Abstract

In addition to the visual information contained in intensity and color, imaging polarimetry allows visual information to be extracted from the polarization of light. However, a major challenge of imaging polarimetry is image degradation due to noise. This paper investigates the mitigation of noise through denoising algorithms and compares existing denoising algorithms with a new method, based on BM3D (Block Matching 3D). This algorithm, Polarization-BM3D (PBM3D), gives visual quality superior to the state of the art across all images and noise standard deviations tested. We show that denoising polarization images using PBM3D allows the degree of polarization to be more accurately calculated by comparing it with spectral polarimetry measurements.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2016 (2)

K. H. Britten, T. D. Thatcher, and T. Caro, “Zebras and biting flies: Quantitative analysis of reflected light from Zebra coats in their natural habitat,” PLoS ONE 11, e0154504 (2016).
[Crossref]

H. Sadreazami, M. O. Ahmad, and M. N. S. Swamy, “A study on image denoising in contourlet domain using the alpha-stable family of distributions,” Signal Process. 128, 459–473 (2016).
[Crossref]

2015 (2)

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[Crossref]

R.-Q. Xia, X. Wang, W.-Q. Jin, and J.-A. Liang, “Optimization of polarizer angles for measurements of the degree and angle of linear polarization for linear polarizer-based polarimeters,” Opt. Commun. 353, 109–116 (2015).
[Crossref]

2014 (3)

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

2013 (1)

M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “Nonlocal transform-domain filter for volumetric data denoising and reconstruction,” IEEE Trans. Image Process. 22, 119–133 (2013).
[Crossref]

2012 (3)

M. Maggioni, G. Boracchi, A. Foi, and K. Egiazarian, “Video denoising, deblocking, and enhancement through separable 4-D nonlocal spatiotemporal transforms,” IEEE Trans. Image Process. 21, 3952–3966 (2012).
[Crossref]

M. Lebrun, “An analysis and implementation of the BM3D image denoising method,” Image Process. On Line 2, 175–213 (2012).
[Crossref]

S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. 29, 2028–2037 (2012).
[Crossref]

2011 (1)

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. 28, 465–474 (2011).
[Crossref]

2009 (1)

2007 (3)

J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83–96 (2007).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

D. Miyazaki and K. Ikeuchi, “Shape estimation of transparent objects by using inverse polarization ray tracing,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2018–2029 (2007).
[Crossref]

2006 (4)

R. Wehner and M. Müller, “The significance of direct sunlight and polarized skylight in the ant’s celestial system of navigation,” Proc. Natl. Acad. Sci. USA 103, 12575–12579 (2006).
[Crossref]

Y.-Q. Zhao, Q. Pan, and H.-C. Zhang, “New polarization imaging method based on spatially adaptive wavelet image fusion,” Opt. Eng. 45, 123202 (2006).
[Crossref]

J. Zallat, S. Aïnouz, 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. 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]

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]

2003 (1)

2000 (1)

S. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process. 9, 1522–1531 (2000).
[Crossref]

1990 (1)

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[Crossref]

Achilefu, S.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Ahmad, M. O.

H. Sadreazami, M. O. Ahmad, and M. N. S. Swamy, “A study on image denoising in contourlet domain using the alpha-stable family of distributions,” Signal Process. 128, 459–473 (2016).
[Crossref]

Aïnouz, S.

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

Boracchi, G.

M. Maggioni, G. Boracchi, A. Foi, and K. Egiazarian, “Video denoising, deblocking, and enhancement through separable 4-D nonlocal spatiotemporal transforms,” IEEE Trans. Image Process. 21, 3952–3966 (2012).
[Crossref]

Britten, K. H.

K. H. Britten, T. D. Thatcher, and T. Caro, “Zebras and biting flies: Quantitative analysis of reflected light from Zebra coats in their natural habitat,” PLoS ONE 11, e0154504 (2016).
[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]

Caldwell, R. L.

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

Caro, T.

K. H. Britten, T. D. Thatcher, and T. Caro, “Zebras and biting flies: Quantitative analysis of reflected light from Zebra coats in their natural habitat,” PLoS ONE 11, e0154504 (2016).
[Crossref]

Chang, S.

S. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process. 9, 1522–1531 (2000).
[Crossref]

Charanya, T.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Chenault, D. B.

Christy, J. H.

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[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]

Collett, E.

E. Collett, Field Guide to Polarization (SPIE, 2005).

Craven-Jones, J.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Cronin, T. W.

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space,” in IEEE International Conference on Image Processing (2007), Vol. 1, pp. I-313–I-316.

Danielyan, A.

A. Danielyan, A. Foi, V. Katkovnik, and K. Egiazarian, “Denoising of multispectral images via nonlocal groupwise spectrum-PCA,” in Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 261–266.

Davis, P.

J. Taylor, P. Davis, and L. Wolff, “Underwater partial polarization signatures from the shallow water real-time imaging polarimeter (shrimp),” in OCEANS’02 MTS/IEEE (2002), pp. 1526–1534.

de Jong, W.

W. de Jong, J. Schavemaker, and A. Schoolderman, “Polarized light camera; a tool in the counter-IED toolbox,” in Prediction and Detection of Improvised Explosive Devices (IED) (SET-117) (RTO, 2007).

De Martino, A.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Egiazarian, K.

M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “Nonlocal transform-domain filter for volumetric data denoising and reconstruction,” IEEE Trans. Image Process. 22, 119–133 (2013).
[Crossref]

M. Maggioni, G. Boracchi, A. Foi, and K. Egiazarian, “Video denoising, deblocking, and enhancement through separable 4-D nonlocal spatiotemporal transforms,” IEEE Trans. Image Process. 21, 3952–3966 (2012).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

A. Danielyan, A. Foi, V. Katkovnik, and K. Egiazarian, “Denoising of multispectral images via nonlocal groupwise spectrum-PCA,” in Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 261–266.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space,” in IEEE International Conference on Image Processing (2007), Vol. 1, pp. I-313–I-316.

Engheta, N.

S.-S. Lin, K. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization enhanced visual surveillance techniques,” in Proceedings of IEEE International Conference on Networking, Sensing and Control (2004), pp. 216–221.

Escuti, M.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Faisan, S.

S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. 29, 2028–2037 (2012).
[Crossref]

Feller, K. D.

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

Fessler, J.

Fineschi, S.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Fofi, D.

A. E. R. Shabayek, O. Morel, and D. Fofi, “Bio-inspired polarization vision techniques for robotics applications,” in Handbook of Research on Advancements in Robotics and Mechatronics (IGI Global, 2015), pp. 81–117.

Foi, A.

M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “Nonlocal transform-domain filter for volumetric data denoising and reconstruction,” IEEE Trans. Image Process. 22, 119–133 (2013).
[Crossref]

M. Maggioni, G. Boracchi, A. Foi, and K. Egiazarian, “Video denoising, deblocking, and enhancement through separable 4-D nonlocal spatiotemporal transforms,” IEEE Trans. Image Process. 21, 3952–3966 (2012).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

A. Danielyan, A. Foi, V. Katkovnik, and K. Egiazarian, “Denoising of multispectral images via nonlocal groupwise spectrum-PCA,” in Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 261–266.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space,” in IEEE International Conference on Image Processing (2007), Vol. 1, pp. I-313–I-316.

Galatsanos, N.

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. 28, 465–474 (2011).
[Crossref]

Gao, S.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Goldstein, D. L.

Gruev, V.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Harrington, D.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Heinrich, C.

S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. 29, 2028–2037 (2012).
[Crossref]

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. 28, 465–474 (2011).
[Crossref]

J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83–96 (2007).
[Crossref]

Hemmi, J. M.

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[Crossref]

Horváth, G.

G. Horváth and D. Varju, Polarized Light in Animal Vision: Polarization Patterns in Nature (Springer, 2004).

How, M. J.

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[Crossref]

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

Ikeuchi, K.

D. Miyazaki and K. Ikeuchi, “Shape estimation of transparent objects by using inverse polarization ray tracing,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2018–2029 (2007).
[Crossref]

Jin, W.-Q.

R.-Q. Xia, X. Wang, W.-Q. Jin, and J.-A. Liang, “Optimization of polarizer angles for measurements of the degree and angle of linear polarization for linear polarizer-based polarimeters,” Opt. Commun. 353, 109–116 (2015).
[Crossref]

Kahan, L.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Katkovnik, V.

M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “Nonlocal transform-domain filter for volumetric data denoising and reconstruction,” IEEE Trans. Image Process. 22, 119–133 (2013).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

A. Danielyan, A. Foi, V. Katkovnik, and K. Egiazarian, “Denoising of multispectral images via nonlocal groupwise spectrum-PCA,” in Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 261–266.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space,” in IEEE International Conference on Image Processing (2007), Vol. 1, pp. I-313–I-316.

Lake, S. P.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Lallement, A.

S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. 29, 2028–2037 (2012).
[Crossref]

Lebrun, M.

M. Lebrun, “An analysis and implementation of the BM3D image denoising method,” Image Process. On Line 2, 175–213 (2012).
[Crossref]

Liang, J.-A.

R.-Q. Xia, X. Wang, W.-Q. Jin, and J.-A. Liang, “Optimization of polarizer angles for measurements of the degree and angle of linear polarization for linear polarizer-based polarimeters,” Opt. Commun. 353, 109–116 (2015).
[Crossref]

Lin, S.-S.

S.-S. Lin, K. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization enhanced visual surveillance techniques,” in Proceedings of IEEE International Conference on Networking, Sensing and Control (2004), pp. 216–221.

Maggioni, M.

M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “Nonlocal transform-domain filter for volumetric data denoising and reconstruction,” IEEE Trans. Image Process. 22, 119–133 (2013).
[Crossref]

M. Maggioni, G. Boracchi, A. Foi, and K. Egiazarian, “Video denoising, deblocking, and enhancement through separable 4-D nonlocal spatiotemporal transforms,” IEEE Trans. Image Process. 21, 3952–3966 (2012).
[Crossref]

Marshall, J.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Marshall, N. J.

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[Crossref]

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

Mawet, D.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Miyazaki, D.

D. Miyazaki and K. Ikeuchi, “Shape estimation of transparent objects by using inverse polarization ray tracing,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2018–2029 (2007).
[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]

Morel, O.

A. E. R. Shabayek, O. Morel, and D. Fofi, “Bio-inspired polarization vision techniques for robotics applications,” in Handbook of Research on Advancements in Robotics and Mechatronics (IGI Global, 2015), pp. 81–117.

Müller, M.

R. Wehner and M. Müller, “The significance of direct sunlight and polarized skylight in the ant’s celestial system of navigation,” Proc. Natl. Acad. Sci. USA 103, 12575–12579 (2006).
[Crossref]

Narasimhan, S. G.

Nayar, S. K.

Nikou, C.

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. 28, 465–474 (2011).
[Crossref]

Pan, Q.

Y.-Q. Zhao, Q. Pan, and H.-C. Zhang, “New polarization imaging method based on spatially adaptive wavelet image fusion,” Opt. Eng. 45, 123202 (2006).
[Crossref]

Porter, M. L.

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

Powell, S. B.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Pugh, E. N.

S.-S. Lin, K. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization enhanced visual surveillance techniques,” in Proceedings of IEEE International Conference on Networking, Sensing and Control (2004), pp. 216–221.

Radford, A. N.

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

Raman, B.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Riedi, J.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Roberts, N. W.

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[Crossref]

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Rousseau, F.

S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. 29, 2028–2037 (2012).
[Crossref]

Sadreazami, H.

H. Sadreazami, M. O. Ahmad, and M. N. S. Swamy, “A study on image denoising in contourlet domain using the alpha-stable family of distributions,” Signal Process. 128, 459–473 (2016).
[Crossref]

Saha, D.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Schavemaker, J.

W. de Jong, J. Schavemaker, and A. Schoolderman, “Polarized light camera; a tool in the counter-IED toolbox,” in Prediction and Detection of Improvised Explosive Devices (IED) (SET-117) (RTO, 2007).

Schechner, Y. Y.

Schoolderman, A.

W. de Jong, J. Schavemaker, and A. Schoolderman, “Polarized light camera; a tool in the counter-IED toolbox,” in Prediction and Detection of Improvised Explosive Devices (IED) (SET-117) (RTO, 2007).

Sfikas, G.

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. 28, 465–474 (2011).
[Crossref]

Shabayek, A. E. R.

A. E. R. Shabayek, O. Morel, and D. Fofi, “Bio-inspired polarization vision techniques for robotics applications,” in Handbook of Research on Advancements in Robotics and Mechatronics (IGI Global, 2015), pp. 81–117.

Shaw, J. A.

Snik, F.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Stoll, M. P.

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

Swamy, M. N. S.

H. Sadreazami, M. O. Ahmad, and M. N. S. Swamy, “A study on image denoising in contourlet domain using the alpha-stable family of distributions,” Signal Process. 128, 459–473 (2016).
[Crossref]

Taylor, J.

J. Taylor, P. Davis, and L. Wolff, “Underwater partial polarization signatures from the shallow water real-time imaging polarimeter (shrimp),” in OCEANS’02 MTS/IEEE (2002), pp. 1526–1534.

Temple, S. E.

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[Crossref]

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

Thatcher, T. D.

K. H. Britten, T. D. Thatcher, and T. Caro, “Zebras and biting flies: Quantitative analysis of reflected light from Zebra coats in their natural habitat,” PLoS ONE 11, e0154504 (2016).
[Crossref]

Tyo, J. S.

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

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]

Valenzuela, J.

Varju, D.

G. Horváth and D. Varju, Polarized Light in Animal Vision: Polarization Patterns in Nature (Springer, 2004).

Vetterli, M.

S. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process. 9, 1522–1531 (2000).
[Crossref]

Wang, X.

R.-Q. Xia, X. Wang, W.-Q. Jin, and J.-A. Liang, “Optimization of polarizer angles for measurements of the degree and angle of linear polarization for linear polarizer-based polarimeters,” Opt. Commun. 353, 109–116 (2015).
[Crossref]

Wehner, R.

R. Wehner and M. Müller, “The significance of direct sunlight and polarized skylight in the ant’s celestial system of navigation,” Proc. Natl. Acad. Sci. USA 103, 12575–12579 (2006).
[Crossref]

Wolff, L.

J. Taylor, P. Davis, and L. Wolff, “Underwater partial polarization signatures from the shallow water real-time imaging polarimeter (shrimp),” in OCEANS’02 MTS/IEEE (2002), pp. 1526–1534.

Wolff, L. B.

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[Crossref]

Xia, R.-Q.

R.-Q. Xia, X. Wang, W.-Q. Jin, and J.-A. Liang, “Optimization of polarizer angles for measurements of the degree and angle of linear polarization for linear polarizer-based polarimeters,” Opt. Commun. 353, 109–116 (2015).
[Crossref]

Yemelyanov, K.

S.-S. Lin, K. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization enhanced visual surveillance techniques,” in Proceedings of IEEE International Conference on Networking, Sensing and Control (2004), pp. 216–221.

York, T.

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Yu, B.

S. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process. 9, 1522–1531 (2000).
[Crossref]

Zallat, J.

S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. 29, 2028–2037 (2012).
[Crossref]

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. 28, 465–474 (2011).
[Crossref]

J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83–96 (2007).
[Crossref]

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

Zhang, H.-C.

Y.-Q. Zhao, Q. Pan, and H.-C. Zhang, “New polarization imaging method based on spatially adaptive wavelet image fusion,” Opt. Eng. 45, 123202 (2006).
[Crossref]

Zhao, Y.-Q.

Y.-Q. Zhao, Q. Pan, and H.-C. Zhang, “New polarization imaging method based on spatially adaptive wavelet image fusion,” Opt. Eng. 45, 123202 (2006).
[Crossref]

Appl. Opt. (2)

Curr. Biol. (1)

M. J. How, J. H. Christy, S. E. Temple, J. M. Hemmi, N. J. Marshall, and N. W. Roberts, “Target detection is enhanced by polarization vision in a fiddler crab,” Curr. Biol. 25, 3069–3073 (2015).
[Crossref]

IEEE Trans. Image Process. (4)

S. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process. 9, 1522–1531 (2000).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “Nonlocal transform-domain filter for volumetric data denoising and reconstruction,” IEEE Trans. Image Process. 22, 119–133 (2013).
[Crossref]

M. Maggioni, G. Boracchi, A. Foi, and K. Egiazarian, “Video denoising, deblocking, and enhancement through separable 4-D nonlocal spatiotemporal transforms,” IEEE Trans. Image Process. 21, 3952–3966 (2012).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[Crossref]

D. Miyazaki and K. Ikeuchi, “Shape estimation of transparent objects by using inverse polarization ray tracing,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2018–2029 (2007).
[Crossref]

Image Process. On Line (1)

M. Lebrun, “An analysis and implementation of the BM3D image denoising method,” Image Process. On Line 2, 175–213 (2012).
[Crossref]

J. Exp. Biol. (1)

M. J. How, M. L. Porter, A. N. Radford, K. D. Feller, S. E. Temple, R. L. Caldwell, N. J. Marshall, T. W. Cronin, and N. W. Roberts, “Out of the blue: The evolution of horizontally polarized signals in Haptosquilla (Crustacea, Stomatopoda, Protosquillidae),” J. Exp. Biol. 217, 3425–3431 (2014).
[Crossref]

J. Opt. A (1)

J. Zallat, S. Aïnouz, 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. (2)

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Recovery of polarimetric Stokes images by spatial mixture models,” J. Opt. Soc. Am. 28, 465–474 (2011).
[Crossref]

S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. 29, 2028–2037 (2012).
[Crossref]

J. Opt. Soc. Am. A (1)

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. Commun. (1)

R.-Q. Xia, X. Wang, W.-Q. Jin, and J.-A. Liang, “Optimization of polarizer angles for measurements of the degree and angle of linear polarization for linear polarizer-based polarimeters,” Opt. Commun. 353, 109–116 (2015).
[Crossref]

Opt. Eng. (1)

Y.-Q. Zhao, Q. Pan, and H.-C. Zhang, “New polarization imaging method based on spatially adaptive wavelet image fusion,” Opt. Eng. 45, 123202 (2006).
[Crossref]

Opt. Express (1)

PLoS ONE (1)

K. H. Britten, T. D. Thatcher, and T. Caro, “Zebras and biting flies: Quantitative analysis of reflected light from Zebra coats in their natural habitat,” PLoS ONE 11, e0154504 (2016).
[Crossref]

Proc. IEEE (1)

T. York, S. B. Powell, S. Gao, L. Kahan, T. Charanya, D. Saha, N. W. Roberts, T. W. Cronin, J. Marshall, S. Achilefu, S. P. Lake, B. Raman, and V. Gruev, “Bioinspired polarization imaging sensors: from circuits and optics to signal processing algorithms and biomedical applications,” Proc. IEEE 102, 1450–1469 (2014).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

R. Wehner and M. Müller, “The significance of direct sunlight and polarized skylight in the ant’s celestial system of navigation,” Proc. Natl. Acad. Sci. USA 103, 12575–12579 (2006).
[Crossref]

Proc. SPIE (1)

F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014).
[Crossref]

Signal Process. (1)

H. Sadreazami, M. O. Ahmad, and M. N. S. Swamy, “A study on image denoising in contourlet domain using the alpha-stable family of distributions,” Signal Process. 128, 459–473 (2016).
[Crossref]

Other (10)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space,” in IEEE International Conference on Image Processing (2007), Vol. 1, pp. I-313–I-316.

A. Danielyan, A. Foi, V. Katkovnik, and K. Egiazarian, “Denoising of multispectral images via nonlocal groupwise spectrum-PCA,” in Conference on Colour in Graphics, Imaging, and Vision (2010), pp. 261–266.

E. Collett, Field Guide to Polarization (SPIE, 2005).

W. de Jong, J. Schavemaker, and A. Schoolderman, “Polarized light camera; a tool in the counter-IED toolbox,” in Prediction and Detection of Improvised Explosive Devices (IED) (SET-117) (RTO, 2007).

S.-S. Lin, K. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization enhanced visual surveillance techniques,” in Proceedings of IEEE International Conference on Networking, Sensing and Control (2004), pp. 216–221.

E. Hecht, Optics (Addison-Wesley, 2002).

G. Horváth and D. Varju, Polarized Light in Animal Vision: Polarization Patterns in Nature (Springer, 2004).

G. Horváth, ed., Polarized Light and Polarization Vision in Animal Sciences (Springer, 2014).

A. E. R. Shabayek, O. Morel, and D. Fofi, “Bio-inspired polarization vision techniques for robotics applications,” in Handbook of Research on Advancements in Robotics and Mechatronics (IGI Global, 2015), pp. 81–117.

J. Taylor, P. Davis, and L. Wolff, “Underwater partial polarization signatures from the shallow water real-time imaging polarimeter (shrimp),” in OCEANS’02 MTS/IEEE (2002), pp. 1526–1534.

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Figures (12)

Fig. 1.
Fig. 1. Simulation of an unpolarized scene with and without noise ( σ = 0.02 ). Black represents a value of 0, white of 1. The error is large for the noisy DoP image.
Fig. 2.
Fig. 2. Basic outline of the CBM3D/PBM3D denoising algorithm.
Fig. 3.
Fig. 3. I 0 of each image in the set.
Fig. 4.
Fig. 4. PSNR for denoised images as a function of σ , the standard deviation of noise. Above each plot is the name of the image denoised; line colors correspond to different denoising algorithms. For the top row, PSNR values are shown in Table 3. It can be seen that, for all images and all values of σ , PBM3D produces images with the greatest PSNR.
Fig. 5.
Fig. 5. “Oranges” image with noise of standard deviation σ = 0.15 (a high noise level), denoised using BM3D Stokes (S) and PBM3D (P) (G, ground truth). For BM3D Stokes and PBM3D, the S 0 images are visually similar to the ground truth. For both methods, however, the S 1 images are notably different, and the S 2 images are almost unrecognizable.
Fig. 6.
Fig. 6. Polarization components of “oranges” image after application of several denoising methods. G, ground truth; N, noisy; B, BM3D; S, BM3D Stokes; P, PBM3D; Z, Zhao; F, Faisan. Noise standard deviation, σ = 0.026 . Note that the DoP images have been scaled such that black represents DoP = 0 and white represents DoP = 0.5 .
Fig. 7.
Fig. 7. Polarization components of “cars” image after application of several denoising methods. G, ground truth; N, noisy; B, BM3D; S, BM3D Stokes; P, PBM3D; Z, Zhao; F, Faisan. Noise standard deviation, σ = 0.026 .
Fig. 8.
Fig. 8. Polarization components of “window” image after application of several denoising methods. G, ground truth; N, noisy; B, BM3D; S, BM3D Stokes; P, PBM3D; Z, Zhao; F, Faisan. Noise standard deviation, σ = 0.026 .
Fig. 9.
Fig. 9. Close-up of “windows” image from Fig. 8 (G, ground truth; S, BM3D Stokes; P, PBM3D). The DoP component of the image denoised using PBM3D exhibits fewer artifacts than the imaged denoised using BM3D Stokes, especially underneath the window. In the AoP components, the lower windows are much more faithfully represented by the image denoised using PBM3D than BM3D Stokes.
Fig. 10.
Fig. 10. Close-up of “cars” image from Fig. 7 (G, ground truth; S, BM3D Stokes; P, PBM3D). DoP components are similar for the images denoised using BM3D Stokes and PBM3D, with slight differences noticeable in the car’s bumper. Detail around the number plate of the car and panels on the right side of the image are more faithfully denoised using PBM3D than BM3D Stokes.
Fig. 11.
Fig. 11. DoP image of a collection of lab objects, taken with an exposure of 0.0222s. (a) Image without denoising. (b) Image denoised using PBM3D. The circles indicate where the true DoP value was measured using a spectrometer. It can be seen that the noisy image tends to show much larger DoP values. DoP values measured at each point are shown in Fig. 12.
Fig. 12.
Fig. 12. Absolute value of the difference between DoP values measured using the spectrometer and the noisy images, | n s | , and spectrometer and the denoised images, | d s | . The noisy and denoised images had varying noise values, as given in Table 4. The locations where the DoP values were calculated are shown in Fig. 11. Black line indicates where | d s | = | n s | ; it can be seen that using the denoised images for DoP measurements results in much greater agreement with measurements taken using the noisy image as for most measurements | d s | < | n s | . Table 4 uses this data to show that denoising significantly reduces the error due to noise.

Tables (8)

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Algorithm 1 BM3D, single step

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Algorithm 2 CBM3D, single step

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Algorithm 3 PBM3D, single step

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Algorithm 4 Pattern search method

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Table 1. PSNR Values for Images (Street, Dome, Building) Denoised Using the Following Matrices: I, Identity Matrix; S, Stokes Matrix; O, Opponent Matrix; P, Pattern Search Optimala

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Table 2. Optimal Matrices Computed Using the Pattern Search Method for 10 Values of σ , the Noise Standard Deviation

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Table 3. PSNR for Denoising of Four Images (“Oranges,” “Cars,” “Windows,” “Statue”) Using Several Methods (B, BM3D; S, BM3D Stokes; P, PBM3D; Z, Zhao; F, Faisan) and Several Values of σ , the Standard Deviation of the Noise, Addeda

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Table 4. Estimated σ , the Standard Deviation of Noise, and Wilcoxon Test Results for the Data in Fig. 12a

Equations (14)

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S 0 = I ,
S 1 = I 0 I 90 ,
S 2 = I 45 I 135 .
S 0 = I 0 + I 90 ,
S 1 = I 0 I 90 ,
S 2 = I 0 + 2 I 45 I 90 .
DoP = S 1 2 + S 2 2 S 0 ,
AoP = 1 2 arctan ( S 2 S 1 ) .
T opt = arg min T i MSE ( I i , PBM 3 D T ( I i ) ) ,
T stokes = ( 1 2 0 1 2 1 2 0 1 2 1 4 1 2 1 4 ) .
T opp = ( 1 3 1 3 1 3 1 2 0 1 2 1 4 1 2 1 4 ) .
T opt = ( 0.3133 0.3833 0.3033 0.4800 0.0300 0.5100 0.2600 0.5200 0.2200 ) .
PSNR = 10 log 10 ( 1 MSE ) ,
MSE = 1 3 M N x Ω ( ( S 0 ( x ) S 0 ( x ) ) 2 + 1 2 ( S 1 ( x ) S 1 ( x ) ) 2 + 1 2 ( S 2 ( x ) S 2 ( x ) ) 2 ) .