Abstract

The polarization properties of any medium are completely described by the sixteen element Mueller matrix that relates the polarization parameters of the light incident on the medium to that emerging from it. Measurement of all the elements of the matrix requires a minimum of sixteen measurements involving both linear and circularly polarized light. However, for many diagnostic applications, it would be useful if the polarization parameters can be quantified with linear polarization measurements alone. In this paper, we present a method based on polar decomposition of Mueller matrix for quantification of the polarization parameters of a scattering medium using the nine element (3×3) Mueller matrix that requires linear polarization measurements only. The methodology for decomposition of the 3×3 Mueller matrix is based on the previously developed decomposition process for sixteen element (4×4) Mueller matrix but with an assumption that the depolarization of linearly polarized light due to scattering is independent of the orientation angle of the incident linear polarization vector. Studies conducted on various scattering samples demonstrated that this assumption is valid for a turbid medium like biological tissue where the depolarization of linearly polarized light primarily arises due to the randomization of the field vector’s direction as a result of multiple scattering. For such medium, polar decomposition of 3×3 Mueller matrix can be used to quantify the four independent polarization parameters namely, the linear retardance (δ), the circular retardance (ψ), the linear depolarization coefficient (Δ) and the linear diattenuation (d) with reasonable accuracy. Since this approach requires measurements using linear polarizers only, it considerably simplifies measurement procedure and might find useful applications in tissue diagnosis using the retrieved polarization parameters.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. V. Sankaran, J.T. Walsh, Jr., and D.J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
    [CrossRef] [PubMed]
  4. N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93-100 (2003).
    [CrossRef]
  5. N. Ghosh, H.S. Patel, and P.K. Gupta, "Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers," Opt. Express 11, 2198-2205 (2003).
    [CrossRef] [PubMed]
  6. N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [PubMed]
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2006

2005

2004

N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

E . Gar cia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

O. Kostyuk and R.A. Brown, "Novel Spectroscopic Technique for In Situ Monitoring of Collagen Fibril Alignment in Gels," Biophys. J. 87, 648-655 (2004).
[CrossRef] [PubMed]

B.L. Boulesteix, A. De Martino, B. Dre villon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystal," Appl. Opt. 43, 2824-2832 (2004).
[CrossRef]

2003

2002

G.L. Liu, Y. Li, and B.D. Cameron, "Polarization based optical imaging and processing techniques with application to the cancer diagnostics," Proceedings SPIE 4617, 208 - 220 (2002).
[CrossRef]

V. Sankaran, J.T. Walsh, Jr., and D.J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

K.C. Hadley and I.A. Vitkin, "Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racimic and achiral turbid media," J. Biomed. Opt. 7, 291-299 (2002).
[CrossRef] [PubMed]

X. Wang, G. Yao, and L.V. Yang, "Monte Carlo model and single scattering approx. Of the propagation of polarized light in turbid media containing glucose," Appl. Opt. 41, 792 - 801, (2002).
[CrossRef] [PubMed]

I. Vitkin, R.D. Laszlo, and C.L. Whyman, "Effects of molecular asymmetry of optically active molecules on the polarization properties of multiply scattered light," Opt. Express 10, 222 - 229 (2002).
[PubMed]

C.W. Sun, L.S. Lu, C.C. Yang, Y.W. Kiang, and M.J. Su, "Myocardial tissue characterization based on the time-resolved Stokes-Mueller formalism," Opt. Express 10, 1347 - 1353 (2002).
[PubMed]

2001

I.A. Vitkin and R.C.N Studinski, "Polarization preservation in diffusive scattering from in-vivo turbid biological media: Effects of tissue optical absorption in the exact backscattering direction," Opt. Commun. 190, 37-43 (2001).
[CrossRef]

M.H. Smith, "Interpreting Mueller matrix images of tissues," Proceedings SPIE 4257, 82 - 89 (2001).
[CrossRef]

A.D. Kim and M. Moscoso, "Influence of the refractive index on the depolarization of multiply scattered waves," Phys. Rev. E 64, 026612, 1-4 (2001).
[CrossRef]

2000

R.J. McNichols and G.L. Cote, "Optical glucose sensing in biological fluids: an overview," J. Biomed. Opt. 5, 5 - 16 (2000).
[CrossRef] [PubMed]

1997

B.D. Cameron and G.L. Cote, "Noninvasive glucose sensing utilizing a digital closed loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-227 (1997).
[CrossRef] [PubMed]

D.J. Maitland and J.T. WalshJr., "Quantitative measurement of linear birefringence during heating of native collagen," Lasers Surg. Med. 20, 310-318 (1997).
[CrossRef] [PubMed]

J.F. de Boer, T.E. Milner, M.J.C. van Gemert, and J.S. Nelson, "Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence Tomography," Opt. Lett. 22, 934-936 (1997).
[CrossRef] [PubMed]

1996

S. Yau Lu and R.A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996).
[CrossRef]

E. Collett and V. Gazerro, "Polarization measurements in a spectrofluorophotometer," Opt. Commun. 129, 229-236 (1996).
[CrossRef]

1994

D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, "Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter," Phys. Rev. E 49, 1767-1770 (1994).
[CrossRef]

1984

M. Wong, M.J. Hendrix, K. Von der Mark, C. Little and R. Stern, "Collagen in the egg shell membranes of the hen," Dev. Biol. 104 (1), 28-36 (1984).
[CrossRef]

Bicout, D.

D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, "Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter," Phys. Rev. E 49, 1767-1770 (1994).
[CrossRef]

Boulbry, B.

F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, "Analysis of the depolarization properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21 - 28 (2005).
[CrossRef]

Boulesteix, B.L.

Boulvert, F.

F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, "Analysis of the depolarization properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21 - 28 (2005).
[CrossRef]

Brosseau, C.

D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, "Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter," Phys. Rev. E 49, 1767-1770 (1994).
[CrossRef]

Brown, R.A.

O. Kostyuk and R.A. Brown, "Novel Spectroscopic Technique for In Situ Monitoring of Collagen Fibril Alignment in Gels," Biophys. J. 87, 648-655 (2004).
[CrossRef] [PubMed]

Buddhiwant, P.

Cameron, B.D.

G.L. Liu, Y. Li, and B.D. Cameron, "Polarization based optical imaging and processing techniques with application to the cancer diagnostics," Proceedings SPIE 4617, 208 - 220 (2002).
[CrossRef]

B.D. Cameron and G.L. Cote, "Noninvasive glucose sensing utilizing a digital closed loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-227 (1997).
[CrossRef] [PubMed]

Cariou, J.

F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, "Analysis of the depolarization properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21 - 28 (2005).
[CrossRef]

Chen, Z.

Chipman, R.A.

Collett, E.

E. Collett and V. Gazerro, "Polarization measurements in a spectrofluorophotometer," Opt. Commun. 129, 229-236 (1996).
[CrossRef]

Cote, D.

Cote, G.L.

R.J. McNichols and G.L. Cote, "Optical glucose sensing in biological fluids: an overview," J. Biomed. Opt. 5, 5 - 16 (2000).
[CrossRef] [PubMed]

B.D. Cameron and G.L. Cote, "Noninvasive glucose sensing utilizing a digital closed loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-227 (1997).
[CrossRef] [PubMed]

de Boer, J.F.

De Martino, A.

E . Gar cia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

B.L. Boulesteix, A. De Martino, B. Dre villon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystal," Appl. Opt. 43, 2824-2832 (2004).
[CrossRef]

Drevillon, B.

E . Gar cia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

Gar cia-Caurel, E

E . Gar cia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
[CrossRef]

Gazerro, V.

E. Collett and V. Gazerro, "Polarization measurements in a spectrofluorophotometer," Opt. Commun. 129, 229-236 (1996).
[CrossRef]

Ghosh, N.

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 14, 190-202 (2006).
[CrossRef] [PubMed]

N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

N. Ghosh, H.S. Patel, and P.K. Gupta, "Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers," Opt. Express 11, 2198-2205 (2003).
[CrossRef] [PubMed]

N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93-100 (2003).
[CrossRef]

Guo, S.

Gupta, P. K.

Gupta, P.K.

N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

N. Ghosh, H.S. Patel, and P.K. Gupta, "Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers," Opt. Express 11, 2198-2205 (2003).
[CrossRef] [PubMed]

N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93-100 (2003).
[CrossRef]

Gupta, S.

N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

Hadley, K.C.

K.C. Hadley and I.A. Vitkin, "Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racimic and achiral turbid media," J. Biomed. Opt. 7, 291-299 (2002).
[CrossRef] [PubMed]

Hendrix, M.J.

M. Wong, M.J. Hendrix, K. Von der Mark, C. Little and R. Stern, "Collagen in the egg shell membranes of the hen," Dev. Biol. 104 (1), 28-36 (1984).
[CrossRef]

Jain, B.

N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93-100 (2003).
[CrossRef]

Jaiswal, V.

N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

Jung, W.

Kiang, Y.W.

Kim, A.D.

A.D. Kim and M. Moscoso, "Influence of the refractive index on the depolarization of multiply scattered waves," Phys. Rev. E 64, 026612, 1-4 (2001).
[CrossRef]

Kostyuk, O.

O. Kostyuk and R.A. Brown, "Novel Spectroscopic Technique for In Situ Monitoring of Collagen Fibril Alignment in Gels," Biophys. J. 87, 648-655 (2004).
[CrossRef] [PubMed]

Laszlo, R.D.

Le Brun, G

F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, "Analysis of the depolarization properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21 - 28 (2005).
[CrossRef]

Li, Y.

G.L. Liu, Y. Li, and B.D. Cameron, "Polarization based optical imaging and processing techniques with application to the cancer diagnostics," Proceedings SPIE 4617, 208 - 220 (2002).
[CrossRef]

Little, C.

M. Wong, M.J. Hendrix, K. Von der Mark, C. Little and R. Stern, "Collagen in the egg shell membranes of the hen," Dev. Biol. 104 (1), 28-36 (1984).
[CrossRef]

Liu, G.L.

G.L. Liu, Y. Li, and B.D. Cameron, "Polarization based optical imaging and processing techniques with application to the cancer diagnostics," Proceedings SPIE 4617, 208 - 220 (2002).
[CrossRef]

Lu, L.S.

Maitland, D.J.

V. Sankaran, J.T. Walsh, Jr., and D.J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

D.J. Maitland and J.T. WalshJr., "Quantitative measurement of linear birefringence during heating of native collagen," Lasers Surg. Med. 20, 310-318 (1997).
[CrossRef] [PubMed]

Manhas, S.

Martinez, A.S.

D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, "Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter," Phys. Rev. E 49, 1767-1770 (1994).
[CrossRef]

McNichols, R.J.

R.J. McNichols and G.L. Cote, "Optical glucose sensing in biological fluids: an overview," J. Biomed. Opt. 5, 5 - 16 (2000).
[CrossRef] [PubMed]

Milner, T.E.

Moscoso, M.

A.D. Kim and M. Moscoso, "Influence of the refractive index on the depolarization of multiply scattered waves," Phys. Rev. E 64, 026612, 1-4 (2001).
[CrossRef]

Nelson, J.S.

Patel, H.S.

N. Ghosh, H.S. Patel, and P.K. Gupta, "Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers," Opt. Express 11, 2198-2205 (2003).
[CrossRef] [PubMed]

N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93-100 (2003).
[CrossRef]

Pradhan, A.

N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

Rivet, S.

F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, "Analysis of the depolarization properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21 - 28 (2005).
[CrossRef]

Sankaran, V.

V. Sankaran, J.T. Walsh, Jr., and D.J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

Schmitt, J.M.

D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, "Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter," Phys. Rev. E 49, 1767-1770 (1994).
[CrossRef]

Singh, B.N.

N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93-100 (2003).
[CrossRef]

Singh, K.

Singh, R.P.

N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

Smith, M.H.

M.H. Smith, "Interpreting Mueller matrix images of tissues," Proceedings SPIE 4257, 82 - 89 (2001).
[CrossRef]

Stern, R.

M. Wong, M.J. Hendrix, K. Von der Mark, C. Little and R. Stern, "Collagen in the egg shell membranes of the hen," Dev. Biol. 104 (1), 28-36 (1984).
[CrossRef]

Studinski, R.C.N

I.A. Vitkin and R.C.N Studinski, "Polarization preservation in diffusive scattering from in-vivo turbid biological media: Effects of tissue optical absorption in the exact backscattering direction," Opt. Commun. 190, 37-43 (2001).
[CrossRef]

Su, M.J.

Sun, C.W.

Swami, M. K.

van Gemert, M.J.C.

Vitkin, I.

Vitkin, I.A.

K.C. Hadley and I.A. Vitkin, "Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racimic and achiral turbid media," J. Biomed. Opt. 7, 291-299 (2002).
[CrossRef] [PubMed]

I.A. Vitkin and R.C.N Studinski, "Polarization preservation in diffusive scattering from in-vivo turbid biological media: Effects of tissue optical absorption in the exact backscattering direction," Opt. Commun. 190, 37-43 (2001).
[CrossRef]

Von der Mark, K.

M. Wong, M.J. Hendrix, K. Von der Mark, C. Little and R. Stern, "Collagen in the egg shell membranes of the hen," Dev. Biol. 104 (1), 28-36 (1984).
[CrossRef]

Walsh, J.T.

V. Sankaran, J.T. Walsh, Jr., and D.J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1.
Fig. 1.

A schematic of the nine element Mueller matrix measurement set-up

Fig. 2.
Fig. 2.

The values for linear retardance δ (solid line), diattenuation d (dashed line) and optical rotation ψ (dotted line) obtained from polar decomposition of single scattering Mueller matrix (δ and ψ are in radian).

Fig. 3.
Fig. 3.

The wavelength variation of linear retardance δ obtained by decomposing the measured nine element spectral Mueller matrix of a linear retarder (symbol asteryx). The theoretical variation of δ with wavelength is shown by squares.

Fig. 4.
Fig. 4.

(a) The nine elements of spectral Mueller matrix recoded from Type I collagen extracted from eggshell membrane. The wavelength variation of (b) linear depolarization [Δ (λ)], (c) linear retardance [δ(λ)] and (d) linear diattenuation [d (λ)] obtained following the polar decomposition of the spectral Mueller matrix.

Tables (2)

Tables Icon

Table 1. The measured 3×3 Mueller matrix (M) and the decomposed components for the combination of the linear retarder and the turbid scattering sample.

Tables Icon

Table 2. The measured 3×3 Mueller matrix (M) and the decomposed components for the combination of linear retarder kept in between the two turbid scattering samples.

Equations (15)

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M = M M D 1 = M Δ M R
M Δ = [ 1 0 0 0 a 0 0 0 b ]
M R = [ 1 0 0 0 cos 2 2 θ + sin 2 2 θ cos δ sin 2 θ cos 2 θ ( 1 cos δ ) 0 sin 2 θ cos 2 θ ( 1 cos δ ) sin 2 2 θ + cos 2 2 θ cos δ ] [ 1 0 0 0 cos 2 ψ sin 2 ψ 0 sin 2 ψ cos 2 ψ ]
δ = cos 1 [ { ( M 22 + M 33 ) 2 + ( M 23 M 32 ) 2 } 1 2 1 ]
Ψ = 1 2 tan 1 [ ( M 23 M 32 ) ( M 22 + M 33 ) ]
M DR = M ( M ) T
M Δ = ( 1 0 0 0 Δ 0 0 0 Δ )
M R = M Δ 1 M
PSA = [ 1 1 0 1 1 0 1 0 1 ] PSG = [ 1 1 1 1 1 0 0 0 1 ]
M i = PSA . M S . PSG
M i vec = W M S vec
W = PSA PSG T
W = [ 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 ]
M = [ 1.000 0.007 0.051 0.012 0.966 0.201 0.067 0.199 0.018 ]
M = [ 1.000 0.071 0.005 0.053 0.748 0.034 0.008 0.001 0.738 ]

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