Abstract

Conventional estimation techniques of Stokes images from observed radiance images through different polarization filters suffer from noise contamination that hampers correct interpretation or even leads to unphysical estimated signatures. This paper presents an efficient restoration technique based on nonlocal means, permitting accurate estimation of smoothly variable polarization signatures in the Stokes image while preserving sharp transitions. The method is assessed on simulated data as well as on real images.

© 2012 Optical Society of America

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References

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  1. R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.
  2. J. Zallat, S. Ainouz, and M.-P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A 8, 807–814 (2006).
    [CrossRef]
  3. J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83–96 (2007).
    [CrossRef]
  4. J. Zallat, C. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express 16, 7119–7133 (2008).
    [CrossRef]
  5. J. Valenzuela and J. Fessler, “Joint reconstruction of Stokes images from polarimetric measurements,” J. Opt. Soc. Am. A 26, 962–968 (2009).
    [CrossRef]
  6. G. Sfikas, C. Heinrich, J. Zallat, and C. Nikou, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. A 28, 465–474 (2011).
    [CrossRef]
  7. 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]
  8. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
    [CrossRef]
  9. R. Coifman and D. Donoho, “Translation-invariant denoising,” in Wavelets and Statistics (Springer Verlag, 1995), pp. 125–150.
  10. S. Kinderman, S. Osher, and P. Jones, “Deblurring and denoising of images by nonlocal functionals,” Multiscale Model. Simul. 4, 1091–1115 (2005).
    [CrossRef]
  11. G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7, 1005–1028 (2008).
    [CrossRef]
  12. M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212 (2008).
    [CrossRef]
  13. F. Rousseau, “A non-local approach for image super-resolution using intermodality priors,” Medical Image Anal. 14, 594–605 (2010).
    [CrossRef]
  14. V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
    [CrossRef]
  15. P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
    [CrossRef]
  16. T. Gasser, L. Sroka, and C. Steinmetz, “Residual variance and residual pattern in nonlinear regression,” Biometrika 73, 625–633 (1986).
    [CrossRef]
  17. B. Goossens, H. Luong, A. Pizurica, and W. Philips, “An improved non-local means algorithm for image denoising,” presented at the 2008 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA2008), Lausanne, Switzerland, August23–242008.
  18. C.-A. Deledalle, V. Duval, and J. Salmon, “Non-local methods with shape-adaptive patches (NLM-SAP),” J. Math. Imaging Vision 43, 103–120 (2012).
    [CrossRef]
  19. J. Zallat, M. Torzynski, and A. Lallement, “Double-pass self-spectral-calibration of a polarization state analyzer,” Opt. Lett. 37, 401–403 (2012).
    [CrossRef]
  20. M. Hanson, “Invexity and the Kuhn–Tucker theorem,” J. Math. Anal. Appl. 236, 594–604 (1999).
    [CrossRef]

2012

C.-A. Deledalle, V. Duval, and J. Salmon, “Non-local methods with shape-adaptive patches (NLM-SAP),” J. Math. Imaging Vision 43, 103–120 (2012).
[CrossRef]

J. Zallat, M. Torzynski, and A. Lallement, “Double-pass self-spectral-calibration of a polarization state analyzer,” Opt. Lett. 37, 401–403 (2012).
[CrossRef]

2011

2010

F. Rousseau, “A non-local approach for image super-resolution using intermodality priors,” Medical Image Anal. 14, 594–605 (2010).
[CrossRef]

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

2009

2008

J. Zallat, C. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express 16, 7119–7133 (2008).
[CrossRef]

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7, 1005–1028 (2008).
[CrossRef]

M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212 (2008).
[CrossRef]

2007

2006

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

2005

S. Kinderman, S. Osher, and P. Jones, “Deblurring and denoising of images by nonlocal functionals,” Multiscale Model. Simul. 4, 1091–1115 (2005).
[CrossRef]

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]

1999

M. Hanson, “Invexity and the Kuhn–Tucker theorem,” J. Math. Anal. Appl. 236, 594–604 (1999).
[CrossRef]

1992

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

1986

T. Gasser, L. Sroka, and C. Steinmetz, “Residual variance and residual pattern in nonlinear regression,” Biometrika 73, 625–633 (1986).
[CrossRef]

Ainouz, S.

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

Astola, J.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

Barillot, C.

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

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]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.

Coifman, R.

R. Coifman and D. Donoho, “Translation-invariant denoising,” in Wavelets and Statistics (Springer Verlag, 1995), pp. 125–150.

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]

Coupé, P.

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

Deledalle, C.-A.

C.-A. Deledalle, V. Duval, and J. Salmon, “Non-local methods with shape-adaptive patches (NLM-SAP),” J. Math. Imaging Vision 43, 103–120 (2012).
[CrossRef]

Donoho, D.

R. Coifman and D. Donoho, “Translation-invariant denoising,” in Wavelets and Statistics (Springer Verlag, 1995), pp. 125–150.

Duval, V.

C.-A. Deledalle, V. Duval, and J. Salmon, “Non-local methods with shape-adaptive patches (NLM-SAP),” J. Math. Imaging Vision 43, 103–120 (2012).
[CrossRef]

Egiazarian, K.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Fessler, J.

Foi, A.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

Gasser, T.

T. Gasser, L. Sroka, and C. Steinmetz, “Residual variance and residual pattern in nonlinear regression,” Biometrika 73, 625–633 (1986).
[CrossRef]

Gilboa, G.

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7, 1005–1028 (2008).
[CrossRef]

Goossens, B.

B. Goossens, H. Luong, A. Pizurica, and W. Philips, “An improved non-local means algorithm for image denoising,” presented at the 2008 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA2008), Lausanne, Switzerland, August23–242008.

Hanson, M.

M. Hanson, “Invexity and the Kuhn–Tucker theorem,” J. Math. Anal. Appl. 236, 594–604 (1999).
[CrossRef]

Heinrich, C.

Hellier, P.

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

Jones, P.

S. Kinderman, S. Osher, and P. Jones, “Deblurring and denoising of images by nonlocal functionals,” Multiscale Model. Simul. 4, 1091–1115 (2005).
[CrossRef]

Katkovnik, V.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

Kervrann, C.

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

Kinderman, S.

S. Kinderman, S. Osher, and P. Jones, “Deblurring and denoising of images by nonlocal functionals,” Multiscale Model. Simul. 4, 1091–1115 (2005).
[CrossRef]

Lallement, A.

Luong, H.

B. Goossens, H. Luong, A. Pizurica, and W. Philips, “An improved non-local means algorithm for image denoising,” presented at the 2008 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA2008), Lausanne, Switzerland, August23–242008.

Mignotte, M.

M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212 (2008).
[CrossRef]

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]

Nikou, C.

Osher, S.

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7, 1005–1028 (2008).
[CrossRef]

S. Kinderman, S. Osher, and P. Jones, “Deblurring and denoising of images by nonlocal functionals,” Multiscale Model. Simul. 4, 1091–1115 (2005).
[CrossRef]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Petremand, M.

Philips, W.

B. Goossens, H. Luong, A. Pizurica, and W. Philips, “An improved non-local means algorithm for image denoising,” presented at the 2008 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA2008), Lausanne, Switzerland, August23–242008.

Pizurica, A.

B. Goossens, H. Luong, A. Pizurica, and W. Philips, “An improved non-local means algorithm for image denoising,” presented at the 2008 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA2008), Lausanne, Switzerland, August23–242008.

Prima, S.

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

Rousseau, F.

F. Rousseau, “A non-local approach for image super-resolution using intermodality priors,” Medical Image Anal. 14, 594–605 (2010).
[CrossRef]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Salmon, J.

C.-A. Deledalle, V. Duval, and J. Salmon, “Non-local methods with shape-adaptive patches (NLM-SAP),” J. Math. Imaging Vision 43, 103–120 (2012).
[CrossRef]

Sfikas, G.

Sroka, L.

T. Gasser, L. Sroka, and C. Steinmetz, “Residual variance and residual pattern in nonlinear regression,” Biometrika 73, 625–633 (1986).
[CrossRef]

Steinmetz, C.

T. Gasser, L. Sroka, and C. Steinmetz, “Residual variance and residual pattern in nonlinear regression,” Biometrika 73, 625–633 (1986).
[CrossRef]

Stoll, M.-P.

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

Torzynski, M.

Valenzuela, J.

Yger, P.

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

Zallat, J.

Biometrika

T. Gasser, L. Sroka, and C. Steinmetz, “Residual variance and residual pattern in nonlinear regression,” Biometrika 73, 625–633 (1986).
[CrossRef]

IEEE Trans. Med. Imag.

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, “An optimized blockwise nonlocal means denoising filter for 3D magnetic resonance images,” IEEE Trans. Med. Imag. 27, 425–441 (2008).
[CrossRef]

Int. J. Comput. Vis.

V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” Int. J. Comput. Vis. 86, 1–32 (2010).
[CrossRef]

J. Math. Anal. Appl.

M. Hanson, “Invexity and the Kuhn–Tucker theorem,” J. Math. Anal. Appl. 236, 594–604 (1999).
[CrossRef]

J. Math. Imaging Vision

C.-A. Deledalle, V. Duval, and J. Salmon, “Non-local methods with shape-adaptive patches (NLM-SAP),” J. Math. Imaging Vision 43, 103–120 (2012).
[CrossRef]

J. Opt. A

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

J. Opt. Soc. Am. A

Medical Image Anal.

F. Rousseau, “A non-local approach for image super-resolution using intermodality priors,” Medical Image Anal. 14, 594–605 (2010).
[CrossRef]

Multiscale Model. Simul.

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]

S. Kinderman, S. Osher, and P. Jones, “Deblurring and denoising of images by nonlocal functionals,” Multiscale Model. Simul. 4, 1091–1115 (2005).
[CrossRef]

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7, 1005–1028 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Pattern Recogn. Lett.

M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212 (2008).
[CrossRef]

Physica D

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Other

R. Coifman and D. Donoho, “Translation-invariant denoising,” in Wavelets and Statistics (Springer Verlag, 1995), pp. 125–150.

B. Goossens, H. Luong, A. Pizurica, and W. Philips, “An improved non-local means algorithm for image denoising,” presented at the 2008 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA2008), Lausanne, Switzerland, August23–242008.

R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.

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Figures (5)

Fig. 1.
Fig. 1.

Example of three patches of size 3 (left), 5 (middle), and 7 (right): S1 is in white, and S2 is in gray.

Fig. 2.
Fig. 2.

Representation of the simulated data. The ith column is associated to the i component of the Stokes vector. The first row represents the ground truth, whereas the other lines represent the Stokes vectors that are estimated from the most noisy simulated images (σ2=0.05) with the pseudoinverse solution M1 (second row), with the proposed approach MP (third row), and with M4 (fourth row).

Fig. 3.
Fig. 3.

Poincaré sphere representation of the original image Sgt (top left), of its estimation from I (σ2=0.01) with method M1 (top right), method M4 (bottom left), and the proposed approach MP (bottom right).

Fig. 4.
Fig. 4.

DOP (first row), ellipticity (second row), and orientation (third row) of the Stokes vectors estimated with the proposed approach (left) and by using the pseudoinverse (right).

Fig. 5.
Fig. 5.

DOP of the Stokes vectors estimated with the proposed approach (left) and by using the pseudoinverse (right). The zoom in on the foot of the figurine reveals the denoising efficiency of the proposed approach.

Tables (4)

Tables Icon

Table 1. PSNR and Stokes Vector Estimation Error Obtained with the Five Different Methods and for Different Variances σ2 of Noisea

Tables Icon

Table 2. PSNR and Stokes Vector Estimation Error Obtained with Four Simplified Versions of the Proposed Approach (T1, T2, T3, and T4) and with the Proposed Approach for Different Variances σ2 of Noisea

Tables Icon

Table 3. PSNR and Stokes Vector Estimation Error Obtained with the GF- and MF-Based Approaches and with the Proposed Approach (MP) for Different Variances σ2 of Noisea

Tables Icon

Table 4. PSNR and Stokes Vector Estimation Error Obtained with Two Simplified Versions of the Proposed Approach (NLMPI, NLMPROJ) and with the Proposed Approach for Different Variances σ2 of Noise

Equations (25)

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

I(x)PS(x),
I(x)=PS(x)+n(x),
S0S12+S22+S32
S^(x)=argminSBI(x)P·S(x)2,
I(x)=f(x)+n(x),
xΩ,Inlm(x)=yΩw(x,y)I(y)yΩw(x,y),
Px={I(x+τ),τ[k,k]2},
w(x,y)=ϕ(dP(x,y)22Nβσ^2),
ϵx=45(I(x)14yN(x)I(y)),
σ^=1.4826medi(|ϵimedj(ϵj)|).
N.β.dP(x,y)22Nβσ2χ2(N).
dP(x,y)2=minSτ[k,k]2S(x,y)(τ)(I(x+τ)I(y+τ))2,
xΩ,Inlm(x)=yΩxw(x,y)I(y)=argminayΩxw(x,y)(I(y)a)2.
xΩ,Inlm(x)=yΩxDx(y)·I(y)=argminayΩx(I(y)a)T·Dx(y)·(I(y)a),
xΩ,S^(x)=argminSyΩx(I(y)P.S)T·Dx(y)·(I(y)P·S).
=2·PTyΩxDx(y)·(I(y)P·S)=2·PT(yΩxDx(y)·I(y)yΩxDx(y)P·S)=2·PT(yΩxDx(y)I(y)P·S),
=2·PT(Inlm(x)P·S).
xΩ,S^(x)=argminSInlm(x)P·S2,
PSNR(Igt,I^)=10log10(d214Pj=14αj2x(Ijgt(x)I^j(x))2),
e(Sgt,S^)=1001PxSgt(x)S^(x)2Sgt(x)2.
S^=argminSIP·S2,
S^=argminSBIP·S2,
minS1,S2,S3IP[S12+S22+S32,S1,S2,S3]T2.
minSIPS2,
={S|S0>S12+S22+S32}.

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