Abstract

We define a geodesic distance associated with the polarization space of non-singular coherency matrices. Its introduction on HPD(2) (the manifold of Hermitian positive definite matrices of dimension 2) can be directly related to the Jones calculus. The expression of distance and related notion of mean value in this particular metric space are also presented. We investigate the properties of this geodesic distance and the classical Euclidean one and their appropriateness for interpixel comparisons in a context of imaging polarimetry. Finally, results are presented for a geodesic version of the classical K-means clustering algorithm with simulated data and real data. The results demonstrate the advantages of the geodesic approach.

© 2010 Optical Society of America

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References

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  1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
    [CrossRef] [PubMed]
  2. V. Nourrit, J. M. Bueno, B. Vohnsen, and P. Arta, “Nonlinear registration for scanned retinal images: application to ocular polarimetry,” Appl. Opt. 47, 5341–5347 (2008).
    [CrossRef] [PubMed]
  3. A. N. Yaroslavsky, V. Neel, and R. R. Anderson, “Fluorescence polarization imaging for delineating nonmelanoma skin cancers,” Opt. Lett. 29, 2010–2012 (2004).
    [CrossRef] [PubMed]
  4. F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
    [CrossRef]
  5. S. Guyot, M. Anastasiadou, E. Deléchelle, and A. De Martino, “Registration scheme suitable to Mueller matrix imaging for biomedical applications,” Opt. Express 15, 7393–7400 (2007).
    [CrossRef] [PubMed]
  6. A. Kreuter, M. Zangerl, M. Schwarzmann, and M. Blumthaler, “All-sky imaging: a simple, versatile system for atmospheric research,” Appl. Opt. 48, 1091–1097 (2009).
    [CrossRef]
  7. P. Terrier, V. Devlaminck, and J. M. Charbois, “Segmentation of rough surfaces using a polarization imaging system,” J. Opt. Soc. Am. A 25, 423–430 (2008).
    [CrossRef]
  8. O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” Appl. Opt. 45, 4062–4068 (2006).
    [CrossRef] [PubMed]
  9. A. Bénière, F. Goudail, M. Alouini, and D. Dolf, “Degree of polarization estimation in the presence of nonuniform illumination and additive Gaussian noise,” J. Opt. Soc. Am. A 25, 919–929 (2008).
    [CrossRef]
  10. J. E. Ahmad and Y. Takakura, “Error analysis for rotating active Stokes–Mueller imaging polarimeters,” Opt. Lett. 31, 2858–2860 (2006).
    [CrossRef] [PubMed]
  11. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  12. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 37, 488–493 (1947).
  13. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
    [CrossRef]
  14. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  15. E. S. Fry and G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
    [CrossRef] [PubMed]
  16. S. R. Cloude, “Group theory and polarisation algebra,” Optik (Jena) 75, 26–36 (1986).
  17. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [CrossRef]
  18. W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).
  19. P. T. Fletcher and S. C. Joshi, “Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors,” in Proceedings of Computer Vision Approaches to Medical Image Analysis (CVAMIA) Workshops (Springer, 2004), pp. 87–98.
  20. S. Helgason, Differential Geometry and Symmetric Spaces (Academic, 1962).
  21. R. Verma, P. Khurd, and C. Davatzikos, “On analyzing diffusion tensor images by identifying manifold structure using Isomaps,” IEEE Trans. Med. Imaging 26, 772–778 (2007).
    [CrossRef] [PubMed]
  22. J. B. Tenenbaum, V. de Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science 290, 2319–2323 (2000).
    [CrossRef] [PubMed]
  23. C. J. C. Burges, “Geometric methods for feature extraction and dimensional reduction,” in Data Mining and Knowledge Discovery Handbook: A Complete Guide for Practitioners and Researchers, L.Rokach and O.Maimon, eds. (Kluwer, 2005), pp. 59–92.
    [CrossRef]
  24. M. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace distancié,” Ann. Inst. Henri Poincare 10, 215–310 (1948).
  25. X. Pennec, “Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements,” J. Math. Imaging Vision 25, 127–154 (2006).
    [CrossRef]
  26. H. Karcher, “Riemannian center of mass and mollifier smoothing,” Commun. Pure Appl. Math. 30, 509–541 (1977).
    [CrossRef]
  27. U. Grenander, Probabilities on Algebraic Structures (Wiley, 1963).
  28. C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras, “Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing,” J. Math. Imaging Vision 25, 423–444 (2006).
    [CrossRef]
  29. J. A. Hartigan, Clustering Algorithms, Wiley Series in Probability and Mathematical Statistics (Wiley, 1975).

2009 (1)

2008 (3)

2007 (2)

R. Verma, P. Khurd, and C. Davatzikos, “On analyzing diffusion tensor images by identifying manifold structure using Isomaps,” IEEE Trans. Med. Imaging 26, 772–778 (2007).
[CrossRef] [PubMed]

S. Guyot, M. Anastasiadou, E. Deléchelle, and A. De Martino, “Registration scheme suitable to Mueller matrix imaging for biomedical applications,” Opt. Express 15, 7393–7400 (2007).
[CrossRef] [PubMed]

2006 (5)

2005 (1)

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

2004 (1)

2000 (1)

J. B. Tenenbaum, V. de Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science 290, 2319–2323 (2000).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Jena) 75, 26–36 (1986).

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

1981 (1)

1977 (1)

H. Karcher, “Riemannian center of mass and mollifier smoothing,” Commun. Pure Appl. Math. 30, 509–541 (1977).
[CrossRef]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

1948 (1)

M. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace distancié,” Ann. Inst. Henri Poincare 10, 215–310 (1948).

1947 (1)

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 37, 488–493 (1947).

Ahmad, J. E.

Alouini, M.

Anastasiadou, M.

Anderson, R. R.

Arta, P.

Bénière, A.

Blumthaler, M.

Boothby, W. M.

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).

Boulbry, B.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Boulvert, F.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Bueno, J. M.

Burges, C. J. C.

C. J. C. Burges, “Geometric methods for feature extraction and dimensional reduction,” in Data Mining and Knowledge Discovery Handbook: A Complete Guide for Practitioners and Researchers, L.Rokach and O.Maimon, eds. (Kluwer, 2005), pp. 59–92.
[CrossRef]

Cariou, J.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Charbois, J. M.

Chenault, D. B.

Cloude, S. R.

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Jena) 75, 26–36 (1986).

Davatzikos, C.

R. Verma, P. Khurd, and C. Davatzikos, “On analyzing diffusion tensor images by identifying manifold structure using Isomaps,” IEEE Trans. Med. Imaging 26, 772–778 (2007).
[CrossRef] [PubMed]

De Martino, A.

de Silva, V.

J. B. Tenenbaum, V. de Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science 290, 2319–2323 (2000).
[CrossRef] [PubMed]

Deléchelle, E.

Deriche, R.

C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras, “Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing,” J. Math. Imaging Vision 25, 423–444 (2006).
[CrossRef]

Devlaminck, V.

Dolf, D.

Faugeras, O.

C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras, “Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing,” J. Math. Imaging Vision 25, 423–444 (2006).
[CrossRef]

Fletcher, P. T.

P. T. Fletcher and S. C. Joshi, “Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors,” in Proceedings of Computer Vision Approaches to Medical Image Analysis (CVAMIA) Workshops (Springer, 2004), pp. 87–98.

Fréchet, M.

M. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace distancié,” Ann. Inst. Henri Poincare 10, 215–310 (1948).

Fry, E. S.

Goldstein, D. L.

Gorria, P.

Goudail, F.

Grenander, U.

U. Grenander, Probabilities on Algebraic Structures (Wiley, 1963).

Guyot, S.

Hartigan, J. A.

J. A. Hartigan, Clustering Algorithms, Wiley Series in Probability and Mathematical Statistics (Wiley, 1975).

Helgason, S.

S. Helgason, Differential Geometry and Symmetric Spaces (Academic, 1962).

Jones, R. C.

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 37, 488–493 (1947).

Joshi, S. C.

P. T. Fletcher and S. C. Joshi, “Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors,” in Proceedings of Computer Vision Approaches to Medical Image Analysis (CVAMIA) Workshops (Springer, 2004), pp. 87–98.

Karcher, H.

H. Karcher, “Riemannian center of mass and mollifier smoothing,” Commun. Pure Appl. Math. 30, 509–541 (1977).
[CrossRef]

Kattawar, G. W.

Khurd, P.

R. Verma, P. Khurd, and C. Davatzikos, “On analyzing diffusion tensor images by identifying manifold structure using Isomaps,” IEEE Trans. Med. Imaging 26, 772–778 (2007).
[CrossRef] [PubMed]

Kim, K.

Kreuter, A.

Langford, J. C.

J. B. Tenenbaum, V. de Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science 290, 2319–2323 (2000).
[CrossRef] [PubMed]

Le Brun, G.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Le Jeune, B.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Lenglet, C.

C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras, “Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing,” J. Math. Imaging Vision 25, 423–444 (2006).
[CrossRef]

Mandel, L.

Meriaudeau, F.

Morel, O.

Neel, V.

Nourrit, V.

Pennec, X.

X. Pennec, “Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements,” J. Math. Imaging Vision 25, 127–154 (2006).
[CrossRef]

Rivet, S.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Rousson, M.

C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras, “Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing,” J. Math. Imaging Vision 25, 423–444 (2006).
[CrossRef]

Schwarzmann, M.

Shaw, J. A.

Simon, R.

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Stolz, C.

Takakura, Y.

Tenenbaum, J. B.

J. B. Tenenbaum, V. de Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science 290, 2319–2323 (2000).
[CrossRef] [PubMed]

Terrier, P.

Tyo, J. S.

Verma, R.

R. Verma, P. Khurd, and C. Davatzikos, “On analyzing diffusion tensor images by identifying manifold structure using Isomaps,” IEEE Trans. Med. Imaging 26, 772–778 (2007).
[CrossRef] [PubMed]

Vohnsen, B.

Wolf, E.

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Yaroslavsky, A. N.

Zangerl, M.

Ann. Inst. Henri Poincare (1)

M. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace distancié,” Ann. Inst. Henri Poincare 10, 215–310 (1948).

Appl. Opt. (5)

Commun. Pure Appl. Math. (1)

H. Karcher, “Riemannian center of mass and mollifier smoothing,” Commun. Pure Appl. Math. 30, 509–541 (1977).
[CrossRef]

IEEE Trans. Med. Imaging (1)

R. Verma, P. Khurd, and C. Davatzikos, “On analyzing diffusion tensor images by identifying manifold structure using Isomaps,” IEEE Trans. Med. Imaging 26, 772–778 (2007).
[CrossRef] [PubMed]

J. Math. Imaging Vision (2)

X. Pennec, “Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements,” J. Math. Imaging Vision 25, 127–154 (2006).
[CrossRef]

C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras, “Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing,” J. Math. Imaging Vision 25, 423–444 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 37, 488–493 (1947).

J. Opt. Soc. Am. A (3)

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Opt. Commun. (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (Jena) (1)

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Jena) 75, 26–36 (1986).

Science (1)

J. B. Tenenbaum, V. de Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science 290, 2319–2323 (2000).
[CrossRef] [PubMed]

Other (7)

C. J. C. Burges, “Geometric methods for feature extraction and dimensional reduction,” in Data Mining and Knowledge Discovery Handbook: A Complete Guide for Practitioners and Researchers, L.Rokach and O.Maimon, eds. (Kluwer, 2005), pp. 59–92.
[CrossRef]

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).

P. T. Fletcher and S. C. Joshi, “Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors,” in Proceedings of Computer Vision Approaches to Medical Image Analysis (CVAMIA) Workshops (Springer, 2004), pp. 87–98.

S. Helgason, Differential Geometry and Symmetric Spaces (Academic, 1962).

J. A. Hartigan, Clustering Algorithms, Wiley Series in Probability and Mathematical Statistics (Wiley, 1975).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

U. Grenander, Probabilities on Algebraic Structures (Wiley, 1963).

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Figures (8)

Fig. 1
Fig. 1

3D representations ( S 3 = 0 ) of point locus of outgoing Stokes vectors from a polarizer (dashed curve) and from a rotator (solid curve) for a fully polarized input vector with unitary intensity. The star marker is the Euclidean mean vector and the filled circle marker is the geodesic mean vector.

Fig. 2
Fig. 2

Euclidean distance (dashed curve) and geodesic distance (solid curve) as function of DOP with S 0 = 2 .

Fig. 3
Fig. 3

Examples of Poincaré sphere representation of samples with generalized Gaussian distribution for a variance fixed at 0.005. The mean point has a DOP of 0.1 (black), 0.6 (dark gray) and 0.9 (pale gray). (a) Intensity of the mean point is fixed at 1. (b) Intensity of the mean point is variable.

Fig. 4
Fig. 4

Estimated values (circle markers) of (a) DOP and (b) polarization angle for the Stokes vector [1, DOP, 0, 0] as function of DOP values The theoretical values are plotted as a solid curve. Estimations with the classical Euclidean distance are plotted as square markers.

Fig. 5
Fig. 5

Estimated values (circle markers) of (a) DOP and (b) polarization angle for the Stokes vector [0.3742/DOP 0.1, 0.2, 0.3] as function of DOP values The theoretical values are plotted as a solid curve. Estimations with the classical Euclidean distance are plotted as square markers.

Fig. 6
Fig. 6

Images of the Stokes parameters of a thin dielectric deposit (text) on a metallic object.

Fig. 7
Fig. 7

Euclidean cluster representation in the image using a three-gray-level scale.

Fig. 8
Fig. 8

Geodesic cluster representation in the image using a three-gray-levels scale.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

Φ = E E = [ E 1 E 1 * E 1 E 2 * E 2 E 1 * E 2 E 2 * ] ,
Φ = 1 2 j = 0 3 S j σ j = 1 2 [ S 0 + S 1 S 2 i S 3 S 2 + i S 3 S 0 S 1 ] ,
Φ out = J Φ in J .
DOP = { 1 4 det ( Φ ) [ Tr ( Φ ) ] 2 } 1 2 = ( S 1 2 + S 2 2 + S 3 2 ) 1 2 S 0 ,
S 0 2 ( S 1 2 + S 2 2 + S 3 2 ) 0 ,
s 1 = S 1 S 0 , s 2 = S 2 S 0 , s 3 = S 3 S 0
x M , φ ( e , x ) = x ;
( g 1 , g 2 ) G × G , x M , φ ( g 1 , φ ( g 2 , x ) ) = φ ( g 1 g 2 , x ) ;
L = 0 1 d γ ( t ) d t d t = 0 1 X p d t = X p ,
y = φ [ g 1 , γ ( p , X ) ( 1 ) ] = γ ( I d , g 1 x g ) ( 1 ) = exp ( g 1 X g ) ,
y = exp ( g 1 X g ) X = log p ( x ) = g log ( y ) g ,
X p , Y p p = d φ ( g 1 , X p ) , d φ ( g 1 , Y p ) I d .
d 2 ( p , x ) = X , X p = log p ( x ) ,
d 2 ( p , x ) = d φ ( g 1 , X p ) d φ ( g 1 , X p ) I d = g 1 X p g , g 1 X p g I d = log ( y ) , log ( y ) I d .
d ( p , x ) = [ Tr ( log ( y ) 2 ) ] 1 2 .
exp p ( X ) = x = p 1 2 exp [ p 1 2 X ( p 1 2 ) ] ( p 1 2 ) ,
log p ( x ) = X = p 1 2 log [ p 1 2 x ( p 1 2 ) ] ( p 1 2 ) .
[ S 0 2 [ 1 + cos ( 2 θ ) ] S 0 2 [ 1 + cos ( 2 θ ) ] cos ( 2 θ ) S 0 2 [ 1 + cos ( 2 θ ) ] sin ( 2 θ ) 0 ] T ,
σ x 2 ( p ) = E [ d ( p , x ) 2 ] ,
μ = arg min p M ( E [ d ( p , x ) 2 ] ) .
μ i + 1 = exp μ i ( 1 n j = 1 n log μ i ( x j ) ) .
d ( Id , x ) = [ log 2 [ S 0 ( 1 + DOP ) 2 ] + log 2 [ S 0 ( 1 DOP ) 2 ] ] 1 2 ,
g GL ( 2 , C ) , x HDP ( 2 ) , φ ( g , x ) = g x g
g GL ( 2 , C ) , x HDP ( 2 ) ,
( g x g ) = ( g ) x g = g x g .
( g 1 , g 2 ) GL ( 2 , C ) × GL ( 2 , C ) , x HDP ( 2 ) ,
φ ( g 1 , φ ( g 2 , x ) ) = φ ( g 1 , g 2 x g 2 ) = g 1 g 2 x g 2 g 1 = ( g 1 g 2 ) x ( g 1 g 2 ) = φ ( g 1 g 2 , x ) .

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