Abstract

The Kubelka–Munk theory, although it provides an equation that relates the reflection of a sample under diffuse illumination to certain of its properties, does not take boundary reflectance into account. Boundary reflection is always present because there is always a difference between the refractive indices of the sample and of the surrounding medium. We describe how a half-sphere is used to achieve diffuse illumination, and we present and exemplify equations that correct for boundary reflection with measurements of four composite restorative dental materials. The refractive index of the sample is measured with a matching technique that employs a glycerol–water mixture. Edge loss errors are estimated.

© 1999 Optical Society of America

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References

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  1. P. S. Mudgett, L. W. Richards, “Simple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [CrossRef] [PubMed]
  2. G. Kortüm, Reflexionsspektroskopie, Grundlagen, Methodik, Anwendungen (Springer-Verlag, Heidelberg, 1969).
  3. K. H. Englmeier, R. Herpers, R. S. Jacoby, F. M. Zwiebel, “A method for the estimation of the hemoglobin distribution in gastroscopic images,” Int. J. Bio-Med. Comput. 41, 153–165 (1996).
    [CrossRef]
  4. W. M. Johnston, M. H. Reisbick, “Color and translucency changes during and after curing of esthetic restorative materials,” Dent. Mater. 13, 89–97 (1997).
    [CrossRef] [PubMed]
  5. R. G. Kuehni, Color, An Introduction to Practice and Principles (Wiley, New York, 1997), Chap. 8.
  6. H. G. Völz, B. Teague, Industrial Color Testing, Fundamentals and Techniques (VCH, Weinheim, Germany, 1994), Chap. 3.3.
  7. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
  8. S. Q. Duntley, “The optical properties of diffusing materials,” J. Opt. Soc. Am. 32, 61–70 (1942).
    [CrossRef]
  9. A. Brockes, “Der einfluss glänzender Oberflächen auf Remissionsmessungen,” Farbe 9, 53–62 (1960).
  10. A. L. Lathrop, “Diffuse scattered radiation theories of Duntley and of Kubelka–Munk,” J. Opt. Soc. Am. 55, 1097–1104 (1965).
    [CrossRef]
  11. S. E. Orchard, “Reflection and transmission of light by diffusing suspensions,” J. Opt. Soc. Am. 59, 1584–1597 (1969).
    [CrossRef]
  12. A. Reule, “Der Grenzflächeneinfluss bei Remissionsmessungen,” Optik 34, 387–405 (1971).
  13. M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
    [CrossRef]
  14. R. Graaff, “Tissue optics applied to reflectance pulse oximetry,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1993), p. 21.
  15. M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
    [CrossRef]
  16. W. E. Meador, W. R. Weaver, “Diffusion approximation for large absorption in radiative transfer,” Appl. Opt. 18, 1204–1208 (1979).
    [CrossRef] [PubMed]
  17. D. Spitzer, J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
    [CrossRef] [PubMed]
  18. P. Kubelka, “New contributions to the optics of intensely light scattering materials. II. Non-homogeneous layers,” J. Opt. Soc. Am. 44, 330–335 (1954).
    [CrossRef]
  19. J. W. Pickering, S. A. Prahl, N. van Wieringen, J. F. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “Double-integrating-sphere system for measuring the optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
    [CrossRef] [PubMed]
  20. K. Rottmann, Mathematische Formelsammlung (Bibliographisches Institut, Mannheim, Germany, 1960).
  21. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1. Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  22. T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]

1997

W. M. Johnston, M. H. Reisbick, “Color and translucency changes during and after curing of esthetic restorative materials,” Dent. Mater. 13, 89–97 (1997).
[CrossRef] [PubMed]

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

1996

K. H. Englmeier, R. Herpers, R. S. Jacoby, F. M. Zwiebel, “A method for the estimation of the hemoglobin distribution in gastroscopic images,” Int. J. Bio-Med. Comput. 41, 153–165 (1996).
[CrossRef]

1993

1992

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1991

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

1987

M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
[CrossRef]

1983

1979

1975

D. Spitzer, J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
[CrossRef] [PubMed]

1971

P. S. Mudgett, L. W. Richards, “Simple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[CrossRef] [PubMed]

A. Reule, “Der Grenzflächeneinfluss bei Remissionsmessungen,” Optik 34, 387–405 (1971).

1969

1965

1960

A. Brockes, “Der einfluss glänzender Oberflächen auf Remissionsmessungen,” Farbe 9, 53–62 (1960).

1954

1942

Beek, J. F.

Brockes, A.

A. Brockes, “Der einfluss glänzender Oberflächen auf Remissionsmessungen,” Farbe 9, 53–62 (1960).

Cheong, W. F.

M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
[CrossRef]

Duntley, S. Q.

Englmeier, K. H.

K. H. Englmeier, R. Herpers, R. S. Jacoby, F. M. Zwiebel, “A method for the estimation of the hemoglobin distribution in gastroscopic images,” Int. J. Bio-Med. Comput. 41, 153–165 (1996).
[CrossRef]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Ferwerda, H. A.

Graaff, R.

R. Graaff, “Tissue optics applied to reflectance pulse oximetry,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1993), p. 21.

Groenhuis, R. A. J.

Herpers, R.

K. H. Englmeier, R. Herpers, R. S. Jacoby, F. M. Zwiebel, “A method for the estimation of the hemoglobin distribution in gastroscopic images,” Int. J. Bio-Med. Comput. 41, 153–165 (1996).
[CrossRef]

Jacoby, R. S.

K. H. Englmeier, R. Herpers, R. S. Jacoby, F. M. Zwiebel, “A method for the estimation of the hemoglobin distribution in gastroscopic images,” Int. J. Bio-Med. Comput. 41, 153–165 (1996).
[CrossRef]

Johnston, W. M.

W. M. Johnston, M. H. Reisbick, “Color and translucency changes during and after curing of esthetic restorative materials,” Dent. Mater. 13, 89–97 (1997).
[CrossRef] [PubMed]

Kortüm, G.

G. Kortüm, Reflexionsspektroskopie, Grundlagen, Methodik, Anwendungen (Springer-Verlag, Heidelberg, 1969).

Kubelka, P.

Kuehni, R. G.

R. G. Kuehni, Color, An Introduction to Practice and Principles (Wiley, New York, 1997), Chap. 8.

Lathrop, A. L.

Meador, W. E.

Motamedi, M.

M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
[CrossRef]

Mudgett, P. S.

Niklasson, G. A.

Orchard, S. E.

Patterson, M. S.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

Pickering, J. W.

Prahl, S. A.

Reisbick, M. H.

W. M. Johnston, M. H. Reisbick, “Color and translucency changes during and after curing of esthetic restorative materials,” Dent. Mater. 13, 89–97 (1997).
[CrossRef] [PubMed]

Reule, A.

A. Reule, “Der Grenzflächeneinfluss bei Remissionsmessungen,” Optik 34, 387–405 (1971).

Richards, L. W.

Rottmann, K.

K. Rottmann, Mathematische Formelsammlung (Bibliographisches Institut, Mannheim, Germany, 1960).

Spitzer, D.

D. Spitzer, J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
[CrossRef] [PubMed]

Star, W. M.

M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
[CrossRef]

Sterenborg, H. J. C. M.

Teague, B.

H. G. Völz, B. Teague, Industrial Color Testing, Fundamentals and Techniques (VCH, Weinheim, Germany, 1994), Chap. 3.3.

Ten Bosch, J. J.

van Gemert, M. J. C.

J. W. Pickering, S. A. Prahl, N. van Wieringen, J. F. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “Double-integrating-sphere system for measuring the optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
[CrossRef] [PubMed]

M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
[CrossRef]

van Wieringen, N.

Vargas, W. E.

Völz, H. G.

H. G. Völz, B. Teague, Industrial Color Testing, Fundamentals and Techniques (VCH, Weinheim, Germany, 1994), Chap. 3.3.

Weaver, W. R.

Welch, A. J.

M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
[CrossRef]

Wilson, B.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Wilson, B. C.

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

Wyman, D. R.

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

Zwiebel, F. M.

K. H. Englmeier, R. Herpers, R. S. Jacoby, F. M. Zwiebel, “A method for the estimation of the hemoglobin distribution in gastroscopic images,” Int. J. Bio-Med. Comput. 41, 153–165 (1996).
[CrossRef]

Appl. Opt.

Calcif. Tissue Res.

D. Spitzer, J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
[CrossRef] [PubMed]

Dent. Mater.

W. M. Johnston, M. H. Reisbick, “Color and translucency changes during and after curing of esthetic restorative materials,” Dent. Mater. 13, 89–97 (1997).
[CrossRef] [PubMed]

Farbe

A. Brockes, “Der einfluss glänzender Oberflächen auf Remissionsmessungen,” Farbe 9, 53–62 (1960).

Int. J. Bio-Med. Comput.

K. H. Englmeier, R. Herpers, R. S. Jacoby, F. M. Zwiebel, “A method for the estimation of the hemoglobin distribution in gastroscopic images,” Int. J. Bio-Med. Comput. 41, 153–165 (1996).
[CrossRef]

J. Opt. Soc. Am.

Lasers Med. Sci.

M. J. C. van Gemert, A. J. Welch, W. M. Star, M. Motamedi, W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295–302 (1987).
[CrossRef]

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue. I. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

Med. Phys.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Optik

A. Reule, “Der Grenzflächeneinfluss bei Remissionsmessungen,” Optik 34, 387–405 (1971).

Other

R. G. Kuehni, Color, An Introduction to Practice and Principles (Wiley, New York, 1997), Chap. 8.

H. G. Völz, B. Teague, Industrial Color Testing, Fundamentals and Techniques (VCH, Weinheim, Germany, 1994), Chap. 3.3.

G. Kortüm, Reflexionsspektroskopie, Grundlagen, Methodik, Anwendungen (Springer-Verlag, Heidelberg, 1969).

R. Graaff, “Tissue optics applied to reflectance pulse oximetry,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1993), p. 21.

K. Rottmann, Mathematische Formelsammlung (Bibliographisches Institut, Mannheim, Germany, 1960).

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Figures (6)

Fig. 1
Fig. 1

Specular reflectances r e and r i as functions of refractive index n. The area between the curves represents the values that r i can have. The lower boundary of r i is equal to r e when the internal radiation distribution is obtained by external diffuse incident radiation (which has passed the surface of the sample). At the upper boundary are the values of r i obtained when the internal radiation distribution is diffuse.

Fig. 2
Fig. 2

Kubelka–Munk sample with specularly reflecting boundaries. (a) Ideal Kubelka–Munk sample; (b), (c) a specularly reflecting boundary is added to the top, indicated by a dashed line; (d) reflecting boundaries are added to both the top and the bottom; (e) a background is added; (f) real sample against black background; (g) real sample against white background; (h) real sample placed against an index-matched black background. Thick arrows, illuminating light; thin arrows, reflected and transmitted light. On diffuse illumination from above the reflectance and transmittance are (a) R 0, T; (b) Rabove, Tabove; (d) R0, T′; (e) R′, T′; (f) Rbl; (g) Rwh; (h) Rimb. On diffuse illumination from below the reflectance and transmittance for (c) are Rbelow and Tbelow.

Fig. 3
Fig. 3

Setup for reflectance measurement. The hemisphere has an inner diameter of 240 mm and a thickness of 30 mm. Light enters rather diffusely from below over an area of 4.4 mm × 104 mm2. The sample is placed upon a sample holder of area 1.5 mm × 103 mm. The detector is mounted upon the detector opening with area A δ = 1.4 mm × 103 mm2. Area A s of the sample is variable.

Fig. 4
Fig. 4

Light distribution inside the hemisphere. Relative irradiance as a function of the angle with respect to the normal of the sample surface. The relative irradiance was calculated by division of all measurements by the highest measured value.

Fig. 5
Fig. 5

Internal reflection coefficient r i as a function of maximal angle of incidence inside the sample.

Fig. 6
Fig. 6

Errors in S and in K as functions of (S + K)d for collimated and diffuse incidence. These errors arise when the Kubelka–Munk theory is used when its conditions have not been met.

Tables (7)

Tables Icon

Table 1 Examples of Errors in S and K from Assumption of Absence of Specular Reflection for Various Values of Thickness d and Refractive Index n

Tables Icon

Table 2 Names, Thicknesses, and Areas of the Samples Measured

Tables Icon

Table 3 Refractive Indices and Their Standard Errors for the Measured Samples at Three Wavelengths

Tables Icon

Table 4 Data for Kerr Prodigy A2

Tables Icon

Table 5 Data for Enamel plus HFO Dentina A2

Tables Icon

Table 6 Data for Enamel plus HFO Smalto Clear 1

Tables Icon

Table 7 Data for Enamel plus HFO Smalto White 1

Equations (31)

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

b=1SKK+2S, R=SS+K+KK+2S,
T=1-R2exp-bSd1-R2 exp-2bSd, R0=expSd1R-R-1expSd1R-RR-R.
S=3s1-g-a4.
ρθ=12n2-sin2 θ-cos θn2-sin2 θ+cos θ2+n2 cos θ-n2-sin2 θn2 cos θ+n2-sin2 θ2.
re=2π 0π/2dθ sin θ cos θ ρθ2π 0π/2dθ sin θ cos θ=2 0π/2dθ sin θ cos θ ρθ.
re=12+n-13n+16n+12+n2n2-12n2+13lnn-1n+1-2n3n2+2n-1n2+1n4-1+8n4n4+1n2+1n4-12lnn.
ri,max=1-1-ren2.
rns=2n20arcsin1/ndθ sin θ cos θ ρθ.
Rabove=re+1-re1-riR01-riR0,
Tabove=1-reT1-riR0.
Rbelow=R0+T2ri1-riR0,
Tbelow=1-riT1-riR0.
R0=re+1-re1-ri1-riR0R0+T2ri1-riR02-T2ri2,
T=1-re1-riT1-riR02-T2ri2.
R=R0+T2Rg1-R0Rg.
R0=1-1-re1-ririR0-re+1-re1-ririR0-re+1-re1-ri2-ri2T2ri,
T=T1-re1-ririR0-re+1-re1-ri2-ri2T2.
R=R-re.
Detsample, tot = Detsmooth outer surface + Detsample, diffuse.
Detsmooth outer surface=ρDetmirror,
Detsample, diffuse=R-reRstandardDetstandard.
Detsample,tot=ρ Detmirror+R-reRstandardDetstandard.
R=re+Detsample,tot-ρ DetmirrorRstandardDetstandard.
ri=1R0R0-Rabove-re1-re1-Rabove-re1-re.
XT1-re1-ri;
T=-1+1+4riX1-riR021/22ri2X.
YRblack-re1-riR01-re1-ri-R0,
T=Y1-riR02ri+Yri21/2.
R=1+R02-T2-1+R02-T22-4R021/22R0.
S=Rd1-R2lnR1-R0RR-R0,
K=12d1-R1+RlnR1-R0RR-R0.

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