Calibration method of microgrid polarimeters with image interpolation Zhenyue Chen,1,2 Xia Wang,1,* and Rongguang Liang2 1
Beijing Institute of Technology, Key Laboratory of Photoelectronic Imaging Technology and Systems of Ministry of Education of China, School of Optoelectronics, Beijing 100081, China 2
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA *Corresponding author: [email protected]
Received 28 October 2014; revised 18 December 2014; accepted 18 December 2014; posted 22 December 2014 (Doc. ID 225817); published 2 February 2015
Microgrid polarimeters have large advantages over conventional polarimeters because of the snapshot nature and because they have no moving parts. However, they also suffer from several error sources, such as fixed pattern noise (FPN), photon response nonuniformity (PRNU), pixel cross talk, and instantaneous field-of-view (IFOV) error. A characterization method is proposed to improve the measurement accuracy in visible waveband. We first calibrate the camera with uniform illumination so that the response of the sensor is uniform over the entire field of view without IFOV error. Then a spline interpolation method is implemented to minimize IFOV error. Experimental results show the proposed method can effectively minimize the FPN and PRNU. © 2015 Optical Society of America OCIS codes: (110.5405) Polarimetric imaging; (120.5410) Polarimetry; (230.5440) Polarizationselective devices; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.54.000995
1. Introduction
Imaging polarimetry is an emerging technology with increasing applications in the fields of remote sensing, biosciences, medicine, machine vision, and astronomy across a wide range of wavelengths, from ultraviolet to infrared [1]. Generally, imaging polarimeters can be divided into two categories: division of time (DoT) polarimeters and snapshot polarimeters. Conventional DoT polarimeters with rotating retarders and polarizers require motorized stages that often introduce vibrations, while increasing system volume and complexity. The snapshot polarimeters can be further categorized into channeled imaging polarimeters (CIPs) [1,2], division of amplitude (DoAM) polarimeters [3,4], division of aperture (DoA) polarimeters [5], and division of focal plane (DoFP) polarimeters [6–8].
1559-128X/15/050995-07$15.00/0 © 2015 Optical Society of America
For the DoFP polarimeters, also known as microgrid polarimeters, the snapshot nature and lack of moving parts eliminate the problems in conventional polarimetry, such as vibration, spurious signals, beam wander, and the need for registration routines. However, their performance is often limited by several error sources, such as the fixed pattern noise (FPN), photon response nonuniformity (PRNU), nonuniformity in the micropolarizer extinction ratio, micropolarizer orientation misalignment, pixel cross talk, and instantaneous field-of-view (IFOV) error. All of these factors contribute to polarimetric inaccuracy; particularly, pixel crosstalk will contribute to polarimetric loss and IFOV error will cause strong edge artifacts. Thus, calibration is needed to minimize these errors. Powell and Gruev described two calibration techniques tailored to mitigate variations in the optical response between pixelated-polarization filters across an imaging array [9]. Bowers et al. investigated the calibration method for long wavelength infrared microgrid polarimeters [10]. Ratliff 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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and co-workers investigated the IFOV error of microgrid polarimeters and the interpolation strategies to reduce it [11,12]. York and Gruev presented a system of measurements to study the alignment, transmission, and contrast ratios of the micropolarizer array and showed how these factors affect microgrid polarimeter performance [8]. In this paper, we propose a calibration method to minimize PRNU and FPN noise, as well as the interference pattern in the microgrid polarimeter. For a 2 × 2 super pixel, the pixel offsets and correction coefficient are calculated pixel by pixel so that the polarimeter has an almost ideal response to the incident light with a non-ideal micro-polarizer array. For each single pixel, a weighted average is carried out to its surrounding four super pixels. Spline interpolation is then implemented to remove the IFOV error. Experimental results show that the proposed method can remove FPN and PRNU noise, as well as blind and dead pixels. In addition, it can reduce the IFOV error while maintaining high spatial resolution. 2. Principles A.
FPN and PRNU Noise Calibration
The microgrid polarimeter in this study is the PolarCam from 4D Technology Inc. The micropolarizer array in this camera utilizes nanoscale patterning to form a metal grating with subwavelength spacing on a thin transparent glass substrate [13]. The micropolarizer array has the pattern shown in Fig. 1. The size of the linear polarizer is the same as that of the detector pixel. Linear polarizers with four different transmission axis orientations, i.e., 0, 45, 90, and 135 deg, are adjacent to each other and combine together to form a superpixel. The polarization state of the incident light can be represented by Stokes vector Sin S0 ; S1 ; S2 ; S3 T. Since the microgrid polarization camera is an incomplete Stokes imaging polarimeter, only linear polarized light can be analyzed. The input Stokes vector is revised as Sin S0 ; S1 ; S2 ; 0T . The Mueller matrix of the linear analyzer with its transmission axis oriented at angle θ can be described by LPθ:
2
03 sin 2θ cos 2θ 0 7 cos2 2θ 16 6 cos 2θ 7 LPθ 6 7: (1) 2 4 sin 2θ sin 2θ cos 2θ 05 sin2 2θ 1
cos 2θ
sin 2θ
0
0
0
0
Since the detector can measure only the flux, only the first row in the analyzer’s Mueller matrix is used to calculate the transmitted flux. For a superpixel, the ideal measurement matrix of the four linear analyzers, W ideal , can be described by 2
W ideal
1 16 1 6 4 2 1 1
1 0 −1 0
3 0 07 7: 05 0
0 1 0 −1
(2)
The output intensity for a superpixel is described by P W ideal Sin :
(3)
Taking the FPN and PRUN noise into consideration, the measurement matrix W is revised by 2
w01 6 w451 W; d 6 4 w901 w1351
w02 w452 w902 w1352
w03 w453 w903 w1353
w04 w454 w904 w1354
3 d0 d45 7 7; d90 5 d135 (4)
in which w0, w45, w90, and w135 are the first rows of the corresponding Mueller matrix of each linear analyzer, and d0, d45, d90, and d135 are the corresponding pixel offsets in the superpixel. The input Stokes vector is consequently revised by Sˆ in S0 ; S1 ; S2 ; 0; 1T :
(5)
S0 can be estimated by the mean value of the S0 Stokes vector image. Consequently, S1 and S2 are calculated as S1 DoLP · S0 · cos2θ; S2 DoLP · S0 · sin2θ;
(6)
in which DoLP is the degree of linear polarization of the incident light and θ is the orientation of the polarizer from the x axis. Substituting Eqs. (4) and (5) into Eq. (3), we get P W; d · Sˆ in :
(7)
W and d can then be acquired by W; d P · S−1 in p ; Fig. 1. Schematic of the micropolarizer array. 996
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(8)
in which S−1 in p represents the pseudo-inverse of Sin . If we use the subscript i; j to describe the location of
Fig. 2. RMSE of different interpolation methods.
the superpixel on the focal plane array (FPA), Eq. (8) can be revised by W; di; j Pi; j · S−1 in p i; j; i 1; 2; 3…m − 1;
j 1; 2; 3…n − 1; (9)
where m, n are the height and width of the FPA, respectively. For an overlapped single pixel with a certain polarization orientation, the offset values in the surrounding superpixels are the same. With known illumination polarization state and the measured matrix W of a superpixel, the correction matrix C can be acquired by Ci; j W ideal · W −1 p i; j; i 1; 2; 3…m − 1;
j 1; 2; 3…n − 1;
(10)
where W −1 p represents the pseudo-inverse of W. The corrected super pixel value can be described by ~ j Ci; j · Pi; j − di; j; Pi; i 1; 2; 3…m − 1;
j 1; 2; 3…n − 1:
(11)
Each single pixel with a certain analyzer orientation has four neighbor superpixels that all contribute to its output intensity, as shown in Fig. 1, so the final pixel value is acquired by employing a weighted average:
there will be aliasing errors. Tyo and co-workers demonstrated that spatially band-limited polarization images could be reconstructed with no IFOV error by using a linear system framework [11,12]. A series of targets with spatially varying intensity sinusoidal patterns are generated to investigate the performance of different interpolation methods to remove the IFOV error [14]. Since we focus on the impact of the IFOV error on the retrieved polarization information, the DoLP of the target is uniform and the angle of linear polarization (AoLP) is set at a constant value. The spatial frequencies of the sinusoidal patterns range from 0 to the highest Nyquist sampling frequency in microgrid polarimeter. For traditional imaging sensors, the highest Nyquist sampling frequency is 0.5 cycles per pixel. However for the microgrid polarimeter, one superpixel is comprised of 2 × 2 pixels, so its Nyquist frequency is reduced by a half. Although a spatial frequency slightly higher than 0.25 cycles per pixel can be reconstructed, it contributes less to IFOV error [11]. Therefore, we focus on the spatial frequency from 0 to 0.25 cycles per pixel. When the DoLP is set at 1 and the AoLP is arbitrarily set at 40 deg, root mean square errors (RMSEs) are calculated between the truth images and interpolated images; the results are shown in Fig. 2. The three most commonly used interpolation methods, i.e., the “spline,” “bicubic,” and “bilinear” methods, are implemented on the four
~ − 1; j − 1 Pi ~ − 1; j Ii; j 1∕4Pi ~ j − 1 Pi; ~ j; Pi; i 2; 3…m; B.
j 2; 3…n:
(12)
IFOV Error Calibration
The four pixels in a superpixel have different IFOVs, i.e., the four micropolarizers analyze the polarization states of different spatial points. If these values are employed to compute the polarization information,
Fig. 3. Experimental setup. 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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Fig. 4. Data plots before and after calibration. (a) and (b) are the mean pixel value in each polarization channel before and after calibration, (c) is the DoLP, (d) is the standard deviation of the DoLP, (e) is the AoLP, and (f) is the standard deviation of the AoLP. (g) is the reconstructed S0 , and (h) is the standard deviation of S0 .
linear polarization images extracted from the raw microgrid image, and the reconstructed S0 and DoLP images are compared. Figure 2(a) shows the RMSE of the reconstructed S0 image, and Fig. 2(b) shows the 998
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RMSE of the reconstructed DoLP image. It can be seen that the reconstructed images without interpolation have higher RMSE due to IFOV error. Among the three interpolation methods, the spline method
Fig. 5. Uniform images before and after calibration. (a) and (b) correspond to S0 image before and after calibration. (c) and (d) correspond to the DoLP image before and after calibration. (e) and (f) correspond to the AoLP image before and after calibration.
has the smallest RMSE and, thus, has highest accuracy. Therefore, in this paper, spline interpolation is implemented to remove IFOV error. 3. Experimental Setup and Results
To calibrate the microgrid polarimeter, a series of uniform illumination with specified input Stokes vectors is needed. The input Stokes vectors should cover different AoLP. In addition, the S0 element in the Stokes vector should have a large dynamic range
so that the calibration can be implemented under different illumination. The experimental setup is shown in Fig. 3. The light from the xenon arc lamp first passes through a variable neutral density filter and then a bandpass filter before entering the 4 in. integrating sphere. A linear polarizer with an extinction ratio of 10; 000∶1 is employed to generate the input Stokes vector. It is mounted on a Thorlabs motorized rotation stage with absolute on-axis accuracy of 0.1% and bidirectional repeatability of 0.1 deg . 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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The calibration is carried out with seven groups of input Stokes vectors; the illumination intensity decreases for each group. The microgrid polarimeter captures the images with 12 bit raw data. In each group, the linear polarizer is rotated from 90 to −90 deg with an increment of 15 deg. We use pixel value to represent the relative intensity of incident light. When a narrowband light source is used, the light reflected from the micropolarizer array and the sensor may introduce interference patterns. In the experiment, a 650 nm bandpass filter with 10 nm FWHM is employed to show that the calibration method can effectively overcome the interference pattern. Figures 4(a) and 4(b) show the mean pixel values of the four polarization channels before and after calibration, respectively. The falloff of the mean pixel values reflects the illumination level in each group. It can be seen that, after calibration, all four channels have the same responses and the crosspolarization channels have almost ideal responses. Figures 4(c) and 4(d) show DoLP and its standard deviation, respectively. A DoLP slightly over 1 means
that the images are slightly overcalibrated. Normalization can be further employed to solve the problem. Figures 4(e) and 4(f) show the AoLP and its standard deviation, respectively. All measured parameters are improved after calibration. The error between the acquired DoLP and its ground truth value is in a range from 0.06 to 0.16 before calibration. It drops to within a range from −0.02 to 0.02 after calibration. The error between the acquired AoLP and its ground truth value before calibration is in a range from −2 to 2 deg. It drops to within a range from −0.5 to 0.5 deg after calibration. The fluctuations in Figs. 4(c), 4(d), and 4(f) are due to the orientations of the four pixel polarizers in the polarizer array, i.e., 0, 45, 90, and 135 deg. We have demonstrated this phenomenon in our previous study [15]. The increases in standard deviation in Figs. 4(d) and 4(f) are due to the decreasing image SNR because the illumination intensity decreases gradually for the measurements from group 1 to group 7. Figures 4(g) and 4(h) show the intensity of the reconstructed S0 image and its standard deviation, respectively. The deviation is greatly reduced after calibration, which means the
Fig. 6. Human tonsil tissue images captured with the microgrid polarimeter. (a) and (b) are the S0 images, and (c) and (d) are the DoLP images before and after calibration. 1000
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noise of the camera is reduced. After calibration, both the correction matrix for each superpixel and the offset for each single pixel are acquired. Figures 5(a) and 5(b) are S0 images before and after calibration. Almost all of the interference pattern is removed with calibration; the difference between the flat illumination and the interference pattern is reduced to less than 2% compared to the original 30% before calibration. Figures 5(c) and 5(d) correspond to DoLP images before and after calibration. The corresponding error compared to its ground truth image is in a range from −0.4 to 0.3 before calibration. After calibration, it is within a range from −0.06 to 0.03. Figures 5(e) and 5(f) are AoLP images before and after calibration. The corresponding error compared to its ground truth image is in a range from −15 to 15 deg before calibration. After calibration it is within a range from −1.5 to 1.5 deg. It can be seen that, after calibration, the PRNU and FPN are greatly minimized. Due to pixel cross talk and low extinction ratio of the micropolarizer array, the DoLP value is approximately 0.9. Our calibration method is able to recover DoLP to approximately 1.0. For AoLP images, the mean value is almost the same before and after calibration. However, the AoLP image is much more uniform after calibration. To demonstrate the effectiveness of the proposed calibration methods, we image a human tonsil microscope slide with a 0.2 NA objective. A uniform linearly polarized white light illuminates the microscope slide in transmission mode. Figures 6(a) and 6(c) are the S0 image and the DoLP image, respectively before calibration; Figs. 6(b) and 6(d) are the corresponding S0 image and DoLP after calibration. It can be seen that the top left region in the S0 image has fluctuation in intensity distribution due to the nonuniformity response; the calibration method is able to remove this fluctuation. Meanwhile, the contrast in the top right region in the S0 image is improved, as well. For the DoLP image, the contrast after calibration is greatly enhanced. The features in the top and center regions can be detected clearly after calibration. 4. Conclusions
In this paper we propose a calibration method for microgrid polarimeters in the visible waveband. Pixel value offsets and correction matrices for each superpixel are computed to remove the PRNU and FPN noise of the polarimeter. Comparisons of S0, DoLP, and AoLP images before and after calibration show that the proposed method is effective and useful. Three interpolation methods, i.e., spline, bicubic,
and bilinear, are investigated to remove the IFOV error. The final experimental results show that the image contrast is enhanced and the sensor has a more uniform response with the proposed calibration method. This work is partially supported by a grant of the China Scholarship Council (No. 201306030018) and the National Natural Science Foundation of China (No. 61231014). We thank Dr. S. Bear Powell and Shaun Pacheco for helpful discussions about the calibration method. References 1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006). 2. M. W. Kudenov, M. J. Escuti, E. L. Dereniak, and K. Oka, “White-light channeled imaging polarimeter using broadband polarization gratings,” Appl. Opt. 50, 2283–2293 (2011). 3. E. Compain and B. Drevillon, “Broadband division-ofamplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998). 4. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982). 5. J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE 5888, 58880V (2005). 6. W. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22, 3063–3074 (2014). 7. G. Myhre, W. Hsu, A. Peinado, C. LaCasse, N. Brock, R. A. Chipman, and S. Pau, “Liquid crystal polymer full-Stokes division of focal plane polarimeter,” Opt. Express 20, 27393–27409 (2012). 8. T. York and V. Gruev, “Characterization of a visible spectrum division-of-focal-plane polarimeter,” Appl. Opt. 51, 5392–5400 (2012). 9. S. B. Powell and V. Gruev, “Calibration methods for divisionof-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013). 10. D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, W. T. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47, 46403 (2008). 11. B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express 17, 9112–9125 (2009). 12. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. 34, 3187–3189 (2009). 13. N. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE 8160, 81600W (2011). 14. S. Gao and V. Gruev, “Bilinear and bicubic interpolation methods for division of focal plane polarimeters,” Opt. Express 19, 26161–26173 (2011). 15. Z. Chen, X. Wang, S. Pacheco, and R. Liang, “Impact of CCD camera SNR on polarimetric accuracy,” Appl. Opt. 53, 7649–7656 (2014).
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Beijing Institute of Technology, Key Laboratory of Photoelectronic Imaging Technology and Systems of Ministry of Education of China, School of Optoelectronics, Beijing 100081, China 2
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA *Corresponding author: [email protected]
Received 28 October 2014; revised 18 December 2014; accepted 18 December 2014; posted 22 December 2014 (Doc. ID 225817); published 2 February 2015
Microgrid polarimeters have large advantages over conventional polarimeters because of the snapshot nature and because they have no moving parts. However, they also suffer from several error sources, such as fixed pattern noise (FPN), photon response nonuniformity (PRNU), pixel cross talk, and instantaneous field-of-view (IFOV) error. A characterization method is proposed to improve the measurement accuracy in visible waveband. We first calibrate the camera with uniform illumination so that the response of the sensor is uniform over the entire field of view without IFOV error. Then a spline interpolation method is implemented to minimize IFOV error. Experimental results show the proposed method can effectively minimize the FPN and PRNU. © 2015 Optical Society of America OCIS codes: (110.5405) Polarimetric imaging; (120.5410) Polarimetry; (230.5440) Polarizationselective devices; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.54.000995
1. Introduction
Imaging polarimetry is an emerging technology with increasing applications in the fields of remote sensing, biosciences, medicine, machine vision, and astronomy across a wide range of wavelengths, from ultraviolet to infrared [1]. Generally, imaging polarimeters can be divided into two categories: division of time (DoT) polarimeters and snapshot polarimeters. Conventional DoT polarimeters with rotating retarders and polarizers require motorized stages that often introduce vibrations, while increasing system volume and complexity. The snapshot polarimeters can be further categorized into channeled imaging polarimeters (CIPs) [1,2], division of amplitude (DoAM) polarimeters [3,4], division of aperture (DoA) polarimeters [5], and division of focal plane (DoFP) polarimeters [6–8].
1559-128X/15/050995-07$15.00/0 © 2015 Optical Society of America
For the DoFP polarimeters, also known as microgrid polarimeters, the snapshot nature and lack of moving parts eliminate the problems in conventional polarimetry, such as vibration, spurious signals, beam wander, and the need for registration routines. However, their performance is often limited by several error sources, such as the fixed pattern noise (FPN), photon response nonuniformity (PRNU), nonuniformity in the micropolarizer extinction ratio, micropolarizer orientation misalignment, pixel cross talk, and instantaneous field-of-view (IFOV) error. All of these factors contribute to polarimetric inaccuracy; particularly, pixel crosstalk will contribute to polarimetric loss and IFOV error will cause strong edge artifacts. Thus, calibration is needed to minimize these errors. Powell and Gruev described two calibration techniques tailored to mitigate variations in the optical response between pixelated-polarization filters across an imaging array [9]. Bowers et al. investigated the calibration method for long wavelength infrared microgrid polarimeters [10]. Ratliff 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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and co-workers investigated the IFOV error of microgrid polarimeters and the interpolation strategies to reduce it [11,12]. York and Gruev presented a system of measurements to study the alignment, transmission, and contrast ratios of the micropolarizer array and showed how these factors affect microgrid polarimeter performance [8]. In this paper, we propose a calibration method to minimize PRNU and FPN noise, as well as the interference pattern in the microgrid polarimeter. For a 2 × 2 super pixel, the pixel offsets and correction coefficient are calculated pixel by pixel so that the polarimeter has an almost ideal response to the incident light with a non-ideal micro-polarizer array. For each single pixel, a weighted average is carried out to its surrounding four super pixels. Spline interpolation is then implemented to remove the IFOV error. Experimental results show that the proposed method can remove FPN and PRNU noise, as well as blind and dead pixels. In addition, it can reduce the IFOV error while maintaining high spatial resolution. 2. Principles A.
FPN and PRNU Noise Calibration
The microgrid polarimeter in this study is the PolarCam from 4D Technology Inc. The micropolarizer array in this camera utilizes nanoscale patterning to form a metal grating with subwavelength spacing on a thin transparent glass substrate [13]. The micropolarizer array has the pattern shown in Fig. 1. The size of the linear polarizer is the same as that of the detector pixel. Linear polarizers with four different transmission axis orientations, i.e., 0, 45, 90, and 135 deg, are adjacent to each other and combine together to form a superpixel. The polarization state of the incident light can be represented by Stokes vector Sin S0 ; S1 ; S2 ; S3 T. Since the microgrid polarization camera is an incomplete Stokes imaging polarimeter, only linear polarized light can be analyzed. The input Stokes vector is revised as Sin S0 ; S1 ; S2 ; 0T . The Mueller matrix of the linear analyzer with its transmission axis oriented at angle θ can be described by LPθ:
2
03 sin 2θ cos 2θ 0 7 cos2 2θ 16 6 cos 2θ 7 LPθ 6 7: (1) 2 4 sin 2θ sin 2θ cos 2θ 05 sin2 2θ 1
cos 2θ
sin 2θ
0
0
0
0
Since the detector can measure only the flux, only the first row in the analyzer’s Mueller matrix is used to calculate the transmitted flux. For a superpixel, the ideal measurement matrix of the four linear analyzers, W ideal , can be described by 2
W ideal
1 16 1 6 4 2 1 1
1 0 −1 0
3 0 07 7: 05 0
0 1 0 −1
(2)
The output intensity for a superpixel is described by P W ideal Sin :
(3)
Taking the FPN and PRUN noise into consideration, the measurement matrix W is revised by 2
w01 6 w451 W; d 6 4 w901 w1351
w02 w452 w902 w1352
w03 w453 w903 w1353
w04 w454 w904 w1354
3 d0 d45 7 7; d90 5 d135 (4)
in which w0, w45, w90, and w135 are the first rows of the corresponding Mueller matrix of each linear analyzer, and d0, d45, d90, and d135 are the corresponding pixel offsets in the superpixel. The input Stokes vector is consequently revised by Sˆ in S0 ; S1 ; S2 ; 0; 1T :
(5)
S0 can be estimated by the mean value of the S0 Stokes vector image. Consequently, S1 and S2 are calculated as S1 DoLP · S0 · cos2θ; S2 DoLP · S0 · sin2θ;
(6)
in which DoLP is the degree of linear polarization of the incident light and θ is the orientation of the polarizer from the x axis. Substituting Eqs. (4) and (5) into Eq. (3), we get P W; d · Sˆ in :
(7)
W and d can then be acquired by W; d P · S−1 in p ; Fig. 1. Schematic of the micropolarizer array. 996
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(8)
in which S−1 in p represents the pseudo-inverse of Sin . If we use the subscript i; j to describe the location of
Fig. 2. RMSE of different interpolation methods.
the superpixel on the focal plane array (FPA), Eq. (8) can be revised by W; di; j Pi; j · S−1 in p i; j; i 1; 2; 3…m − 1;
j 1; 2; 3…n − 1; (9)
where m, n are the height and width of the FPA, respectively. For an overlapped single pixel with a certain polarization orientation, the offset values in the surrounding superpixels are the same. With known illumination polarization state and the measured matrix W of a superpixel, the correction matrix C can be acquired by Ci; j W ideal · W −1 p i; j; i 1; 2; 3…m − 1;
j 1; 2; 3…n − 1;
(10)
where W −1 p represents the pseudo-inverse of W. The corrected super pixel value can be described by ~ j Ci; j · Pi; j − di; j; Pi; i 1; 2; 3…m − 1;
j 1; 2; 3…n − 1:
(11)
Each single pixel with a certain analyzer orientation has four neighbor superpixels that all contribute to its output intensity, as shown in Fig. 1, so the final pixel value is acquired by employing a weighted average:
there will be aliasing errors. Tyo and co-workers demonstrated that spatially band-limited polarization images could be reconstructed with no IFOV error by using a linear system framework [11,12]. A series of targets with spatially varying intensity sinusoidal patterns are generated to investigate the performance of different interpolation methods to remove the IFOV error [14]. Since we focus on the impact of the IFOV error on the retrieved polarization information, the DoLP of the target is uniform and the angle of linear polarization (AoLP) is set at a constant value. The spatial frequencies of the sinusoidal patterns range from 0 to the highest Nyquist sampling frequency in microgrid polarimeter. For traditional imaging sensors, the highest Nyquist sampling frequency is 0.5 cycles per pixel. However for the microgrid polarimeter, one superpixel is comprised of 2 × 2 pixels, so its Nyquist frequency is reduced by a half. Although a spatial frequency slightly higher than 0.25 cycles per pixel can be reconstructed, it contributes less to IFOV error [11]. Therefore, we focus on the spatial frequency from 0 to 0.25 cycles per pixel. When the DoLP is set at 1 and the AoLP is arbitrarily set at 40 deg, root mean square errors (RMSEs) are calculated between the truth images and interpolated images; the results are shown in Fig. 2. The three most commonly used interpolation methods, i.e., the “spline,” “bicubic,” and “bilinear” methods, are implemented on the four
~ − 1; j − 1 Pi ~ − 1; j Ii; j 1∕4Pi ~ j − 1 Pi; ~ j; Pi; i 2; 3…m; B.
j 2; 3…n:
(12)
IFOV Error Calibration
The four pixels in a superpixel have different IFOVs, i.e., the four micropolarizers analyze the polarization states of different spatial points. If these values are employed to compute the polarization information,
Fig. 3. Experimental setup. 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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Fig. 4. Data plots before and after calibration. (a) and (b) are the mean pixel value in each polarization channel before and after calibration, (c) is the DoLP, (d) is the standard deviation of the DoLP, (e) is the AoLP, and (f) is the standard deviation of the AoLP. (g) is the reconstructed S0 , and (h) is the standard deviation of S0 .
linear polarization images extracted from the raw microgrid image, and the reconstructed S0 and DoLP images are compared. Figure 2(a) shows the RMSE of the reconstructed S0 image, and Fig. 2(b) shows the 998
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RMSE of the reconstructed DoLP image. It can be seen that the reconstructed images without interpolation have higher RMSE due to IFOV error. Among the three interpolation methods, the spline method
Fig. 5. Uniform images before and after calibration. (a) and (b) correspond to S0 image before and after calibration. (c) and (d) correspond to the DoLP image before and after calibration. (e) and (f) correspond to the AoLP image before and after calibration.
has the smallest RMSE and, thus, has highest accuracy. Therefore, in this paper, spline interpolation is implemented to remove IFOV error. 3. Experimental Setup and Results
To calibrate the microgrid polarimeter, a series of uniform illumination with specified input Stokes vectors is needed. The input Stokes vectors should cover different AoLP. In addition, the S0 element in the Stokes vector should have a large dynamic range
so that the calibration can be implemented under different illumination. The experimental setup is shown in Fig. 3. The light from the xenon arc lamp first passes through a variable neutral density filter and then a bandpass filter before entering the 4 in. integrating sphere. A linear polarizer with an extinction ratio of 10; 000∶1 is employed to generate the input Stokes vector. It is mounted on a Thorlabs motorized rotation stage with absolute on-axis accuracy of 0.1% and bidirectional repeatability of 0.1 deg . 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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The calibration is carried out with seven groups of input Stokes vectors; the illumination intensity decreases for each group. The microgrid polarimeter captures the images with 12 bit raw data. In each group, the linear polarizer is rotated from 90 to −90 deg with an increment of 15 deg. We use pixel value to represent the relative intensity of incident light. When a narrowband light source is used, the light reflected from the micropolarizer array and the sensor may introduce interference patterns. In the experiment, a 650 nm bandpass filter with 10 nm FWHM is employed to show that the calibration method can effectively overcome the interference pattern. Figures 4(a) and 4(b) show the mean pixel values of the four polarization channels before and after calibration, respectively. The falloff of the mean pixel values reflects the illumination level in each group. It can be seen that, after calibration, all four channels have the same responses and the crosspolarization channels have almost ideal responses. Figures 4(c) and 4(d) show DoLP and its standard deviation, respectively. A DoLP slightly over 1 means
that the images are slightly overcalibrated. Normalization can be further employed to solve the problem. Figures 4(e) and 4(f) show the AoLP and its standard deviation, respectively. All measured parameters are improved after calibration. The error between the acquired DoLP and its ground truth value is in a range from 0.06 to 0.16 before calibration. It drops to within a range from −0.02 to 0.02 after calibration. The error between the acquired AoLP and its ground truth value before calibration is in a range from −2 to 2 deg. It drops to within a range from −0.5 to 0.5 deg after calibration. The fluctuations in Figs. 4(c), 4(d), and 4(f) are due to the orientations of the four pixel polarizers in the polarizer array, i.e., 0, 45, 90, and 135 deg. We have demonstrated this phenomenon in our previous study [15]. The increases in standard deviation in Figs. 4(d) and 4(f) are due to the decreasing image SNR because the illumination intensity decreases gradually for the measurements from group 1 to group 7. Figures 4(g) and 4(h) show the intensity of the reconstructed S0 image and its standard deviation, respectively. The deviation is greatly reduced after calibration, which means the
Fig. 6. Human tonsil tissue images captured with the microgrid polarimeter. (a) and (b) are the S0 images, and (c) and (d) are the DoLP images before and after calibration. 1000
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noise of the camera is reduced. After calibration, both the correction matrix for each superpixel and the offset for each single pixel are acquired. Figures 5(a) and 5(b) are S0 images before and after calibration. Almost all of the interference pattern is removed with calibration; the difference between the flat illumination and the interference pattern is reduced to less than 2% compared to the original 30% before calibration. Figures 5(c) and 5(d) correspond to DoLP images before and after calibration. The corresponding error compared to its ground truth image is in a range from −0.4 to 0.3 before calibration. After calibration, it is within a range from −0.06 to 0.03. Figures 5(e) and 5(f) are AoLP images before and after calibration. The corresponding error compared to its ground truth image is in a range from −15 to 15 deg before calibration. After calibration it is within a range from −1.5 to 1.5 deg. It can be seen that, after calibration, the PRNU and FPN are greatly minimized. Due to pixel cross talk and low extinction ratio of the micropolarizer array, the DoLP value is approximately 0.9. Our calibration method is able to recover DoLP to approximately 1.0. For AoLP images, the mean value is almost the same before and after calibration. However, the AoLP image is much more uniform after calibration. To demonstrate the effectiveness of the proposed calibration methods, we image a human tonsil microscope slide with a 0.2 NA objective. A uniform linearly polarized white light illuminates the microscope slide in transmission mode. Figures 6(a) and 6(c) are the S0 image and the DoLP image, respectively before calibration; Figs. 6(b) and 6(d) are the corresponding S0 image and DoLP after calibration. It can be seen that the top left region in the S0 image has fluctuation in intensity distribution due to the nonuniformity response; the calibration method is able to remove this fluctuation. Meanwhile, the contrast in the top right region in the S0 image is improved, as well. For the DoLP image, the contrast after calibration is greatly enhanced. The features in the top and center regions can be detected clearly after calibration. 4. Conclusions
In this paper we propose a calibration method for microgrid polarimeters in the visible waveband. Pixel value offsets and correction matrices for each superpixel are computed to remove the PRNU and FPN noise of the polarimeter. Comparisons of S0, DoLP, and AoLP images before and after calibration show that the proposed method is effective and useful. Three interpolation methods, i.e., spline, bicubic,
and bilinear, are investigated to remove the IFOV error. The final experimental results show that the image contrast is enhanced and the sensor has a more uniform response with the proposed calibration method. This work is partially supported by a grant of the China Scholarship Council (No. 201306030018) and the National Natural Science Foundation of China (No. 61231014). We thank Dr. S. Bear Powell and Shaun Pacheco for helpful discussions about the calibration method. References 1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006). 2. M. W. Kudenov, M. J. Escuti, E. L. Dereniak, and K. Oka, “White-light channeled imaging polarimeter using broadband polarization gratings,” Appl. Opt. 50, 2283–2293 (2011). 3. E. Compain and B. Drevillon, “Broadband division-ofamplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998). 4. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982). 5. J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE 5888, 58880V (2005). 6. W. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22, 3063–3074 (2014). 7. G. Myhre, W. Hsu, A. Peinado, C. LaCasse, N. Brock, R. A. Chipman, and S. Pau, “Liquid crystal polymer full-Stokes division of focal plane polarimeter,” Opt. Express 20, 27393–27409 (2012). 8. T. York and V. Gruev, “Characterization of a visible spectrum division-of-focal-plane polarimeter,” Appl. Opt. 51, 5392–5400 (2012). 9. S. B. Powell and V. Gruev, “Calibration methods for divisionof-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013). 10. D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, W. T. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47, 46403 (2008). 11. B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express 17, 9112–9125 (2009). 12. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. 34, 3187–3189 (2009). 13. N. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE 8160, 81600W (2011). 14. S. Gao and V. Gruev, “Bilinear and bicubic interpolation methods for division of focal plane polarimeters,” Opt. Express 19, 26161–26173 (2011). 15. Z. Chen, X. Wang, S. Pacheco, and R. Liang, “Impact of CCD camera SNR on polarimetric accuracy,” Appl. Opt. 53, 7649–7656 (2014).
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