Optical Engineering 58(8), 082408 (August 2019)

Calibration and performance assessment of microgrid
polarization cameras
Nathan Hagen,a,* Shuhei Shibata,a,b and Yukitoshi Otania,baUtsunomiya University, Center for Optical Research and Engineering, Department of Optical Engineering, Utsunomiya, Japan
Utsunomiya University, Graduate School of Engineering, Utsunomiya, Japanb

Abstract. We provide a method for calibrating microgrid polarization cameras that is simpler and easier to set
up than existing methods. Applying this method to three different commercially available cameras, we compare
the mean values and variances in their diattenuation and orientation properties. We derive formulas giving theaccuracy with which the pixel polarization properties can be calibrated in both the Gaussian and Poisson noise
regimes and demonstrate the statistical instability of the extinction ratio as a parameter. In a series of calibration
measurements, we estimate the pixel-to-pixel variation of polarization properties and show how to separate theeffects of temporal noise from manufacturing variation. © 2019 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.58.8.082408]

Keywords: polarization; imaging polarimetry; noise.
Paper 181690SS received Nov. 28, 2018; accepted for publication Feb. 1, 2019; published online Feb. 23, 2019.

1 Introduction
Polarization cameras are division of focal plane imaging
polarimeters1 that use an array of micropolarizer filters aligned
to the detector array pixels, typically with the micropolarizers
oriented at angles 0 deg, 45 deg, 90 deg, and 135 deg. We
provide a polarization camera calibration approach that is simpler than existing methods, and which does not require
a motorized rotation stage or the use of highly uniform flat
field illumination, such as that produced by an integrating
sphere.2–7 Taking advantage of this calibration approach,
we show how to separate the effects of temporal noise frommanufacturing variation when measuring the camera—a separation that is essential if we wish to fairly compare one camera’s performance to another. As more polarization cameras
become commercially available, it becomes increasingly
important to have a practical and unbiased method for evaluating and comparing these cameras.
Polarization cameras have conventionally been assembled
by manufacturing the detector array and the micropolarizer
array separately, then aligning them to one another and fixing
them in place. As a result, there is a small vertical displacement between the polarizer layer and the detection layer thatallows for cross talk between neighboring pixels when thelight is incident from nonzero angles of incidence (seeFig. 1).4 More recently, attempts have been made to manufacture the sensor and polarization filters together as part ofan integrated process to minimize cross talk and improvealignment.8 From a user’s standpoint, reduced cross talkhere will appear as an increase in the diattenuation valueof the individual pixels.
Micropolarizers use finely spaced wire grid patterns thatcan be difficult to manufacture, so that micropolarizer filters
have historically had trouble achieving the polarization
purity that monolithic polarization devices readily achieve.
Although monolithic polarizers can readily achieve extinction ratios of better than 103 , or even 106 , micropolarizers

have generally been limited to extinction ratios of <30.2,4Moreover, previous reports in the literature have indicatedthat micropolarizer orientation accuracy can vary by asmuch as 0.5 deg from pixel to pixel, whereas mounts formonolithic polarizers can readily achieve angular orientation
accuracy of better than 0.01 deg.4,9After introducing our approach for calibrating microgrid
polarization cameras, we show measurements obtained fromthree commercial polarization cameras. Using parametervariance formulas derived from a measurement model, weshow how one can separate the effects of measurement
noise from manufacturing variations, allowing for quantitative performance assessment and fair comparison amongcameras. This procedure is demonstrated on three commercially available polarization cameras, showing the behaviorof not just a select number of pixels in the center region of theimage, but the entire set of pixels across the array.

*Address all correspondence to Nathan Hagen, E-mail: nh@hagenlab.org

0091-3286/2019/$25.00 © 2019 SPIE

Optical Engineering

2 Calibration Procedure
A nonideal linear polarizer is a diattenuator defined by its
maximum and minimum transmission values q and r as
the diattenuator is rotated with respect to incident linearlypolarized light. Thus, while an ideal polarizer achieves
100% transmission and 0% transmission at 0-deg and
90-deg orientation angles, the nonideal diattenuator willachieve q and r. The Mueller matrix for a diattenuator
element with the q axis oriented at angle θ can be written
as Mðq; r; θÞ, as given in Eq. (1), where the diattenuationis D ¼ ðq − rÞ∕ðq þ rÞ, and A ¼ 12 ðq þ rÞ is the mean transmission. For a given diattenuator, it is possible to achieve an
ideal diattenuation of D ¼ 1 even when the peak transmission of the polarizer is very low A ≪ 1. The parameter A canthus be considered the “efficiency” of the polarizing element.
The extinction ratio X is derived from the diattenuationas X ¼ q∕r ¼ ð1 þ DÞ∕ð1 − DÞ.

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August 2019 • Vol. 58(8)Hagen, Shibata, and Otani: Calibration and performance assessment of microgrid polarization cameras

(a)

(b)

Fig. 1 The detection layer of a polarization camera, with micropolarizers (a) attached above the sensor
layer and (b) integrated into the sensor layer. Optical rays shown in blue are cross talk rays from one pixel
to its neighbor.

EQ-TARGET;temp:intralink-;e001;63;5060qþrðq − rÞ cosð2αÞðq − rÞ sinð2αÞ
pﬃﬃﬃﬃﬃ 2
pﬃﬃﬃﬃﬃ
12
ðq þ rÞcos ð2αÞ þ 2 qr sin ð2αÞ2 ðq þ r − 2 qrÞ sinð4αÞ
pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
1
2 qr cos2 ð2αÞ þ ðq þ rÞsin2 ð2αÞ2 ðq þ r − 2 qrÞ sinð4αÞ

B
1B
B ðq − rÞ cosð2αÞMld ðq; r; αÞ ¼ B
2 B ðq − rÞ sinð2αÞ
@
0
0
0
1
D cosð2αÞD sinð2αÞ
B
1
B D cosð2αÞ 1 − D sin2 ð2αÞB
4A D sinð4αÞ
¼ AB
1
B D sinð2αÞ
1 − D cos2 ð2αÞ4A D sinð4αÞ@000

To measure the diattenuation properties of a polarization
camera, we generate linearly polarized light sequentiallyoriented at four different θ angles and measure the detected
intensity at each of the four positions (see Fig. 2). Thus,
the source light Stokes vector is ssrc ¼ ðI p ; 0;0; 0ÞT , andthe Mueller matrix of the generating polarizer Mlp ðθÞ,which together generate a fully polarized state, sin ¼ I p ·ð1; cos θ; sin θ; 0ÞT for use in calibrating the camera pixels.
The quantity I p is the light flux in photons/sec incident on
the given pixel of interest.
The behavior of a single pixel in the polarization camera
can be modeled as a linear diattenuator Mld ðq; r; αÞ followed

1
0
0
0
1−D

0

C
C
C
C:
C
A

g ¼ d · Mld ðq; r; αÞ · Mlp ðθÞ · sin :EQ-TARGET;temp:intralink-;e002;326;330

(1)

(2)

g0 ¼ I e ½1 þ D cosð2αÞ þ n0 ;(3)

g45 ¼ I e ½1 þ D sinð2αÞ þ n45 ;(4)

g90 ¼ I e ½1 − D cosð2αÞ þ n90 ;(5)g135 ¼ I e ½1 − D sinð2αÞ þ n135 ;(6)

EQ-TARGET;temp:intralink-;e003;326;265

EQ-TARGET;temp:intralink-;e006;326;179

where gθ represents the number of detected electrons, and nθ
the number of noise electrons, for incident light polarized atangle θ. To work with both Gaussian and Poisson noise, wewill assume that all measurements are scaled to photoelectron units—the same units as the noise terms nθ . Since theincident light level is082408-2

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0
pﬃﬃﬃﬃﬃ
2 qr

C
C
C
C
C
A

Setting the generating polarizer to orientations
θ ¼ 0 deg, 45 deg, 90 deg, and 135 deg, we obtain fourmeasurements:

EQ-TARGET;temp:intralink-;e005;326;206

Optical Engineering

01

by detection vector d ¼ ðη; 0;0; 0Þ with quantum efficiency
η. Thus, the above generated state will be detected as

EQ-TARGET;temp:intralink-;e004;326;233

Fig. 2 Experimental layout for the calibration polarization state generator and the microgrid polarization camera. A diffuser is used toensure that the light source is unpolarized. The inset at rightshows a 2 × 2 section of the detector array.

0

August 2019 • Vol. 58(8)Hagen, Shibata, and Otani: Calibration and performance assessment of microgrid polarization cameras

1
I;
Ip ¼
ηA e



EQ-TARGET;temp:intralink-;e012;326;752(7)hI^ 2e i ¼1
ðg0 þ g45 þ g90 þ g135 Þ2
42intensity I e can be considered as the idealized photoelectron
number that would be detected if the polarizer were removed(i.e., A ¼ 1). Since the detector element’s quantum efficiency η and the polarizing element efficiency A appear
as a product, we can define ηext ¼ Aη as the external quantum efficiency of the pixel. If the illumination is known to beuniform a priori, then it is possible to estimate ηext at eachpixel to within an arbitrary constant. To measure the missing
constant, however, it is necessary to obtain an independentmeasurement of I p , such as with a radiometer.
Whereas previous calibration methods fitted pixel diattenuation parameters using images taken at a large number ofdifferent input polarization angles,4,5,10 the choice of the fourangles (0 deg, 45 deg, 90 deg, and 135 deg) produces simple
formulas for estimating the incident intensity I e and thepolarization properties at each pixel α, D, and X:¼

1
hðI ½1 þ D cosð2αÞ þ n0
16 e
þ I e ½1 þ D sinð2αÞ þ n45

EQ-TARGET;temp:intralink-;e007;63;752

1I^ e ¼ ðg0 þ g45 þ g90 þ g135 Þ;
4

(8)

1
α^ ¼ arctan½ðg0 − g90 Þ∕ðg45 − g135 Þ;
2(9)

EQ-TARGET;temp:intralink-;e008;63;541

EQ-TARGET;temp:intralink-;e009;63;499

22 1∕2^ ¼ 2½ðg0 − g90 Þ þ ðg45 − g135 Þ  ;D
g0 þ g45 þ g90 þ g135

(10)

^
^
X^ ¼ ð1 þ DÞ∕ð1
− DÞ:

(11)

EQ-TARGET;temp:intralink-;e010;63;462

EQ-TARGET;temp:intralink-;e011;63;420þ I e ½1 − D cosð2αÞ þ n90
þ I e ½1 − D sinð2αÞ þ n135 Þ2 i
¼ I 2e þ1½hn2 i þ hn245 i þ hn290 i þ hn2135 i;
16 0

(12)

where we have used the assumption that the noise terms are
zero mean: hnθ i ¼ 0. This causes no difficulties for Poissondistributed noise, since our definition of the measurement
and noise (g and n) result in the mean value of the
Poisson-distributed variable to be incorporated into g whilen retains the zero-mean stochastic portion.
Since hI^ e i ¼ I e , we can write that varðI^ e Þ ¼ hI^ 2e i − I 2e .
For uniformly distributed independent Gaussian (IG) noise,
this gives
1
varIG ðI^ e Þ ¼ vg ;4

(13)

EQ-TARGET;temp:intralink-;e013;326;514

where vg is the Gaussian noise variance at each pixel.
The 14 factor comes from the process of averaging overfour measurements to obtain I^ e . In the pure Poisson (PP)shot noise regime, we can substitute:hn20 i ¼ varðn0 Þ ¼ hg0 i ¼ hI e ½1 þ D cosð2αÞi;

EQ-TARGET;temp:intralink-;sec3;326;424

From these equations, we can see that the estimates for the
pixel polarization properties do not depend explicitly on theincident light intensity I e. As a result, as long as the lightintensity does not vary significantly from one pixel to its
neighboring pixels, calibrating the camera does not require
spatially uniform illumination. While differences in lightlevel at the camera will produce differences in noise ateach pixel, this can be made small in comparison to thedifferences produced by manufacturing variations.
3 Estimating Parameter Variances
From Eqs. (8)–(11), we can estimate the polarization properties of a pixel from the calibration measurements gθ . It isalso useful to assess how much individual pixels vary fromthe overall mean—the parameter nonuniformity. Differences
in manufacturing process between, say, the center of thedetector array and its edges, may result in the pixel diattenuation and orientation at the edge being different than at thecenter. If the manufacturing differences are small in comparison to the error induced by measurement noise, then weshould be able to ignore them. Otherwise, we need tomake sure to calibrate for the individual variations in pixel
properties.
Deriving formulas for the parameter variances involves
the application of the well-known expression for parametervariance: varð^xÞ ¼ h^x2 i − h^xi2 , where h·i indicates taking
the mean value. To proceed, we insert the equation for theestimator x^ and solve. Starting with I^ e , we obtain
Optical Engineering

for each of the noise terms in Eq. (12), producing
1
varPP ðI^ e Þ ¼ I e :4

EQ-TARGET;temp:intralink-;e014;326;381

Next, we can calculate the variance of α^ by inserting
Eq. (9) into the variance formula. While the result is a nonlinear equation, we can obtain a second-order power series
representation by taking the Maclaurin series expansion
in the noise variables nθ and extract the second-orderterms.11,12 This produces a lengthy polynomial expression
in the four noise variables, but if we assume that thenoise terms are independent of one another, then the ensemble average of all mixed-noise terms (i.e., terms having n0 n45
as factors) becomes zero. This greatly simplifies the expression. Finally, we substitute the IG noise model into the result,giving^ ¼
varIG ðαÞEQ-TARGET;temp:intralink-;e015;326;208vgðradians2 Þ;
8D2 I 2e(15)

or the PP noise model, giving^ ¼
varPP ðαÞEQ-TARGET;temp:intralink-;e016;326;155

1ðradians2 Þ:
8D2 I e(16)

A similar series-expansion procedure allows for calculat^ and X^ as well:ing the variances of D

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Table 1 Formulas for the mean and variance of detector array polarization properties.

Parameter

IG noise variance

PP noise variance

1
4 vg

1
4 Ie

varðI^e Þ^varðαÞ

^varðDÞ

hvarðX^ Þ^ ¼
varIG ðDÞEQ-TARGET;temp:intralink-;e017;63;600

1
8D 2





3D 2 þ4
16



vg
I 2e
vg
I 2e


i

2D 3 þ2D 2 þ4Dþ1 v g
I 2e
4ð1−DÞ4 D

h

1
8D 2



4−D 2
16

1
Ie



1
Ie

i

−D 4 þ6D 3 þ3D 2 þ4Dþ1 1
Ie
4ð1−DÞ4 D

ð3D2 þ 4Þvg;
16I 2e^ ¼4−D ;varPP ðDÞ
16I e

(17)

2

EQ-TARGET;temp:intralink-;e018;63;552

(18)

^ ¼
varIG ðXÞ

ð2D3 þ 2D2 þ 4D þ 1Þvg;4ð1 − DÞ4 DI 2e

(19)

^ ¼
varPP ðXÞ−D4 þ 6D3 þ 3D2 þ 4D þ 1:
4ð1 − DÞ4 DI e

(20)

EQ-TARGET;temp:intralink-;e019;63;522

EQ-TARGET;temp:intralink-;e020;63;479

The parameter variance equations are summarized inTable 1.
4 Experimental Results
To verify that we can characterize polarization cameras andseparate manufacturing variability from noise in the results,
we collected calibration data from three polarization cameras(“A,” “B,” and “C”) from three different manufacturers(Table 2). The experimental setup uses a linearly polarizedincoherent light source, generated using a white light LED,
diffuser, rotatable linear polarizer (Glan–Thompson type),
and a narrowband spectral filter: 532  10 nm for cameras
A and B, 520  20 nm for camera C (Fig. 2). The spectral
filters are needed in order to make the camera comparison asfair as possible, since camera C uses a nonremovable filter infront of its detector array.
The initial datasets were collected by setting the generating polarizer to 0 deg and summing over many frames inorder to reduce the measurement noise. The same procedure
was then used for the 45 deg, 90 deg, and 135 deg orientations of the generating polarizer, and for the estimation of the
background. After subtracting the background, the summed
images were scaled from digital counts to photoelectron
units using previous calibration of the cameras’ gain values.
The radiometric gain values used for the four cameras were
2.2, 8.0, and 1.3 photoelectrons per digital count for cameras
A, B, and C, respectively. Taking the sum of the images,
rather than mean as is more common, is done in order tomaintain Poisson statistics in the data.
Unlike previous calibration methods, we do not use an
integrating sphere in order to make the illumination field uniform, as exact uniformity is unnecessary for the parameter
estimates. The main purpose of the diffuser in the experimental setup is to remove any residual polarization from the light
Optical Engineering

Table 2 The four polarization cameras measured.

Camera A

Camera B

Camera C

2464 × 2056

1200 × 1800

1164 × 874

Pixel size (μm)

3.45

7.4

4.65

Frame rate (fps)

90

110

20

Bit depth

12

12

12

VIS-NIR

VIS-NIR

520  20 nm

PixelsWavelength range

source. With the diffuser in place, the estimated degree of
linear polarization of the light source is measured to
be <0.1%.
First, we sum together 2000 calibration images for thecase of camera A, 100 images for cameras B and C, intoa single “sum image.” With the sum image, we implement
the parameter estimation Eqs. (8)–(11) at each pixel.
Taking the histograms of the resulting parameter images,
we calculate the mean and standard deviations of eachparameter, giving the results shown in Figs. 3 and 4 and summarized in Table 3. Thus, these parameter histograms have a
distribution determined by a mix of (temporal) noise at eachpixel as well as pixel-to-pixel variation in polarization properties. Note that since there is an easy-to-adjust degree of
freedom in the orientation angle, we have defined the meanof 0-deg micropolarizers’ orientation angle to be exactly
zero, so that the remaining orientation angles are definedwith respect to it.
The measurement results show a substantial range of performance for parameter mean and spatial variation among
the three measured cameras, especially in the estimatedmicropolarizer extinction ratios. However, it is importantto keep in mind that each camera uses pixels of different
sizes, with different quantum efficiencies, different readout
electronics, and different integration times. Thus, the measurement variation alone is not in itself sufficient; for faircomparison, it is necessary to separate the variation due tomeasurement noise from variation due to manufacturing
differences of the pixels.
Looking at the distributions for I^ e in Fig. 4, we findthat the measured standard deviations for I^ e are far larger
than the theoretical noise-only standard deviations, i.e.,varðI^ e Þ ≫ 14 hI^ e i. This is an indication that the illumination
is significantly nonuniform, or that there is significant
pixel-to-pixel variation in the external quantum efficiency
ηA [defined in Eq. (7)].
^ and X^ in Fig. 4, we can
Looking at the histograms for D
see that the two distributions are generally similar, but with^the value and the width of the probability distribution prðXÞ
^increasing rapidly as hDi approaches 1. This close association of the two distributions is also evident in Eq. (20) for^ where a factor of ð1 − DÞ4 appears in the denomivarðXÞ,^ is determined primarily bynator. Whereas the variance for D
the factor of I e in the denominator Eq. (18), we see that thevariance for X^ will be amplified by ð1 − DÞ−4 while themean value as given in Eq. (11) will be amplified by only

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August 2019 • Vol. 58(8)Hagen, Shibata, and Otani: Calibration and performance assessment of microgrid polarization cameras

Fig. 3 Spatial histogram results for the three cameras, calculated after summing 2000 frames for cameraA, 100 frames for cameras B and C. Note that all of the orientation angles are adjusted so that the mean of
α0 is exactly zero. The calculated mean at the top left of each subfigure is given for each of the four pixel
types, followed by the measured standard deviation and the estimated noise-only standard deviation
values [obtained from Eq. (16)] in parentheses, i.e., mean (meas_std) (est_std). Gaussian curves forthe fitted histogram mean and standard deviations are shown as solid curves overlying the histograms.

ð1 − DÞ−1 . If we write the signal-to-noise ratio for the estimate of the extinction coefficient, we obtain

EQ-TARGET;temp:intralink-;sec4;63;237^
meanðXÞ
^ ¼q
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ≈ ð1 − DÞI 1∕2SNRðXÞe :^varðXÞ

Thus, for a fixed number of measurement photoelectrons,
the SNR declines as D approaches 1, and this behavior is theproperty we see exhibited most clearly in the histogram forcamera A—the camera where the diattenuation is the highest.
The tail on the right hand side of the extinction ratio dis^ moves rapidly to higher values as D comestribution prðXÞ
closer to 1. This is a well-known property for “Gaussian ratio
distributions”—in this case the ratio of ðD þ 1Þ to ðD − 1Þ,for which the mean becomes undefined and the variancebecomes infinite.13,14 While this behavior poses no serious
Optical Engineering

problems for cameras B and C, camera A’s diattenuation
is sufficiently close to 1 that after background subtractiona small fraction of pixels are left with zero or negative values
when in the crossed-polarization condition. As a result,^ extends past 1, so that for
the tail of the distribution for D
these pixels the extinction coefficient becomes infinite(for D ¼ 1), then wraps around to negative infinity andapproaches zero from the negative number side (for D > 1).
These values are unphysical, of course, so that in Fig. 4 wehave truncated the distribution for D at 1, and the distribution
for X at 1250. Only a small portion of the distribution tail lies
above these extreme values, so that without truncation thedata distribution mean will be dominated by these rare values
and will produce mean and variance values that are highly
unstable, just as theory predicts.
^ the
As a result of the non-Gaussian shape for prðXÞ,median and mode of the distribution are more useful summary metrics of the distribution than the mean, and082408-5

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August 2019 • Vol. 58(8)Hagen, Shibata, and Otani: Calibration and performance assessment of microgrid polarization cameras

Fig. 4 Spatial histogram results for the three cameras, calculated after summing 2000 frames for cameraA, 100 frames for cameras B and C. Gaussian curves for the histogram mean and standard deviations
are shown as solid curves overlying the histograms. In camera A’s data, the histograms for D^ have beentruncated to 1, and those X have been truncated at 2000 to prevent unphysical values. The four colors
are coded to the four pixel orientation types: 0 deg, 45 deg, 90 deg, and 135 deg.

Table 3 Measured pixel properties for the four polarization cameras
obtained from the 2000-frame-sum images (camera A) or 100-framesum images (cameras B and C). Each entry shows the mean value ofeach parameter taken across the entire array, together with the temporal-noise-removed estimate of the spatial standard deviation (thevariation due to differences in manufacture) in square brackets.

Parameter

Camera A

Camera B

Camera C

α0 (deg)

0 [0.11]0 [0.65]

0 [0.36]α45 (deg)

44.96 [0.11]

45.20 [0.52]

41.75 [0.42]

α90 (deg)

89.81 [0.13]

89.50 [0.20]

86.34 [0.34]

α135 (deg)135.10 [0.12]

134.88 [0.23]

125.70 [0.43]

D0

0.9927 [0.0025]

0.8495 [0.0152]

0.6621 [0.0082]

D 45

0.9928 [0.0029]

0.8495 [0.0152]

0.6545 [0.0092]

D 90

0.9894 [0.0031]

0.8945 [0.0117]

0.6984 [0.0067]

D 135

0.9913 [0.0025]

0.8853 [0.0108]

0.6010 [0.0123]

4.1 Using Temporal Noise Estimates to Validate the
Variance Estimates

distribution quantiles can be used in place of the variance inorder to describe the width. In Table 4, we compare theextinction ratio mean, median, and mode for camera A.
Whereas for cameras B and C these three metrics are almost
the same, we can see that for the case of camera A the mean
Optical Engineering

is biased toward high values by the long one-sided tail of the
distribution. In this situation, the median is probably themost useful single metric for camera users to use in evaluating camera pixel properties. For example, if we take themean of the diattenuation distribution prior to applying
the equation for the extinction coefficient, X ¼ ð1 þ DÞ∕
ð1 − DÞ, we obtain a result (column 4 in Table 4) that closely
approximates the median value.

To validate the variance formulas, we capture a longsequence of frames and analyze the behavior of the polarization parameters for individual pixels over time. This
removes the effect of spatial variability so that only temporalnoise is present. For cameras A and B, we collected asequence of 2000 frames, while for camera C, we were
only able to capture 750 frames during a single calibration
period. At each individual frame, we calculate the pixel properties I e , α, D, and X, and from the resulting set of 2000 (or750) estimates, we calculate the parameter variance. This isshown as the first point at the upper left of the plots in Fig. 5,
corresponding to N ¼ 1, where N is the number of framessummed before calculating the parameters.
Next, we take the same dataset and sum every pair offrames (i.e., N ¼ 2) before calculating the parameters. Fora shot-noise-limited measurement, this is equivalent to082408-6

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August 2019 • Vol. 58(8)Hagen, Shibata, and Otani: Calibration and performance assessment of microgrid polarization cameras
Table 4 Extinction coefficient summary statistics for camera A, using
the data shown in Fig. 4.
Pixel orientation(deg)

Mean (X^ )Median (X^ )Mode (X^ )

^
1þhDi
^
1−hDi

0

327

275

236

275

45

342

289

246

278

90

208

190

172

187

135

254

232

216

230

doubling the integration time and thus collecting twice thenumber of photoelectrons. Calculating the variance forthis new set of 1000 (or 375) parameter estimates, we obtainthe second point shown in the plots of Fig. 5.
Following this procedure for increasing values of N, we
simulate the effect of measuring with steadily improving
SNR. Using hI^ e i in place of I e in each of the variance
Eqs. (14)–(20), we plot the corresponding noise-only variances predicted from theory as a solid curve.
4.2 Separating Spatial Variability from Temporal
Noisevariance of all the pixels in the image (Figs. 3 and 4) aremuch larger than the values predicted from the variance formulas. This is an indication that pixel-to-pixel (deterministic)
variability is dominating the measured variation, not temporal (stochastic) noise. To confirm this conclusion, we showa single row of pixels from each of the calibration images(Fig. 6), selecting every other pixel in order to avoid issuesof differences between micropolarizer orientations. Here, we^ and X^ variances are dominated by a broad^ D,
see that the α,systematic variation across the row rather than by uncorrelated noise.
With the effects of stochastic and deterministic variation
now clear, we can use our variance formulas to remove thestochastic portion from a measured variance value, leaving
only the spatial pixel-to-pixel differences. If we assume
that the two sources of variation are Gaussian-distributed
and uncorrelated, then their combined effect is given by aconvolution of the two distributions: for stochastic variables
x and y, the variance of the sum z ¼ x þ y is given as
EQ-TARGET;temp:intralink-;sec4.2;326;542

varðzÞ ¼ varðxÞ þ varðyÞ:Therefore, if we know varðzÞ and varðyÞ and want tosolve for the standard deviation of x, thenpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃstdðxÞ ¼ varðzÞ − varðyÞ:EQ-TARGET;temp:intralink-;sec4.2;326;495

Although the temporal noise measurements of Fig. 5 show aclose fit to predictions, we can also see that the measured

If we take the data for the estimated extinction coefficientα^ 0 from camera C in Fig. 3, then varðzÞ ¼ ½0.3652 andFig. 5 Temporal standard deviation of each parameter for a single pixel in each camera. The horizontal
axis indicates the number N of frames summed prior to calculating the parameter standard deviation. The
dots indicate measurements, whereas the curve indicates the theoretical standard deviation value calculated from the measured light level. Note that the camera A subfigure for X^ uses a semilogarithmic plot,
while all of the other subfigures use a linear plot.

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Fig. 6 Parameter estimates for a row through the camera images, calculated after summing 2000 frames
for camera A, 100 frames for cameras B and C. The inset numbers give the measured mean and standard deviation, and the theoretical noise-only standard deviation, of the row data.

varðyÞ ¼ ½0.0652 , so that stdðxÞ ¼ 0.359. In this case, we
see that the observed spatial variation is almost entirely
due to pixel-to-pixel manufacturing differences rather thanto random noise. This and the corresponding results for α^^ for each of the four cameras is given in square brackets
and D
as the “manufacturing variation” in Table 3. By removing
the stochastic component, we now have a direct means ofcomparing the spatial variability in the pixel polarization
parameters.
5 Conclusions
To assess the performance of a microgrid polarization camera, it is natural to look first for the mean angle of orientationand the mean extinction coefficient of each of the four orientations of detector array micropolarizers. The spatial variation in these parameters, however, is also an importantconcern: any calibration that does not estimate each pixel
individually is subject to increasing error as the spatial variability increases. To eliminate this source of error, we cancalibrate each pixel individually, but the inability to averageover an ensemble of many samples means that the calibration
measurements will have a much lower SNR. Using the variance formulas of Eqs. (14)–(20), we have a simple method of
estimating parameter calibration accuracy, so that users candetermine how many calibration frames of data are needed.
We have also shown that the most popular metric forquantifying the performance of polarizers, the mean of theextinction coefficient, becomes problematic when the polarizer approaches perfection. As the diattenuation value
approaches 1, the tail of the distribution for X^ lengthens,
Optical Engineeringso that the mean becomes biased and highly unstable.
Unless the mean of the diattenuation distribution is many
standard deviations below 1, i.e.,^^ ≫ 1;
Q ¼ ð1 − hDiÞ∕stdðDÞ

EQ-TARGET;temp:intralink-;sec5;326;363

the noise will dominate, producing mean and variance estimates that are of little utility. As a result, the meaning of
extinction ratios of 103 , or even 106 reported in the literature
can be unclear without knowing the conditions of the measurement. If these reported values are taken from the meancalculated from the data, then they are only useful if theexperimenter has made sure that Q ≫ 1, a condition thatcan be difficult to achieve in practice. For the calibrationof camera A, we summed together 2000 frames of data
and found that even this quantity of data (amounting to
95 GB) was not nearly sufficient to achieve this condition.
Using our variance formulas, we can estimate that increasing
the number of frames summed to about 2 × 105 (i.e., 9.5 TB)
should be sufficient. In general, for high values of X, themedian and mode are much more robust metrics than themean, allowing them to be used at much more reasonable
signal-to-noise ratios.
It is also important to keep in mind that the micropolarizer
characteristics will, in general, depend on the wavelengthand the incident angle of incidence (and therefore on thelens numerical aperture), so that a user may need to recalibrate the sensor for each spectrum at which it is to be used,
and to a lesser extent, with each lens.15

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The four cameras examined in Sec. 4 above show a wide
spread in estimated performance, and it is natural to ask just
how important it is to work with a camera that has pixels of
extinction ratio 300 versus one with, say, an extinction ratioof 10. Tyo and Wei16 and Roussel et al.17 have shown thateven for an extinction ratio as low as X ¼ 5, the SNR in the
Stokes vector elements increases by a factor of only ∼2.7
relative to an ideal diattenuator (X ¼ ∞). This would
seem to argue that there is little benefit to be had oncethe extinction ratio exceeds 10 or 20, but most researchersand engineers persist in pushing hard to get the highest X
values. It seems likely that the reason for this gap between
the theoretical value of high X and the practical value placedon it by users lies with the ease of calibration. For a polarimeter employing high extinction ratio elements, one canapproximate the polarizers as ideal diattenuators withouta heavy cost in error. A polarimeter with low extinction
ratio elements, on the other hand, requires a high accuracy
calibration, and the polarimetric estimation equations willneed to be more complex in order to accommodate the nonideal diattenuation values.
Another place where the higher extinction ratio provides
tangible benefits is for imaging polarimetry of natural outdoor scenes. In this situation, many pixels will be seeing
light with a degree of polarization close to zero, so thatthe polarimetric SNR will be poor for measuring the spatiallyresolved angle of polarization.17 Using a higher extinction
ratio will allow for seeing smaller polarization features
above the noise level.

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