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Depth-wise differentiation of Jones
matrices obtained from Mueller
optical coherence tomography
Milos Todorovic, Shuliang Jiao, George Stoica, Lihong V.
Wang
Milos Todorovic, Shuliang Jiao, George Stoica, Lihong V. Wang, "Depth-
wise differentiation of Jones matrices obtained from Mueller optical coherence
tomography," Proc. SPIE 5316, Coherence Domain Optical Methods and
Optical Coherence Tomography in Biomedicine VIII, (1 July 2004); doi:
10.1117/12.531024
Event: Biomedical Optics 2004, 2004, San Jose, CA, United States
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Depth-wise differentiation of Jones matrices obtained from Mueller
optical coherence tomography
Miloš Todorovi
ć
a
, Shuliang Jiao
a
, George Stoica
b
, Lihong V. Wang
*a
a
Optical Imaging Laboratory, Department of Biomedical Engineering, Texas A&M University,
College Station, TX 77843-3120
b
Department of Pathobiol
ogy, Texas A&M University
, College Station, TX 77843-5547
ABSTRACT
A unique feature of polarization-sensitive Mueller optical coherence tomography (Mueller-OCT) is that it can
reveal various polarization properties of biological samples that are not observable using conventional OCT. One of the
most important polarization parameters is birefringence, which can be measured in its integrated form using existing
Mueller-OCT systems. We present a new method that uses the least squares algorithm to differentiate measured
integrated Jones matrices so that the samples can be obser
ved layer-by-layer. We tested the algorithm using simulated
data with variable additive white Gaussian noise (AWGN) levels. We further verified the algorithm using
in vitro
measurements of the porcine tendon and the septum of the rat heart. This least squares-based algorithm has the potential
to reveal structures previously hidden by the inherent masking properties of the integrated images and provide localized
phase retardation and orientation information.
Keywords:
differentiation, polar decomposition, polarization, OCT, birefringence, orientation, phase retardation, Jones
matrix, Mueller matrix.
1. INTRODUCTION
Ever since its introduction, optical coherence tomography
(OCT) has been establishing itself as one of the most
promising noninvasive imaging modalities offering high resolution and multiple contrast mechanisms. In addition to
conventional OCT, which uses the amplitude based contrast,
several other branches of OCT were developed based on
detecting changes in polarization of light
1-5
or Dopler shifts
6-8
resulting from the flow of fluids. The contrast in an OCT
image can result from the optical properties of a sample th
at modify the amplitude or the polarization state of the
incident light field.
Birefringence is a description of the anisotropy of the phase velocity of light in a sample. It is inherent in a variety
of biological components such as collagen, keratin, myelin,
and elastic fibers. Many denaturalization processes, such as
thermal denaturalization, alter birefringence in biological tissues and polarization-based OCT (PS-OCT) systems can
detect these changes
9-11
. Hence, the ability to detect the birefringent properties of structures within tissues can enhance
diagnosis.
Jiao
et al
have shown
4
that a scattering sample acts as a nondepolarizing medium because of the heterodyne
detection scheme used in OCT. Therefore, the polarization properties of biological samples can be described equally by
either Jones or Mueller matrices. In order to provide complete information about polarization properties of a sample,
PS-OCT should measure one of those matrices. We used the fiber-based polarization-sensitive Mueller-OCT system
12
developed in the Optical Imaging Laboratory at Texas A&M University to obtain Jones matrices used in this paper.
Previous papers
2,4
reporting on birefringent properties of biological tissues used Jones and Mueller calculus to
extract information about phase retardation and orientation fr
om integrated round-trip matrices. Those matrices can give
an insight into combined values of parameters of interest but cannot characterize individual layers precisely. We
propose a new differentiation algorithm that is capable of extracting single-trip Jones matrices of individual layers. It is
based on polar decomposition of matrices and employs a least squares algorithm to extract parameters, such as phase
*
LWang@tamu.edu; http://oilab.tamu.edu
Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VIII,
edited by Valery V. Tuchin, Joseph A. Izatt, James G. Fujimoto, Proc. of SPIE Vol. 5316
(SPIE, Bellingham, WA, 2004) · 1605-7422/04/$15 · doi: 10.1117/12.531024
370
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retardation and orientation of the optical axes, from Jones matrices. We have tested the algorithm using both simulated
data and
in vitro
measurements.
2. DIFFERENTIATION ALGORITHM
The measured Jones matrices along one A-scan obtained from Mueller-OCT encompass the integrated effect of all
of the layers preceding the layer whose local polariza
tion properties we are interested in. Therefore, the
i
-th measured
Jones matrix from any A-scan is influenced by all pr
eceding layers starting from the sample surface up to, and
including, the
i
-th layer as shown in equation (1),
,
1111
,
21221121
.
3 123321 12321
.........
T
em
st
st
rt
TT
T
em
st
st
st
st
st
rt
st
TT T
TT
e
m stststststst ststrtstst
==
==
==
JJJJ
JJJJJJJJ
J
JJJJJJ JJJJJ
(1)
In equation (1), indices
emi
denote the expected measured matrices;
rti
denote the round-trip matrices of a single layer;
and
sti
represent the one-way or actual Jones matrices of a give
n layer. We can recover every round-trip single-layer
matrix by recovering one-way single-layer matrices one by
one and eliminating their effects on measured matrices of
deeper layers, as shown by a generalized formula
11
1
1
(
)
...(
)
(
)
...(
)
(1)
1
1
(1)
TT
rti
st i
st
emi
st
st i
−−
−
−
=
−−
JJ
J JJ
J
. (2)
We need to assume a model of a tissu
e layer to proceed with characterizing
its properties. A sequence of a linear
retarder and linear diattenuator approximates a thin layer of biol
ogical tissue well. It is assume
d that the fast axes of the
retarder and diattenuator are collinear. The polar decomposition theorem, when used with Jones matrices, states that any
non-depolarizing polarization element described by a Jones matr
ix can be represented as a
cascade of a diattenuator and
a retarder
13
. The polar decomposition yields two matrices – a unitary matrix
U
corresponding to a retarder and a
nonnegative definite Hermitian matrix
H
corresponding to a diattenuator. Once round-trip matrices of individual layers
are calculated using equation (2), we can apply the polar decomposition to obtain matrices
U
and
H
. In order to
calculate parameters of an individual layer we fitted in the least–squares sense the matrices
U
and
H
to the model Jones
matrices of a linear retarder
and diattenuator, respectively.
Equations (3) represent final expressions for calculating the orientation, retardation, and transmission parameters of
Jones matrices of individual layers consisting
of a sequence of a retarder and diattenuator.
12
21
tan 2
11
22
11
22
cot
()cos2()sin2
11
22
12
21
22
'
cos
sin
(
) sin cos
11
22
12
21
22
'
sin
cos
(
) sin cos ,
11
22
12
21
ii
UU
mm
ii
UU
mm
rr
UU
mm
ii
i i
UU
U U
mm
m m
rr rr
pp H
H
H H
xx m
m
m m
rr rr
pp H
H
H H
yy m
m
m m
θ
φ
θθ
θθ
θθ
θθ
θθ
+
=
−
+
=
−++
==
+
+ +
==
+
− +
(3)
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where
φ
is the phase retardation;
θ
is the orientation angle of the fast eigenvector for both the diattenuator and retarder;
and
p
x
and
p
y
are the transmittances of the fast and slow eigenvectors, respectively.
3. RESULTS
To evaluate the validity of the algorithm, we conducted a computer simulation with an artificially generated data
set. A white Gaussian noise of different power levels was added to the signal in order to mimic the actual experimental
environment more accurately. Figure 1 depicts the ability of
the algorithm to recover the original values for retardation
when the generated data set had SNR levels of 20 and 50 dB. It is obvious that the number of layers that can be
recovered successfully drops as the SNR worsens. We can
see in Figure 1.a that the algorithm achieved a perfect
recovery of the phase retardation values for noise-free and 50 dB SNR cases; when the SNR dropped to 20 dB, the
algorithm was able to recover approximately first 35 layers with little or no error. Even more illustrative is Figure 1.b
where we show differences between the simulated retarda
tion values and values obtained using the differentiation
algorithm. Little or no deviation from the original values is observed for noise-free and 50 dB SNR. The only obvious
departure occurs for SNR of 20 dB after the 35
th
layer. SNR levels above 20 dB can be achieved in present day OCT
systems, especially in the most superficial layers of biological samples. Therefore, this differentiation algorithm can
perform correctly in actual systems.
Figure 1. Simulation results: a. recovered phase retardation – original phase retardation and
retardation recovered for noise free and 50 dB SNR environments overlap; b. deviations from the
original values.
In order to show the achievability of extracting parameters of one-way Jones matrices the algorithm was applied to
calculate polarization parameters of biological samples. Tendon is the one of the samples that is extensively used to
show the ability of OCT to image birefringence in biological tissue. Its highly pronounced birefringence originates from
the high concentration of collagen fibers. The integrated phase retardation image of a typical tendon shows a regularly
banded structure, which is a result of the 180 degrees phase wrapping. We imaged the fresh porcine tendon using the
Mueller-OCT system that will be reported in the paper by our group
12
. The algorithm for calculating Jones matrices
12
and the dynamic calibration technique for eliminating the effects of birefringence of the single mode fiber
14
can be
found in references. Figure 2 shows the integrated and differentiated phase retardation images together with the profiles
of phase retardation along one A-scan. Integrated image was obtained from the measured Jones matrices without
eliminating the effects of preceding layers. We can observe th
e banded structure in the integrated image and the phase
0
10
20
30
40
50
0
2
4
6
8
10
Retardation
φ
layers
retardation [
O
]
original
noise free
50dB SNR
20dB SNR
0
10
20
30
40
50
-2
0
2
4
6
8
10
Retardation error
∆
φ
layers
retardation error [
O
]
original
noise free
50dB SNR
20dB SNR
a.
b
.
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retardation values span the range between 0 and 180 degrees. This clearly indicates that the previously mentioned 180
degrees phase wrapping is present in the image. On the ot
her hand, differentiated image obtained using the algorithm
presented in this paper is much flatter; there are regions with more pronounced retardation but the range of retardation
values is only between 0 and 50 degrees, with majority of pixels having values up to 10 degrees. Higher values are
observed deeper in the sample where the signal-to-noise ratio (SNR) is much lower than in the superficial layers of the
sample. Profile of the integrated phase retardation values in
Figure 2.c shows the oscillatory nature resulting from phase
wrapping while the profile of the differentiated
values does not exhibit any abrupt changes.
Figure 2. Phase retardation of the porcine tendon: a. differentiated image; b. integrated image; c. profiles of
integrated and differentiated retardation along one averaged A-scan. Banded structure in the integrated phase
retardation image results from 180 degrees phase wrapping. There is no phase wrapping present in the
differentiated image. Image dimensions are 1mm x 1mm (width x height).
Finally, we investigated the orientation of muscle fibers in the septum of the rat heart. The heart tissue was fixed in
10% buffered neutral formalin before imaging. Following imaging, the paraffin-embedded tissue was sectioned at 20
microns and stained with hematoxylin and eosin (HE). The stained sections were examined under an Olympus light
microscope with 40X magnification. Figure
3 presents results of the analysis of
the septum from the surface up to the
depth of 680
μ
m along one A-scan. In order to improve SNR, the A-scan used to calculate the orientation and phase
retardation is a result of averaging over
50 A-scans taken at the same spot on the surface of the septum. Orientation of
fibers was observed from the histological images at different depths and is plotted in Figure 3 using a solid line. We can
see that the calculated values are in good agreement (within standard deviation) with the data from histology. As
expected, standard deviation increases with depth and this can
be attributed to lower SNR in the deeper regions of the
sample. The linear change of orientation observed in our study agrees with the results of a study
15
where the authors
observed that fiber orientation linearly changes from the surface of the septum towards the deeper layers.
a.
b.
c.
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-30
-20
-10
0
10
20
30
0
200
400
600
800
depth [μm]
orientation [deg] .
algorithm
histology
Figure 3. Orientation of muscle fibers in the septum of the rat heart. Data points from histological images are
spaced at 20 μ m intervals while the data points obtained using the differentiation algorithm are 80 μm apart.
4. CONCLUSIONS
Polarization properties of samples that do not depolarize in
cident light can be characterized either by Jones or
Mueller matrices. Because of the detection scheme used in
OCT, scattering samples such as biological tissue can be
considered a nondepolarizing medium. Jones matrices fully
characterize polarization properties, such as birefringence
and diattenuation, of a sample. Changes in values of magnitude and orientation of these polarization parameters provide
polarization-based contrast in PS-OCT. Previously reported algorithms for calculating the phase retardation and
orientation of birefringence worked on integrated round-
trip Jones matrices that are a direct outcome of OCT
reconstruction algorithms.
In this work, we introduced a novel algorithm that uses polar decomposition and least squares fitting to recover
single-trip Jones matrices of individual layers within scatte
ring media. The algorithm starts with the round-trip Jones
matrix of the most superficial layer and eliminates the effect
of preceding layers on the consecutive round-trip matrices.
This enables the use of polar decomposition, which decomposes the round-trip matrix of each layer into a unitary matrix
corresponding to a retarder and a nonnegative definite Hermitian matrix corresponding to a diattenuator. Each layer is
assumed to be a series of a linear retarder and diattenuato
r. Although the accuracy of the algorithm heavily depends on
the SNR level, in its current form it is capable of extracting useful information from the depths up to a half millimeter in
the sample. Future work will include the implementation of a de-noising algorithm that should aid the differentiation by
increasing the SNR.
We have shown the capabilities of this method using the computer simulation and
in vitro
measurements of the
porcine tendon and the septum of the rat heart. OCT images of the differentiated phase retardation have shown that we
were able to eliminate the integrating effect that is pr
esent in images where phase re
tardation is calculated from
integrated round-trip Jones matrices. We were also able to recover the orientation of muscle fibers in the septum. With
further improvements, this depth-wise differentiation algorithm promises to enhance the polarization-based contrast in
OCT by revealing structures previously hidden by the inherent masking properties of the integrated images.
ACKNOWLEDGMENTS
This project was sponsored by the National Institutes of Health grant R01 CA092415.
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