Abstract

The Mueller matrix describes all the polarizing properties of a sample and, therefore, the optical differences between noncancerous and precancerous tissue that may be present within the matrix elements. A high-speed polarimetry system that generates 16 (4×4) full Mueller matrices to characterize tissues is presented. Feature extraction is done on the Mueller matrix elements resulting in depolarization and retardance images by polar decomposition. These are used to detect and classify early oral cancers and precancerous changes in epithelium such as dysplasia. These images are compared with orthogonal polarization images and analyzed in an attempt to identity useful factors for the differentiation between cancerous lesions and their benign counterparts. Our results indicate that polarimetry has potential as a method for the in vivo early detection and diagnosis of oral premalignancy.

© 2007 Optical Society of America

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  29. J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).
  30. R. Sridhar and R. Simon, "Normal form for Mueller matrices in polarization optics," J. Mod. Opt. 41, 1903-1915 (1994).
  31. G. L. Liu, Y. Li, and B. D. Cameron, "Polarization-based optical imaging and processing techniques with application to cancer diagnostics," in International Biomedical Optics Conference,Proc. SPIE 4617, 208-220 (2002).
    [CrossRef]
  32. M. H. Smith, A Lompado, and P. Burke, "Mueller matrix imaging polarimetry in dermatology," in Proc. SPIE 3911, 210-216 (2000).
    [CrossRef]
  33. J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, "Development and calibration of an automated Mueller matrix polarization imaging system," J. Biomed. Opt. 7, 341-349 (2002).
    [CrossRef]
  34. L. Wang and S. Jacques, "Non-invasive detection of skin cancers by measuring optical properties of tissue," in Proc. SPIE 2395, 548-558 (1995).
    [CrossRef]
  35. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elseveir, 1987).
  36. J. J. Salley, "Experimental carcinogenesis in the cheek pouch of the Syrian hamster," J. Dent. Res. 33, 253-262 (1954).
  37. A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
    [CrossRef]
  38. P. Wilder-Smith, A. Ebihara, L. H. Liaw, T. B. Krasieva, and D. Messadi, "Detection of dysplasia and malignancy in oral mucosa using autofluorescence," Lasers Surg. Med. 12, 264 (2000).

2004

A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
[CrossRef]

2002

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

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, "Development and calibration of an automated Mueller matrix polarization imaging system," J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef]

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, "Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures," J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef]

2001

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

M. Mujat and A. Dogariu, "Real-time measurement of the polarization transfer function," Appl. Opt. 40, 34-44 (2001).

2000

S. G. Demos, H. B. Radousky, and R. R. Alfano, "Deep surface imaging in tissue using spectral and polarization filtering," Opt. Express 7, 23-28 (2000).

S. L. Jacques and R. J. Ramella-Roman, "Propagation of polarized light beams through biological tissues," in Proc. SPIE 3914, 345 (2000).
[CrossRef]

J. M. Bueno, "Polarimetry using liquid-crystal variable retarders: theory and calibration," J. Opt. A 2, 216-222 (2000).

S. L. Jacques, J. R. Roman, and K. Lee, "Imaging superficial tissue with polarized light," Lasers Surg. Med. 26, 119-129 (2000).
[CrossRef]

M. H. Smith, A Lompado, and P. Burke, "Mueller matrix imaging polarimetry in dermatology," in Proc. SPIE 3911, 210-216 (2000).
[CrossRef]

P. Wilder-Smith, A. Ebihara, L. H. Liaw, T. B. Krasieva, and D. Messadi, "Detection of dysplasia and malignancy in oral mucosa using autofluorescence," Lasers Surg. Med. 12, 264 (2000).

1999

1998

1997

1996

S. L. Jacques, M. Ostemeyer, L. V. Wang, and D. Stephens, "Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers," in Proc. SPIE 2671, 199-210 (1996).
[CrossRef]

M. Dogariu and T. Asakura, "Photon path length distribution from polarized backscattering in random media," Opt. Eng. 35, 2234 (1996).
[CrossRef]

1995

L. Wang and S. Jacques, "Non-invasive detection of skin cancers by measuring optical properties of tissue," in Proc. SPIE 2395, 548-558 (1995).
[CrossRef]

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

1994

R. Sridhar and R. Simon, "Normal form for Mueller matrices in polarization optics," J. Mod. Opt. 41, 1903-1915 (1994).

1993

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

1990

W. Denk, J. H. Strickler, and W. W. Webb, "2-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef]

1987

J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).

1976

W. S. Bickel, J. F. Davidson, D. R. Huffman, and R. Kilkson, "Application of polarization effects in light scattering: a new biophysical tool," Proc. Natl. Acad. Sci. USA 73, 486-490 (1976).
[CrossRef]

1954

J. J. Salley, "Experimental carcinogenesis in the cheek pouch of the Syrian hamster," J. Dent. Res. 33, 253-262 (1954).

Aida, T.

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, "Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures," J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef]

Alfano, R. R.

Asakura, T.

M. Dogariu and T. Asakura, "Photon path length distribution from polarized backscattering in random media," Opt. Eng. 35, 2234 (1996).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elseveir, 1987).

Baba, J. S.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, "Development and calibration of an automated Mueller matrix polarization imaging system," J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elseveir, 1987).

Bernabeu, E.

J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).

Bickel, W. S.

W. S. Bickel, J. F. Davidson, D. R. Huffman, and R. Kilkson, "Application of polarization effects in light scattering: a new biophysical tool," Proc. Natl. Acad. Sci. USA 73, 486-490 (1976).
[CrossRef]

Bigio, I. J.

Bueno, J. M.

J. M. Bueno, "Polarimetry using liquid-crystal variable retarders: theory and calibration," J. Opt. A 2, 216-222 (2000).

Burke, P.

M. H. Smith, A Lompado, and P. Burke, "Mueller matrix imaging polarimetry in dermatology," in Proc. SPIE 3911, 210-216 (2000).
[CrossRef]

Cameron, B. D.

Carpenter, S.

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, "Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures," J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Chipman, R. A.

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

Chung, J. R.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, "Development and calibration of an automated Mueller matrix polarization imaging system," J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef]

Cote, G. L.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, "Development and calibration of an automated Mueller matrix polarization imaging system," J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef]

Coté, G. L.

Davidson, J. F.

W. S. Bickel, J. F. Davidson, D. R. Huffman, and R. Kilkson, "Application of polarization effects in light scattering: a new biophysical tool," Proc. Natl. Acad. Sci. USA 73, 486-490 (1976).
[CrossRef]

DeLaughter, A. H.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, "Development and calibration of an automated Mueller matrix polarization imaging system," J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef]

Delplancke, F.

Demos, S. G.

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, "2-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef]

Dogariu, A.

Dogariu, M.

M. Dogariu and T. Asakura, "Photon path length distribution from polarized backscattering in random media," Opt. Eng. 35, 2234 (1996).
[CrossRef]

Drezek, R.

Drezek, R. A.

Ebihara, A.

A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
[CrossRef]

P. Wilder-Smith, A. Ebihara, L. H. Liaw, T. B. Krasieva, and D. Messadi, "Detection of dysplasia and malignancy in oral mucosa using autofluorescence," Lasers Surg. Med. 12, 264 (2000).

Eick, A. A.

Fago, S.

A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Freyer, J. P.

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, "Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures," J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef]

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, "Diffuse backscattering Mueller matrices of highly scattering media," Opt. Express 1, 441-453 (1997).

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Gil, J. J.

J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).

Gossage, K.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Guerra, A.

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, "Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures," J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef]

Guo, Y. C.

Harris, D.

Hee, W. M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Hielscher, A. H.

Ho, P. P.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Huffman, D. R.

W. S. Bickel, J. F. Davidson, D. R. Huffman, and R. Kilkson, "Application of polarization effects in light scattering: a new biophysical tool," Proc. Natl. Acad. Sci. USA 73, 486-490 (1976).
[CrossRef]

Jacques, S.

L. Wang and S. Jacques, "Non-invasive detection of skin cancers by measuring optical properties of tissue," in Proc. SPIE 2395, 548-558 (1995).
[CrossRef]

Jacques, S. L.

S. L. Jacques, J. R. Roman, and K. Lee, "Imaging superficial tissue with polarized light," Lasers Surg. Med. 26, 119-129 (2000).
[CrossRef]

S. L. Jacques and R. J. Ramella-Roman, "Propagation of polarized light beams through biological tissues," in Proc. SPIE 3914, 345 (2000).
[CrossRef]

S. L. Jacques and K. Lee, "Imaging tissues with a polarized light video camera," in Proc. SPIE 3863, 68-74 (1999).
[CrossRef]

S. L. Jacques, M. Ostemeyer, L. V. Wang, and D. Stephens, "Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers," in Proc. SPIE 2671, 199-210 (1996).
[CrossRef]

Johnson, T. M.

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, "Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures," J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef]

Kattawar, G. W.

Kilkson, R.

W. S. Bickel, J. F. Davidson, D. R. Huffman, and R. Kilkson, "Application of polarization effects in light scattering: a new biophysical tool," Proc. Natl. Acad. Sci. USA 73, 486-490 (1976).
[CrossRef]

Krasieva, T. B.

A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
[CrossRef]

P. Wilder-Smith, A. Ebihara, L. H. Liaw, T. B. Krasieva, and D. Messadi, "Detection of dysplasia and malignancy in oral mucosa using autofluorescence," Lasers Surg. Med. 12, 264 (2000).

Kriete, Andres

Kukulies, Jorg

Lee, K.

S. L. Jacques, J. R. Roman, and K. Lee, "Imaging superficial tissue with polarized light," Lasers Surg. Med. 26, 119-129 (2000).
[CrossRef]

S. L. Jacques and K. Lee, "Imaging tissues with a polarized light video camera," in Proc. SPIE 3863, 68-74 (1999).
[CrossRef]

Li, Y.

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

Liaw, L. H.

A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
[CrossRef]

P. Wilder-Smith, A. Ebihara, L. H. Liaw, T. B. Krasieva, and D. Messadi, "Detection of dysplasia and malignancy in oral mucosa using autofluorescence," Lasers Surg. Med. 12, 264 (2000).

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Liu, F.

Liu, G. L.

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

Lompado, A

M. H. Smith, A Lompado, and P. Burke, "Mueller matrix imaging polarimetry in dermatology," in Proc. SPIE 3911, 210-216 (2000).
[CrossRef]

Masters, Barry R.

Mehrubeoglu, M.

Messadi, D.

A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
[CrossRef]

P. Wilder-Smith, A. Ebihara, L. H. Liaw, T. B. Krasieva, and D. Messadi, "Detection of dysplasia and malignancy in oral mucosa using autofluorescence," Lasers Surg. Med. 12, 264 (2000).

Mourant, J. R.

Mujat, M.

Osann, K.

A. Ebihara, T. B. Krasieva, L. H. Liaw, S. Fago, D. Messadi, K. Osann, and P. Wilder-Smith, "Detection and diagnosis of oral cancer by light-induced fluorescence," Lasers Surg. Med. 32, 17-24 (2004).
[CrossRef]

Ostemeyer, M.

S. L. Jacques, M. Ostemeyer, L. V. Wang, and D. Stephens, "Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers," in Proc. SPIE 2671, 199-210 (1996).
[CrossRef]

Pezzaniti, J. L.

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef]

Radousky, H. B.

Rakovic, M. J.

Ramella-Roman, R. J.

S. L. Jacques and R. J. Ramella-Roman, "Propagation of polarized light beams through biological tissues," in Proc. SPIE 3914, 345 (2000).
[CrossRef]

Rastegar, S.

Richards-Kortum, R.

Richards-Kortum, R. R.

Roman, J. R.

S. L. Jacques, J. R. Roman, and K. Lee, "Imaging superficial tissue with polarized light," Lasers Surg. Med. 26, 119-129 (2000).
[CrossRef]

Sacks, P.

Salley, J. J.

J. J. Salley, "Experimental carcinogenesis in the cheek pouch of the Syrian hamster," J. Dent. Res. 33, 253-262 (1954).

Savage, H.

Schantz, S.

Schuman, J. S.

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Figures (4)

Fig. 1
Fig. 1

(Color online) Mueller matrix polarimetry system.

Fig. 2
Fig. 2

(Color online) (a) In situ Microscopic image of hamster cheek pouch tissue with precancer. Dysplasia and normal region are marked by the dotted circles and the lined circles, respectively, (b) 2D depolarization image, (c) 2D retardance image. Scare bars, 1   mm . (d) Histology in the healthy tissue and (e) histology in the dysplastic region. Scale bars, 0.5 mm.

Fig. 3
Fig. 3

(a) Orthogonal polarization images, (b) depolarization images, and (c) retardance images for the M 33 element of the Mueller matrix of healthy tissues. The scale bars are 0.5 mm.

Fig. 4
Fig. 4

(Color online) (a) Orthogonal polarization images, (b) depolarization images, and (c) retardance images for the M 33 element of the Mueller matrix of precancerous tissues. The scale bars are 0.5   mm .

Tables (1)

Tables Icon

Table 1 Calculation of the 16 Full Mueller Matrix a

Equations (18)

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   S o u t = [ S 0 S 1 S 2 S 3 ] o u t = [ M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 21 M 22 M 23 M 30 M 31 M 32 M 33 ] s a m p l e [ S 0 S 1 S 2 S 3 ] i n .
I O P = I p a r I p e r I p a r + I p e r ,
M = M Δ × M R × M D .
D = m 01 2 + m 02 2 + m 03 2 m 00 .
M D = T u [ 1 D ¯ T D ¯ m D ] ,
m D = 1 D 2 I + ( 1 1 D 2 ) D ^ D ^ T ,
T u = m 00 , D ¯ = 1 m 00 [ m 01 m 02 m 03 ] .
m 1 m 00 [ m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ] ,
M = m 00 [ 1 D ¯ T P ¯ m ] ,
M M M D 1 .
M = M Δ M R = [ 1 0 ¯ T P ¯ Δ m Δ ] [ 1 0 ¯ T 0 ¯ m R ] = [ 1 0 ¯ T P ¯ Δ m Δ m R ] = [ 1 0 ¯ T P ¯ Δ m ] .
P ¯ Δ = P ¯ m D ¯ 1 D 2 ,
m = m Δ m R .
m Δ 2 = m ( m ) T .
m Δ = ± { m ( m ) T + ( λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 ) I } 1 × { ( λ 1 + λ 2 + λ 3 ) m ( m ) T + λ 1 λ 2 λ 3 I } .
m R = m Δ 1 m .
Δ = 1 | t r a c e ( m Δ ) | 3 , 0 Δ 1 .
R = cos 1 [ t r a c e ( M R ) 2 1 ] .

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