Abstract

Measurement of absorption and reduced-scattering optical coefficients μa and μs′ is possible when a steady-state backscattered image is used on a sample surface. A new method for processing the backscattered image, acquired with a CCD, has been developed. The image is integrated to decrease sensitivity to noise. The resulting curve is defined as the integral reflectance. The curve is then fitted with a relaxation model to evaluate μa and μs′. We have validated the method with calibrated scattering and absorption phantoms. The integral reflectance method is then applied to measurements of the μa and μs′ coefficients of human skin in vivo.

© 1999 Optical Society of America

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    [CrossRef]

1997

1996

1995

1994

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]

1990

1983

1941

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

Aarnouds, J. G.

Avrillier, S.

B. Gélébart, J. M. Tualle, E. Tinet, S. Avrillier, J. P. Ollivier, “Time and space resolved reflectance from multilayered turbid media,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 40–50 (1998).
[CrossRef]

Baillargeon, G.

G. Baillargeon, J. Rainville, Statistique appliquée, Tome 2, Tests statistiques, régression et corrélations (Sciences Mathématiques Gestion, Trois Rivières, Québec, 1980).

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Blanchot, L.

L. Voisin, L. Blanchot, D. Kalinowski, J. Virmont, G. Ledanois, H. Saint-Jalmes, “2D CCD imaging in oblique incidence for noninvasive measurements of biological tissues optical coefficients,” in Photon Propagation in Tissues III, D. A. Benaron, B. Chance, M. Ferrari, eds., Proc. SPIE3194, 279–284 (1998).
[CrossRef]

Bolt, R.

R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
[CrossRef]

Dassel, A. C. M.

de Mul, F. F. M.

Dögnitz, N.

N. Dögnitz, G. Wagnières, A. Kienle, H. van den Berg, “Determination of the absorption and reduced scattering coefficients of human skin and bladder by spatial frequency domain reflectometry,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 102–109 (1997).
[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]

Feng, T. C.

Ferwerda, H. A.

Gélébart, B.

B. Gélébart, J. M. Tualle, E. Tinet, S. Avrillier, J. P. Ollivier, “Time and space resolved reflectance from multilayered turbid media,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 40–50 (1998).
[CrossRef]

Graff, R.

Greenstein, J. L.

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

Groenhius, R. A. J.

Gutsche, A.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to specify optical properties of tissue in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 211–226.

Haskell, R. C.

Henyey, L. G.

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

Hibst, R.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).

Jacques, S. L.

L. Wang, S. L. Jacques, “Use of a laser beam with an oblique incidence to measure the reduced scattering coefficient of a turbid medium,” Appl. Opt. 34, 2362–2366 (1995).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport Multilayered Tissues in Standard C, (Laser Biology Research Laboratory, University of Texas, M. D. Anderson Cancer Center, 1515 Holcombe Boulevard, Houston, Tex. 77030, 1992), e-mail: sljaser.mda.uth.tmc.edu.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to specify optical properties of tissue in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 211–226.

Kalinowski, D.

L. Voisin, L. Blanchot, D. Kalinowski, J. Virmont, G. Ledanois, H. Saint-Jalmes, “2D CCD imaging in oblique incidence for noninvasive measurements of biological tissues optical coefficients,” in Photon Propagation in Tissues III, D. A. Benaron, B. Chance, M. Ferrari, eds., Proc. SPIE3194, 279–284 (1998).
[CrossRef]

Kienle, A.

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for the reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).

N. Dögnitz, G. Wagnières, A. Kienle, H. van den Berg, “Determination of the absorption and reduced scattering coefficients of human skin and bladder by spatial frequency domain reflectometry,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 102–109 (1997).
[CrossRef]

Koelink, M. H.

Ledanois, G.

L. Voisin, L. Blanchot, D. Kalinowski, J. Virmont, G. Ledanois, H. Saint-Jalmes, “2D CCD imaging in oblique incidence for noninvasive measurements of biological tissues optical coefficients,” in Photon Propagation in Tissues III, D. A. Benaron, B. Chance, M. Ferrari, eds., Proc. SPIE3194, 279–284 (1998).
[CrossRef]

Lilge, L.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).

McAdams, M.

Ollivier, J. P.

B. Gélébart, J. M. Tualle, E. Tinet, S. Avrillier, J. P. Ollivier, “Time and space resolved reflectance from multilayered turbid media,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 40–50 (1998).
[CrossRef]

Patterson, M. S.

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for the reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

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]

A. Kienle, L. Lilge, M. S. Patterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).

Rainville, J.

G. Baillargeon, J. Rainville, Statistique appliquée, Tome 2, Tests statistiques, régression et corrélations (Sciences Mathématiques Gestion, Trois Rivières, Québec, 1980).

Saint-Jalmes, H.

L. Voisin, L. Blanchot, D. Kalinowski, J. Virmont, G. Ledanois, H. Saint-Jalmes, “2D CCD imaging in oblique incidence for noninvasive measurements of biological tissues optical coefficients,” in Photon Propagation in Tissues III, D. A. Benaron, B. Chance, M. Ferrari, eds., Proc. SPIE3194, 279–284 (1998).
[CrossRef]

Schmitt, J. M.

Schwartz, J.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to specify optical properties of tissue in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 211–226.

Steiner, R.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).

Svaasand, L. O.

ten Bosch, J.

R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
[CrossRef]

Ten Bosch, J. J.

Tinet, E.

B. Gélébart, J. M. Tualle, E. Tinet, S. Avrillier, J. P. Ollivier, “Time and space resolved reflectance from multilayered turbid media,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 40–50 (1998).
[CrossRef]

Tittel, F. K.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to specify optical properties of tissue in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 211–226.

Tromberg, B. J.

Tsay, T. T.

Tualle, J. M.

B. Gélébart, J. M. Tualle, E. Tinet, S. Avrillier, J. P. Ollivier, “Time and space resolved reflectance from multilayered turbid media,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 40–50 (1998).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van den Berg, H.

N. Dögnitz, G. Wagnières, A. Kienle, H. van den Berg, “Determination of the absorption and reduced scattering coefficients of human skin and bladder by spatial frequency domain reflectometry,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 102–109 (1997).
[CrossRef]

Van der Zee, P.

P. Van der Zee, “Methods for measuring the optical properties of tissue samples in the visible and near infrared wavelength range,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 166–192.

Virmont, J.

L. Voisin, L. Blanchot, D. Kalinowski, J. Virmont, G. Ledanois, H. Saint-Jalmes, “2D CCD imaging in oblique incidence for noninvasive measurements of biological tissues optical coefficients,” in Photon Propagation in Tissues III, D. A. Benaron, B. Chance, M. Ferrari, eds., Proc. SPIE3194, 279–284 (1998).
[CrossRef]

Voisin, L.

L. Voisin, L. Blanchot, D. Kalinowski, J. Virmont, G. Ledanois, H. Saint-Jalmes, “2D CCD imaging in oblique incidence for noninvasive measurements of biological tissues optical coefficients,” in Photon Propagation in Tissues III, D. A. Benaron, B. Chance, M. Ferrari, eds., Proc. SPIE3194, 279–284 (1998).
[CrossRef]

Wagnières, G.

N. Dögnitz, G. Wagnières, A. Kienle, H. van den Berg, “Determination of the absorption and reduced scattering coefficients of human skin and bladder by spatial frequency domain reflectometry,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 102–109 (1997).
[CrossRef]

Walker, E. C.

Wall, R. T.

Wang, L.

L. Wang, S. L. Jacques, “Use of a laser beam with an oblique incidence to measure the reduced scattering coefficient of a turbid medium,” Appl. Opt. 34, 2362–2366 (1995).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport Multilayered Tissues in Standard C, (Laser Biology Research Laboratory, University of Texas, M. D. Anderson Cancer Center, 1515 Holcombe Boulevard, Houston, Tex. 77030, 1992), e-mail: sljaser.mda.uth.tmc.edu.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to specify optical properties of tissue in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 211–226.

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.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).

Zhou, G. X.

Zijlstra, W. G.

Appl. Opt.

Astrophys. J.

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

J. Opt. Soc. Am. A

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]

Waves Random Media

R. Bolt, J. ten Bosch, “On determination of optical parameters for turbid materials,” Waves Random Media 4, 233–242 (1994).
[CrossRef]

Other

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to specify optical properties of tissue in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 211–226.

P. Van der Zee, “Methods for measuring the optical properties of tissue samples in the visible and near infrared wavelength range,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. Van der Zee, eds., Vol. ISII of SPIE Institute Series (SPIE, Bellingham, Wash., 1993), pp. 166–192.

B. Gélébart, J. M. Tualle, E. Tinet, S. Avrillier, J. P. Ollivier, “Time and space resolved reflectance from multilayered turbid media,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 40–50 (1998).
[CrossRef]

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport Multilayered Tissues in Standard C, (Laser Biology Research Laboratory, University of Texas, M. D. Anderson Cancer Center, 1515 Holcombe Boulevard, Houston, Tex. 77030, 1992), e-mail: sljaser.mda.uth.tmc.edu.

A. Kienle, L. Lilge, M. S. Patterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).

N. Dögnitz, G. Wagnières, A. Kienle, H. van den Berg, “Determination of the absorption and reduced scattering coefficients of human skin and bladder by spatial frequency domain reflectometry,” in Laser–Tissue Interaction, Tissue Optics, and Laser Welding, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, L. O. Svaasand, eds., Proc. SPIE3195, 102–109 (1997).
[CrossRef]

L. Voisin, L. Blanchot, D. Kalinowski, J. Virmont, G. Ledanois, H. Saint-Jalmes, “2D CCD imaging in oblique incidence for noninvasive measurements of biological tissues optical coefficients,” in Photon Propagation in Tissues III, D. A. Benaron, B. Chance, M. Ferrari, eds., Proc. SPIE3194, 279–284 (1998).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

G. Baillargeon, J. Rainville, Statistique appliquée, Tome 2, Tests statistiques, régression et corrélations (Sciences Mathématiques Gestion, Trois Rivières, Québec, 1980).

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Figures (23)

Fig. 1
Fig. 1

Zoom on a backscattered experimental image made on human skin (25° incidence). The maximum image cannot be shown plainly.

Fig. 2
Fig. 2

Geometry definition of the coordinates system: {O, x, y}, plane defining the sample surface; O, cylindrical coordinates system; {r, ψ}, incidence point of the collimated light beam; {O, z}, axis perpendicular to the sample; θ, angle between the {O, z} axis and the incident light direction.

Fig. 3
Fig. 3

Comparisons of the radius r of the dynamic range of the integral reflectance R int(r), the image intensity profile R(r, ψ), and the averaged profile (r). The number of points on the curves is N = 300. The experimental data are given in arbitrary units (camera gray level). R(r, ψ) covers 3 decades, but in the last decade the pixel value can be only 0 or 1. (r) covers 4.5 decades, but for a low-signal pixel the SNR is still lower than 1. The R int(r) curve allows one to see the whole signal with a very good SNR. The left-hand scale corresponds to R(r, 0) and (r). The right-hand scale corresponds to Rint(r). R(r, 0) and (r) are in cm-2. Rint(r) has no dimensions because it is the integration of R(r) across a surface.

Fig. 4
Fig. 4

Simulation-averaged reflectance profile (r) (dotted curve) compared with the analytical profile (solid curve) for n = 1, μ a = 0.1 cm-1, μ s ′ = 10 cm-1, g = 0.9. The misfit between the model and the simulation curve exists until all photons have scattered at least once, i.e., for r < 1/µ s ′ = 0.1 cm. R(r) is in cm-2.

Fig. 5
Fig. 5

Simulation-averaged reflectance profile (r) (cm-2) (solid curve) compared with analytical profiles (dashed and dotted curves) for n = 1.4, μ a = 0.1 cm-1, μ s ′ = 10 cm-1, and g = 0.9. The misfit between Farrell’s model and the simulation curve spreads to r > 1/µ s ′ = 0.1 cm. The Kienle model is more appropriate.

Fig. 6
Fig. 6

Simulation-averaged reflectance profile (r) (cm-2) (solid curve) compared with analytical profiles (dashed and dotted curves) for n = 1.4, μ a = 0.1 cm-1, μ s ′ = 10 cm-1, g = 0.9 for a range other than that in Fig. 5. The misfit between Farrell’s model and the simulation curve disappears remotely from the incidence point.

Fig. 7
Fig. 7

R int(r) as given by Eq. (4) for n = 1, μ s ′ = 70 cm-1, and μ a = 1 cm-1. For n = 1 the relative contributions of the buried source g(r, z 0) and the external source g(r, z 0′) [Eq. (5)] are well adapted for evaluating the entire integral reflectance. The slight misfit between the model and the simulation is due to the inadequacy of Farrell’s model near the incidence point. The legends for the curves are indicated in the inset. R int(r) is dimensionless.

Fig. 8
Fig. 8

R int(r) given by Eq. (4) for n = 1.55, μ s ′ = 70 cm-1, and μ a = 1 cm-1. For n = 1.55 the buried source contribution g(r, z 0) [Eq. (5)] has not changed. The weight given to the external source g(r, z 0′) [Eq. (5)] is too important to ensure precise measurements of μ a and μ s ′. R int(r) is dimensionless.

Fig. 9
Fig. 9

Model of k and z. The k function of 1/δ is for μ a s ′ < 0.2. The dashed curve is from Eq. (8). The crosses are measured data from simulation results for which μ a and μ s ′ are known. The data have been measured by fitting R int(r) curves of simulation images with Eq. (7).

Fig. 10
Fig. 10

Model of k and z. The z function of 1/(2μ a + μ s ′/n) for a for μ a s ′ < 0.2. The dashed curve is from Eq. (9). The crosses are measured data from simulation results for which μ a and μ s ′ are known. The data were measured by fitting simulation images R int(r) curves with Eq. (7).

Fig. 11
Fig. 11

Plain dependence of b on μeff-1, which cannot be found. b has been measured by fitting the R int(r) curves of simulation images with Eq. (10).

Fig. 12
Fig. 12

Determination of a with respect to μ a and μ s ′. a is a function of μ s ′/μ a . For all curves the fits show correlation coefficients equal to 0.999; a is from fitting simulation R int(r) with Eq. (10). The lines are the best fit of a a s ′) for the refractive index ranging from 1 to 1.55.

Fig. 13
Fig. 13

Determination of b × μ s ′ with respect to μ a and μ s ′. b × μ s ′ is a decreasing function of μ s ′/μ a . For all curves the fits show correlation coefficients equal to 0.999. b is from fitting simulation R int(r) with Eq. (10). The lines are the best fit of b a s ′) for a refractive index ranging from 1 to 1.55.

Fig. 14
Fig. 14

Experimental setup. The source collimated beam lights the sample surface, and the backscattered intensity is acquired by the camera.

Fig. 15
Fig. 15

Experimental integral reflectance from measured backscattered images of white thick paper and concentrated suspended latex beads. For clarity the 200 points of the curves are not represented but are replaced with lines.

Fig. 16
Fig. 16

Example of μ s ′ = 70 cm-1, μ a = 1 cm-1 of the performances of the models. Farrell’s integrated formula gives very imprecise measurements for n 2 > 1.33. The k and z model shows good results for n 2 > 1.2, but its performances are scattered. The a and b relaxation model is well designed to cover the whole range of refractive indices.

Fig. 17
Fig. 17

Comparison of the performances of Farrell’s integrated model and the a and b model for simulation results in which n = 1, μ s ′ = 30 cm-1. The y axis is the error (%) in the measurement of μ a with Farrell’s integrated model (dashed curve) and with the a and b model (solid curve).

Fig. 18
Fig. 18

Comparison of the performances of Farrell’s integrated model and the a and b model for simulation results in which n = 1, μ s ′ = 30 cm-1. The y axis is the error (%) in the measurement of μ s ′ with Farrell’s integrated model (dashed curve) and with the a and b model (solid curve).

Fig. 19
Fig. 19

Experimental integral reflectance obtained with suspended latex beads. The right scale is for R int(r) and its fit a[1 -exp(-r/b)]. The difference between the experimental integral reflectance curve and its fit is also plotted (on the left scale).

Fig. 20
Fig. 20

Experimental results. μ s ′ has been measured for 17 different suspended latex beads. Horizontal error bars represent μ s ′ uncertainty due to dispersion of the latex beads’s diameter. The vertical error bars are uncertainty in the repeatability of the measurements. The correlation coefficient is 0.993. The method allows measurement of the scattering coefficient μ s ′ with a confidence level of better than 99%. The dashed line is the best fit for the data.

Fig. 21
Fig. 21

Accuracy of the method with respect to the incidence angle (simulation results for which μ a = 1 cm-1, μ s ′ = 70 cm-1, n = 1.47, g = 0.9). Symbols are for the data. They are drawn to simulate the oblique incidence fit of the a and b model. The curves are the best fits for data with a cosine function.

Fig. 22
Fig. 22

R int(r) curves for two different camera lenses. One lens provides a 14-µm pixel resolution (thick solid curve), the other a 60-µm resolution. The two curves overlap. Lines join the experimental data points of R int(r).

Fig. 23
Fig. 23

The a with respect to the anisotropy coefficient g on the left scale, b with respect to the anisotropy coefficient g on the right scale. The lines are the best data fit. Symbols are simulation measurements of a or b made with fitting simulation R int(r) curves with a constant μ s ′ and g ranging from 0.1 to 0.9: μ s ′ = 70 cm-1, μ a = 1 cm-1, n = 1.

Tables (4)

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Table 1 a and b Model-Parameter Relation with Respect to na

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Table 2 Description of Solutions of Latex Beads Mixed with India Ink

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Table 3 Results from Suspensions Defined in Table 2

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Table 4 Comparison of our Results with Those Published by Other Teams

Equations (15)

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R¯r= Rr, ψdψ/2π,
R intr=0r02π Rρ, ψdψdρ=0r Rρ2πρdρ.
RFarrellr=z04π1ρ1+μeffexp-μeffρ1ρ12+z04π×1ρ2+μeffexp-μeffρ2ρ22,
RKienler=0.118 14πDexp-μeffρ1ρ1-exp-μeffρ2ρ2+0.306RFarrellr.
R intr=0r Rρ2πρdρ=gr, z0+gr, z0,
gr, z=exp-μeffz-zr2+z2-0.5×exp-μeffr2+z20.5.
Rr=zk+r2+z2-0.5exp-kr2+z20.5r2+z2cm-2.
R intr=exp-kz1-z×r2+z2-0.5×exp-kr2+z20.5.
k  μeff,
z  2μa+μs/n-1.
R intr=a1-exp-r/b.
a=a1 exp-a21+μs/μa-0.5,
b=b1+b2 lnμa/μsμs-1.
t=ρ*-ρ1-ρ*2-0.5N-20.5,
tα/2;N-2=t0.005;15=2.95.

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