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Analysis and Evaluation of Measurement Errors in Mueller Matrix
Microscopy
Tongyu Huanga,b, Zheng Hua, Hui Maa,b,*
aShenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China
bDepartment of Biomedical Engineering, Tsinghua University, Beijing 100084, China
ABSTRACT
Mueller matrix microscopy can provide a comprehensive description of sample’s polarization properties non-invasively,
and has shown attractive applications in many different fields such as biomedicine, marine particle monitoring, and
material identification. Mueller matrix measurements are based on multiple light intensity measurements under
illuminations of different polarization states, therefore have more error sources compared to other imaging methods, which
reduce the image quality and hinder efforts for quantitatively characterizing the polarization features of weak polarization
biological samples such as living cells. Measurement errors of a Mueller matrix microscope are mainly classified into three
categories including systematical error, random noise, and mismatch error. In this paper, based on the physical realizability
of Mueller matrix and image frequency analysis, different categories and sources of measurement errors in Mueller matrix
microscopy are analyzed in detail, a no-reference Mueller matrix image quality evaluator (MIQE) is also proposed to
estimate the error level of biomedical sample’s Mueller matrix.
Keywords: Mueller matrix microscopy, measurement error, image quality evaluator
1. INTRODUCTION
Polarization measurement technology has many advantages, such as sensitivity to the anisotropic microstructure of samples,
non-destructive to the samples, and high optical structural compatibility, therefore it has played an important role in marine
particle detection [1], material analysis [2], and biomedical fields [3]. Among different polarization measurement methods,
Mueller matrix microscopy can obtain the full-polarization properties of the sample and has gradually received more and
more attention in recent years. In practical applications, biomedical samples -- the observation target of Mueller matrix
microscopes, often has relatively weak polarization properties with complex and limited measurement environment
compared to standard polarized components, which puts higher requirements on the signal-to-noise ratio and error level of
Mueller matrix microscopy. However, since the reconstruction of the Mueller matrix is based on multiple intensity
measurements under different polarization modulation states, various error sources may interfere with the measurement
results of Mueller matrix, such as the imperfection of polarization components within the Mueller matrix microscope, the
random noise of the detected light intensity, and the mismatch between the measurement resolution of Mueller matrix
microscope and the polarization properties of the sample itself. Those measurement errors follow complex error transfer
rules, making it difficult to analyze and eliminate, which seriously affects the improvement of Mueller matrix image quality
of the biomedical samples, and hinder the promotion of Mueller matrix microscopy in biomedical applications.
The traditional method for evaluating the Mueller matrix microscope’s measurement error level is to measure standard
polarization components such as linear polarizer and then conduct quantitative comparison with its theoretical value.
However, this method cannot fully characterize the measurement performance of microscopes on real biomedical samples,
therefore, it is critical to develop image quality assessment method based on biological sample’s Mueller matrix. In the
field of digital image processing, many image quality evaluators have been proposed including peak signal to ratio (PSNR)
[4], structural similarity index (SSIM) [5], feature similarity index (FSIM) [6], gradient magnitude similarity deviation
(GMSD) [7], natural image quality evaluator (NIQE) [8] and so on. Based on whether the reference image is required,
these criteria can be divided into reference image quality assessment (R-IQA) methods and no-reference image quality
assessment (NR-IQA) methods. In polarization measurement fields, however, the reference polarization images of
biomedical samples are difficult to obtained in practice, which limits the application of R-IQA methods. Besides, most
current NR-IQA methods such as NIQE which based on the human visual perception model does not conform to the
characteristics of polarized images. A no-reference polarization image quality assessment method which is based on self-
consistency of redundant linear polarization measurement has been proposed, however, this method is not applicable for
the quality evaluation of Mueller matrix image data which is more mathematically complex.
In this paper, we proposed a no-reference Mueller matrix image quality evaluator (MIQE) to estimate the image
quality of biomedical sample’s Mueller matrix image. MIQE can provide a reference criterion and help to analyze the
sources of various measurement errors in Mueller matrix images biomedical samples, which is beneficial for continuously
optimize the imaging performance of Mueller matrix microscopes.
2. MUELLER MATRIX MICROSCOPY
As shown in Fig. 1, in this paper, the Mueller matrix microscope is established by adding polarization states generator
(PSG) and polarization states analyzer (PSA) on the transmission biological microscope, which is capable of measuring
Mueller matrix images of transparent biomedical samples such as live cells and tissue sections.
Figure 1. The simplified structural diagram of Mueller matrix microscope
The monochromatic illuminating light (λ= 633 nm) is emitted from the LED and then undergoes polarization
modulation by PSG. PSG consists of a fixed linear polarizer P1 and a rotatable quarter wave plate R1, whose fast axis
angle is parallel to the polarization orientation of P1 at the initial stage. During the Mueller matrix measurement, R1 in
PSG rotates to 4 different preset angles to modulate the polarization state of the illuminating light, then the 4×4
instrument matrix of PSG can be described as:
[ ]
PSG R1 P1 LED in in in in
= (1), (2), (3), (4)=AMMS SSSS
(1)
Where
in
S
(i) (i = 1,2,3,4) represents the input Stokes vector corresponding to the i’th rotation angle of R1.
R1
M
and
P1
M
represents the Mueller matrices of R1 and P1, respectively.
LED
S
= [1,0,0,0] represents the Stokes vector of the initial
natural light emitted by LED. After the sample is illuminated by the polarized light from PSG, the scattered light from the
sample is modulated and captured by the PSA. PSA adopts dual division of focal plane (DoFP) polarimeters method [9],
each DoFP polarimeter has the ability to obtain images of four linear polarization channels including 0°, 45°, 90°, and
135° in a single shot. In PSA, two DoFP polarimeters named DoFP1 and DoFP2 are fixed at the transmission end and
reflection end of a non-polarized beam splitter (NPBS), respectively, and a fixed quarter wave plate R2 is fixed between
DoFP1 and NPBS, then the 8×4 instrument matrix of the PSA can be represented as:
DoFP R2 R2 R2
PSA
DoFP
(, )
θδ
=
AM
A
A
(2)
Where
R2
M
is the Mueller matrix of the quarter wave plate R2 with fast axis orientation
R2
θ
and retardance
R2
δ
,
DoFP
A
is
the instrument matrix of the DoFP polarimeter.
Finally, Eq. (3) is applied to reconstruct sample’s Mueller matrix
sample
M
,
11
sample PSA sample PSG
[]
−−
=M AI A
(3)
or
-1
sample sample
( ) [ ( )]vec vec=M AI
(4)
In which
PSG PSA
=
T
⊗AA A
(5)
A represents the 16×4 effective instrument matrix of Mueller matrix microscopy,
sample
I
is the intensity matrix directly
captured by DoFP1 and DoFP2, the superscript
1−
represents the inverse or pseudoinverse of the matrix.
3. MUELLER MATRIX MEASUREMENT ERROR (MME)
According to Eq. (4), sample’s Mueller matrix
sample
M
can be reconstructed by the measured light
sample
I
through the
instrument matrix A of the Mueller matrix microscope. When their real values are denoted as
real
M
,
real
I
,
real
A
, and the
corresponding measured values are denoted as
exp
M
,
exp
I
,
exp
A
, then the Mueller matrix measurement error (MME) can be
expressed as the following equation:
-1 -1
sample r eal exp real real exp exp
( )= [ ( )] [ ( )]MME vec vec−= −M MM A I A I
(6)
Therefore, according to Eq. (6), MME of the sample originates from the measured light intensity
sample
I
by the detector
and the instrument matrix A composed of PSG and PSA. Meanwhile, according to the definition of measurement accuracy
and precision, MME can be divided into the systematical error (
sample
∆I
, ΔA) and random noise (
sample
δ
I
,
δ
A). As the
inherent error with in Mueller matrix microscopy, the systematical error will affect measurement accuracy, which mainly
manifested in the following forms such as the nonlinear response of photodetectors, and the fixed deviation of PSG and
PSA’s instrument matrices. While random noise refers to the noise of the photodetectors and the instability of the
instrument matrices. In the practical applications, the random noise of the instrument matrix and the systematic error of
the measured light intensity are often ignored (
sample
∆I
0,
δ
A 0), then Eq. (6) can be expanded as the following form:
-1 -1
sample sample sample real real real real
( )= ( ) [( ) ][ ( )]MME vec vec
δδ
∆ + = − +∆ +M M M A I AA I I
(7)
-1 -1
sample real real real
[ ( )] ( )vec∆ = − +∆M A AA I
(8)
-1
sample real
( ) ()vec
δδ
=− +∆
M AA I
(9)
Where
sample
∆M
denotes the systematical error of the measured Mueller matrix,
sample
∆M
only related to the systematical
error of the instrument matrix
∆A
. While
sample
δ
M
denotes the random noise of the measured Mueller matrix,
sample
δ
M
is
influenced by both
∆A
and the random noise of the detected light
δ
I
, both
δ
I
and
sample
δ
M
shares the same type of
probability distribution. In this case, other than low-light conditions, the dominant type of random noise is additive
Gaussian noise.
For the Mueller matrix point-measurement, MME mainly includes two categories including systematical error
sample
∆M
and random noise
sample
δ
M
. However, for Mueller matrix imaging techniques, during the measurement process
of the Mueller matrix, the polarization properties of the sample may vary in either spatial or temporal dimensions. When
the spatial resolution or the temporal resolution of the Mueller matrix microscope cannot capture these variations,
unneglectable mismatch error
sample
ε
M
will occur in the results. Therefore, the overall measurement error of Mueller matrix
is modified as:
sample sample sample sample
( )=MME
δε
∆+ +MM MM
(10)
4. MUELLER MATRIX IMAGE QUALITY EVALUATOR (MIQE)
Since Mueller matrix microscopy is a quantitative imaging technique, when the measurement conditions remain unchanged,
there only exits one set of true value of the sample’s Mueller matrix, therefore, the quality of the Mueller matrix image
can be quantitatively characterized by Mueller matrix’s measurement error MME. However, due to the difficulty in
obtaining the true value of biomedical sample’s Mueller matrix
real
M
, it is hard to directly characterize the MME.
Unlike Jones matrix, Mueller matrix comply to various physical constrains, also known as physical realizability. As
an important Mueller matrix decomposition method, Cloude decomposition can obtain the Mueller matrix
PR
M
which is
not only closest to the measured Mueller matrix
exp
M
but also meets the physical realizability requirement [10]. Therefore,
we can define the Mueller matrix physical realizability error (MPRE):
sample PR exp
( )=MPRE −M MM
(11)
Mueller matrix can be decomposed into a superposition of three independent polarization characteristics based on
Mueller matrix polarization decomposition (MMPD) method: depolarization , diattenuation D and retardance [11]. For
the Mueller matrix image of biomedical sample with weak polarization properties such as tissue sections and living cells,
we make the following assumption: (a) their average values of polarization properties are relatively small (
,
, 0,
sample
M
air
M
) , (b) their physical feasibility error should be positively correlated with the measurement error
(
sample sample
() ()MPRE MME∝MM
).
Based on the above two assumptions, we can quantitatively characterize the error level and image quality of
biomedical sample’s Mueller matrix through MPRE, without the need for any reference images or pre-dividing the
foreground and background of the Mueller matrix image, which enhances the operability of Mueller matrix image quality
evaluation. Besides, since Mueller matrix contains both the polarization information (
12
m
,…,
44
m
) and the non-
polarization intensity information
11
m
,
11
m
’s image quality also play an important role in Mueller matrix image quality
evaluation. Here we apply non-reference image clarity parameter based on Tenengrad operator to evaluate the image
quality of the intensity image
11
m
. Finally, similar to the concept of signal-noise-ratio (SNR) in the fields of digital signal
processing, we proposed a Mueller matrix image quality evaluator (MIQE) as follows:
11
sample 10 int ensity polarization
sample
()
( )=10log +
()
Clarity m
MIQE MIQE MIQE
MPRE
=
M
M
(12)
Where
11
()Clarity m
is the image clarity of
11
m
.
sample
()MPRE M
is the average value of
sample
()MPRE M
.
16
sample sample
1
1
()= (())
16 i
MPRE MPRE i
=
∑
MM
(13)
In addition to quantitatively evaluating the image quality of the Mueller matrix, it is also important to classify and
evaluate the different categories of Mueller matrix’s measurement errors, in order to select targeted error-elimination-
method. Here we also consider the separation of three categories of measurement errors: systematical error
sample
()MPRE∆M
, random noise
sample
()MPRE
δ
M
and mismatch error
sample
()
MPRE
ε
M
. The steps are as follows: firstly
we perform 2D Fourier transform (2D-FT) to convert MPRE matrix into the spatial frequency domain F(MPRE).
According to the assumption (a) in section 4, systematical error
sample
()MPRE∆M
can be separated first from the the DC
component of F(MPRE). This is because
sample
M
air
M
, therefore
sample
()MPRE∆M
MPRE
. However, separating
random noise
sample
()MPRE
δ
M
and mismatch error
sample
()MPRE
ε
M
would be more complex. Here we measure the air
using Mueller matrix microscope, as shown in Fig. 2 (a), whose Mueller matrix should not contain any mismatch errors
since the image of air has no details. Therefore, after separating the systematical error
sample
()
MPRE∆M
, the remaining
frequency components should all belong to the random noise
sample
()MPRE
δ
M
(Fig. 2 (c)). Besides, we also found that
the frequency distribution of random noise is similar to the form of the Gaussian function (Fig. 2 (d)).
Figure 2. Air’s (a) Mueller matrix, (b) MPRE matrix, (c) Spatial frequency of MPRE, (d) Spatial frequency of MPRE along
the centered row of data marked in (c). The red solid curve represents the original value and the blue dashed curve
represents the fitted Gaussian function corresponds to the random noise.
As shown in Fig. 3, when the sample turns to other biomedical samples such as H-E stained sections while other
experimental conditions remain the same, mismatch error
sample
()
MPRE
ε
M
will appears due to the mismatch between the
polarization properties of the sample and the measurement resolution of Mueller matrix microscopy. However, the noise
level of random noise
sample
()MPRE
δ
M
will not change, which means in Fig. 3 (b), the increase in MPRE is mainly due
to the generation of mismatch error
sample
()MPRE
ε
M
. Similar phenomenon can also be observed in spatial frequency
domain of MPRE (Fig. 3 (c)). In Fig. 3 (d), the red solid curve represents the superposition of random noise
sample
()MPRE
δ
M
manifested as a Gaussian distribution and mismatch error
sample
()
MPRE
ε
M
manifested as other types of
function. Based on Gaussian Mixture Module (GMM),
sample
()MPRE
δ
M
can be separated (blue dashed curve) and the
difference between the red curve and blue curve belongs to the mismatch error
sample
()
MPRE
ε
M
(green dashed curve). In
this way, three different categories of measurement errors can be split and quantified.
Figure 3. Under same experimental conditions, H-E stained tissue section’s (a) Mueller matrix, (b) MPRE matrix, (c) Spatial
frequency of MPRE, (d) Spatial frequency of MPRE along the centered row of data marked in (c). The red solid curve
represents the original value and the blue dashed curve represents the fitted Gaussian function corresponds to the random
noise, the green dashed curve represents the difference between the red curve and the blue curve, which corresponds to the
mismatch error.
5. CONCLUSIONS
This work has several potential values. Firstly, a no-reference Mueller matrix image quality quantitative assessment
method is provided, which is capable to comprehensively and conveniently characterizes the measurement performance
of Mueller matrix microscopes; Secondly, through quantitatively separating different types of measurement errors can help
to select targeted methods to improve the system's capabilities; Finally, conducting experiments under different
experimental conditions on the same sample is beneficial to the further study of Mueller matrix’s error transfer rule.
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