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Mueller matrix retinal imager with optimized
polarization conditions
K. M. Twietmeyer1* and R. A. Chipman1, A. E. Elsner2, Y. Zhao2, and D. VanNasdale2
1University of Arizona College of Optical Sciences, 1630 E. University Boulevard,
Tucson, Arizona 85721, USA
2Indiana University School of Optometry, 800 E. Atwater Avenue,
Bloomington, Indiana 47405, USA
*Corresponding author: karen.twietmeyer@yahoo.com
Abstract: A new Mueller matrix polarimeter was used to image the retinas
of normal subjects. Light from a linearly polarized 780 nm laser was
passed through a system of variable retarders and scanned across the retina.
Light returned from the eye passed through a second system of retarders
and a polarizing beamsplitter to two confocal detection channels.
Optimization of the polarimetric data reduction matrix was via a condition
number metric. The accuracy and repeatability of polarization parameter
measurements were within ± 5%. The magnitudes and orientations of
retardance and diattenuation, plus depolarization, were measured over 15°
of retina for 15 normal eyes.
©2008 Optical Society of America
OCIS codes: (120.5410) Polarimetry; (170.5755) Retina scanning; (170.3880) Medical and
biological imaging; (170.3890) Medical optics instrumentation; (170.4470) Ophthalmology
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1. Introduction
Glaucoma, age-related macular degeneration, and diabetic retinopathy are responsible for
25% of blindness worldwide [1]. These diseases can cause complex changes to the retina that
are often difficult to detect early enough to ensure effective treatment [2-4]. With disease, the
retinal tissue becomes disordered, in contrast to the well-ordered layers of the healthy retina,
and thus they differ in their interaction with polarized light [5].
More sensitive diagnostic tests could potentially be developed using the interaction of
polarized light with retinal tissue, exploiting subtle changes in retinal and choroidal
microstructure. The more ordered a structure, the larger the expected diattenuation and
retardance. As these cellular structures become disordered in certain disease states, the
diattenuation and retardance would be expected to decrease and the depolarization would be
expected to increase. Previously, incomplete polarimeters have been used to probe the
increase of scattered light concomitant with retinal disease [5-8], demonstrating the utility of
depolarization images to provide contrast for deep tissue scattering abnormalities in age-
related macular degeneration, central serous chorioretinopathy, and other maculopathies.
Polarimeters such as the Carl Zeiss Meditec (formerly Laser Diagnostics) GDx have been
used to quantify changes to the retinal ganglion cell axons in glaucoma, by measuring the
linear retardance of the nerve fiber layer. The GDx, as an incomplete polarimeter, does not
measure all three forms of polarization behavior: depolarization, diattenuation, and
retardance. Thus its retardance measurements are increasingly inaccurate as depolarization
and diattenuation increase [9].
The overall thickness of the retina ranges from 50 μm at the foveal pit to 600 μm at the
optic nerve head, with different layers changing in thickness as a function of distance from the
fovea. At the center of the fovea, the superficial retinal layers are displaced eccentrically,
leaving only cone cells with their axons radiating outwards in the Henle fiber layer towards
synapses with bipolar cells. Where the retinal tissue has at least partial structural regularity
over the measurement volume, collective magnitude and phase modifications to incident
polarized light accrue and are measured as (respectively) diattenuation and retardance. The
measured values are specific to the tissue structure. However, when the structure has low
structural regularity over the measurement volume, the magnitude and phase modifications
vary significantly. Lacking a collective magnitude and phase change, the measurement
indicates depolarization. Multiple tissue layers in the light path with multiple polarization
effects all contribute to the polarization interaction.
Structures with the most regular microstructure and thus most likely to generate non-
depolarizing polarization effects are: the stroma in the cornea; the retinal nerve fiber layer; the
lamina cribrosa and scleral crescent at the ONH; Henle’s fiber layer in the retina; the rod and
cone photoreceptors; Bruch’s membrane; and the sclera [8]. The strongest polarization effects
are found in the cornea, retinal nerve fiber layer, and Henle’s layer [10]. These anisotropic
structures contain long thin parallel-oriented cylinders (such as collagen fibrils or
microtubules), uniformly distributed within the surrounding medium, and with dimension
smaller than the wavelength of visible light. Models have been developed to describe
diattenuation and retardance effects when polarized light interacts with structured ocular
tissues [11,12]. Though models of this type are useful for predicting the general types of
polarization effects to be found in ocular tissues based on their structure, they have limited
application in quantifying the magnitude of the effects due to strong dependencies on ocular
variables which are difficult to specify accurately, such as spatial distributions of refractive
indices, cylinder size, and packing density.
Direct measurements of retinal polarization properties have been made using a variety of
techniques. The Mueller matrix ellipsometry technique reported in 1985 by van Blokland
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[13] was the first method to use a complete Mueller matrix polarimeter together with analysis
methods based on the Mueller calculus and Poincaré sphere trajectories; this work set the
stage for later research efforts. Though non-imaging, this approach provided a full Mueller
matrix of individual retinal locations and demonstrated significant retinal depolarization. To
determine polarization properties over a wider retinal region, imaging methods have since
been developed including camera-based polarimeters [9] and scanning laser polarimetry
(SLP) [14-16]. The accuracy of these methods depends on the configuration of the
polarimeter generator (polarization components in the outgoing path) and analyzer
(polarization components in the incoming path). An incomplete polarimeter configuration
must be inaccurate when the collected light has polarization content in a direction lying
outside the polarimeter’s measurement space; this energy is coupled as error into the
measurable polarization components [17]. For example, the Carl Zeiss Meditec GDx is an
incomplete polarimeter with a measurement space which includes retardance but not
depolarization or diattenuation. Given that ocular interactions are partially depolarizing,
particularly in older and surgically altered eyes [9, 18], the GDx retardance measurement
must be inaccurate to some degree. Bueno has observed this inaccuracy to be significant and
correlated with degree of polarization in a study involving subjects displaying varying levels
of ocular depolarization [9].
Mueller matrix imaging polarimetry (MMIP) uses a polarimeter that is complete so that a
full Mueller matrix is obtained at every point in a retinal image. MMIP in a camera-based
configuration has been used to quantitatively measure polarization in the in vitro cornea [19]
and lens [20] and in the in vivo full eye [21]. MMIP in the SLP configuration has been
reported in the context of image quality improvement and enhanced contrast for retinal
features [22,23]. Lara and Dainty [24] have reported a complete polarimeter configuration
using multiple detection channels; this instrument is in early development and requires
manual translation to produce a two-dimensional image. MMIP techniques require the
collection of at least 16 images at different polarimeter states, typically over a wide intensity
range within the image set. Reported MMIP instruments have been limited by the need to
manually adjust polarization components, slow data acquisition time (lens paralysis or pupil
dilation has been required), difficulty in co-registering images, and inadequate dynamic range.
The imaging Mueller matrix retinal polarimeter described here was developed to obtain
high quality retinal images in vivo to measure the full 16 degrees of freedom in the Mueller
matrix. This polarimeter, referred to as the GDx-MM (MM for Mueller Matrix), is based on
modifications to a commercially available scanning laser polarimeter, the GDx (the GDx used
in this study was manufactured by Laser Diagnostics). Using the GDx-MM all polarization
parameters (retardance, diattenuation, and depolarization) are measured independently and
simultaneously. The GDx-MM is completely automated, recording 72 polarization images
with 20 μm resolution over a 15° visual field in 4 seconds. Lens paralysis and pupil dilation
are not required. The polarimeter design is optimized for good performance in the presence
of known error sources. In this paper, the GDx-MM instrument design is discussed, including
design requirements, optimization methods, calibration, data processing, and validation.
Representative polarization parameter images from a study on normal human subjects are
presented and discussed.
2. Methods
2.1 Experimental system
The original GDx is a SLP with a linearly polarized 780 nm laser source, a rotating half wave
plate, and a polarizing beamsplitter which directs the co- and cross- polarized return light to
two detection channels. A two-dimensional retinal scan is performed for each of 20
waveplate orientations. The signals from the co- and cross-polarized detectors are processed
to produce a linear retardance map of the scanned region. A constant birefringence factor
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converts the retardance map to a thickness map of the retinal nerve fiber layer. The resolution
at the retina is about 15 μm, and the scan covers a visual field of 15°. Scanning is via a slow
scan galvanometer driven at 26 Hz coupled to a fast resonant scanner oscillating at 4 kHz.
The data presented in this paper is from the co-polarized channel only.
Complete Mueller matrix polarimetry requires at least 16 measurements, utilizing linearly
independent combinations of generator and analyzer polarization states. Typical methods to
create these states include polarizers in combination with retardation elements, which may
include retarders on rotary stages, liquid crystal variable retarders, electro-optic modulators,
or photoelastic modulators. Space is very limited within the GDx and thus our choice was to
install two liquid crystal variable retarders (LCVRs), one in the generator path and one in the
analyzer path. The generator and analyzer states are then determined by combinations of the
LCVR retardances and the rotating waveplate orientation. Fig. 1 shows the GDx-MM optical
and polarization path with modifications from the original GDx. The instrument is more fully
described in [25].
The use of LCVRs in polarimetric applications is well established [26,27]. Although
LCVRs have small size and adjust retardance at low voltage, they have significant
disadvantages which adversely affect the polarimeter accuracy. (1) The retardance drift with
temperature is large, approximately 0.5% per ºC. (2) The retardance varies rapidly with angle
of incidence. (3) The response time is slow, tens of milliseconds. (4) There is significant
scatter and nonuniformity across the liquid crystal aperture which introduces noticeable
amounts of depolarization. Our LCVRs have a 3º change in retardance per degree in angle of
incidence. The response time is 30-80 ms, and depolarization index is 0.01.
Fig. 1. GDx-MM optical and polarization path. NPBS: non-polarizing beamslitter; PBS:
polarizing beamslitter; APD: avalanche photodiode; LCR: liquid crystal retarder.
Modifications from the GDx path are emphasized in red and include the following: insertion of
LCR-G in generator path; insertion of LCR-A in analyzer path; and changing the retardance
and rotational increment of the rotating waveplate.
2.2. Optimization of polarimeter design
The GDx-MM generator and analyzer states were selected through consideration of Poincaré
sphere trajectories and by minimization of the polarimetric measurement matrix condition
number [17]. A Mueller matrix polarimeter illuminates the sample with a series of generator
states (Stokes vectors) gi and analyzes the light which has interacted with the sample with a
set of analyzer states ai recording a set of flux measurements arranged in a vector P. The
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polarimeter is represented by the polarimetric measurement matrix W where each row wi is
given by:
][ 3,3,0,1,3,0,2,0,1,0,0,0, iiiiiiiiiiiii gagagagagagaw = . (1)
The sample Mueller matrix M is calculated from P as a linear least squares estimation process
using the established polarimetry equations [28]
MPWPWM -1 == , (2)
where the pseudo-inverse is preferred when W is not square.
For the GDx-MM the optimizable parameters are the LCVR retardances and orientations
and the rotating waveplate retardance and angular increment. To perform this optimization,
the generator and analyzer were expressed as composite Mueller matrices with the retardances
and orientations as optimization parameters. A polarimeter state sequence of rotating the
waveplate several increments for each of several LCVR retardance values was established by
considering that the LCVR response time is significantly longer than the waveplate rotation
time. The sequence duration is limited by consideration of typical eye movement and
blinking patterns; a maximum measurement time of four seconds was selected. The full
sequence consists of four sets of fixed LCVR retardances, with 18 rotating waveplate
positions per set, for a total of 72 states. Optimization was begun via a Poincaré sphere
analysis identifying parameter values which minimize the maximum distance from any point
on the sphere to that trajectory. The analysis showed that to form a rank 16 polarimetric
measurement matrix with good coverage of the sphere: (1) the GDx rotating half wave plate
should be replaced by one with a lower retardance; (2) this waveplate should rotate twice as
far (through 90° rather than 45°); and (3) the LCVRs should be oriented at 45°.
A stability analysis utilizing condition number was performed to determine optimal
retardances; this is a standard technique used by many researchers to design polarimeters with
improved stability in the presence of error [17,29,30]. To perform our condition number
analysis each variable in W was varied, and the values yielding the W with lowest L2
condition number κ2(W) was selected. κ2(W) is defined as the product of the L2 norms of W
and its inverse:
2
1
2
2)( −
=WWW
κ
, where
2
2
2sup x
x
x
A
A= . (3)
The L2 condition number also equals the ratio of the largest to smallest singular values in the
singular value decomposition of W.
Our optimized polarimeter configuration is summarized in Table 1. For each of four
groups, the generator and analyzer LCVRs have fixed retardance combinations selected from
3/8, 5/8, and 7/8 wave (multiple combinations giving minimum condition number exist).
Within each group, the waveplate rotates in 4.864° increments for a total rotation of 87.5° (the
increment size is constrained by the GDx motor drive). The optimized rotating waveplate
retardance was 145°. This configuration gives a rank 16 polarimeter with an L2 condition
number of 14.5 (sixteen singular values ranging from 9.4 to 0.6), comparable to that of a dual
rotating retarder polarimeter [29]. By comparison, the GDx is a rank 8 polarimeter with eight
singular values ranging from 7.4 to 0.4. A comparison of the trajectories of each
polarimeter’s generator on the Poincaré sphere demonstrates that the accessible portion of the
sphere is significantly greater for the GDx-MM (Fig. 2).
LCVRs were supplied by Bolder Vision Optik, Boulder, CO. A zero-order 145°
birefringent polymer waveplate was obtained from Meadowlark Optics, Frederick, CO. The
GDx electronics were replaced with a custom data acquisition and control system [National
Instruments, Austin, TX]. More detail may be found in [25].
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Table 1 GDx-MM Polarimeter States
Group:States
Generator
LCVR
retardance
Analyzer
LCVR
retardance
LCVRs
orientation
Rotating
waveplate
retardance
Rotating
waveplate
orientation
1: 1-18 7/8 wave 5/8 wave 45° 145° 0°-87.5°
2: 19-36 7/8 wave 5/8 wave 45° 145° 90°-177.5°
3: 37-54 5/8 wave 3/8 wave 45° 145° 180°-267.5°
4: 55-72 3/8 wave 7/8 wave 45° 145° 270°-357.5°
(a) (b)
Fig. 2. Generator trajectories on the Poincare sphere. (a): GDx trajectory is limited to one half
of the equator, and thus cannot measure elliptical or circular polarization. (b): GDx-MM
trajectory covers a substantially larger portion of the sphere; shown are different trajectories
corresponding to the set of LCVR retardances.
2.3. Calibration
Calibration was performed by direct measurement of each polarimeter state using external
calibrated polarimeters. In a novel approach to calibration, data was collected for all possible
combinations of rotating waveplate orientation (from 0° to 87.5° in 4.864° increments) and
LCVR retardance (in the set {3/8, 5/8, 7/8 wave}). All possible polarimetric measurement
matrices were formed from the large number of calibrated states, and the associated L2
condition number of each was calculated. The W with the lowest L2 condition number (of
about 15.1) was identified, and the corresponding combination of states was chosen. This
approach provides improved measurement accuracy in the presence of error.
A data reduction algorithm converted raw data into sets of 72 retinal images. These
images were co-registered to compensate for saccadic eye movement and/or blinking during
the scan. A Mueller matrix was then calculated at each pixel via W. Individual images could
be dropped from the data set due to blinking, excessive eye movement, or the presence of
artifacts such as tear films; in this case the corresponding rows from W are deleted, WP-1 is
recomputed, and data reduction proceeds with a reduced image set.
Due to noise, M is often not a physically realizable (“physical”) Mueller matrix, i.e. for
some input Stokes vectors the degree of polarization is less than zero or greater than one, or
the intensity is negative. To ensure accurate extraction of polarization parameters such as
retardance, the nearest physical Mueller matrix M is first found. Two methods have been
applied with similar results. In both methods M is first converted to a Hermitian matrix H:
(C) 2008 OSA 22 December 2008 / Vol. 16, No. 26 / OPTICS EXPRESS 21345
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∑∑
==
⊗= 3
0
3
0
2
1
ij jiij
m
σσ
H , (4)
where mij are the elements of M, ⊗indicates the outer product and the ’s are the normalized
Pauli spin matrices:
−
=
−
=
== 10
01
,
0
0
,
01
10
,
10
01 3210
σσσσ
i
i . (5)
Physical M yield H with positive eigenvalues; non physical M have one or more negative
eigenvalues. When M is non physical, the eigenvalues of H are used as the starting point for
a new Hermitian matrix H. In one method based on [31], the normalized Cholesky
decomposition of H is used to determine a non-negative eigenvalue set. In an alternate
method having faster processing time but lower accuracy in some cases, all negative
eigenvalues of H are zeroed and H is formed per the eigen decomposition theorem from the
eigenvectors of H. M is reconstructed from H as
[
]
{
}
*
jiij Trm
σσ
⊗
′
=
′H , (6)
where Tr is the trace function.
Each M is decomposed into diattenuation, retardance, and depolarization component
matrices via the Lu-Chipman algorithm [32]. Polarization properties of interest such as
retardance magnitude and orientation are calculated on a per-pixel basis from the appropriate
component matrix, and then assembled into “polarization parameter” images showing how
each parameter varies spatially across the imaged portion of the retina. All data processing
steps were performed using custom applications written in Labview, Mathematica [Wolfram
Research Inc., Champaign, IL], and Matlab [The Mathworks, Natick, MA].
2.4. Validation
Known error sources reduce the accuracy of the GDx-MM. Systematic (nonzero mean)
sources include uncertainty in the rotating waveplate start position (± 1 motor step: ± 2.432º),
temperature-induced retardance drift of the LCVRs (0.5% per ºC), and temperature-induced
gain drift in the avalanche photodiode detectors (3% - 5% per ºC). Random (zero mean)
sources include uncertainty in the waveplate incremental rotation (± 0.1º) and the Poisson-
distributed shot noise in the avalanche photodiode detectors. For a temperature range of
± 5ºC, a simulation predicts that the GDx-MM should have absolute worst case error of 0.25
in diattenuation magnitude, 6º in diattenuation orientation, 0.15 rad in retardance magnitude,
5º in retardance orientation, and 0.2 in depolarization index. Highest error contribution is
from the waveplate start position and the LCVR retardance drift with temperature. In an ideal
case where the waveplate starts properly at position zero and the instrument is used within ±
1º of the calibration temperature, the predicted error is about five times lower than these worst
case values.
A validation procedure was performed by measuring the Mueller matrices of known
standards including a polarizer, two waveplates, and a depolarizer. This was done to
demonstrate typical accuracy of the GDx-MM, in support of the error model predictions. An
optical system mounted to the GDx-MM collimated the output beam with a pivot point at the
external pupil, giving a telecentric scan over a 12 mm field of view. A plane mirror reflected
this beam directly back to the GDx-MM for measurement. Validation targets mounted in the
collimated region between the lens system and mirror included a Polarcor polarizer [Corning
Specialty Materials, Corning, NY] on a rotation stage (diattenuation magnitude and angle), a
precision polymer quarter wave plate [Meadowlark Optics, Frederick CO] on a rotation stage
(retardance angle), and a LCVR [Bolder Vision Optik, Boulder CO] oriented at 60º
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(retardance magnitude). In another test a Spectralon 99% reflectance standard [Labsphere,
North Sutton, NH], a highly diffuse material, replaced the mirror (depolarization). Data was
collected for the polarizer with axis varying from 0º to 90º in 15º increments, for the quarter
wave plate with axis varying from 0º to 90º in 15º increments, for the LCVR with retardance
varying from 3/16 to 12/16 wave in 1/16 wave increments, and for the Spectralon. Table 2
shows the observed accuracy of polarization parameter measurement. Diattenuation
orientation error is higher than model predictions due to a strong sensitivity to motor start
position. GDx-MM accuracy could be improved in a future design through use of
temperature control, improved motor control, and replacing the LCVRs with more stable
retardance modulators.
Table 2 Observed accuracy of GDx-MM measurement of polarization standards
Polarization Parameter Measured Accuracy
Diattenuation magnitude -0.05 ± 0.03
Diattenuation orientation -9.9º ± 0.9º
Retardance magnitude ±0.09 ± 0.1 rad
Retardance orientation -3º ± 2º
Depolarization index 0.05 ± 0.03
3. Use on human subjects
Retinal images of the optic nerve head (ONH) and macula of 15 human subject volunteers
were obtained under the direction of Dr. Ann Elsner at Indiana University School of
Optometry. Subjects were of Caucasian or Asian descent and in the approximate age range
20-55. All subjects received human subject training and gave informed consent. One subject
has an epiretinal membrane and one subject is hyper-myopic; otherwise all subjects are
normal and free of retinal disease. Several datasets were recorded for each subject; only the
highest quality macula and ONH set were retained for further analysis.
3.1 Mueller matrix images
Representative normalized Mueller matrix images (“MM images”) which show the spatial
variation across the retina of each of the 16 Mueller matrix elements are shown in Fig. 3. All
elements except the m00 element have been divided by m00 to remove variations due to
intensity and highlight variations due to diattenuation, retardance, and depolarization. These
MM images provide a quick assessment of polarization properties; values near zero in the first
row and column indicate low diattenuation and polarizance, and values near one on the
diagonal indicate moderately low depolarization. A bowtie-shaped pattern is clearly visible in
the macular data in the off-diagonal elements in the second through fourth rows and columns,
due to the interaction of the anterior segment and retinal retardance distributions.
The Frobenius error F when converting from a measured Mueller matrix M to a physically
realizable Mueller matrix M is an indicator of the accuracy of the polarization data at each
pixel. F is defined as:
MM MM ′
+
′
−
=F . (7)
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Fig. 3. Mueller matrix images: (a) optic nerve head of right eye (for one subject); and (b)
macula of right eye (for a different subject).
Typical Frobenius error images are shown in Fig. 4, where a threshold of 0.25 (i.e. 25%
difference between measured and physical) was selected to indicate excessive error. The
mean error over all pixels in all collected datasets is approximately 0.1. The error along blood
vessels is higher due to: (1) low light levels, and (2) larger intensity gradients, thus making
polarization measurements more sensitive to eye motion than in other retinal areas. The
central circular-shaped regions of higher error are due to laser back-reflections within the
instrument from optical interfaces near normal incidence. A single such back-reflection is
observed in conventional GDx images, but in the GDx-MM there are multiple back-reflection
spots due to the angle of the inserted LCVRs.
Fig. 4. Frobenius error images for one subject. Yellow regions indicate an error greater than
0.25; mean error is approximately 0.1. (a): optic nerve head for right eye (for one subject); and
(b): macula for right eye (for a different subject).
3.2 Anterior segment compensation
The anterior segment (AS), in particular the cornea, may exhibit retardance ranging from near
zero to several times the retinal retardance [33,34]. This AS retardance component should be
compensated in each dataset to obtain accurate retinal retardance images. This is done here
by forming an estimate of the AS retardance, and then removing this retardance component
using a Mueller matrix inverse algorithm. The estimation process is underdetermined,
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involving two “equations” (the measured retardance magnitude and orientation), and four
unknowns (AS retardance magnitude and orientation and retinal retardance magnitude and
orientation). The cornea may be modeled in general as a biaxial crystal with the fast axis
normal to the surface [35]. Significant differences in the absolute magnitude and spatial
variation of corneal birefringence have been observed among subjects [36]. Thus, the AS
retardance component will have significant dependence on the location and incident angle of
the probing beam, complicating the estimation process.
Two methods, “bowtie” and “screen”, have been described to estimate AS retardance from
ocular polarization data. The experimental basis for the bowtie method was established by
klein Brink and van Blokland [37]. Retardation images of a normal macula exhibit a distinct
bowtie-shaped pattern. The bowtie method models the AS as a uniform linear retarder with
fixed retardance magnitude and fixed axis, and the retinal Henle layer as a uniform linear
retarder with fixed retardance magnitude and radial slow axis centered on the fovea. Where
the axes are aligned, the overall retardance is a maximum equal to the sum, and where the
axes are orthogonal the overall retardance is a minimum equal to the difference. The second
harmonic of a Fourier series representing the intensity variation around a circle centered on
the fovea provides an estimate of the constant AS retardance magnitude and orientation. The
accuracy of bowtie method estimates is variable, depending on image quality and individual
retardance variations in the macula region. The screen method suggested by Knighton and
Huang [38] is an alternative, particularly when the bowtie pattern is weak or indeterminate. It
is assumed that the AS retardance may be approximated by averages over a large area of the
macula region (i.e. assumes macular retardance is negligible). Since the averaging area is
larger, the effect of noise is reduced as compared to the bowtie method. Accuracy depends on
the retardance variability in the nerve fiber layer, which may be significant.
After evaluating the bowtie and screen methods, we selected the screen method to perform
AS compensation. To extract the estimated AS retardance magnitude δA and orientation φA
from the Mueller matrix image, the original Mueller matrix M at each pixel is decomposed
into component matrices for diattenuation (Mdiatt), retardance (Mret), and depolarization
(Mdep) using the Lu-Chipman algorithm. Retinal retardance matrices Mret with AS
component removed are calculated:
11 ,
2
,
2
−−
′
=
′′ A
A
A
ALRLR
φ
δ
φ
δ
retret MM , (8)
where LR[δ,φ] indicates a linear retarder with retardance δ and orientation φ. A new Mueller
matrix is reassembled at each pixel as
diattretdep MMMM ′′′′
=
′′ . (9)
The composite MM image assembled from the M matrices is then the full retinal
polarization image with anterior segment retardance extracted.
Elsner has observed that retinal nerve fiber layer thickness and the related retardance
magnitude are lowest and most uniform within a visual angle of about 1.25° radius from the
fovea [39], implying that AS estimation would have best accuracy in this region. Our
algorithm averages over a circular region within a 2.5° radius (5 pixel radius) of the fovea,
irrespective of the bowtie appearance (back-reflection pixels were excluded). This method
usually resulted in a uniform circularly symmetric (i.e. “donut-like”) residual foveal area
retardance pattern after compensation (Fig. 5). The importance of corneal compensation is
clearly illustrated in this figure, which shows that retardance is significantly over-estimated
when compensation is not performed.
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(a) (b)
Fig. 5. Anterior segment compensation using the “screen” method within a 2.5° radius of the
fovea of a right eye. (a): uncompensated linear retardance image exhibits a bowtie pattern; (b)
post-compensation linear retardance image exhibits a donut pattern. The spots in the center of
the images are due to laser back-reflections within the instrument.
AS compensation was problematic for some datasets. In three cases our compensation
method did not produce a circularly symmetric pattern near the fovea. Of the remaining 12
datasets, three had well defined donut-like patterns, and nine had irregular circular patterns.
The wide variety of AS compensation results among these 15 normal subjects clearly
illustrates the difficulty in reliably compensating the anterior segment. This is a shortcoming
of the GDx-MM and similar retinal polarimeters, and might be mitigated for example through
use of an independent means to estimate AS retardance [40].
3.3 Retinal polarization parameter images
Example polarization parameter images showing linear retardance, retardance orientation,
diattenuation magnitude, diattenuation orientation, and depolarization index are shown in Fig.
6 for the ONH and in Fig. 7 for the macula. These images derive from the Mueller matrix
images of Fig. 3, with use of the technique of Section 3.2 to perform corneal compensation.
A simple statistical analysis of our linear retardance, diattenuation, and depolarization
index values was performed over full images for each of the 15 subjects. This analysis is
intended to facilitate comparison to previously reported results, and to illustrate the typical
range of values seen. Pixels with high Frobenius error were excluded from the calculations.
Table 3 tabulates lowest and highest values from each image-averaged parameter, along with
the mean and standard deviation of the distributions of individual pixel values over the fifteen
images.
Table 3: Polarization parameter statistical summary
Parameter Minimum Maximum Mean, Std Dev
ONH linear retardance 23.3º 34.9º 28.4º ± 15.5º
Macula linear retardance 9.2º 21.1º 17.3º ± 9.4º
ONH diattenuation magnitude 0.07 0.11 0.09 ± 0.05
Macula diattenuation magnitude 0.07 0.10 0.08 ± 0.05
ONH depolarization index 0.21 0.29 0.25 ± 0.07
Macula depolarization index 0.21 0.32 0.25 ± 0.07
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Fig. 6. Polarization parameter images for ONH of right eye, for one subject. These images
derive from the Mueller matrix image of Fig. 3(a), with use of the technique of Secion 3.2 to
perform corneal compensation. (a): normalized average intensity; (b): linear retardance; (c):
retardance orientation; (d): depolarization index; (e) diattenuation magnitude; (f) diattenuation
orientation.
Fig. 7. Polarization parameter images for macula of right eye, for one subject. These images
derive from the Mueller matrix image of Fig. 3(b), with use of the technique of Secion 3.2 to
perform corneal compensation. Details are the same as for Fig. 6.
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Linear retardance images of the ONH show highest magnitude in superior and inferior
locations (particularly near large vessels) and lowest magnitude in nasal and temporal regions.
These findings are consistent with the typical thickness variation of a normal retinal nerve
fiber layer. The mean retardance averaged over the 15 subjects is 28.4º ± 15.5º for ONH and
17.3º ± 9.4º for macula. The variation around the ONH was approximately 10° to 60°,
consistent with the results of Huang et al. (mean retardance of 33 ± 3°, variation from 10° to
50°) [41]. Retardance orientation images show a fast axis orientation that is horizontal at 12
and 6 o’clock, and vertical at 3 and 9 o’clock, with smooth variation between. The fast axis
thus exhibits approximate circular symmetry about a center point (the fovea or the ONH), and
the slow axis is radially symmetric about this center point, as would be expected given the
radial orientation of the retinal fibers in these areas. The “pinwheel” patterns in the macula
region are consistently well defined and symmetric over the 15 datasets. The patterns about
the ONH exhibit lower symmetry, as the fibers in this area are not as precisely radial as the
Henle fibers in the macula.
Depolarization index images show weak features with mean value of approximately 25%,
indicating for polarized light incident, the average degree of polarization is 75%. Higher
depolarization of up to 40% is seen particularly in annular regions around the ONH and in
localized regions near the macula. Depolarization along major blood vessels is higher than in
surrounding regions; this may be due to higher scattering from vessels, but may also be a
result of lower accuracy due to the lower signal to noise ratio. These results are in reasonable
agreement with degree of polarization measurements by van Blokland and Norren (80%-90%)
[42] and Bueno (67% - 92%) [9]. Localized higher depolarization in the macula has also
been observed by Elsner [43].
Diattenuation magnitude images show weak features and exhibit mean value of
approximately 8%, in reasonable agreement with measurements by Park et al (6%) [44],
Kemp et al (2%) [12], and Bueno and Artal (10%) [26]. Diattenuation magnitude is
consistently higher superior and inferior to the ONH, but with qualitatively somewhat
different appearance from retardance magnitude, suggesting that the diattenuation profile may
be partly but not completely due to the arrangement of fibers in the nerve fiber layer. The
diattenuation orientation images (though noisy due to low diattenuation magnitude) are
qualitatively similar to the retardance orientation images, also suggesting a significant
contribution from the nerve fiber layer. These results are in general agreement with models
[12,45] which predict that the structural arrangement of fibers in the nerve fiber layer should
give rise to dichroism (and hence diattenuation) as well as birefringence. Bueno and Artal
observed no correlation between ocular retardance and diattenuation axes using a nonimaging
Mueller matrix polarimeter [26], but as no corneal compensation was performed the
birefringence axis measurement was dependent on both retinal and corneal contributions.
Diattenuation magnitude images for the macula region show variation about the fovea, which
is most likely due to Henle layer variations. The orientation images are variable among
subjects, showing either no preferred orientation, localized regions of preferred orientation as
in Fig. 7, or a single preferred orientation. The measured diattenuation necessarily includes
contributions from both anterior segment and retina. Bueno and Jaronski measured single-
pass diattenuation of 0.03±0.01 in bovine and porcine corneas [19], less than but comparable
to the diattenuation reported here. For accurate retinal diattenuation measurement anterior
segment diattenuation may need to be compensated using an analog to the retardance
technique discussed in Section 3.2.
To compare GDx-MM and GDx measurements of linear retardance (the GDx is
demonstrably less accurate due to depolarization and diattenuation coupling into the linear
retardance measurement), data was collected at Indiana University on three study subjects
using a GDx. Raw image sets were processed to produce retardance values according to the
algorithm in [38]. To avoid ambiguities from the different AS compensation methods, GDx-
MM post-processing compensation was not performed and the corneal compensator retarder
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was removed from the GDx. The data from the two instruments were similar (Fig. 8), but the
retardance measured with the GDx-MM was consistently higher. For the ONH shown in Fig.
8, linear retardance measured using the GDx-MM was M = 48.1° ± 16.8°, depolarization
index using the GDx-MM was a = 0.21 ± 0.06, and linear retardance using the GDx over
approximately the same retinal field of view was G = 39.3° ± 16.9°. By modeling the nerve
fiber layer as an ideal linear retarder of retardance M combined with a pure depolarizer which
imparts a degree of polarization (1-a) to all states, the approximate expected GDx error in
retardance measurement may be calculated [25]. For the data of Fig. 8 the calculated error is
12°, in good agreement with the observed error of M – G 9°. Our results compare
favorably with the error of approximately 10° measured by Bueno in an eye with a 0.2 [9].
Fig. 8. Comparison of uncompensated linear retardance measurement for the right eye of same
subject with (a) GDx and (b) GDx-MM.
4. Conclusions
A complete imaging retinal polarimeter, the GDx-MM, has been designed, optimized,
calibrated, validated, and used in a clinical setting to obtain retinal data for normal human
subjects. Images showing the spatial variation of polarization parameters were generated and
analyzed. Collected data was demonstrated to be consistent with previously reported results.
Space limitations within the GDx mandated the use of non optimal polarization components,
liquid crystal variable retarders, which limited the polarization accuracy of the instrument.
No subjects with retinal disease were evaluated for this study; however, it is important to
consider how each polarization parameter image might provide useful information in disease
diagnosis and monitoring. Conjectures can be made based on the demonstrated utility of
similar instruments such as the Carl Zeiss Meditec GDx and knowledge of polarization effects
in the eye. Retardance images may provide insight into the Henle layer, retinal nerve fiber
layer, cornea, and lens. The most apparent use is in monitoring the thickness of the nerve
fiber and Henle layers, as an indicator of the presence and progression of glaucoma.
Disruptions in or absence of the bowtie in the uncompensated linear retardance plot and/or the
pinwheel in the retardance orientation plot may indicate disruptions of the Henle layer.
Based on qualitative comparisons of the 15 datasets, diattenuation orientation and
retardance orientation have similar appearance in the ONH, suggesting that much of the
diattenuation arises from the nerve fiber layer. The diattenuation has a higher spatial
frequency component than the retardance and so may be simultaneously indicating the
configuration of other retinal tissues. These results suggest that retinal diattenuation
measurements may be used to probe the nerve fiber layer structure with significantly lower
anterior segment effect than is observed in the retardance measurement. This hypothesis is
supported by a study by Naoun et al. [46] in which significant differences in dichroism were
observed between normal and glaucomatous eyes. Huang and Knighton [11] have modeled
the nerve fiber layer fibrils as an array of cylinders. They predict a thickness-independent
diattenuation with magnitude similar to our observations, but with orientation that is parallel
to the fibrils at normal incidence, in contradiction to our observations (Fig. 6(f)). Our results
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may derive from deeper back-scattering events, in which case diattenuation magnitude would
be thickness dependent. The GDx-MM, with its unique ability to measure a full Mueller
matrix in direct backscatter, is well suited to explore the validity of the hypotheses set forth in
scientific models.
Depolarization images are of ongoing interest since they emphasize multiple scattering
and deeper scattering structures [5-9]. Elsner and colleagues have generated depolarization
images by reprocessing GDx data to find the minimum intensity in the cross polarized
channel. This metric, while not an exact calculation of the depolarization index, responds
strongly to depolarization and weakly to elliptical retardance, elliptical diattenuation, and
inhomogeneous (non aligned) linear retardance and diattenuation. By this method Elsner
observed that many disease-related features have higher contrast in the depolarization image,
including drusen, pools of fluid and leakage points, vessel abnormalities, edema, exudative
lesions, peripapillary hyperpigmentation, and neovascular membranes. Naoun et al. have
measured significant differences in degree of polarization between normal and glaucomatous
eyes, attributed to the loss of ganglion cells and axons [46]. It is thus reasonable to predict
that depolarization images would facilitate detection and monitoring of disease processes
which disrupt the regular structure of retinal tissue.
Acknowledgments
The authors express their appreciation to Paula Smith, Tiffany Lam, Karlton Crabtree, Brian
Daugherty, Phillip McCulloch, Charles Burkhart, Tom Zobrist, and Noah Gilbert for their
assistance in modifying the GDx-MM optics, electronics, and mechanics. They also thank
Drs. Steve Burns, Jim Schwiegerling, John Greivenkamp, Jose Sasian, and Stanley Pau for
helpful discussions on the instrument design. This work was funded by NIH NEI
RO1EY007624 to Dr. Ann Elsner, a grant from the Boye Foundation, and graduate
scholarships from the University of Arizona Biomedical Imaging Scholarship program and
from Achievement Rewards for College Scientists (ARCS).
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