Alternative linear microgrid polarimeters: design, analysis, and demosaicing considerations

Bradley M. Ratliff and Garrett C. Sargent

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Bradley M. Ratliff and Garrett C. Sargent, "Alternative linear microgrid polarimeters: design, analysis, and demosaicing considerations," Appl. Opt. 60, 5805-5818 (2021)

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Table of Contents Category
- Imaging Systems, Image Processing, and Displays
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About this Article
History
- Original Manuscript: April 12, 2021
- Revised Manuscript: June 4, 2021
- Manuscript Accepted: June 7, 2021
- Published: July 6, 2021

Abstract

Linear division of focal plane (DoFP), or integrated microgrid polarimeters, provide a measurement strategy for obtaining time-synchronized polarized intensity measurements across a scene. This is accomplished by masking pixels in the focal plane array sensor with a repeating pattern of different linear polarizers. The convention in industry has been to use a repeating $2 \times 2$ pattern of four linear polarizers with chosen polarizer orientation angles of ${\{0,45,90,135\} ^ \circ}$. Alternative designs based upon other $P \times Q$ modulation patterns have been proposed that demonstrate improved performance over conventional microgrid arrays due to better utilization of bandwidth in the frequency domain. Here, we develop a model for linear DoFP snapshot polarimeters that provides an in-depth understanding of these devices in both the spatial and frequency domains and relate this model to previously reported generalized DoFP channeled polarimeter models. We then use the model to identify practical modulation patterns and study their performance through empirical simulations based upon data collected from real polarimeters. We demonstrate the validity of the developed model and compare the performance of the identified modulation schemes against a common set of ground truth images. We find that choosing alternative sets of polarizer angles, in conjunction with modulators that improve bandwidth usage, result in the best overall designs that can improve performance over conventional microgrid polarimeters.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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$P \times Q$	$C$	$(u, v)$
$2 \times 2$	$[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 1 & - 1 & 1 & - 1 \\ 1 & - 1 & - 1 & 1 \end{array}]$	$(\begin{matrix} (0, 0) \\ (0, π) \\ (π, 0) \\ (π, π) \end{matrix})$
$3 \times 2$	$[\begin{array}{cccccc} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & - 1 & - 1 & - 1 \\ 1 & e^{- j \frac{2 π}{3}} & e^{j \frac{2 π}{3}} & 1 & e^{- j \frac{2 π}{3}} & e^{j \frac{2 π}{3}} \\ 1 & e^{j \frac{2 π}{3}} & e^{- j \frac{2 π}{3}} & 1 & e^{j \frac{2 π}{3}} & e^{- j \frac{2 π}{3}} \\ 1 & e^{- j \frac{2 π}{3}} & e^{j \frac{2 π}{3}} & - 1 & e^{j \frac{2 π}{3}} & e^{- j \frac{2 π}{3}} \\ 1 & e^{j \frac{2 π}{3}} & e^{- j \frac{2 π}{3}} & - 1 & e^{- j \frac{2 π}{3}} & e^{j \frac{2 π}{3}} \end{array}]$	$(\begin{matrix} (0, 0) \\ (0, π) \\ (- \frac{2 π}{3}, 0) \\ (\frac{2 π}{3}, 0) \\ (- \frac{2 π}{3}, π) \\ (\frac{2 π}{3}, π) \end{matrix})$
$4 \times 2$	$[\begin{array}{cccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & - 1 & - 1 & - 1 & - 1 \\ 1 & - 1 & 1 & - 1 & 1 & - 1 & 1 & - 1 \\ 1 & - 1 & 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 1 & - j & - 1 & j & 1 & - j & - 1 & j \\ 1 & j & - 1 & - j & 1 & j & - 1 & - j \\ 1 & - j & - 1 & j & - 1 & j & 1 & - j \\ 1 & j & - 1 & - j & - 1 & - j & 1 & j \end{array}]$	$(\begin{matrix} (0, 0) \\ (0, π) \\ (π, 0) \\ (π, π) \\ (- \frac{π}{2}, 0) \\ (\frac{π}{2}, 0) \\ (- \frac{π}{2}, π) \\ (\frac{π}{2}, π) \end{matrix})$

$3 \times 2$ Permutations		$4 \times 2$ Permutations
$S_{1} \pm j S_{2}$	$- S_{1} \pm j S_{2}$	$S_{1} \pm j S_{2}$	$- S_{1} \pm j S_{2}$
$[\begin{array}{ccc} 0 & 60 & 120 \\ # 2 & 150 & 30 \end{array}]$	$[\begin{array}{ccc} 45 & 105 & 165 \\ # 2 & 15 & 75 \end{array}]$	$[\begin{array}{cccc} 0 & 45 & 90 & 135 \\ # 2 & 135 & 0 & 45 \end{array}]$	$[\begin{array}{cccc} 90 & 45 & 0 & 135 \\ # 2 & 135 & 90 & 45 \end{array}]$
$S_{1} \pm j S_{2}$	$- S_{1} \pm j S_{2}$	$S_{1} \pm j S_{2}$	$- S_{1} \pm j S_{2}$
$[\begin{array}{ccc} 0 & 120 & 60 \\ # 2 & 30 & 150 \end{array}]$	$[\begin{array}{ccc} 45 & 165 & 105 \\ # 2 & 75 & 15 \end{array}]$	$[\begin{array}{cccc} 0 & 135 & 90 & 45 \\ # 2 & 45 & 0 & 135 \end{array}]$	$[\begin{array}{cccc} 90 & 135 & 0 & 45 \\ # 2 & 45 & 90 & 135 \end{array}]$
$S_{2} \pm j S_{1}$	$- S_{2} \pm j S_{1}$	$S_{2} \pm j S_{1}$	$- S_{2} \pm j S_{1}$
${[\begin{array}{ccc} 90 & 30 & 150 \\ # 2 & 120 & 60 \end{array}]}^{†}$	$[\begin{array}{ccc} 135 & 15 & 75 \\ # 2 & 105 & 165 \end{array}]$	$[\begin{array}{cccc} 45 & 0 & 135 & 90 \\ # 2 & 90 & 45 & 0 \end{array}]$	$[\begin{array}{cccc} 135 & 0 & 45 & 90 \\ # 2 & 90 & 135 & 0 \end{array}]$
$S_{2} \pm j S_{1}$	$- S_{2} \pm j S_{1}$	$S_{2} \pm j S_{1}$	$- S_{2} \pm j S_{1}$
$[\begin{array}{ccc} 90 & 150 & 30 \\ # 2 & 60 & 120 \end{array}]$	$[\begin{array}{ccc} 135 & 75 & 15 \\ # 2 & 165 & 105 \end{array}]$	${[\begin{array}{cccc} 45 & 90 & 135 & 0 \\ # 2 & 0 & 45 & 90 \end{array}]}^{† †}$	$[\begin{array}{cccc} 135 & 90 & 45 & 0 \\ # 2 & 0 & 135 & 90 \end{array}]$

	PSNR			SSIM			GMSD
Image	$θ_{4}^{(0, 45)}$	$θ_{4}^{(22.5, 45)}$	$θ_{6}^{(0, 30)}$	$θ_{4}^{(0, 45)}$	$θ_{4}^{(22.5, 45)}$	$θ_{6}^{(0, 30)}$	$θ_{4}^{(0, 45)}$	$θ_{4}^{(22.5, 45)}$	$θ_{6}^{(0, 30)}$
$s_{0}$	64.731	66.558	68.148	0.9998	0.9999	0.9999	0.0005	0.0005	0.0001
$s_{1}$	51.734	54.827	55.896	0.9933	0.9937	0.9966	0.0048	0.0030	0.0023
$s_{2}$	53.121	54.905	57.347	0.9929	0.9933	0.9962	0.0037	0.0027	0.0021
DoLP	37.292	38.256	40.217	0.9053	0.9142	0.9455	0.0194	0.0168	0.0122
AoP	14.141	14.743	15.726	0.5324	0.5597	0.6496	0.1974	0.1835	0.1611

	PSNR				SSIM				GMSD
Image	$(2 \times 2)_{1}$	$(2 \times 2)_{2}$	$3 \times 2$	$4 \times 2$	$(2 \times 2)_{1}$	$(2 \times 2)_{2}$	$3 \times 2$	$4 \times 2$	$(2 \times 2)_{1}$	$(2 \times 2)_{2}$	$3 \times 2$	$4 \times 2$
$s_{0}$	50.054	50.070	50.075	50.071	0.9952	0.9953	0.9953	0.9953	0.0020	0.0019	0.0018	0.0019
$s_{1}$	49.117	51.835	51.160	50.703	0.9864	0.9936	0.9924	0.9911	0.0191	0.0072	0.0094	0.0114
$s_{2}$	49.095	49.120	51.774	51.385	0.9848	0.9847	0.9916	0.9903	0.0206	0.0204	0.0098	0.0114
DoLP	36.672	37.148	38.443	38.176	0.9153	0.9242	0.9359	0.9331	0.0490	0.0445	0.0340	0.0362
AoP	14.715	14.929	15.674	15.573	0.5153	0.5410	0.5679	0.5606	0.2037	0.1947	0.1747	0.1779

	PSNR				SSIM				GMSD
Image	$(2 \times 2)_{1}$	$(2 \times 2)_{2}$	$3 \times 2$	$4 \times 2$	$(2 \times 2)_{1}$	$(2 \times 2)_{2}$	$3 \times 2$	$4 \times 2$	$(2 \times 2)_{1}$	$(2 \times 2)_{2}$	$3 \times 2$	$4 \times 2$
$s_{0}$	49.910	49.990	49.991	49.926	0.9952	0.9952	0.9952	0.9952	0.0022	0.0022	0.0023	0.0021
$s_{1}$	47.233	50.430	49.785	48.180	0.9843	0.9918	0.9908	0.9890	0.0209	0.0091	0.0108	0.0138
$s_{2}$	47.836	48.009	50.657	49.583	0.9829	0.9831	0.9899	0.9883	0.0220	0.0231	0.0111	0.0132
DoLP	34.723	35.476	36.485	35.662	0.8816	0.8988	0.9098	0.8982	0.0615	0.0565	0.0478	0.0512
AoP	13.412	14.030	14.518	13.874	0.4475	0.4884	0.5084	0.4814	0.2407	0.2190	0.2076	0.2252

Alternative linear microgrid polarimeters: design, analysis, and demosaicing considerations

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Tables (6)

Equations (22)

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