef]

Other (8)

H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

D. P. Trauernicht, R. Van Metter, “The measurement of conversion noise in x-ray intensifying screens,” in Medical Imaging II: Image Formation, Detection, Processing, and Interpretation, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 100–116 (1988).
[CrossRef]

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).

N. Graham, “Complex channels, early nonlinearities, and normalization in texture segmentation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1990), pp. 273–290.

C. Berg, J. P. Reus Christensen, P. Russel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions (Springer-Verlag, New York, 1984).

M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, Vol. 15 of the American Mathematical Society Colloquium (American Mathematical Society, New York, 1932).

E. B. Cargill, “A mathematical liver model and its application to system optimization and texture analysis,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1989).

H. H. Barrett, R. F. Wagner, K. J. Myers, “Correlated point processes in radiological imaging,” in Physics of Medical Imaging, R. L. Van Metter, J. Beutel, eds., Proc. SPIE3032, 110–124 (1998).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Layout of the model system.

Fig. 2
Fig. 2

Two samples of the clustered lumpy background.

Fig. 3
Fig. 3

SNR plots at different aperture sizes. The signal position is measured from the optical axis on the entrance surface of the screen. The image plane position is unchanged in all cases. The aperture size is in terms of the numerical aperture in object space.

Fig. 4
Fig. 4

Spot diagrams at four fields of different aperture sizes. The field position is in terms of the radial distance from the optical axis on the image plane.

Fig. 5
Fig. 5

SNR plots at different working distances. The signal position is measured from the optical axis on the entrance surface of the screen. The aperture size is unchanged in all cases, with a numerical aperture of 0.055. The working distance is in terms of the magnification.

Fig. 6
Fig. 6

Spot diagrams at four fields of different working distances. The field position is in terms of the radial distance from the optical axis on the image plane.

Fig. 7
Fig. 7

SNR plots at different image plane positions. The signal position is measured from the optical axis on the entrance surface of the screen. The aperture size is unchanged in all cases. The defocus is in terms of the distance from the ideal image plane in image space.

Fig. 8
Fig. 8

Spot diagrams at four fields of different image plane positions. The field position is in terms of the radial distance from the optical axis on the image plane.

Equations (107)

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

yx(R)=n=1Nδ(R-Rn),
N¯(f)=suppd2Rf(R).
prR(Rn|f)=f(Rn)N¯(f)=f(Rn)suppd2Rf(R).
prZ(zn)=1dxexp(-zn/dx)1-exp(-d/dx),  0znd.
yv,in(r)=n=1Nk=1Knδ(r-rnk)=n=1Nk=1Knδ(r-Rn-Δrnk),
Ltrs(r, sˆ)=2πdΩBTDF(r, sˆ, sˆ)Linc(r, sˆ)cos(θ),
Ltrs(r, sˆ|z)=2πdΩd2RBTDF×(r, sˆ, R, sˆ|z)Linc(R, sˆ)cos(θ),
Linc(R, sˆ)=Minc(R)δ(sˆ-sˆ0)cos(θ)=Minc(R)δ(sˆ-zˆ),
Ltrs(r, sˆ|z)=d2RMinc(R)BTDF(r, sˆ, r, zˆ|z).
Mtrs(r|z)=2πdΩLtrs(r, sˆ|z)cos(θ)=d2RMinc(R)2πdΩBTDF×(r, sˆ, R, zˆ|z)cos(θ).
PRF(r, R|z)=2πdΩBTDF(r, sˆ, R, zˆ|z)cos(θ).
Minc(R)=f(R),  Mtrs(r|z)=f(r).
pd(r, R|z)=2πdΩBTDF(r, sˆ, R, zˆ|z)cos(θ)=k¯(R|z)·prΔr(Δr|R, z).
prΔr(Δr|R, z)=pd(R+Δr, R|z)Kn¯(R|z),
Kn¯(R|z)=d2rpd(r, R|z),
PRF(r, sˆ, R, zˆ|z)=BTDF(r, sˆ, R, zˆ|z)cos(θ).
pr(r, sˆ|R, z)=BTDF(r, sˆ, R, zˆ|z)cos(θ)K¯(R|z).
pr(sˆ|Δr, R, z)=pr(R+Δr, sˆ|R, z)pr(Δr|R, z)=BTDF(R+Δr, sˆ, R, zˆ|z)cos(θ)2πdΩBTDF(R+Δr, sˆ, R, zˆ|z)cos(θ).
pass(R+Δr, R|z)=Ω(R+Δr)dΩpr(sˆ|Δr, R, z)=Ω(R+Δr)dΩBTDF(R+Δr, sˆ, R, zˆ|z)cos(θ)2πdΩBTDF(R+Δr, sˆ, R, zˆ|z)cos(θ),
yv,out(r)=n=1Nk=1Knβnkδ(r-Rn-Δrnk),
Pr(βnk|Rn, Δrnk, zn)=pass(Rn+Δrnk, Rn|zn),βnk=11-pass(Rn+Δrnk, Rn|zn),βnk=0.
yg(r)=n=1Nk=1Knβnkδ(r-rnk),
yg(r)=n=1Nk=1Knβnkδ(r-rnk)=n=1Nk=1Knβnkδ(r-Rn-Δrnk-Δrnk),
pg(R+Δr+Δr, R+Δr|R, z)=prΔr(Δr|R, Δr, z).
Δr=Δr(sˆ, r)=Δr(sˆ, R+Δr).
prsˆ(sˆ|R, Δr, z)=BTDF(R+Δr, sˆ, R, zˆ|z)cos(θ)Ω(R+Δr)dΩBTDF(R+Δr, sˆ, R, zˆ|z)cos(θ).
d2Δru(Δr)prΔr(Δr|R, Δr, z)=Ω(R+Δr)dΩu[Δr(sˆ, r)]pr(sˆ|R, Δr, z)=Ω(R+Δr)dΩu[Δr(sˆ, r)]BTDF(R+Δr, sˆ, R, zˆ|z)cos(θ)Ω(R+Δr)dΩBTDF(R+Δr, sˆ, R, zˆ|z)cos(θ),
g={gm, m={1, 1}, {1, 2},, {M, M}}, gm=suppd2rg(r)hm(r),
g=Dyg,
[Dyg]m=suppd2ryg(r)hm(r).
w=Kg-1Δg¯,
Kg=12(K1+K0),
Δg¯=g¯1-g¯0,
SNRHot2=Δg¯tKg-1Δg¯.
SNRHot=2 erf-1(2×AUC-1).
wch=T(TtKT)-1TtΔg¯,
SNRch2=Δg¯tT(TtKT)-1TΔg¯.
g¯=Dyg¯,
yg¯(r)=[H1f¯](r),
Kg=DKgD,
Kg(r1, r2)=[H1f¯](r2)δ(r1-r2)+[H2f¯](r1, r2)+[H1KfH1](r1, r2),
[H1f¯](r)=suppd2Rf¯(R)ptot(r, R),
ptot(r, R)=1dx 0ddz exp(-d/dx)ptot(r, R|z),
ptot(r, R|z)=d2rpd(r, R|z)pass×(r, R|z)pg(r, r|R, z),
[H2f¯](r1, r2)=1dx suppd2Rf¯(R)0ddz×exp(-z/dx)Q(R, z)ptot×(r1, R|z)ptot(r2, R|z),
Q(R, z)=m2¯-m¯m¯2,
[H1KfH1](r1, r2)=suppsuppd2R1d2R2ptot×(r1, R1)Kf(R1, R2)ptot(r2, R2),
Q=1S-1m¯.
SNRch2=Δg¯tT(TtKT)-1TΔg¯,
Δg¯=DΔyg¯=DH1Δf,
Δf=s,
K=D12 (K1+K0)D,
12(K1+K0)=[H1(b¯+12s)]δ(r1-r2)+[H2(b¯+12s)]+H1KbH1.
[Δg¯]i=d2r[H1s](r)hi(r)=d2Rs(R)d2rptot(r, R)hi(r).
Kij=d2rH1b¯+12 s(r)hi(r)hj(r)+d2r1d2r2H2b¯+12 s×(r1, r2)hi(r1)hj(r2)+d2r1d2r2(H1KbH1)×(r1, r2)hi(r1)hj(r2)=d2Rb¯(R)+12 s(R)×d2rptot(r, R)hi(r)hj(r)+d2Rb¯(R)+12 s(R) 1dx×0ddz exp(-z/dx)Q(R, z)×d2r1ptot(r1, R|z)hi(r1)×d2r2ptot(r2, R|z)hj(r2)+d2R1d2R2Kb(R1, R2)×d2r1ptot(r1, R1)hi(r1)×d2r2ptot(r2, R2)hj(r2).
d2rptot(r, R|z)hi(r)=d2rd2rpd(r, R|z)×pass(r, R|z)pg(r, r|R, z)=d2rpd(r, R|z)pass(r, R|z)×d2ΔrprΔr(Δr|r, R, z)hi(r+Δr)=d2rpd(r, R|z)pass(r, R|z)×Ω(r)dΩprsˆ(sˆ|r, R, z)hi(r+Δr(r+R, sˆ)).
d2rptot(r, R|z)hi(r)=K¯(R, z)×d2r 2πdΩBTDF(r, sˆ, R, zˆ|z)d2r2πdΩBTDF(r, sˆ, R, zˆ|z)×Ω(r)dΩ BTDF(r, sˆ, R, zˆ|z)2πdΩBTDF(r, sˆ, R, zˆ|z)×hi(r+Δr(sˆ, r)).
Kb(R1-R2)=d2RΦ(R1-R)Φ(R-R2).
Kij(3)=d2Rd2r1d2R1Φ×(R1-R)ptot(r1, R1)hi(r1)×d2r2d2R2Φ(R2-R)ptot(r2, R2)hj(r2),
Cj(ρ)=exp-12 ρAσj2-exp-12 ρσj2,
g¯=suppyg(r)h(r)d2r=suppyg(r)h(r)d2r=D(yg(r)),
yg(r){Δrnk}=nkd2rnkβnkδ×(r-Rn-Δrnk-Δrnk)×pg(Rn+Δrnk+Δrnk, Rn+Δrnk|Rn, zn)=nkβnkpg(r, Rn+Δrnk|Rn, zn).
yg(r)(βnk)=nkβnkpg(r, Rn+Δrnk|Rn, zn)=nkpass(Rn+Δrnk, Rn|zn)pg(r, Rn+Δrnk|Rn, zn).
yg(r)(Δrnk)=nkd2rnk pass (Rn+Δrnk, Rn|zn)×pg(r, Rn+Δrnk|Rn, zn)×Kn¯(Rn, zn)-1pd(Rn+Δrnk, Rn|zn)=nKn¯(Rn, zn)-1kd2rpd(r, Rn|zn)×pass(r, Rn|zn)pg(r, r|Rn, zn)=nKn¯(Rn, zn)-1Kn(Rn, zn)d2rpd(r, Rn|zn)×pass(r, Rn|zn)pg(r, r|Rn, zn).
yg(r){Kn}=nd2rpd(r, Rn|zn)×pass(r, Rn|zn)pg(r, r|Rn, zn).
yg(r)(zn)=n0ddzn 1dx·exp(-zn/dx)1-exp(-d/dx)×d2rpd(r, Rn|zn)×pass(r, Rn|zn)pg(r, r|Rn, zn).
yg(r)(Rn)=nsuppd2Rn f(Rn)f(R)d2R×0ddzn 1dx·exp(-zn/dx)1-exp(-d/dx)×d2rpd(r, Rn|zn)×pass(r, Rn|zn)pg(r, r|Rn, zn)=1dxN¯(f) nsuppd2Rnf(Rn)0ddzn×exp(-zn/dx)×d2rpd(r, Rn|zn)×pass(r, Rn|zn)pg(r, r|R, zn)=N(f)dxN¯(f) suppd2Rf(R)0ddz exp(-z/dx)×d2rpd(r, R|z)pass(r, R|z)pg(r, r|R, z).
yg(r)N=suppd2Rf(R)0ddz 1dxexp(-z/dx)×d2rpd(r, R|z)×pass(r, R|z)pg(r, r|R, z).
yg(r)N=1dx suppd2Rf¯(R)0ddz exp(-z/dx)×d2rpd(r, Rn|z)×pass(r, Rn|z)pg(r, r|R, z).
ptot(r, R|z)=d2rpd(r, R|z)×pass(r, R|z)pg(r, r|R, z),
ptot(r, R)=1dx 0ddz exp(-d/dx)ptot(r, R),
[H1f](r)=suppd2Rf¯(R)ptot(r, R).
g¯=[DH1f¯].
Kg=Rg-ggt.
Rg=ggt=d2r1yg(r1)h(r1)d2r2yg(r2)ht(r2)=d2r1d2r2h(r1)ht(r2)yg(r1)yg(r2)=[DRg(r1, r2)D].
Rg(r1, r2)=n1k1βn1k1δ(r1-Rn1-Δrn1k1-Δrn1k1)n2k2βn2k2δ×(r2-Rn2-Δrn2k2-Δrn2k2).
[Rg(r1, r2)]1=[H1b](r2)δ(r1-r2).
[Rg(r1, r2)]2=n Kn2-KnKn2 ptot(r1, Rn|zn)ptot(r2, Rn|zn).
PrKn(Kn)=mKnAKn(1-A)m-Kn,
Kn¯=mA¯,
Kn2|m=mA(1-A)+m2A2,
Kn2=m¯A(1-A)+m2¯A2,
Q(Rn, zn)Kn2¯-Kn¯Kn¯2,=m¯A(1-A)+m2¯A2-m¯Am¯2A2,=m2¯-m¯m¯2.
[Rg(r1, r2)]2=n Kn2¯-Kn¯Kn¯2 ptot×(r1, Rn|zn)ptot(r2, Rn|zn)=nQ(Rn, zn)ptot(r1, Rn|zn)ptot×(r2, Rn|zn).
[Rg(r1, r2)]2suppd2Rf¯(R)0ddz 1dx×exp(-z/dx)Q(R, z)×ptot(r1, R|z)ptot(r2, R|z).
[H2f¯](r1, r2)=suppd2Rf¯(R)0ddz 1dx×exp(-z/dx)Q(R, z)ptot×(r1, R|z)ptot(r2, R|z).
[Rg(r1, r2)]2=[H2f¯](r1, r2).
[Rg(r1, r2)]3-N2(f)-N(f)N¯2(f) [H1f](r1)[H1f](r2).
N2-N|b=Var(N|f)+N¯2(f)-N¯(f)=N¯2(f).
[Rg(r1, r2]3=[H1f](r1)[H1f](r2).
[Rg(r1, r2)]3=[H1RfH1](r1, r2).
Rg(r1, r2)=[H1f¯](r2)δ(r1-r2)+[H2f¯](r1, r2)+[H1RfH1](r1, r2).
Kg(r1, r2)=Rg(r1, r2)-yg(r1)yg(r2)=[H1f¯](r2)δ(r1-r2)+[H2f¯](r1, r2)+[H1KfH1](r1, r2).
K(R1-R2)=NK¯A [Rb(R1-R2)+KR¯s(R-1-R2)],
b(R, Rθ)=b(R, ϕ, Rθ)=exp-αRβ×[Ly2 cos2(θ+ϕ)+Lx2 sin2(θ+ϕ)]1/2LxLy,
Rb(R1-R2)=12π 02πdθAd2Rb(R1-R, Rθ)b×(R2-R, Rθ),
Rs=12π 02πdθAd2RSθ(R1-R, Rθ)Sθ×(R2-R, Rθ),
Sθ(R)=d2RΦ(R)b(R-R, Rθ),
[F2Rb](ρ)=12π 02πdθ[F2b](ρ, θ)2.
[F2Rs](ρ)=12π 02πdθ[F2Sθ](ρ, θ)2,
[F2Sθ](ρ, θ)=exp(-2π2σϕ2ρ2)[F2b](ρ, θ).
[F2Kb](ρ)=N¯K¯A [1+K¯ exp(-4π2σϕ2ρ2)] 12π 02πdθF2b(ρ, θ)2.
|F(ρ)|=N¯K¯A[1+K¯ exp(-4π2σϕ2ρ2)]1/2×[F2b(ρ, θ)2θ]1/2,
[F2b](ρ, θ)=[F2b](ρ, ϕ, θ)=002πdrdϕrb(r, ϕ, θ)×exp[-j2πρr cos(ϕ-ϕ)]=002πdrdϕr×exp-αrβ [Ly2 cos2(θ+ϕ)+Lx2 sin2(θ+ϕ)]1/2LxLy×exp[-j2πρr cos(ϕ-ϕ)]=0θ2π+θdrdϕr×exp-αrβ [Ly2 cos2(ϕ)+Lx2 sin2(ϕ)]1/2LxLy×exp[-j2πρr cos(ϕ+θ-ϕ)].
[F2b](ρ, ϕ, θ)=002πdrdϕr×exp-αrβ [Ly2 cos2(ϕ)+Lx2 sin2(ϕ)]1/2LxLy×exp[-j2πρr cos(ϕ+θ-ϕ)]=[F2b](ρ, ϕ+θ, 0).
F2b(ρ, θ)2θ=F2b(ρ, ϕ+θ, 0)2θ=F2b(ρ, θ, 0)2θ.
f(R)=F2-1NK¯A [1+K¯ exp(-4π2σϕ2ρ2)]1/2[F2b(ρ, θ, 0)2θ]1/2.

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