Abstract

An inverse algorithm is presented for tomographically imaging morphologic characteristics of nonspherical particles in heterogeneous turbid media. The particles are assumed to have spheroidal shapes with random orientations. The inverse algorithm is based on a relationship of the particle scattering spectra, obtained from multispectral diffuse optical tomography, and the size, concentration, and aspect ratio of spheroidal particles through the T-matrix method. The algorithm is implemented based on Tikhonov–Marquardt regularization techniques that minimize the difference between the observed and calculated scattering spectra. Different statistical models are assumed for the suspended nonspherical particles and the performance of the inverse algorithm is tested using noise-corrupted data up to 50% noise added to the observed scattering spectra.

© 2011 Optical Society of America

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  1. C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
    [CrossRef] [PubMed]
  2. C. Li and H. Jiang, “Measurement of particle size distribution and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Appl. Opt. 44, 1838–1844 (2005).
    [CrossRef] [PubMed]
  3. C. Li and H. Jiang, “Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Opt. Express 12, 6313–6318 (2004).
    [CrossRef] [PubMed]
  4. A. Wax and V. Backman, Biomedical Applications of Light Scattering (McGraw-Hill, 2009).
  5. H. Jiang, Diffuse Optical Tomography: Principles and Applications (CRC Press, 2010).
    [CrossRef]
  6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).
  7. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).
  8. X. Li, Z. Chen, A. Taflove, and V. Backman, “Equiphase-sphere approximation for analysis of light scattering by arbitrarily shaped nonspherical particles,” Appl. Opt. 43, 4497–4505(2004).
    [CrossRef] [PubMed]
  9. D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectrosc. Radiat. Transfer 102, 85–110(2006).
    [CrossRef]
  10. M. Mishchenko, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
    [CrossRef]
  11. M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
    [CrossRef]
  12. A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-matrix computations of light scattering by red blood cells,” Appl. Opt. 37, 2735–2748 (1998).
    [CrossRef]
  13. K. Si, W. Gong, and C. J. R. Sheppard, “Model for light scattering in biological tissue and cells based on random rough nonspherical particles,” Appl. Opt. 48, 1153–1157 (2009).
    [CrossRef]
  14. M. A. Velazco-Roa, E. Dzhongova, and S. N. Thennadil, “Complex refractive index of nonspherical particles in the visible near infrared region—application to Bacillus subtilis spores,” Appl. Opt. 47, 6183–6189 (2008).
    [CrossRef] [PubMed]
  15. A. M. K. Enejder, J. Swartling, P. Aruna, and S. Andersson-Engels, “Influence of cell shape and aggregate formation on the optical properties of flowing whole blood,” Appl. Opt. 42, 1384–1394 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  18. C. Amoozegar, M. G. Giacomelli, J. D. Keener, K. J. Chalut, and A. Wax, “Experimental verification of T-matrix-based inverse light scattering analysis for assessing structure of spheroids as models of cell nuclei,” Appl. Opt. 48, D20–D25 (2009).
    [CrossRef] [PubMed]
  19. K. J. Chalut, M. G. Giacomelli, and A. Wax, “Application of Mie theory to assess structure of spheroidal scattering in backscattering geometries,” J. Opt. Soc. Am. A 25, 1866–1874 (2008).
    [CrossRef]
  20. J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 2006).
  21. A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interf. Sci. 240, 78–82 (2001).
    [CrossRef]
  22. W. Yip and X. Li, “Multiple scattering effects on optical characterization of biological tissue using spectroscopic scattering parameters,” Opt. Lett. 33, 2877–2879 (2008).
    [CrossRef] [PubMed]

2010 (3)

H. Jiang, Diffuse Optical Tomography: Principles and Applications (CRC Press, 2010).
[CrossRef]

M. Giacomelli, Y. Zhu, J. Lee, and A. Wax, “Size and shape determination of spheroidal scatterers using two-dimensional angle resolved scattering,” Opt. Express 18, 14616–14626(2010).
[CrossRef] [PubMed]

M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
[CrossRef]

2009 (3)

2008 (4)

2006 (2)

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 2006).

D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectrosc. Radiat. Transfer 102, 85–110(2006).
[CrossRef]

2005 (1)

2004 (2)

2003 (1)

2002 (1)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

2001 (1)

A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interf. Sci. 240, 78–82 (2001).
[CrossRef]

1999 (1)

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

1998 (3)

M. Mishchenko, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[CrossRef]

A. Quirantes and A. Delgado, “Experimental size determination of spheroidal particles via the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 60, 463–474 (1998).
[CrossRef]

A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-matrix computations of light scattering by red blood cells,” Appl. Opt. 37, 2735–2748 (1998).
[CrossRef]

Alsholm, P.

Amoozegar, C.

Andersson-Engels, S.

Arroyo, F.

A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interf. Sci. 240, 78–82 (2001).
[CrossRef]

Aruna, P.

Backman, V.

Chalut, K. J.

Chen, L.

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

Chen, Z.

Delgado, A.

A. Quirantes and A. Delgado, “Experimental size determination of spheroidal particles via the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 60, 463–474 (1998).
[CrossRef]

Dzhongova, E.

Enejder, A. M. K.

Fajardo, L. L.

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

Giacomelli, M.

Giacomelli, M. G.

Gong, W.

Grobmyer, S. R.

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

Jiang, H.

H. Jiang, Diffuse Optical Tomography: Principles and Applications (CRC Press, 2010).
[CrossRef]

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

C. Li and H. Jiang, “Measurement of particle size distribution and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Appl. Opt. 44, 1838–1844 (2005).
[CrossRef] [PubMed]

C. Li and H. Jiang, “Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Opt. Express 12, 6313–6318 (2004).
[CrossRef] [PubMed]

Karlsson, A.

Keener, J. D.

Khlebtsov, N. G.

M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

Lee, J.

Li, C.

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

C. Li and H. Jiang, “Measurement of particle size distribution and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Appl. Opt. 44, 1838–1844 (2005).
[CrossRef] [PubMed]

C. Li and H. Jiang, “Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Opt. Express 12, 6313–6318 (2004).
[CrossRef] [PubMed]

Li, X.

Liang, X.

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

Massol, N.

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

Mishchenko, M.

M. Mishchenko, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

Nilsson, A. M. K.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 2006).

Petrov, D.

D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectrosc. Radiat. Transfer 102, 85–110(2006).
[CrossRef]

Quirantes, A.

A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interf. Sci. 240, 78–82 (2001).
[CrossRef]

A. Quirantes and A. Delgado, “Experimental size determination of spheroidal particles via the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 60, 463–474 (1998).
[CrossRef]

Quirantes-Ros, J.

A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interf. Sci. 240, 78–82 (2001).
[CrossRef]

Sheppard, C. J. R.

Shkuratov, Y.

D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectrosc. Radiat. Transfer 102, 85–110(2006).
[CrossRef]

Si, K.

Swartling, J.

Synelnyk, E.

D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectrosc. Radiat. Transfer 102, 85–110(2006).
[CrossRef]

Taflove, A.

Thennadil, S. N.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

Velazco-Roa, M. A.

Videen, G.

M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
[CrossRef]

D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectrosc. Radiat. Transfer 102, 85–110(2006).
[CrossRef]

Wax, A.

Wriedt, T.

M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
[CrossRef]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 2006).

Yip, W.

Zakharova, N. T.

M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
[CrossRef]

Zhang, Q.

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

Zhu, Y.

Appl. Opt. (7)

C. Li and H. Jiang, “Measurement of particle size distribution and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Appl. Opt. 44, 1838–1844 (2005).
[CrossRef] [PubMed]

X. Li, Z. Chen, A. Taflove, and V. Backman, “Equiphase-sphere approximation for analysis of light scattering by arbitrarily shaped nonspherical particles,” Appl. Opt. 43, 4497–4505(2004).
[CrossRef] [PubMed]

A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-matrix computations of light scattering by red blood cells,” Appl. Opt. 37, 2735–2748 (1998).
[CrossRef]

K. Si, W. Gong, and C. J. R. Sheppard, “Model for light scattering in biological tissue and cells based on random rough nonspherical particles,” Appl. Opt. 48, 1153–1157 (2009).
[CrossRef]

M. A. Velazco-Roa, E. Dzhongova, and S. N. Thennadil, “Complex refractive index of nonspherical particles in the visible near infrared region—application to Bacillus subtilis spores,” Appl. Opt. 47, 6183–6189 (2008).
[CrossRef] [PubMed]

A. M. K. Enejder, J. Swartling, P. Aruna, and S. Andersson-Engels, “Influence of cell shape and aggregate formation on the optical properties of flowing whole blood,” Appl. Opt. 42, 1384–1394 (2003).
[CrossRef] [PubMed]

C. Amoozegar, M. G. Giacomelli, J. D. Keener, K. J. Chalut, and A. Wax, “Experimental verification of T-matrix-based inverse light scattering analysis for assessing structure of spheroids as models of cell nuclei,” Appl. Opt. 48, D20–D25 (2009).
[CrossRef] [PubMed]

J. Colloid Interf. Sci. (1)

A. Quirantes, F. Arroyo, and J. Quirantes-Ros, “Multiple light scattering by spherical particle systems and its dependence on concentration: a T-matrix study,” J. Colloid Interf. Sci. 240, 78–82 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (4)

A. Quirantes and A. Delgado, “Experimental size determination of spheroidal particles via the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 60, 463–474 (1998).
[CrossRef]

D. Petrov, E. Synelnyk, Y. Shkuratov, and G. Videen, “The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size,” J. Quant. Spectrosc. Radiat. Transfer 102, 85–110(2006).
[CrossRef]

M. Mishchenko, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998).
[CrossRef]

M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer 111, 650–658 (2010).
[CrossRef]

Med. Phys. (1)

C. Li, S. R. Grobmyer, N. Massol, X. Liang, Q. Zhang, L. Chen, L. L. Fajardo, and H. Jiang, “Noninvasive in vivo tomographic optical imaging of cellular morphology in the breast: possible convergence of microscopic pathology and macroscopic radiology,” Med. Phys. 35, 2493–2501 (2008).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Other (5)

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 2006).

A. Wax and V. Backman, Biomedical Applications of Light Scattering (McGraw-Hill, 2009).

H. Jiang, Diffuse Optical Tomography: Principles and Applications (CRC Press, 2010).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

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Figures (9)

Fig. 1
Fig. 1

Randomly oriented spheroidal particles with different sizes and aspect ratios.

Fig. 2
Fig. 2

(a) Averaged reconstruction error versus the iteration number. (b) The mesh used in calculation of averaged reconstruction error.

Fig. 3
Fig. 3

Reconstructed two-target images of (a, b) particle size, (c, d) concentration, and (e, f) aspect ratio without noise added.

Fig. 4
Fig. 4

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 20% noise added.

Fig. 5
Fig. 5

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 50% noise added.

Fig. 6
Fig. 6

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 20% noise added.

Fig. 7
Fig. 7

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 50% noise added.

Fig. 8
Fig. 8

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 20% noise added for a log-normal distribution.

Fig. 9
Fig. 9

Reconstructed two-target images of (a), (b) particle size, (c), (d) concentration, and (e), (f) aspect ratio with 50% noise added for a log-normal distribution.

Tables (1)

Tables Icon

Table 1 Reduced Scattering Coefficient at Nine Wavelengths of Interest ( mm 1 )

Equations (22)

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E ¯ inc ( R ¯ ) = n = 1 m = n n [ a m n Rg M ¯ m n ( k 1 R ¯ ) + b m n Rg N ¯ m n ( k 1 R ¯ ) ] ,
E ¯ scat ( R ¯ ) = n = 1 m = n n [ p m n M ¯ m n ( k 1 R ¯ ) + q m n N ¯ m n ( k 1 R ¯ ) ] ,
[ p q ] = T [ a b ] = [ T 11 T 12 T 21 T 22 ] [ a b ] .
C sca = 1 k 1 2 | E 0 inc | 2 n = 1 m = n n [ | p m n | 2 + | q m n | 2 ] ,
C sca = 2 π k 1 2 n = 1 m = n n n = 1 m = n n ( | T m n m n 11 | 2 + | T m n m n 12 | 2 + | T m n m n 21 | 2 + | T m n m n 22 | 2 ) ,
g = cos θ = a 1 1 3 ,
C sca ( r ) = ( 1 g ) C sca ( r ) ,
F ˜ 11 ( θ ) = S = 0 S max a 1 s P 00 s ( cos θ ) .
C sca = r = r min r = r max ε = ε min ε = ε max d r d ε f ( r , ε ) C sca ( r , ε ) ,
g = 1 3 C sca r = r min r = r max ε = ε min ε = ε max d r d ε f ( r , ε ) C sca ( r , ε ) α 1 1 ( r , ε ) ,
r = r min r = r max ε = ε min ε = ε max d r d ε f ( r , ε ) = 1.
μ s ( λ ) = ϕ r = r min r = r max ε = ε min ε = ε max d r d ε f ( r , ε ) ( 1 α 1 1 ( r , ε ) 3 ) C sca ( r , ε ) .
χ 2 = λ j ( μ s ( λ j ) o μ s ( λ j ) c ) 2 ,
μ s ( λ j ) c = ϕ r = r min r = r max ε = ε min ε = ε max e 1 2 ( ( r r m ) 2 δ m 2 + ( ε ε m ) 2 δ ε 2 ) 2 π δ r δ ε ( 1 α 1 1 ( r , ε ) 3 ) C sca ( r , ε ) d r d ε ,
f ( r , ε ) = 1 2 π δ r δ ε e 1 2 [ ( r r m ) 2 δ m 2 + ( ε ε m ) 2 δ ε 2 ] ,
Δ χ = [ μ s ( λ 1 ) o μ s ( λ 1 ) c μ s ( λ 2 ) o μ s ( λ 2 ) c · ] ,
Δ ζ = [ Δ ϕ Δ δ r Δ r m Δ δ ε Δ ε m ] .
J T Δ χ = J T J Δ ζ ,
J = [ ϕ μ s ( λ 1 ) δ r μ s ( λ 1 ) r m μ s ( λ 1 ) δ ε μ s ( λ 1 ) ε m μ s ( λ 1 ) ϕ μ s ( λ 2 ) δ r μ s ( λ 2 ) r m μ s ( λ 2 ) δ ε μ s ( λ 2 ) ε m μ s ( λ 2 ) · · · · · · · · · · · · · · · ] .
( J T J + α I ) Δ ζ = J T Δ χ ,
ζ new = ζ old + Δ ζ .
μ s ( λ j ) c = ϕ r = r min r = r max ε = ε min ε = ε max e 1 2 [ ( log ( r ) r m ) 2 δ m 2 + ( log ( ε ) ε m ) 2 δ ε 2 ] 2 π r ε δ r δ ε ( 1 α 1 1 ( r , ε ) 3 ) C sca ( r , ε ) d r d ε .

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