Abstract

An imaging, variable-retardance, Fourier-transform spectropolarimeter is presented that is capable of creating spectropolarimetric images of scenes with independent characterization of spatial, spectral, and polarimetric information. The device has a spectral resolution of ∼225 cm-1, making it truly hyperspectral in nature. Images of canonical targets such as spheres and cylinders obtained in a laboratory setup are presented. The results demonstrate the capability of developing systems to collect spectropolarimetric data of field images by use of the concept of pushbroom scanning and serial collection of polarimetric information. Further development of a parallelized collection strategy would allow the collection of near-real-time images of real-world targets.

© 2001 Optical Society of America

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References

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  1. L. J. Cheng, J. C. Mahoney, G. Reyes, “Target detection using an AOTF hyperspectral imager,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2237, 251–259 (1994).
    [CrossRef]
  2. J. S. Tyo, M. P. Rowe, E. N. Pugh, N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996).
    [CrossRef] [PubMed]
  3. M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
    [CrossRef]
  4. P.-Y. Gerligand, R. A. Chipman, E. A. Sornsin, M. H. Smith, “Polarization signatures of spherical and conical targets measured by Mueller matrix imaging polarimetry,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 63–73 (1997).
    [CrossRef]
  5. T. S. Turner, M. R. Hawks, “Ruggedized portable Fourier transform spectrometer for hyperspectral imaging applications,” in Remote Sensing for Agriculture, Forestry, and Natural Resources, E. T. Engman, G. Guyot, M. Marino, eds., Proc. SPIE2585, 222–232 (1995).
    [CrossRef]
  6. T. S. Turner, K. W. Peters, J. S. Tyo, “Portable visible imaging spectro-polarimeter for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
    [CrossRef]
  7. J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. Descour, S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
    [CrossRef]
  8. T. H. Barnes, “Photodiode array Fourier transform spectrometer with improved dynamic range,” Appl. Opt. 24, 3702–3706 (1985).
    [CrossRef] [PubMed]
  9. G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
    [CrossRef]
  10. L. J. Rickard, R. Basedow, E. Zalewske, P. R. Silvergate, M. Landers, “HYDICE: an airborne system for hyperspectral imaging,” in Imaging Spectrometry for Terrestrial Environments, G. Vane, ed., Proc. SPIE1937, 173–179 (1993).
    [CrossRef]
  11. See, for example, R. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1996), Vol. 2, pp. 22.10–22.12.
  12. L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
    [CrossRef]
  13. J. S. Tyo, E. N. Pugh, N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15, 367–374 (1998).
    [CrossRef]
  14. D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
    [CrossRef]
  15. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable retardance-polarimeters,” Opt. Lett. 25, 1198–1200 (2000).
    [CrossRef]
  16. G. D. Bernard, R. Wehner, “Functional similarities between polarization and color vision,” Vision Res. 17, 1019–1028 (1977).
    [CrossRef]

2000

1998

1997

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

1996

1993

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

1985

1977

G. D. Bernard, R. Wehner, “Functional similarities between polarization and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef]

Barnes, T. H.

Basedow, R.

L. J. Rickard, R. Basedow, E. Zalewske, P. R. Silvergate, M. Landers, “HYDICE: an airborne system for hyperspectral imaging,” in Imaging Spectrometry for Terrestrial Environments, G. Vane, ed., Proc. SPIE1937, 173–179 (1993).
[CrossRef]

Bernard, G. D.

G. D. Bernard, R. Wehner, “Functional similarities between polarization and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef]

Cheng, L. J.

L. J. Cheng, J. C. Mahoney, G. Reyes, “Target detection using an AOTF hyperspectral imager,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2237, 251–259 (1994).
[CrossRef]

Chipman, R.

See, for example, R. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1996), Vol. 2, pp. 22.10–22.12.

Chipman, R. A.

P.-Y. Gerligand, R. A. Chipman, E. A. Sornsin, M. H. Smith, “Polarization signatures of spherical and conical targets measured by Mueller matrix imaging polarimetry,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 63–73 (1997).
[CrossRef]

Chrien, T. G.

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

Dereniak, E.

Descour, M. R.

Engheta, N.

Enmark, H. T.

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

Franklin, A.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Fronterhouse, D. S.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Gerligand, P.-Y.

P.-Y. Gerligand, R. A. Chipman, E. A. Sornsin, M. H. Smith, “Polarization signatures of spherical and conical targets measured by Mueller matrix imaging polarimetry,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 63–73 (1997).
[CrossRef]

Green, R. O.

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

Grotbeck, R.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Hansen, E. G.

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

Hawks, M. R.

T. S. Turner, M. R. Hawks, “Ruggedized portable Fourier transform spectrometer for hyperspectral imaging applications,” in Remote Sensing for Agriculture, Forestry, and Natural Resources, E. T. Engman, G. Guyot, M. Marino, eds., Proc. SPIE2585, 222–232 (1995).
[CrossRef]

Kemme, S. A.

Landers, M.

L. J. Rickard, R. Basedow, E. Zalewske, P. R. Silvergate, M. Landers, “HYDICE: an airborne system for hyperspectral imaging,” in Imaging Spectrometry for Terrestrial Environments, G. Vane, ed., Proc. SPIE1937, 173–179 (1993).
[CrossRef]

Mahoney, J. C.

L. J. Cheng, J. C. Mahoney, G. Reyes, “Target detection using an AOTF hyperspectral imager,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2237, 251–259 (1994).
[CrossRef]

Meigs, A. D.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Otten, L. J.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Peters, K. W.

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable visible imaging spectro-polarimeter for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

Phipps, G. S.

Porter, W. M.

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

Pugh, E. N.

Rafert, J. B.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Reyes, G.

L. J. Cheng, J. C. Mahoney, G. Reyes, “Target detection using an AOTF hyperspectral imager,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2237, 251–259 (1994).
[CrossRef]

Rickard, L. J.

L. J. Rickard, R. Basedow, E. Zalewske, P. R. Silvergate, M. Landers, “HYDICE: an airborne system for hyperspectral imaging,” in Imaging Spectrometry for Terrestrial Environments, G. Vane, ed., Proc. SPIE1937, 173–179 (1993).
[CrossRef]

Robinson, M. W.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Rowe, M. P.

Sabatke, D. S.

Sears, R. D.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

Silvergate, P. R.

L. J. Rickard, R. Basedow, E. Zalewske, P. R. Silvergate, M. Landers, “HYDICE: an airborne system for hyperspectral imaging,” in Imaging Spectrometry for Terrestrial Environments, G. Vane, ed., Proc. SPIE1937, 173–179 (1993).
[CrossRef]

Silverman, M. P.

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Smith, M. H.

P.-Y. Gerligand, R. A. Chipman, E. A. Sornsin, M. H. Smith, “Polarization signatures of spherical and conical targets measured by Mueller matrix imaging polarimetry,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 63–73 (1997).
[CrossRef]

Sornsin, E. A.

P.-Y. Gerligand, R. A. Chipman, E. A. Sornsin, M. H. Smith, “Polarization signatures of spherical and conical targets measured by Mueller matrix imaging polarimetry,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 63–73 (1997).
[CrossRef]

Strange, W.

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Sweatt, W. C.

Turner, T. S.

J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. Descour, S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
[CrossRef]

T. S. Turner, M. R. Hawks, “Ruggedized portable Fourier transform spectrometer for hyperspectral imaging applications,” in Remote Sensing for Agriculture, Forestry, and Natural Resources, E. T. Engman, G. Guyot, M. Marino, eds., Proc. SPIE2585, 222–232 (1995).
[CrossRef]

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable visible imaging spectro-polarimeter for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

Tyo, J. S.

J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable retardance-polarimeters,” Opt. Lett. 25, 1198–1200 (2000).
[CrossRef]

J. S. Tyo, E. N. Pugh, N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15, 367–374 (1998).
[CrossRef]

J. S. Tyo, M. P. Rowe, E. N. Pugh, N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996).
[CrossRef] [PubMed]

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable visible imaging spectro-polarimeter for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. Descour, S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
[CrossRef]

Vane, G.

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

Wehner, R.

G. D. Bernard, R. Wehner, “Functional similarities between polarization and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef]

Zalewske, E.

L. J. Rickard, R. Basedow, E. Zalewske, P. R. Silvergate, M. Landers, “HYDICE: an airborne system for hyperspectral imaging,” in Imaging Spectrometry for Terrestrial Environments, G. Vane, ed., Proc. SPIE1937, 173–179 (1993).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase-modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Opt. Lett.

Remote Sens. Environ.

G. Vane, R. O. Green, T. G. Chrien, H. T. Enmark, E. G. Hansen, W. M. Porter, “The airborne visible/infrared imaging spectrometer (AVIRIS),” Remote Sens. Environ. 44, 127–143 (1993).
[CrossRef]

Vision Res.

G. D. Bernard, R. Wehner, “Functional similarities between polarization and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef]

Other

L. J. Rickard, R. Basedow, E. Zalewske, P. R. Silvergate, M. Landers, “HYDICE: an airborne system for hyperspectral imaging,” in Imaging Spectrometry for Terrestrial Environments, G. Vane, ed., Proc. SPIE1937, 173–179 (1993).
[CrossRef]

See, for example, R. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1996), Vol. 2, pp. 22.10–22.12.

L. J. Otten, A. D. Meigs, A. Franklin, R. D. Sears, M. W. Robinson, J. B. Rafert, D. S. Fronterhouse, R. Grotbeck, “On board spectral imager data processor,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 86–94 (1999).
[CrossRef]

L. J. Cheng, J. C. Mahoney, G. Reyes, “Target detection using an AOTF hyperspectral imager,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2237, 251–259 (1994).
[CrossRef]

P.-Y. Gerligand, R. A. Chipman, E. A. Sornsin, M. H. Smith, “Polarization signatures of spherical and conical targets measured by Mueller matrix imaging polarimetry,” in Polarization Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 63–73 (1997).
[CrossRef]

T. S. Turner, M. R. Hawks, “Ruggedized portable Fourier transform spectrometer for hyperspectral imaging applications,” in Remote Sensing for Agriculture, Forestry, and Natural Resources, E. T. Engman, G. Guyot, M. Marino, eds., Proc. SPIE2585, 222–232 (1995).
[CrossRef]

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable visible imaging spectro-polarimeter for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. Descour, S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic of the optical setup of the VRFTSP. The polarimeter (VR1, VR2, LP) is placed in front of the spectrometer (I, L1, L2). The plane of the page represents the interferogram dimension. The cylinder lens (L2) forms an image in the direction perpendicular to the page.

Fig. 2
Fig. 2

(a) Representative spatiospectral image obtained with the slit at the center scan location (clear plastic) in Fig. 4(a). The optical system creates an interferogram in the horizontal direction and an image in the vertical, so each row in the image gives the interferogram at that vertical position within the slit. (b) and (c) The value of the interferogram along the black line in (a). (d) Absolute value of the Fourier transform.

Fig. 3
Fig. 3

Wavelength calibration process for the VRFTSP. We calibrated the spectral dimension by removing the cylinder lens L2 in Fig. 1 and allowing the interferogram to take up the entire FPA. (a) and (b) Interferograms obtained by use of 10-nm FWHM interference filters. (c) The white-light spectrum had no filter. (d) The average spectrum for each image.

Fig. 4
Fig. 4

Schematic of the target configurations. The dotted area corresponds to the portion of the scene that is imaged. The solid rectangle indicates the projection of the slit onto the target. The three plastics composing the stack of cylinders were dyed red, clear, and blue from left to right.

Fig. 5
Fig. 5

Spatiospectral images of reconstructed s 0 at three horizontal scan locations on the stack of cylinders. The data correspond to the vertical lines on the inset: (a), (b) red plastic (leftmost location); (c), (d) clear (center location); (e), (f) blue (rightmost location). The bottom row of images is a two-dimensional projection of the three-dimensional data presented in the top row.

Fig. 6
Fig. 6

Transmissivity data. We determined the transmissivity of the three plastics by taking the ratio between the 150th row and the 50th row at a single scan location for each color. The 50th row was well within the background, and the 150th row was well within the cylinder. The curves correspond to spectrophotometric measurements used for verification. Transmissivities greater than 100% result from the fact that data from different vertical positions within the image were used to infer the actual value. To account for the variations in thickness of the sample (because of curvature), a one-parameter least-squares fit was used to normalize the data obtained with the VRFTSP to match the curves from the spectrophotometer.

Fig. 7
Fig. 7

Polarimetric data from the center section of the stack of cylinders. (a) and (b), s 1/s 0; (c) and (d), s 2/s 0; and (e) and (f), s 3/s 0. The data are presented between only 500 and 700 nm because the SNR is too low outside of this range. Note that the polarization signature of the target is wavelength independent, as should be expected.

Fig. 8
Fig. 8

Single-wavelength images of the stack of cylinders. Each row corresponds to the wavelength notations, and each column presents Stokes parameter images. All images are presented with the same gray scale. For s 1, light shades are partially horizontally polarized, and dark shades are partially vertically polarized. For s 2, light shades are 45° linearly polarized. For s 3, light shades are left-circularly polarized, whereas dark shades are right-circularly polarized. Noisy regions of the images correspond to areas with low values of s 0. These small values appear in the denominator of the normalized images.

Fig. 9
Fig. 9

Spatiospectral scans of s 0 on (a) the clear marble and (b) the green marble. The clear data are from the right scan line indicated on the insert, and the green data are from the left scan line. The decrease in intensity at approximately 3 mm in (a) is due to an inclusion in the glass marble.

Fig. 10
Fig. 10

Single-wavelength polarimetric images of the green and clear marbles. The data are from λ = 530 nm. From left to right the images are normalized s 0, s 1/s 0, s 2/s 0, and s 3/s 0.

Fig. 11
Fig. 11

Three-dimensional polarimetric image depicting the angle of linear polarization and the degree of linear polarization by use of the colorimetric parameters of hue and saturation, respectively. The color red corresponds to vertical polarization and cyan blue corresponds to horizontal. Highly saturated colors appear where the degree of polarization is high. These data are from the single-wavelength images at λ = 650 nm.

Equations (6)

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

Sout=s0,out s1,out s2,out s3,outT=MLP·MVR2·MVR1·Si,
Si=s0 s1 s2 s3T
MLP=12q+rq-r00q-rq+r00002qr00002qr,
MVR1Δ=10000cosΔ0-sinΔ00100sinΔ0cosΔ,
MVR2δ=10000121+cosδ121-cosδ-12sinδ0121-cosδ121+cosδ12sinδ012sinδ-12sinδcosδ,
s0,out=s012q+r+s112q-r121+cos δcos Δ-12sin δ sin Δ+s214q-r1-cos δ+s312q-r121+cos δsin Δ+12sin δ sin Δ.

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