Abstract

An instantaneous point source generating a light field in an infinite turbid medium with anisotropic individual scatterers is considered. Single-scattering solutions are sought as the first term of a series expansion of the solution of the radiative transfer equation in successive scattering orders. A simple formula for a single-scattering solution in media with an arbitrary axially symmetric phase function is derived. Applications of this formula are shown for the Henyey–Greenstein phase function, an ellipsoidal phase function, and a linear phase function. In addition, the single-scattering term of the RTE solution derived by Kholin [Zh. Vych. Mat. i Mat. Fys. 4, 1126 (1964)] for media with a phase function represented by a finite series in Legendre polynomials is considered in more detail.

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  1. L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
    [CrossRef]
  2. D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. Am. 69, 464–470 (1979).
    [CrossRef]
  3. V. Pegoraro and S. G. Parker, “An analytical solution to single scattering in homogeneous participating media,” Comput. Graph. Forum 28, 329–335 (2009).
    [CrossRef]
  4. A. S. Monin, “A statistical interpretation of the scattering of microscopic particles,” Theory Probab. Appl. 1, 298–311 (1956).
    [CrossRef]
  5. J. C. J. Paasschens, “Solution of time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135–1141 (1997).
    [CrossRef]
  6. S. A. Kholin, “Certain exact solutions of the nonstationary kinetic equation without taking retardation into account,” Zh. Vych. Mat. i Mat. Fys. 4, 1126–1131 (1964).
  7. S. A. Kholin, I. M. Anisina, and A. Yu. Madyanov, “Green’s functions for non-stationary kinetic equation with constant speed,” Transp. Theory Stat. Phys. 37, 361–376 (2008).
    [CrossRef]
  8. V. N. Fomenko and F. M. Shvarts, “Exact description of photon migration in isotropically scattering media,” Proc. SPIE 3194, 334–342 (1998).
    [CrossRef]
  9. V. N. Fomenko, F. M. Shvarts, and M. A. Shvarts, “Exact description of photon migration in anisotropically scattering media,” Phys. Rev. E 61, 1990–1995 (2000).
    [CrossRef]
  10. B. D. Ganapol, “Solution of the one-group time-dependent neutron transport equation in an infinite medium by polynomial reconstruction,” Nucl. Sci. Eng. 92, 272–279 (1986).
  11. H. Sato, “Energy propagation including scattering effects. Single isotropic scattering approximation,” J. Phys. Earth 25, 27–41 (1977).
    [CrossRef]
  12. H. Sato, “Formulation of the multiple non-isotropic scattering process in 3-D space on the basis of energy transport theory,” Geophys. J. Int. 121, 523–531 (1995).
    [CrossRef]
  13. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  14. L. O. Reynolds and N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206–1212 (1980).
    [CrossRef]
  15. G. D. Pedersen, N. J. McCormick, and L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
    [CrossRef]
  16. Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).
  17. K. Rinzema, B. J. Hoenderst, H. A. Ferwerda, and J. J. Ten Bosch, “Low-degree polynomial phase-functions with high g-value,” Phys. Med. Biol. 38, 1343–1350 (1993).
    [CrossRef]

2009 (2)

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

V. Pegoraro and S. G. Parker, “An analytical solution to single scattering in homogeneous participating media,” Comput. Graph. Forum 28, 329–335 (2009).
[CrossRef]

2008 (1)

S. A. Kholin, I. M. Anisina, and A. Yu. Madyanov, “Green’s functions for non-stationary kinetic equation with constant speed,” Transp. Theory Stat. Phys. 37, 361–376 (2008).
[CrossRef]

2000 (1)

V. N. Fomenko, F. M. Shvarts, and M. A. Shvarts, “Exact description of photon migration in anisotropically scattering media,” Phys. Rev. E 61, 1990–1995 (2000).
[CrossRef]

1998 (1)

V. N. Fomenko and F. M. Shvarts, “Exact description of photon migration in isotropically scattering media,” Proc. SPIE 3194, 334–342 (1998).
[CrossRef]

1997 (1)

J. C. J. Paasschens, “Solution of time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135–1141 (1997).
[CrossRef]

1995 (1)

H. Sato, “Formulation of the multiple non-isotropic scattering process in 3-D space on the basis of energy transport theory,” Geophys. J. Int. 121, 523–531 (1995).
[CrossRef]

1993 (1)

K. Rinzema, B. J. Hoenderst, H. A. Ferwerda, and J. J. Ten Bosch, “Low-degree polynomial phase-functions with high g-value,” Phys. Med. Biol. 38, 1343–1350 (1993).
[CrossRef]

1986 (1)

B. D. Ganapol, “Solution of the one-group time-dependent neutron transport equation in an infinite medium by polynomial reconstruction,” Nucl. Sci. Eng. 92, 272–279 (1986).

1980 (1)

1979 (1)

1977 (1)

H. Sato, “Energy propagation including scattering effects. Single isotropic scattering approximation,” J. Phys. Earth 25, 27–41 (1977).
[CrossRef]

1976 (1)

G. D. Pedersen, N. J. McCormick, and L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef]

1964 (1)

S. A. Kholin, “Certain exact solutions of the nonstationary kinetic equation without taking retardation into account,” Zh. Vych. Mat. i Mat. Fys. 4, 1126–1131 (1964).

1956 (1)

A. S. Monin, “A statistical interpretation of the scattering of microscopic particles,” Theory Probab. Appl. 1, 298–311 (1956).
[CrossRef]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Anisina, I. M.

S. A. Kholin, I. M. Anisina, and A. Yu. Madyanov, “Green’s functions for non-stationary kinetic equation with constant speed,” Transp. Theory Stat. Phys. 37, 361–376 (2008).
[CrossRef]

Ferwerda, H. A.

K. Rinzema, B. J. Hoenderst, H. A. Ferwerda, and J. J. Ten Bosch, “Low-degree polynomial phase-functions with high g-value,” Phys. Med. Biol. 38, 1343–1350 (1993).
[CrossRef]

Florescu, L.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

Fomenko, V. N.

V. N. Fomenko, F. M. Shvarts, and M. A. Shvarts, “Exact description of photon migration in anisotropically scattering media,” Phys. Rev. E 61, 1990–1995 (2000).
[CrossRef]

V. N. Fomenko and F. M. Shvarts, “Exact description of photon migration in isotropically scattering media,” Proc. SPIE 3194, 334–342 (1998).
[CrossRef]

Ganapol, B. D.

B. D. Ganapol, “Solution of the one-group time-dependent neutron transport equation in an infinite medium by polynomial reconstruction,” Nucl. Sci. Eng. 92, 272–279 (1986).

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Guo, D. R.

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hoenderst, B. J.

K. Rinzema, B. J. Hoenderst, H. A. Ferwerda, and J. J. Ten Bosch, “Low-degree polynomial phase-functions with high g-value,” Phys. Med. Biol. 38, 1343–1350 (1993).
[CrossRef]

Kholin, S. A.

S. A. Kholin, I. M. Anisina, and A. Yu. Madyanov, “Green’s functions for non-stationary kinetic equation with constant speed,” Transp. Theory Stat. Phys. 37, 361–376 (2008).
[CrossRef]

S. A. Kholin, “Certain exact solutions of the nonstationary kinetic equation without taking retardation into account,” Zh. Vych. Mat. i Mat. Fys. 4, 1126–1131 (1964).

Madyanov, A. Yu.

S. A. Kholin, I. M. Anisina, and A. Yu. Madyanov, “Green’s functions for non-stationary kinetic equation with constant speed,” Transp. Theory Stat. Phys. 37, 361–376 (2008).
[CrossRef]

Markel, V. A.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

McCormick, N. J.

L. O. Reynolds and N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206–1212 (1980).
[CrossRef]

G. D. Pedersen, N. J. McCormick, and L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef]

Monin, A. S.

A. S. Monin, “A statistical interpretation of the scattering of microscopic particles,” Theory Probab. Appl. 1, 298–311 (1956).
[CrossRef]

Paasschens, J. C. J.

J. C. J. Paasschens, “Solution of time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135–1141 (1997).
[CrossRef]

Parker, S. G.

V. Pegoraro and S. G. Parker, “An analytical solution to single scattering in homogeneous participating media,” Comput. Graph. Forum 28, 329–335 (2009).
[CrossRef]

Pedersen, G. D.

G. D. Pedersen, N. J. McCormick, and L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef]

Pegoraro, V.

V. Pegoraro and S. G. Parker, “An analytical solution to single scattering in homogeneous participating media,” Comput. Graph. Forum 28, 329–335 (2009).
[CrossRef]

Reilly, D. M.

Reynolds, L. O.

L. O. Reynolds and N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206–1212 (1980).
[CrossRef]

G. D. Pedersen, N. J. McCormick, and L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef]

Rinzema, K.

K. Rinzema, B. J. Hoenderst, H. A. Ferwerda, and J. J. Ten Bosch, “Low-degree polynomial phase-functions with high g-value,” Phys. Med. Biol. 38, 1343–1350 (1993).
[CrossRef]

Sato, H.

H. Sato, “Formulation of the multiple non-isotropic scattering process in 3-D space on the basis of energy transport theory,” Geophys. J. Int. 121, 523–531 (1995).
[CrossRef]

H. Sato, “Energy propagation including scattering effects. Single isotropic scattering approximation,” J. Phys. Earth 25, 27–41 (1977).
[CrossRef]

Schotland, J. C.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

Shvarts, F. M.

V. N. Fomenko, F. M. Shvarts, and M. A. Shvarts, “Exact description of photon migration in anisotropically scattering media,” Phys. Rev. E 61, 1990–1995 (2000).
[CrossRef]

V. N. Fomenko and F. M. Shvarts, “Exact description of photon migration in isotropically scattering media,” Proc. SPIE 3194, 334–342 (1998).
[CrossRef]

Shvarts, M. A.

V. N. Fomenko, F. M. Shvarts, and M. A. Shvarts, “Exact description of photon migration in anisotropically scattering media,” Phys. Rev. E 61, 1990–1995 (2000).
[CrossRef]

Ten Bosch, J. J.

K. Rinzema, B. J. Hoenderst, H. A. Ferwerda, and J. J. Ten Bosch, “Low-degree polynomial phase-functions with high g-value,” Phys. Med. Biol. 38, 1343–1350 (1993).
[CrossRef]

Wang, Z. X.

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).

Warde, C.

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Biophys. J. (1)

G. D. Pedersen, N. J. McCormick, and L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef]

Comput. Graph. Forum (1)

V. Pegoraro and S. G. Parker, “An analytical solution to single scattering in homogeneous participating media,” Comput. Graph. Forum 28, 329–335 (2009).
[CrossRef]

Geophys. J. Int. (1)

H. Sato, “Formulation of the multiple non-isotropic scattering process in 3-D space on the basis of energy transport theory,” Geophys. J. Int. 121, 523–531 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. Earth (1)

H. Sato, “Energy propagation including scattering effects. Single isotropic scattering approximation,” J. Phys. Earth 25, 27–41 (1977).
[CrossRef]

Nucl. Sci. Eng. (1)

B. D. Ganapol, “Solution of the one-group time-dependent neutron transport equation in an infinite medium by polynomial reconstruction,” Nucl. Sci. Eng. 92, 272–279 (1986).

Phys. Med. Biol. (1)

K. Rinzema, B. J. Hoenderst, H. A. Ferwerda, and J. J. Ten Bosch, “Low-degree polynomial phase-functions with high g-value,” Phys. Med. Biol. 38, 1343–1350 (1993).
[CrossRef]

Phys. Rev. E (3)

V. N. Fomenko, F. M. Shvarts, and M. A. Shvarts, “Exact description of photon migration in anisotropically scattering media,” Phys. Rev. E 61, 1990–1995 (2000).
[CrossRef]

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

J. C. J. Paasschens, “Solution of time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135–1141 (1997).
[CrossRef]

Proc. SPIE (1)

V. N. Fomenko and F. M. Shvarts, “Exact description of photon migration in isotropically scattering media,” Proc. SPIE 3194, 334–342 (1998).
[CrossRef]

Theory Probab. Appl. (1)

A. S. Monin, “A statistical interpretation of the scattering of microscopic particles,” Theory Probab. Appl. 1, 298–311 (1956).
[CrossRef]

Transp. Theory Stat. Phys. (1)

S. A. Kholin, I. M. Anisina, and A. Yu. Madyanov, “Green’s functions for non-stationary kinetic equation with constant speed,” Transp. Theory Stat. Phys. 37, 361–376 (2008).
[CrossRef]

Zh. Vych. Mat. i Mat. Fys. (1)

S. A. Kholin, “Certain exact solutions of the nonstationary kinetic equation without taking retardation into account,” Zh. Vych. Mat. i Mat. Fys. 4, 1126–1131 (1964).

Other (1)

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989).

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Figures (6)

Fig. 1.
Fig. 1.

Single-scattering geometry. The source S and the observation point R are located at the foci of an ellipsoid, and the scattering event takes place at point P on the ellipsoid’s surface.

Fig. 2.
Fig. 2.

Single-scattering solutions for media with the HG phase function for various anisotropy factors, shown in the figure. Here ls=0.05cm, r=0.1cm, and v=30cm/ns.

Fig. 3.
Fig. 3.

Comparison of single-scattering solutions for media with the HG phase function and the ellipsoidal phase function for anisotropy factor μ¯=0.9, where ls=0.05cm, r=0.1cm, and v=30cm/ns.

Fig. 4.
Fig. 4.

Ratio of angle-averaged intensities I¯1el/I¯1HG for various anisotropy factors. Here lS=0.05cm, r=0.1cm, and v=30cm/ns.

Fig. 5.
Fig. 5.

HG phase function and its approximation with polynomial phase functions, Eq. (39), with N=6 and N=10 and anisotropy factor g=0.7.

Fig. 6.
Fig. 6.

Single-scattering solutions corresponding to the HG phase function and the polynomial phase functions with N=6 and N=10 for two distances from the source, where ls=0.05cm, v=30cm/ns, and g=0.7. The left peak corresponds to r=0.00025cm, and the right peak corresponds to r=0.0025cm.

Equations (42)

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1vI(r⃗,t,s^)t+s^·I(r⃗,t,s^)+(μs+μa)I(r⃗,t,s^)=μsI˜(r⃗,t,s^)+1vS(r⃗,t),
S(r⃗,t)=S0δ(r⃗)δ(t),
S0=N0ωv/(4π)
I˜=4πp(s^·s^)I(r⃗,t,s^)ds^.
4πp(s^·s^)ds^=1
I¯(r,t)=14π4πI(r⃗,t,s^)ds^.
I(r⃗,t,s^)=N=0IN(r⃗,t,s^)
I˜(r⃗,t,s^)=N=0I˜N(r⃗,t,s^),
I¯(r⃗,t)=N=0I¯N(r⃗,t),
(1vt+s·+1ls)IN(r⃗,t,s^)=1lsI˜N1(r⃗,t,s^)
(1vt+s·+1ls)I0(r⃗,t,s^)=1vS(r⃗,t).
IN(r⃗,t,s^)=1ls0er1/lsI˜N1(r⃗r1s^,tr1/v,s^)dr1,
I0(r⃗,t,s^)=1v0er0/lsS(r⃗r0s^,tr0/v)dr0.
I0(r⃗,t,s^)=0er0/lsδ(r⃗r0s^)δ(vtr0)dr0=evt/lsδ(r⃗vts^),
I¯0(r⃗,t)=evt/ls4πr2δ(vtr).
I1(r⃗,t,s^)=1ls0er1/lsI˜0(r⃗r1s^,tr1/v,s^)dr1.
I1(r⃗,t,s^)=1ls04πer1/lsI0(r⃗r1s^,tr1/v,s^)p(s^·s^)ds^dr1=1ls004πe(r1+r0)/lsδ(r⃗r0s^r1s^)δ(vtr0r1)×p(s^·s^)dr0dr1ds^.
I¯1(r,t)=1ls00e(r0+r1)/ls4πr02r12δ(r⃗r⃗0r⃗1)δ(vtr0r1)×p(r^0·r^1)dr⃗0dr⃗1,
I¯1(r,t)=1ls00e(r0+r1)/ls(4πr0r1)2δ(r⃗r⃗0r⃗1)δ(vtr0r1)dr⃗0dr⃗1,
I¯1iso(r,t)=evt/ls4πls11dξ(vt)2+r22vtrξ
I¯1iso(r,t)=evt/ls4πlsvtrln(vt+rvtr)H(vtr),
I¯1(r,t)=evt/lsls11p(cosθ)dξ(vt)2+r22vtrξ.
r0=a+ex,
r1=aex,
ξ=cosα=cxaex,
μ=cosθ=2c2a2e2x2a2e2x2.
pHG(cosθ)=1g24π(1+g22gcosθ)3/2,
I¯1HG(r,t)=(1g2)evt/ls2πls×01(v2t2r2χ2)1/2dχ[(v2t2r2χ2)(1g)2+4g(v2t2r2)]3/2,
4π0vtI¯1r2dr=vtlsevt/ls.
pel(cosθ)=ς2πln(1+ς1ς)(1+ς22ςcosθ),
μ¯=(1+ς2)2ς1ln[(1+ς)/(1ς)].
I¯1el(r,t)=ςevt/ls2πlsr(1ς)ln(1+ς1ς)v2t2(1+ς)24r2ς×ln(v2t2(1+ς)24r2ς+(1ς)rv2t2(1+ς)24r2ς(1ς)r),
ppl(cosθ)=14πk=0NbkPk(cosθ),
I¯1Kh=ievt/ls4π2rvtlsk=0Nbk[Qk2(z+)Qk2(z)],
η=r/(vt).
Qk(z±)=iπ2Pk(η)+12ln(1+η1η)Pk(η)Wk1(η),
Wk1(η)=m=0k1Pm(η)Pkm1(η)km
I¯1Kh=evt/ls2πrvtls{12ln(1+η1η)k=0NbkPk2(η)k=1NbkPk(η)Wk1(η)},
pHG(μ)=14πk=0(2k+1)gkPk(μ).
pHGpl(μ)=14πk=0N(2k+1)gkPk(μ),
plin(cosθ)=(1+3gcosθ)/(4π).
I¯1lin=evt/ls4πrvtls{(1+3gη2)ln(1+η1η)6gη}.

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