Abstract

Optical characterization of biological tissues is of real interest to improve medical diagnosis, in particular in the detection of precancerous tissues. We propose a new, noninvasive method allowing the estimation of the anisotropy factor. This method is based on the image analysis of the Q element of the Stokes vector backscattered from the turbid medium. These Q-element images show specific patterns depending on g. Therefore the use of Fourier descriptors (FDs) on simulated data to discriminate the specific geometrical features of the Q element enabled us to determine a linear relation between the anisotropy factor and six FDs. This method was applied on experimental data obtained with calibrated solutions. The anisotropy factor was estimated with a maximum relative error of 13%.

© 2008 Optical Society of America

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    [CrossRef]
  3. E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
    [CrossRef] [PubMed]
  4. J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, and T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586-3593 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J Biomed. Opt. 11, 064026 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. D. J. Faber, F. van der Meer, M. C. Aalders, M. de Bruin, and T. G. van Leeuwen, “Hematocrit-dependance of the scattering coefficient of blood determined by optical coherence tomography,” in European Congress of Biomedical Optics (O. S. O. American, 2005).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. A. Folkers and H. Samet, “Content-based image retrieval using Fourier descriptors on a logo database,” in 16th International Conference on Pattern Recognition, R. Kasturi, D. Laurendeau, and C. Suen, eds. (IEEE, 2002), p. 30521.
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    [CrossRef]
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2006 (3)

E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
[CrossRef] [PubMed]

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J Biomed. Opt. 11, 064026 (2006).
[CrossRef]

N. Joshi, C. Donner, and H. W. Jensen, “Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination,” Opt. Lett. 31, 936-938 (2006).
[CrossRef] [PubMed]

2003 (1)

2001 (1)

Y. Du, X. H. Hu, M. Cariveau, X. Ma, G. W. Kalmus, and J. Q. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167-181 (2001).
[CrossRef]

1999 (1)

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

1998 (1)

1997 (2)

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, J. R. Mourant, and I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125-135 (1997).
[CrossRef] [PubMed]

1996 (1)

1995 (2)

L. Wang, S. L. Jacques, and Z. Liqiong, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

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

1994 (1)

B. Beauvoit, T. Kitai, and B. Chance, “Contribution of the mitochondrial compartment to the optical properties of rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501-2510 (1994).
[CrossRef] [PubMed]

1982 (1)

O. Bertrand, R. Queval, and H. Maître, “Shape interpolation using Fourier descriptors with application to animation graphics,” Signal Processing 4, 53-58 (1982).
[CrossRef]

1981 (1)

1972 (1)

C. T. Zahn and R. Z. Roskies, “Fourier descriptors for plane closed curves,” IEEE Trans. Comput. C-21, 269-281 (1972).
[CrossRef]

Aalders, M.

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

Aalders, M. C.

D. J. Faber, F. van der Meer, M. C. Aalders, M. de Bruin, and T. G. van Leeuwen, “Hematocrit-dependance of the scattering coefficient of blood determined by optical coherence tomography,” in European Congress of Biomedical Optics (O. S. O. American, 2005).

Abdi, H.

H. Abdi, “Bonferroni and Sidak corrections for multiple comparisons,” in Encyclopedia of Measurement and Statistics, Sage, ed. (N. J. Salkind, 2007), pp. 103-107.

Anumula, H.

A. A. Nezhuvingal, L. Yanfang, H. Anumula, and B. D. Cameron, “Mueller matrix optical imaging with application to tissue diagnostics,” in Laser-Tissue Interaction XIV, S. L. Jacques, D. D. Duncan, S. J. Kirkpatrick, and A. Kriete, eds. (SPIE, 2003), pp. 137-146.

Backman, V.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

Badizadegan, K.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

Baker, K. S.

Beauvoit, B.

B. Beauvoit, T. Kitai, and B. Chance, “Contribution of the mitochondrial compartment to the optical properties of rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501-2510 (1994).
[CrossRef] [PubMed]

Beek, J.

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

Bertrand, O.

O. Bertrand, R. Queval, and H. Maître, “Shape interpolation using Fourier descriptors with application to animation graphics,” Signal Processing 4, 53-58 (1982).
[CrossRef]

Bigio, I. J.

Blokland, P.

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

Cameron, B. D.

A. A. Nezhuvingal, L. Yanfang, H. Anumula, and B. D. Cameron, “Mueller matrix optical imaging with application to tissue diagnostics,” in Laser-Tissue Interaction XIV, S. L. Jacques, D. D. Duncan, S. J. Kirkpatrick, and A. Kriete, eds. (SPIE, 2003), pp. 137-146.

Cariveau, M.

Y. Du, X. H. Hu, M. Cariveau, X. Ma, G. W. Kalmus, and J. Q. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167-181 (2001).
[CrossRef]

Chance, B.

B. Beauvoit, T. Kitai, and B. Chance, “Contribution of the mitochondrial compartment to the optical properties of rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501-2510 (1994).
[CrossRef] [PubMed]

Dasari, R. R.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

de Bruin, M.

D. J. Faber, F. van der Meer, M. C. Aalders, M. de Bruin, and T. G. van Leeuwen, “Hematocrit-dependance of the scattering coefficient of blood determined by optical coherence tomography,” in European Congress of Biomedical Optics (O. S. O. American, 2005).

Donner, C.

Du, Y.

Y. Du, X. H. Hu, M. Cariveau, X. Ma, G. W. Kalmus, and J. Q. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167-181 (2001).
[CrossRef]

Eick, A. A.

Faber, D. J.

D. J. Faber, F. van der Meer, M. C. Aalders, M. de Bruin, and T. G. van Leeuwen, “Hematocrit-dependance of the scattering coefficient of blood determined by optical coherence tomography,” in European Congress of Biomedical Optics (O. S. O. American, 2005).

Falconet, J.

J. Falconet, R. Sablong, F. Jaillon, E. Perrin, and H. Saint-Jalmes, “Towards optical characterization of biological media: analysis of backscattered images in linearly polarized light, simulations and experiments,” in Optics and Optoelectronics, A. Kowalczyk, A. F. Fercher, and V. V. Turchin, eds. (SPIE, 2005), pp. 99-109.

Feld, M. S.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

Folkers, A.

A. Folkers and H. Samet, “Content-based image retrieval using Fourier descriptors on a logo database,” in 16th International Conference on Pattern Recognition, R. Kasturi, D. Laurendeau, and C. Suen, eds. (IEEE, 2002), p. 30521.

Freyer, J. P.

Gurjar, R.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

Hielscher, A. H.

Hu, X. H.

Y. Du, X. H. Hu, M. Cariveau, X. Ma, G. W. Kalmus, and J. Q. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167-181 (2001).
[CrossRef]

Itzkan, I.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

Jacques, S. L.

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

L. Wang, S. L. Jacques, and Z. Liqiong, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Jaillon, F.

F. Jaillon and H. Saint-Jalmes, “Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media,” Appl. Opt. 42, 3290-3296 (2003).
[CrossRef] [PubMed]

J. Falconet, R. Sablong, F. Jaillon, E. Perrin, and H. Saint-Jalmes, “Towards optical characterization of biological media: analysis of backscattered images in linearly polarized light, simulations and experiments,” in Optics and Optoelectronics, A. Kowalczyk, A. F. Fercher, and V. V. Turchin, eds. (SPIE, 2005), pp. 99-109.

Jensen, H. W.

Jiang, B.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J Biomed. Opt. 11, 064026 (2006).
[CrossRef]

Johnson, T. M.

Joshi, N.

Kalmus, G. W.

Y. Du, X. H. Hu, M. Cariveau, X. Ma, G. W. Kalmus, and J. Q. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167-181 (2001).
[CrossRef]

Kienle, A.

Kitai, T.

B. Beauvoit, T. Kitai, and B. Chance, “Contribution of the mitochondrial compartment to the optical properties of rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501-2510 (1994).
[CrossRef] [PubMed]

Liqiong, Z.

L. Wang, S. L. Jacques, and Z. Liqiong, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Lu, J. Q.

Y. Du, X. H. Hu, M. Cariveau, X. Ma, G. W. Kalmus, and J. Q. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167-181 (2001).
[CrossRef]

Ma, X.

Y. Du, X. H. Hu, M. Cariveau, X. Ma, G. W. Kalmus, and J. Q. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167-181 (2001).
[CrossRef]

Maître, H.

O. Bertrand, R. Queval, and H. Maître, “Shape interpolation using Fourier descriptors with application to animation graphics,” Signal Processing 4, 53-58 (1982).
[CrossRef]

Mourant, J. R.

Nezhuvingal, A. A.

A. A. Nezhuvingal, L. Yanfang, H. Anumula, and B. D. Cameron, “Mueller matrix optical imaging with application to tissue diagnostics,” in Laser-Tissue Interaction XIV, S. L. Jacques, D. D. Duncan, S. J. Kirkpatrick, and A. Kriete, eds. (SPIE, 2003), pp. 137-146.

Novak, J.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J Biomed. Opt. 11, 064026 (2006).
[CrossRef]

Ott, L.

Papazoglou, E. S.

E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
[CrossRef] [PubMed]

Patterson, M. S.

Perelman, L. T.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5, 1019-1026 (1999).
[CrossRef]

Perrin, E.

J. Falconet, R. Sablong, F. Jaillon, E. Perrin, and H. Saint-Jalmes, “Towards optical characterization of biological media: analysis of backscattered images in linearly polarized light, simulations and experiments,” in Optics and Optoelectronics, A. Kowalczyk, A. F. Fercher, and V. V. Turchin, eds. (SPIE, 2005), pp. 99-109.

Pickering, J.

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

Posthumus, P.

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

Pourrezaei, K.

E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
[CrossRef] [PubMed]

Queval, R.

O. Bertrand, R. Queval, and H. Maître, “Shape interpolation using Fourier descriptors with application to animation graphics,” Signal Processing 4, 53-58 (1982).
[CrossRef]

Roskies, R. Z.

C. T. Zahn and R. Z. Roskies, “Fourier descriptors for plane closed curves,” IEEE Trans. Comput. C-21, 269-281 (1972).
[CrossRef]

Sablong, R.

J. Falconet, R. Sablong, F. Jaillon, E. Perrin, and H. Saint-Jalmes, “Towards optical characterization of biological media: analysis of backscattered images in linearly polarized light, simulations and experiments,” in Optics and Optoelectronics, A. Kowalczyk, A. F. Fercher, and V. V. Turchin, eds. (SPIE, 2005), pp. 99-109.

Saint-Jalmes, H.

F. Jaillon and H. Saint-Jalmes, “Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media,” Appl. Opt. 42, 3290-3296 (2003).
[CrossRef] [PubMed]

J. Falconet, R. Sablong, F. Jaillon, E. Perrin, and H. Saint-Jalmes, “Towards optical characterization of biological media: analysis of backscattered images in linearly polarized light, simulations and experiments,” in Optics and Optoelectronics, A. Kowalczyk, A. F. Fercher, and V. V. Turchin, eds. (SPIE, 2005), pp. 99-109.

Salomatina, E.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J Biomed. Opt. 11, 064026 (2006).
[CrossRef]

Samet, H.

A. Folkers and H. Samet, “Content-based image retrieval using Fourier descriptors on a logo database,” in 16th International Conference on Pattern Recognition, R. Kasturi, D. Laurendeau, and C. Suen, eds. (IEEE, 2002), p. 30521.

Saporta, G.

G. Saporta, Probabilités, Analyse des Données et Statistique (Editions Technip, 1990).

Shen, D.

Smith, R. C.

Steiner, R.

Sterenborg, H.

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

Tyagi, S.

E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
[CrossRef] [PubMed]

van der Meer, F.

D. J. Faber, F. van der Meer, M. C. Aalders, M. de Bruin, and T. G. van Leeuwen, “Hematocrit-dependance of the scattering coefficient of blood determined by optical coherence tomography,” in European Congress of Biomedical Optics (O. S. O. American, 2005).

van Gemert, M.

J. Beek, P. Blokland, P. Posthumus, M. Aalders, J. Pickering, H. Sterenborg, and M. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255-2261 (1997).
[CrossRef] [PubMed]

van Leeuwen, T. G.

D. J. Faber, F. van der Meer, M. C. Aalders, M. de Bruin, and T. G. van Leeuwen, “Hematocrit-dependance of the scattering coefficient of blood determined by optical coherence tomography,” in European Congress of Biomedical Optics (O. S. O. American, 2005).

Voisin-Gobin, L.

L. Voisin-Gobin, “Quantification de l'interaction lumière-tissus biologiques par la mesure non invasive du coefficient d'absorption et du coefficient réduit de diffusion,” in Génie Biologique et Médical (Université Paris XII, 1999).

Wang, L.

L. Wang, S. L. Jacques, and Z. Liqiong, “Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

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

Weingarten, M. S.

E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
[CrossRef] [PubMed]

Yanfang, L.

A. A. Nezhuvingal, L. Yanfang, H. Anumula, and B. D. Cameron, “Mueller matrix optical imaging with application to tissue diagnostics,” in Laser-Tissue Interaction XIV, S. L. Jacques, D. D. Duncan, S. J. Kirkpatrick, and A. Kriete, eds. (SPIE, 2003), pp. 137-146.

Yaroslavsky, A. N.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J Biomed. Opt. 11, 064026 (2006).
[CrossRef]

Zahn, C. T.

C. T. Zahn and R. Z. Roskies, “Fourier descriptors for plane closed curves,” IEEE Trans. Comput. C-21, 269-281 (1972).
[CrossRef]

Zhu, L.

E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
[CrossRef] [PubMed]

Zubkov, L.

E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu, S. Tyagi, and K. Pourrezaei, “Optical properties of wounds: diabetic versus healthy tissue,” IEEE Trans. Biomed. Eng. 53, 1047-1055 (2006).
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Figures (12)

Fig. 1
Fig. 1

Simulated Q elements of Stokes vector of linearly polarized light backscattered by media of g = 0 to 0.9. μ s = 20 cm 1 and μ a = 0.01 cm 1 . As Q elements of Stokes vectors are obtained by subtracting two images, positive and negative intensities are present. Thus the absolute value of intensities is represented here with a logarithmic scale. The number of lobes increases with g, and the mean size of the figure decreases as g increases.

Fig. 2
Fig. 2

Correlation matrix between the simulated images of backscattered Q elements of the Stokes vector for varying values of g. The correlation coefficients are coded with the scale on the right. μ s = 20 cm 1 and μ a = 0.01 cm 1 .

Fig. 3
Fig. 3

Simulated images (logarithm of the absolute value) of linearly polarized light backscattered by media of g = 0 to 0.9 and magnitude (coded with the scale indicated on the right of the last image) of the corresponding FDs for different radii. μ s = 20 cm 1 and μ a = 0.01 cm 1 . Resolution, 35 μm × 35 μm ; images, 150   pixels × 150   pixels or 5.25 mm × 5.25 mm .

Fig. 4
Fig. 4

Correlation matrix between the FD moduli for different radii of Q elements images for g = 0.006 to 0.9301. μ s = 20 cm 1 and μ a = 0.01 cm 1 . The correlation matrix obtained with the FDs of reflectance images is far closer to the ideal case than the one of the raw reflectance image.

Fig. 5
Fig. 5

(a) f values of each FD represented with the logscale. As the profiles extracted in the images are even functions, the corresponding Fourier series are also even, and only FDs with positive indices are considered. This is why there are only indices 1 to 231. (b) f values greater than 22.2, the critical value of a Fisher–Snedecor variable F ( 2 , 27 ) with a global confidence interval of 5%. The other values appear in white (they are put equal to 0). A lot of FDs have a high f value; 121 are over the threshold of the Fisher–Snedecor variable.

Fig. 6
Fig. 6

Boxplots of some of the FD moduli (expressed as FD ( k , r ) , with r being the radius in micrometers) obtained with simulated media, selected with the analysis of variance, plotted as function of g; all are not bijective, and it is necessary to propose a finer selection.

Fig. 7
Fig. 7

Six FDs kept for the regression plotted as function of the anisotropy factor.

Fig. 8
Fig. 8

Estimated anisotropy factors g ^ versus input anisotropy factors of the simulations g.

Fig. 9
Fig. 9

Estimated anisotropy factors g ^ versus input anisotropy factors of the simulations g for six different Monte Carlo simulations. The continuous line stands for ideal case: g ^ = g .

Fig. 10
Fig. 10

Scheme of the experimental setup.

Fig. 11
Fig. 11

Experimental images of seven calibrated solutions. μ s = 20 cm 1 and μ a = μ a , water = 10 3 cm 1 [23].

Fig. 12
Fig. 12

Estimated anisotropy factors g ^ versus theoretical anisotropy factors g (obtained with the Mie theory). The mean relative error is of 4.5%, and the maximum relative error is of 13.1% with respect to theoretical value. The continuous line is ideal case: g ^ = g .

Tables (4)

Tables Icon

Table 1 Parameters of Twelve Simulated Media, Estimated Anisotropy Factor, and Relative Error a

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Table 2 Parameters of the Calibrated Solutions a

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Table 3 Theoretical and Estimated g Values of the Solutions and Mean and Maximum Relative Error with Respect to the Theoretical Values a

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Table 4 Variability of Theoretical and Evaluated g Values for Each Solution and Absolute Error between Estimates and Theoretical Values a

Equations (8)

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r A , B = i j ( A i j A ¯ ) ( B i j B ¯ ) [ ( i j ( A i j A ¯ ) 2 ) ( i j ( B i j B ¯ ) 2 ) ] 1 / 2 ,
d f ( k ) = 1 N n = 0 N 1 c ( n ) exp ( 2 π j k n N ) ,
d f r ( k ) = 1 N r n = 0 N r 1 c r ( n ) exp ( 2 π j k n N r ) ,
f = S A 2 k 1 S B 2 n k ,
g ^ = 0.874 0.179 × FD ( 3 , 490 ) 0.115 × FD ( 3 , 595 ) 0.183 × FD ( 3 , 665 ) + 0.312 × FD ( 9 , 315 ) + 0.118 × FD ( 9 , 560 ) + 0.159 × FD ( 9 , 770 ) .
e = ( g ^ g ) / g .
I = ( R p t p R n p t n p ) + ( R c t c R n c t n c ) , Q = ( R p t p R n p t n p ) ( R c t c R n c t n c ) .
g ^ = 3.35 · g 2 4.19 · g + 1.95.

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