Abstract

We derive Kubelka–Munk (KM) theory systematically from the radiative transport equation (RTE) by analyzing the system of equations resulting from applying the double spherical harmonics method of order one and transforming that system into one governing the positive- and negative-going fluxes. Through this derivation, we establish the theoretical basis of KM theory, identify all parameters, and determine its range of validity. Moreover, we are able to generalize KM theory to take into account general boundary sources and nonhomogeneous terms, for example. The generalized Kubelka–Munk (gKM) equations are also much more accurate at approximating the solution of the RTE. We validate this theory through comparison with numerical solutions of the RTE.

© 2014 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).
  3. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  4. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef]
  5. B. Philips-Invernizzi, D. Dupont, and C. Cazé, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 40, 1082–1092 (2001).
    [CrossRef]
  6. B. J. Brinkworth, “Interpretations of the Kubelka-Munk coefficients in reflection theory,” Appl. Opt. 11, 1434–1435 (1972).
    [CrossRef]
  7. L. F. Gate, “Comparison of the photon diffusion model and Kubelka-Munk equation with exact solution of the radiative transport equation,” Appl. Opt. 13, 236–238 (1974).
    [CrossRef]
  8. J. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration 15, 66–75 (1985).
    [CrossRef]
  9. W. M. Star, J. P. A. Marijnissen, and M. J. C. Van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
    [CrossRef]
  10. W. E. Vargas and G. A. Niklasson, “Applicability conditions of the Kubelka-Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef]
  11. R. Molenaar, J. ten Bosch, and J. Zijp, “Determination of Kubelka-Munk scattering and absorption coefficients by diffuse illumination,” Appl. Opt. 38, 2068–2077 (1999).
    [CrossRef]
  12. L. Yang and B. Kruse, “Revised Kubelka-Munk theory. I. Theory and application,” J. Opt. Soc. Am. A 21, 1933–1941 (2004).
    [CrossRef]
  13. L. Yang, B. Kruse, and S. J. Miklavcic, “Revised Kubelka-Munk theory. II. Unified framework for homogeneous and inhomogeneous optical media,” J. Opt. Soc. Am. A 21, 1942–1952 (2004).
    [CrossRef]
  14. L. Yang and S. J. Miklavcic, “Revised Kubelka-Munk theory. III. A general theory of light propagation in scattering and absorptive media,” J. Opt. Soc. Am. A 22, 1866–1873 (2005).
    [CrossRef]
  15. P. Edström, “Examination of the revised Kubelka-Munk theory: considerations of modeling strategies,” J. Opt. Soc. Am. A 24, 548–556 (2007).
    [CrossRef]
  16. S. N. Thennadil, “Relationships between the Kubelka-Munk scattering and radiative transfer coefficients,” J. Opt. Soc. Am. A 25, 1480–1485 (2008).
    [CrossRef]
  17. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A 27, 1032–1039 (2010).
    [CrossRef]
  18. M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).
  19. B. Davison, Neutron Transport Theory (Oxford University, 1958).
  20. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  21. E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, 1993).
  22. R. Aronson, “PN vs. double-PN approximations for highly anisotropic scattering,” Transp. Theory Stat. Phys. 15, 829–840 (1986).
    [CrossRef]

2011 (1)

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

2010 (1)

2008 (1)

2007 (1)

2005 (1)

2004 (2)

2001 (1)

B. Philips-Invernizzi, D. Dupont, and C. Cazé, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 40, 1082–1092 (2001).
[CrossRef]

1999 (1)

1997 (1)

1988 (1)

W. M. Star, J. P. A. Marijnissen, and M. J. C. Van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef]

1986 (1)

R. Aronson, “PN vs. double-PN approximations for highly anisotropic scattering,” Transp. Theory Stat. Phys. 15, 829–840 (1986).
[CrossRef]

1985 (1)

J. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration 15, 66–75 (1985).
[CrossRef]

1974 (1)

1972 (1)

1948 (1)

1931 (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Aronson, R.

R. Aronson, “PN vs. double-PN approximations for highly anisotropic scattering,” Transp. Theory Stat. Phys. 15, 829–840 (1986).
[CrossRef]

Baranowski, M.

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

Brinkworth, B. J.

Brooke, H.

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

Case, K. M.

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

Cazé, C.

B. Philips-Invernizzi, D. Dupont, and C. Cazé, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 40, 1082–1092 (2001).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Davison, B.

B. Davison, Neutron Transport Theory (Oxford University, 1958).

Dupont, D.

B. Philips-Invernizzi, D. Dupont, and C. Cazé, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 40, 1082–1092 (2001).
[CrossRef]

Edström, P.

Gate, L. F.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).

Kruse, B.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
[CrossRef]

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, 1993).

Marijnissen, J. P. A.

W. M. Star, J. P. A. Marijnissen, and M. J. C. Van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef]

McCutcheon, J. N.

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

Miklavcic, S. J.

Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, 1993).

Molenaar, R.

Morgan, S. L.

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Myrick, M. L.

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

Neuman, M.

Niklasson, G. A.

Nobbs, J.

J. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration 15, 66–75 (1985).
[CrossRef]

Philips-Invernizzi, B.

B. Philips-Invernizzi, D. Dupont, and C. Cazé, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 40, 1082–1092 (2001).
[CrossRef]

Simcock, M. N.

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

Star, W. M.

W. M. Star, J. P. A. Marijnissen, and M. J. C. Van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef]

ten Bosch, J.

Thennadil, S. N.

Van Gemert, M. J. C.

W. M. Star, J. P. A. Marijnissen, and M. J. C. Van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef]

Vargas, W. E.

Yang, L.

Zijp, J.

Zweifel, P. F.

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

Appl. Opt. (4)

Appl. Spectrosc. Rev. (1)

M. L. Myrick, M. N. Simcock, M. Baranowski, H. Brooke, S. L. Morgan, and J. N. McCutcheon, “The Kubelka-Munk diffuse reflectance formula revisited,” Appl. Spectrosc. Rev. 46, 140–165 (2011).

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

B. Philips-Invernizzi, D. Dupont, and C. Cazé, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 40, 1082–1092 (2001).
[CrossRef]

Phys. Med. Biol. (1)

W. M. Star, J. P. A. Marijnissen, and M. J. C. Van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef]

Rev. Prog. Coloration (1)

J. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration 15, 66–75 (1985).
[CrossRef]

Transp. Theory Stat. Phys. (1)

R. Aronson, “PN vs. double-PN approximations for highly anisotropic scattering,” Transp. Theory Stat. Phys. 15, 829–840 (1986).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (5)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).

B. Davison, Neutron Transport Theory (Oxford University, 1958).

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

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, 1993).

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Figures (6)

Fig. 1.
Fig. 1.

Comparisons of F+(τ) (top) and F(τ) (bottom) computed from solutions of boundary value problem (6.1) using the radiative transport equation (RTE), the generalized Kubelka–Munk (gKM) equations, and the Kubelka–Munk (KM) equations. Here, τ1=0, τ2=1, ϖ0=0.99, and g=0.

Fig. 2.
Fig. 2.

Comparisons of F+(τ) (top) and F(τ) (bottom) computed from solutions of boundary value problem (6.1) using the RTE, the gKM equations, and the KM equations. Here, τ1=0, τ2=1, ϖ0=0.99, and g=0.8.

Fig. 3.
Fig. 3.

Comparisons of F+(τ) (top) and F(τ) (bottom) computed from solutions of boundary value problem (6.1) using the RTE, the gKM equations, and the KM equations. Here, τ1=0, τ2=1, ϖ0=0.5, and g=0.

Fig. 4.
Fig. 4.

Comparisons of F+(τ) (top) and F(τ) (bottom) computed from solutions of boundary value problem (6.1) using the RTE, the gKM equations, and the KM equations. Here, τ1=0, τ2=1, ϖ0=0.5, and g=0.8.

Fig. 5.
Fig. 5.

Comparisons of F+(τ) (top) and F(τ) (bottom) computed from solutions of boundary value problem (6.11) using the RTE, and the gKM equations. Here, τ1=0, τ2=1, ϖ0=0.99, and g=0.

Fig. 6.
Fig. 6.

Comparisons of F+(τ) (top) and F(τ) (bottom) computed from solutions of boundary value problem (6.11) using the RTE and the gKM equations. Here, τ1=0, τ2=1, ϖ0=0.99, and g=0.7.

Equations (73)

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dF+dτ+(K+S)F+=SF,
dFdτ+(K+S)F=SF+,
μIτ+Iϖ0211h(μ,μ)I(μ,τ)dμ=Q(μ,τ)inτ1<τ<τ2,
I(μ,τ1)=f+(μ)on0<μ1,
I(μ,τ2)=f(μ)on1μ<0.
1211h(μ,μ)dμ=1.
1211h(μ,μ)μdμ=gμ,
μI+τ+I+ϖ0201[h(μ,μ)I+(μ,τ)+h(μ,μ)I(μ,τ)]dμ=Q+(μ,τ),
μIτ+Iϖ0201[h(μ,μ)I+(μ,τ)+h(μ,μ)I(μ,τ)]dμ=Q(μ,τ),
I+(μ,τ1)=f+(μ),
I(μ,τ2)=f(μ),
01P˜m(μ)P˜n(μ)dμ=δmn,
I±(μ,τ)=n=0cn±(τ)P˜n(μ).
ddτ[M00M][c+c]=[ϖ0H(1)Iϖ0H(2)ϖ0H(2)ϖ0H(1)I][c+c]+[q+q],
Mm+1,n+1=01P˜m(μ)P˜n(μ)μdμ,=12[m(2m1)(2m+1)δm,n1+δmn+m+1(2m+1)(2m+3)δm,n+1],
Hm+1,n+1(1)=1201P˜m(μ)01h(μ,μ)P˜n(μ)dμdμ,
Hm+1,n+1(2)=1201P˜m(μ)01h(μ,μ)P˜n(μ)dμdμ,
qn+1±(τ)=01Q±(μ,τ)P˜n(μ)dμ,
c+(τ1)=f+,
c(τ2)=f,
fn+1+=01f+(μ)P˜n(μ)dμ,
fn+1=01f(μ)P˜n(μ)dμ,
λ[M00M][uv]=[ϖ0H(1)Iϖ0H(2)ϖ0H(2)ϖ0H(1)I][uv],
λ(N+1)λ1<0<λ1λN+1,
F±(τ)=01I±(μ,τ)μdμ.
F±=12(c0±+13c1±),
G±=12(13c0±+c1±).
[F±G±]=[1212312312][c0±c1±].
[I00I]ddτ[y+y]=[S1S2S2S1][y+y]+[q+q],
y+(τ1)=Mf+,
y(τ2)=Mf,
λ˜[I00I][u˜v˜]=[S1S2S2S1][u˜v˜],
λ˜2λ˜1<0<λ˜1λ˜2,
[y+y]=j=12{[u˜jv˜j]a˜j(τ)+[v˜ju˜j]b˜j(τ)},
λ[M00M][uv]+ϵ2[H(1)H(2)H(2)H(1)][uv]=[H(1)IH(2)H(2)H(1)I][uv].
λ[M00M][uv]=[H(1)IH(2)H(2)H(1)I][uv].
λ1[Me^1Me^1]=[H(1)IH(2)H(2)H(1)I][u1v1].
[u1v1]=λ11g[m1m1].
λ1[M00M][e^1e^1]+λ1[M00M][u1v1]+[H(1)H(2)H(2)H(1)][e^1e^1]=[H(1)IH(2)H(2)H(1)I][u1v1].
λ12=(1g)m12.
[c+c][u1v1]eλ1(ττ2)α1+[v1u1]eλ1(ττ1)β1
[F+F]=[u¯v¯v¯u¯][eλ1(ττ2)00eλ1(ττ1)][α1β1],
ddτ[F+F]=[u¯v¯v¯u¯][λ100λ1][u¯v¯v¯u¯]1[F+F].
K=2(1ϖ0),
S=34(1g)(1ϖ0).
K˜S˜=(1R)22R,
μIτ+I=ϖ0211h(μ,μ)I(μ,τ)dμinτ1<τ<τ2,
I(μ,τ1)=e100(μ1)2on0<μ1,
I(μ,τ2)=1on1μ<0.
h(μ,μ)=2(1g2)π(ab)a+bE(2ba+b),
j=12{u˜jeλ˜j(τ2τ1)α˜j+v˜jβ˜j}=Mf+.
j=12{v˜jα˜j+u˜jeλ˜j(τ2τ1)β˜j}=Mf.
F+(τ)=j=12{U˜1jeλ˜j(ττ2)α˜j+V˜1jeλ˜j(ττ1)β˜j},
F(τ)=j=12{V˜1jeλ˜j(ττ2)α˜j+U˜1jeλ˜j(ττ1)β˜j}.
F+(τ1)=01f+(μ)μdμ,
F(τ2)=01f(μ)μdμ.
μIτ+I=ϖ0211h(μ,μ)I(μ,τ)dμinτ1<τ<τ2,
I(μ,τ1)=δ(μ1)on0<μ1,
I(μ,τ2)=0on1μ<0.
μIriτ+Iri=0,
Iri(μ,τ1)=δ(μ1)on0<μ1,
Iri(μ,τ2)=0on1μ<0.
μIdτ+Id=ϖ0211h(μ,μ)Id(μ,τ)dμ+Qri,
Id(μ,τ1)=0on0<μ1,
Id(μ,τ2)=0on1μ<0.
Fd±(τ)=01Id(μ,τ)μdμ,
[I00I]ddτ[y+y]=[S1S2S2S1][y+y]+ϖ02[h+h]e(ττ1),
hn+1±=01h(±μ,1)Pn(μ)dμ,n=0,1.
[IS1S2S2IS1][η+η]=ϖ02[h+h].
j=12{u˜jeλ˜j(τ2τ1)α˜j+v˜jβ˜j}=η+.
j=12{v˜jα˜j+u˜jeλ˜j(τ2τ1)β˜j}=ηe(τ2τ1).
F+(τ)=j=12{U˜1jeλ˜j(ττ2)α˜j+V˜1jeλ˜j(ττ1)β˜j}+η1+e(ττ1),
F(τ)=j=12{V˜1jeλ˜j(ττ2)α˜j+U˜1jeλ˜j(ττ1)β˜j}+η1e(ττ1),

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