Abstract

This article addresses the estimation of polarization signatures in the Mueller imaging framework by non-local means filtering. This is an extension of previous work dealing with Stokes signatures. The extension is not straightforward because of the gap in complexity between the Mueller framework and the Stokes framework. The estimation procedure relies on the Cholesky decomposition of the coherency matrix, thereby ensuring the physical admissibility of the estimate. We propose an original parameterization of the boundary of the set of Mueller matrices, which makes our approach possible. The proposed method is fully unsupervised. It allows noise removal and the preservation of edges. Applications to synthetic as well as real data are presented.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
    [CrossRef]
  2. J. Zallat, Ch. Collet, and Y. Takakura, “Clustering of polarization-encoded images,” Appl. Opt.43, 283–292 (2004).
    [CrossRef] [PubMed]
  3. Ch. Collet, J. Zallat, and Y. Takakura, “Clustering of Mueller matrix images for skeletonized structure detection,” Opt. Express12, 1271–1280 (2004).
    [CrossRef] [PubMed]
  4. J. Zallat, Ch. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express16, 7119–7133 (2008).
    [CrossRef] [PubMed]
  5. S. Faisan, Ch. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering-estimation of Stokes vector images based on a non-local means approach,” J. Opt. Soc. Am. A29, 2028–2037 (2012).
    [CrossRef]
  6. A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul.4, 490–530 (2005).
    [CrossRef]
  7. Z. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt.39, 461–484 (1992).
    [CrossRef]
  8. C. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt.40, 471–481 (1993).
    [CrossRef]
  9. C. Vandermee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys.34, 5072–5088 (1993).
    [CrossRef]
  10. D. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A11, 2305–2319 (1994).
    [CrossRef]
  11. A. Gopala Rao, K. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics – I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt.45, 955–987 (1998).
  12. S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng.34, 1599–1610 (1995).
    [CrossRef]
  13. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett.31, 817–819 (2006).
    [CrossRef] [PubMed]
  14. C. Deledalle, L. Denis, and F. Tupin, “Iterative weighted maximum likelihood denoising with probabilistic patch-based weights,” IEEE Trans. Image Process.18, 2661–2672 (2009).
    [CrossRef] [PubMed]
  15. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).
  16. G. Stewart, Matrix algorithms. Vol II: eigensystems (SIAM, 2001).
    [CrossRef]
  17. T. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program.67, 189–224 (1994).
    [CrossRef]
  18. T. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim.6, 418–445 (1996).
    [CrossRef]
  19. G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Edition) (The Johns Hopkins University Press, 1996).

2012 (1)

2009 (1)

C. Deledalle, L. Denis, and F. Tupin, “Iterative weighted maximum likelihood denoising with probabilistic patch-based weights,” IEEE Trans. Image Process.18, 2661–2672 (2009).
[CrossRef] [PubMed]

2008 (1)

2006 (2)

A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett.31, 817–819 (2006).
[CrossRef] [PubMed]

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

2005 (1)

A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul.4, 490–530 (2005).
[CrossRef]

2004 (2)

1998 (1)

A. Gopala Rao, K. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics – I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt.45, 955–987 (1998).

1996 (1)

T. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim.6, 418–445 (1996).
[CrossRef]

1995 (1)

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng.34, 1599–1610 (1995).
[CrossRef]

1994 (2)

T. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program.67, 189–224 (1994).
[CrossRef]

D. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A11, 2305–2319 (1994).
[CrossRef]

1993 (2)

C. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt.40, 471–481 (1993).
[CrossRef]

C. Vandermee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys.34, 5072–5088 (1993).
[CrossRef]

1992 (1)

Z. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt.39, 461–484 (1992).
[CrossRef]

Aiello, A.

Anderson, D.

Anouz, S.

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

Barakat, R.

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

Buades, A.

A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul.4, 490–530 (2005).
[CrossRef]

Cloude, S.

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng.34, 1599–1610 (1995).
[CrossRef]

Coleman, T.

T. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim.6, 418–445 (1996).
[CrossRef]

T. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program.67, 189–224 (1994).
[CrossRef]

Coll, B.

A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul.4, 490–530 (2005).
[CrossRef]

Collet, Ch.

Deledalle, C.

C. Deledalle, L. Denis, and F. Tupin, “Iterative weighted maximum likelihood denoising with probabilistic patch-based weights,” IEEE Trans. Image Process.18, 2661–2672 (2009).
[CrossRef] [PubMed]

Denis, L.

C. Deledalle, L. Denis, and F. Tupin, “Iterative weighted maximum likelihood denoising with probabilistic patch-based weights,” IEEE Trans. Image Process.18, 2661–2672 (2009).
[CrossRef] [PubMed]

Faisan, S.

Givens, C.

C. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt.40, 471–481 (1993).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Edition) (The Johns Hopkins University Press, 1996).

Gopala Rao, A.

A. Gopala Rao, K. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics – I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt.45, 955–987 (1998).

Heinrich, Ch.

Kostinski, A.

C. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt.40, 471–481 (1993).
[CrossRef]

Lallement, A.

Li, Y.

T. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim.6, 418–445 (1996).
[CrossRef]

T. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program.67, 189–224 (1994).
[CrossRef]

Mallesh, K.

A. Gopala Rao, K. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics – I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt.45, 955–987 (1998).

Morel, J.

A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul.4, 490–530 (2005).
[CrossRef]

Petremand, M.

Pottier, E.

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng.34, 1599–1610 (1995).
[CrossRef]

Puentes, G.

Rousseau, F.

Stewart, G.

G. Stewart, Matrix algorithms. Vol II: eigensystems (SIAM, 2001).
[CrossRef]

Stoll, M. P.

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

Sudha,

A. Gopala Rao, K. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics – I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt.45, 955–987 (1998).

Takakura, Y.

Tupin, F.

C. Deledalle, L. Denis, and F. Tupin, “Iterative weighted maximum likelihood denoising with probabilistic patch-based weights,” IEEE Trans. Image Process.18, 2661–2672 (2009).
[CrossRef] [PubMed]

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Edition) (The Johns Hopkins University Press, 1996).

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

Vandermee, C.

C. Vandermee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys.34, 5072–5088 (1993).
[CrossRef]

Voigt, D.

Woerdman, J. P.

Xing, Z.

Z. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt.39, 461–484 (1992).
[CrossRef]

Zallat, J.

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

C. Deledalle, L. Denis, and F. Tupin, “Iterative weighted maximum likelihood denoising with probabilistic patch-based weights,” IEEE Trans. Image Process.18, 2661–2672 (2009).
[CrossRef] [PubMed]

J. Math. Phys. (1)

C. Vandermee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys.34, 5072–5088 (1993).
[CrossRef]

J. Mod. Opt. (3)

Z. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt.39, 461–484 (1992).
[CrossRef]

C. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt.40, 471–481 (1993).
[CrossRef]

A. Gopala Rao, K. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics – I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt.45, 955–987 (1998).

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

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

Math. Program. (1)

T. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program.67, 189–224 (1994).
[CrossRef]

Multiscale Model. Simul. (1)

A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul.4, 490–530 (2005).
[CrossRef]

Opt. Eng. (1)

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng.34, 1599–1610 (1995).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

SIAM J. Optim. (1)

T. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim.6, 418–445 (1996).
[CrossRef]

Other (3)

G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Edition) (The Johns Hopkins University Press, 1996).

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

G. Stewart, Matrix algorithms. Vol II: eigensystems (SIAM, 2001).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Mueller image of the synthetic scene. For convenience, 16 images (one for each channel of the Mueller matrix) are represented on a 4 × 4 grid. Row i column j image corresponds to the mij Mueller matrix element image.

Fig. 2
Fig. 2

Mueller image of the synthetic scene estimated with the proposed (NLM) approach (a) and pseudo-inverse (b) from the noisy intensity images (σ = 0.1). Data are presented with the convention used in Fig. 1.

Fig. 3
Fig. 3

Mueller image of the first scene estimated with the proposed (NLM) approach (a) and projected pseudo-inverse (b). Data are presented with the convention used in Fig. 1 except that all channels but m11 have been pixelwise normalized with respect to m11 (mij(x) = mij(x)/m11(x)).

Fig. 4
Fig. 4

Mueller image of the second scene estimated with the proposed (NLM) approach (left column) and projected pseudo-inverse (right column). For convenience, only channels m22 (a) and m44 (c) are presented. Images (b) and (d) correspond to a zoom in of (a) and (c), respectively.

Fig. 5
Fig. 5

Mean diattenuation of the first scene obtained by the polar decomposition applied to the NLM (left column) and to the MPIortho solutions (right column).

Tables (1)

Tables Icon

Table 1 PSNR (left) and Mueller matrix estimation error (right) obtained with the four different methods, and for different values of σ. Bold values correspond to the best results.

Equations (24)

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

x Ω , I nlm ( x ) = y Ω w ( x , y ) I ( y ) = arg min a y Ω w ( x , y ) ( I ( y ) a ) 2 ,
x Ω , I nlm ( x ) = y Ω D x ( y ) I ( y ) = arg min a y Ω ( I ( y ) a ) t D x ( y ) ( I ( y ) a ) ,
x Ω , S ^ ( x ) = arg min S y Ω ( I ( y ) P S ) t D x ( y ) ( I ( y ) P S ) .
x Ω , S ^ ( x ) = arg min S I nlm ( x ) P S 2 .
x Ω , S ^ ( x ) = arg min S B I nlm ( x ) P S 2 ,
I ( x ) = A M ( x ) W ( + noise ) ,
I ( x ) _ = ( W t A ) M ( x ) _ ( + noise ) = P M ( x ) _ ( + noise ) ,
x Ω , M ^ ( x ) _ = arg min M y Ω ( I ( y ) _ P M _ ) t D x ( y ) ( I ( y ) _ P M _ ) ,
x Ω , M ^ ( x ) _ = arg min M I nlm ( x ) _ P M _ 2 ,
x Ω , I nlm ( x ) _ = y Ω D x ( y ) I ( y ) _ .
x Ω , M ^ ( x ) _ = arg min M B I nlm ( x ) _ P M _ 2 ,
M _ = T H _ ,
H = Λ Λ t ,
Λ = ( λ 1 0 0 0 λ 5 + i λ 6 λ 2 0 0 λ 7 + i λ 8 λ 9 + i λ 10 λ 3 0 λ 11 + i λ 12 λ 13 + i λ 14 λ 15 + i λ 16 λ 4 ) .
B = { M : M _ = T Λ Λ t _ with λ 4 = 0 } .
M ^ ( x ) _ = T Λ ^ Λ ^ t _ with Λ ^ = arg min Λ , λ 4 = 0 I nlm ( x ) _ P T Λ Λ t _ 2
P S N R ( I g t , I ^ ) = 10 log 10 ( d 2 1 16. P j = 1 16 α j 2 x ( I j g t ( x ) I ^ j ( x ) ) 2 ) ,
e ( M g t , M ^ ) = 100 1 P x M g t ( x ) M ^ ( x ) 2 M g t ( x ) 2 .
H = ( h 1 h 5 i h 6 h 7 i h 8 h 11 + i h 12 h 5 + i h 6 h 2 h 9 i h 10 h 13 i h 14 h 7 + i h 8 h 9 + i h 10 h 3 h 15 i h 16 h 11 i h 12 h 13 + i h 14 h 15 + i h 16 h 4 ) .
{ λ 1 = h 1 λ i = h i λ 1 ( for i = 5 to 8 and for i = 11 to 12 ) if λ 1 > 0 , and 0 otherwise λ 2 = h 2 λ 5 2 λ 6 2 λ 9 = 1 λ 2 ( h 9 λ 5 λ 7 λ 6 λ 8 ) if λ 2 > 0 , and 0 otherwise λ 10 = 1 λ 2 ( h 18 λ 6 λ 7 λ 5 λ 8 ) if λ 2 > 0 , and 0 otherwise λ 13 = 1 λ 2 ( h 13 λ 5 λ 11 λ 6 λ 12 ) if λ 2 > 0 , and 0 otherwise λ 14 = 1 λ 2 ( h 14 + λ 6 λ 11 λ 5 λ 12 ) if λ 2 > 0 , and 0 otherwise λ 3 = h 3 λ 7 2 λ 8 2 λ 9 2 λ 10 2 λ 15 = 1 λ 3 ( h 15 λ 7 λ 11 λ 8 λ 12 λ 9 λ 13 λ 10 λ 14 ) if λ 3 > 0 , and 0 otherwise λ 16 = 1 λ 3 ( h 16 + λ 8 λ 11 λ 7 λ 12 + λ 10 λ 13 λ 9 λ 14 ) if λ 3 > 0 , and 0 otherwise λ 4 = h 4 λ 11 2 λ 12 2 λ 13 2 λ 14 2 λ 15 2 λ 16 2
Λ = ( 0 0 0 0 0 λ 2 0 0 0 λ 9 + i λ 10 λ 3 0 0 λ 13 + i λ 14 λ 15 + i λ 16 λ 4 ) ,
R = ( 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 ) .
R = ( 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ) ;
R = ( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ) .

Metrics