Abstract

In connection with recent work on remote imaging of random media by light, a straightforward generalization of the proper diffusion boundary conditions is presented that takes into account Fresnel reflection. The Milne problem at exterior boundaries is solved for various values of index of refraction, absorption, and scattering anisotropy parameters to yield extrapolated end points and extrapolation distances. A generalized interface condition is derived to replace the usual condition of continuity of intensity. Benchmark-quality numerical results are given for the extrapolation distance and for the new index-dependent parameter in the interface conditions. Difficulties in using the extrapolated end point when the index is sufficiently large are discussed, and a new image procedure suitable for this case is presented.

© 1995 Optical Society of America

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References

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  1. See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Imaging: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120.
  2. For an overview of much current work, see A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
    [CrossRef]
  3. M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  4. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef] [PubMed]
  5. I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
    [CrossRef] [PubMed]
  6. R. Aronson, “Exact interface conditions for photon diffusion,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. Mang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1641, 72–78 (1992).
    [CrossRef]
  7. R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).
    [CrossRef]
  8. For a comparative study of various proposed boundary conditions, see R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  9. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  10. R. Aronson, “Subcritical problems in spherical geometry,” Nucl. Sci. Eng. 86, 436–449 (1984).
  11. See, for instance, J. R. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1983), Chap. 5.
  12. R. Aronson, “Transfer matrix solution of some one- and two-medium transport problems in slab geometry,” J. Math. Phys. 11, 931–940 (1970).
    [CrossRef]
  13. R. Aronson, “Traveling mode solution of transport-theory problems by inspection,” Transp. Theory Stat. Phys. 2, 181–196 (1972).
    [CrossRef]
  14. G. Caroll, R. Aronson, “One-speed neutron transport problems. Part II: Slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).
  15. M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).
  16. A. Cohen, “Photoelectric determination of the relative oxygenation of blood,” Ph.D. dissertation (Carnegie-Mellon University, Pittsburgh, Pa., 1969).
  17. R. W. Preisendorfer, Hydrologic Optics (U.S. Department of Commerce, National Oceanic and Atmospheric Administration, Washington, D.C., 1976), Section 2.6.
  18. Unpublished report by Walsh quoted by J. W. Ryde, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A131, 451–475 (1931).
    [CrossRef]
  19. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  20. R. L. Barbor, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1993).
    [CrossRef]

1995 (1)

For an overview of much current work, see A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

1994 (1)

1992 (1)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

1991 (1)

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

1984 (1)

R. Aronson, “Subcritical problems in spherical geometry,” Nucl. Sci. Eng. 86, 436–449 (1984).

1973 (1)

G. Caroll, R. Aronson, “One-speed neutron transport problems. Part II: Slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

1972 (1)

R. Aronson, “Traveling mode solution of transport-theory problems by inspection,” Transp. Theory Stat. Phys. 2, 181–196 (1972).
[CrossRef]

1970 (1)

R. Aronson, “Transfer matrix solution of some one- and two-medium transport problems in slab geometry,” J. Math. Phys. 11, 931–940 (1970).
[CrossRef]

Aronson, R.

R. Aronson, “Subcritical problems in spherical geometry,” Nucl. Sci. Eng. 86, 436–449 (1984).

G. Caroll, R. Aronson, “One-speed neutron transport problems. Part II: Slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

R. Aronson, “Traveling mode solution of transport-theory problems by inspection,” Transp. Theory Stat. Phys. 2, 181–196 (1972).
[CrossRef]

R. Aronson, “Transfer matrix solution of some one- and two-medium transport problems in slab geometry,” J. Math. Phys. 11, 931–940 (1970).
[CrossRef]

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Imaging: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120.

R. Aronson, “Exact interface conditions for photon diffusion,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. Mang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1641, 72–78 (1992).
[CrossRef]

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).
[CrossRef]

R. L. Barbor, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1993).
[CrossRef]

Barbor, R. L.

R. L. Barbor, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1993).
[CrossRef]

Barbour, R. L.

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Imaging: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120.

Bonner, R. F.

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

Caroll, G.

G. Caroll, R. Aronson, “One-speed neutron transport problems. Part II: Slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chance, B.

For an overview of much current work, see A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

Cohen, A.

A. Cohen, “Photoelectric determination of the relative oxygenation of blood,” Ph.D. dissertation (Carnegie-Mellon University, Pittsburgh, Pa., 1969).

Feng, T.-C.

Freund, I.

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

Graber, H. L.

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Imaging: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120.

R. L. Barbor, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1993).
[CrossRef]

Haskell, R. C.

Havlin, S.

Keijzer, M.

Lamarsh, J. R.

See, for instance, J. R. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1983), Chap. 5.

Lubowsky, J.

R. L. Barbor, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1993).
[CrossRef]

McAdams, M. S.

Nossal, R.

Pine, D. J.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Preisendorfer, R. W.

R. W. Preisendorfer, Hydrologic Optics (U.S. Department of Commerce, National Oceanic and Atmospheric Administration, Washington, D.C., 1976), Section 2.6.

Ryde, J. W.

Unpublished report by Walsh quoted by J. W. Ryde, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A131, 451–475 (1931).
[CrossRef]

Star, W. M.

Storchi, P. R. M.

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.-T.

Wang, Y.

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Imaging: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120.

Weiss, G. H.

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

Yodh, A.

For an overview of much current work, see A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

Zhu, J. X.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (1)

J. Math. Phys. (1)

R. Aronson, “Transfer matrix solution of some one- and two-medium transport problems in slab geometry,” J. Math. Phys. 11, 931–940 (1970).
[CrossRef]

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

Nucl. Sci. Eng. (2)

R. Aronson, “Subcritical problems in spherical geometry,” Nucl. Sci. Eng. 86, 436–449 (1984).

G. Caroll, R. Aronson, “One-speed neutron transport problems. Part II: Slab transmission and reflection and finite reflector critical problems,” Nucl. Sci. Eng. 51, 166–179 (1973).

Phys. Rev. A (2)

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

Phys. Today (1)

For an overview of much current work, see A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (3), 34–40 (1995).
[CrossRef]

Transp. Theory Stat. Phys. (1)

R. Aronson, “Traveling mode solution of transport-theory problems by inspection,” Transp. Theory Stat. Phys. 2, 181–196 (1972).
[CrossRef]

Other (10)

See, for instance, J. R. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1983), Chap. 5.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

A. Cohen, “Photoelectric determination of the relative oxygenation of blood,” Ph.D. dissertation (Carnegie-Mellon University, Pittsburgh, Pa., 1969).

R. W. Preisendorfer, Hydrologic Optics (U.S. Department of Commerce, National Oceanic and Atmospheric Administration, Washington, D.C., 1976), Section 2.6.

Unpublished report by Walsh quoted by J. W. Ryde, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A131, 451–475 (1931).
[CrossRef]

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Imaging: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

R. Aronson, “Exact interface conditions for photon diffusion,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. Mang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1641, 72–78 (1992).
[CrossRef]

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).
[CrossRef]

R. L. Barbor, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1993).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Extrapolated end point. No absorption, isotropic scattering.

Fig. 2
Fig. 2

Extrapolated end point times c. Isotropic scattering.

Fig. 3
Fig. 3

Extrapolation distance. Isotropic scattering, transport mean free path units.

Fig. 4
Fig. 4

Interface coefficient C(n).

Tables (4)

Tables Icon

Table 1 Extrapolated End Point Times c, Isotropic Scattering

Tables Icon

Table 2 Extrapolation Distance, Isotropic Scattering

Tables Icon

Table 3 Extrapolation Distance, g = 0.8, in Transport Mean Free Pathsa

Tables Icon

Table 4 Interface Coefficient C(n)

Equations (50)

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I as ( z 0 ) = 0 ,
d = I as ( 0 ) / I as ( 0 ) ,
I as ( z ) = exp ( λ z ) + A exp ( λ z ) ,
z 0 = ( 1 / 2 λ ) ln ( A ) ,
d = 1 λ 1 + A 1 A .
μ 0 2 = 1 n 2 + n 2 μ 2 .
R ( μ , μ ) = r ( μ ) δ ( μ μ ) ,
r ( μ ) = 1 2 [ ( μ n μ 0 μ + n μ 0 ) 2 + ( μ 0 n μ μ 0 + n μ ) 2 ] , μ μ c , = 1 , μ μ c .
R i j = ( 2 i + 1 ) 0 1 D i ( μ ) d μ 0 1 R ( μ , μ ) D j ( μ ) d μ = ( 2 i + 1 ) 0 1 r ( μ ) D i ( μ ) D j ( μ ) d μ = δ i j ( 2 i + 1 ) s i j ,
s i j = μ c 1 [ 1 r ( μ ) ] D i ( μ ) D j ( μ ) d μ .
D i ( μ ) = P i ( 2 μ 1 ) .
J k = μ c 1 [ 1 r ( μ ) ] μ k d μ .
J k = 1 μ c k + 1 k + 1 1 2 ( n 2 1 2 n ) k + 1 × p 1 [ t 4 + ( t 2 g 1 g t 2 ) 2 ] ( 1 + t 2 ) k ( 1 t 2 ) t k + 2 d t ,
s i j = ( 1 / n ) 1 1 [ 1 ( μ + n μ 0 ) 2 + 1 ( μ 0 + n μ ) 2 ] × D i ( μ ) D j ( μ ) μ 0 2 d x .
s i , j + 1 = 1 j + 1 { 2 j + 1 2 i + 1 [ ( i + 1 ) s i + 1 , j + i s i 1 , j ] j s i , j 1 } ,
J = ( ϕ / 2 ) 0 1 μ d μ + ( ϕ / 2 t ) 0 1 μ 2 d μ = ϕ / 4 + ϕ / 6 t ,
J + = ( ϕ / 2 ) 0 1 μ d μ ( ϕ / 2 t ) 0 1 μ 2 d μ = ϕ / 4 ϕ / 6 t .
J = J + J .
J = ( 1 / 3 t ) ϕ .
J = D ϕ .
D = 1 / 3 t .
J = ( 1 / 4 ) ϕ + ( 1 / 2 ) D ϕ = ( 1 / 4 ) ϕ ( 1 / 2 ) J ,
J + = ( 1 / 4 ) ϕ ( 1 / 2 ) D ϕ = ( 1 / 4 ) ϕ + ( 1 / 2 ) J .
tr = ( 1 g ) s + a ,
J = ϕ 1 / 2 0 1 [ 1 r ( μ 0 ) ] μ 0 d μ 0 + 3 D 1 ϕ 1 / 2 × 0 1 [ 1 r ( μ 0 ) ] μ 0 2 d μ 0 ,
J + = ϕ 2 / 2 0 1 [ 1 r ( μ ) ] μ d μ 3 D 2 ϕ 2 / 2 × 0 1 [ 1 r ( μ ) ] μ 2 d μ ,
D 1 ϕ 1 = D 2 ϕ 2 = J .
J = ( A 1 ϕ 1 B 1 J ) / 2 ,
J + = ( A 2 ϕ 2 + B 2 J ) / 2 ,
A 1 = 0 1 [ 1 r ( μ 0 ) ] μ 0 d μ 0 ,
A 2 = μ c 1 [ 1 r ( μ ) ] μ d μ ,
B 1 = 3 0 1 [ 1 r ( μ 0 ) ] μ 0 2 d μ 0 ,
B 2 = 3 μ c 1 [ 1 r ( μ ) ] μ 2 d μ .
J = ( A 2 ϕ 2 A 1 ϕ 1 ) / ( 2 B 1 B 2 ) .
ϕ 2 n 2 ϕ 1 = C ( n ) J ,
C ( n ) = ( 2 B 1 B 2 ) / A 2 .
B 1 = 3 μ c 1 [ 1 r ( μ ) ] μ 0 μ d μ .
C ( n ) = 25.6 p 3 [ 1 ( 45 / 32 ) p + ( 83 / 21 ) p 2 ] , 1 < n < 1.82 , = 14.3 ( 1 + 9.35 p 1 + 49.1 p 1 2 + 327 p 1 3 + 1800 p 1 4 ) , 1.82 n 3.73 .
A 1 = 5 n 6 + 8 n 5 + 6 n 4 5 n 3 n 1 3 ( n 2 + 1 ) 2 ( n 2 1 ) ( n + 1 ) 4 n 4 ( n 4 + 1 ) ( n 2 + 1 ) 3 ( n 2 1 ) 2 log n + n 2 ( n 2 1 ) 2 2 ( n 2 + 1 ) 3 log n + 1 n 1 ,
B 1 = 1 3 ( n 2 1 ) 3 / 2 16 ( I 0 I 1 I 2 + I 3 ) ,
B 2 = 1 ( n 2 1 ) 3 / 2 n 3 [ 1 + 3 16 ( I 0 + I 1 I 2 I 3 ) ] ,
I n = ( 1 p 2 n + 1 ) / ( 2 n + 1 ) + J n ,
J 0 = 16 n 4 ( n 2 + 1 ) 4 K 2 8 n 2 ( n 2 1 ) 2 ( n 2 + 1 ) 4 K 1 8 n 2 ( n 2 1 ) ( n 2 + 1 ) 3 × ( 1 p 1 ) + ( n 2 1 ) 2 3 ( n 2 + 1 ) 2 ( 1 p 3 1 ) ,
J 1 = 16 n 4 ( n 2 + 1 ) 3 ( n 2 1 ) K 2 8 n 2 ( n 4 + 1 ) ( n 2 + 1 ) 3 ( n 2 1 ) K 1 + ( n 2 1 ) 2 ( n 2 + 1 ) 2 ( 1 p 1 ) ,
J 2 = 16 n 4 ( n 2 + 1 ) 2 ( n 2 1 ) 2 K 2 8 n 2 ( n 2 + 1 ) 3 ( n 2 1 ) K 1 + ( n 2 + 1 ) 2 ( n 2 1 ) 2 ( 1 p ) ,
J 3 = 16 n 4 ( n 2 + 1 ) ( n 2 1 ) 3 K 2 8 n 2 ( n 4 + 4 n 2 + 1 ) ( n 2 + 1 ) ( n 2 1 ) 3 K 1 + 8 n 2 ( n 2 + 1 ) ( n 2 1 ) 3 ( 1 p ) + ( n 2 + 1 ) 2 3 ( n 2 1 ) 2 ( 1 p 3 ) ,
K 1 = n 2 + 1 n 2 1 ( log n 2 + 1 + n 2 1 n 2 + 1 + n 1 + 1 2 log n ) ,
K 2 = n 2 + 1 4 ( 1 p n ) + 1 2 K 1 ,
p = n 1 n + 1 .
d = 2.94 7.01 n + 5.76 n 2 1.16 n 3 + 0.159 n 4 , c = 1.00 = 2.98 6.96 n + 5.68 n 2 1.16 n 3 + 0.157 n 4 , c = 0.98 = 2.83 6.50 n + 5.30 n 2 1.06 n 3 + 0.144 n 4 , c = 0.96 = 2.63 5.94 n + 4.84 n 2 0.94 n 3 + 0.129 n 4 , c = 0.94 = 2.77 6.16 n + 5.00 n 2 1.03 n 3 + 0.139 n 4 , c = 0.92 = 2.79 6.10 n + 4.95 n 2 1.04 n 3 + 0.140 n 4 , c = 0.90 .

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