Image Anal Stereol 2016;35:149-158 doi:10.5566/ias.1554
OriginalResearchPaper
THECORRELATIONANALYSISOFTHESHAPEPARAMETERSFOR
ENDOTHELIALIMAGECHARACTERISATION
KAROLINA NURZYNSKA
B
;1
AND ADAM PIORKOWSKI
2
1
Institute of Informatics, Faculty of Automatic Control, Electronics and Computer Science, Silesian University
of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland;
2
AGH University of Science and Technology,
Departament of Geoinformatics and Applied Computer Science, Cracow, Poland.
e-mail: Karolina.Nurzynska@polsl.pl, pioro@agh.edu.pl
(ReceivedMay23,2016;revisedOctober5,2016;acceptedOctober20,2016)
ABSTRACT
Microscopic images of corneal endothelium cells are investigated to deliver information about their medical
state. Although this could be achieved automatically, this examination is manual and very time consuming.
Two medical parameters for endothelial layer quality description have been introduced and more are planned.
Yet, since they will exploit image processing, a thoughtful overview of applicable existing shape parameters
is necessary. This work investigates the possibility of exploiting well-known image processing techniques
for describing the endothelial layer by calculating information about shape features using spatial moments or
topological attributes. The comparison concentrates on finding which shape measures could be combined to
improve descriptions, and which cannot due to their high correlation and the fact that they do not contain any
new information. The performed experiments revealed a set of 17 non-correlated features and four groups of
shape parameters that show some correlation, but one representative can always be selected. Moreover, the
investigation proved some correlation between the metrics used in medicine and considered shape features.
Keywords: correlation analysis, endothelial images, shape parameters.
INTRODUCTION
The corneal endothelium is the translucent frontal
part of the eyeball which is responsible for keeping the
cornea clear by draining water from it (Agarwal et al.,
2002). It is a monolayer of cells whose shape in healthy
structures reflects a hexagon. The number of corneal
endothelium cells at birth is around 6,500 cells=mm
2
,
decreasing to 1,700–2,000 cells=mm
2
at the age of 80,
as reported by Koetal. (2000; 2001), because the cells
do not reproduce (Meyer et al., 1988). When a cell
dies, neighboring cells grow and take its place to fill
the layer tightly. Their shape changes as a result.
The structure, number, and composition of cells
in the endothelial layer are observed in vivo by
non-contact specular microscopy or inverse phase
contrast microscopy. The shape and spatial distribution
reflect the state of the corneal endothelium after
surgical procedures and are basic information for a
physician. Although manual annotation of this data
is not a problem, it is tiresome work. Therefore,
several automatic and semi-automatic solutions for
endothelial cell location were developed (Meijering,
2012).
DETERMINATIONOFCELLLOCATION
Although precise determination of cell location
in endothelial images seems simple, it proved to be
a difficult task for a computer program due to the
irregular illumination and artefacts caused by high
amounts of noise and distortion, as indicated by Gavet
and Pinoli (2008). Initial solutions tend to deal with
all problems altogether as present works by Nadachi
and Nunokawa (1992); Vincent and Masters (1992);
Hasegawa et al. (1996); Mahzoun et al. (1996);
Foracchia and Ruggeri (2000); Serra and Mlynarczuk
(2000).
Yet, the trend in algorithm design later changed,
when issues concerning accurate location of
endothelial cell borders were well defined. Therefore,
three stages are clearly noticeable in recently
introduced solutions. The first stage is responsible for
illumination compensation and noise removal. Some
interesting ideas can be found in works by Sanchez-
Marin (1999); Habratetal. (2016).
The second stage prepares a binary image with
located cell borders. This is the most widely researched
area and results in a myriad of solutions, including
a trivial ones based on thresholding as presented
by Sanchez-Marin (1999) or watershed exploited
by Caetano et al. (2000). Bernander et al. (2013);
Malmberg et al. (2014), and Selig et al. (2015a)
developed fast robust stochastic watershed algorithms
based on a probability density function that can locate
silent contours without wrong matches. There is also
a group of more sophisticated approaches: applying
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NURZYNSKA K ET AL: Thecorrelationanalysisoftheshapeparameters
Bayesian framework supported by simulated annealing
for cell border location (Foracchia and Ruggeri, 2003)
improved by statistical description (Foracchia and
Ruggeri, 2007); exploiting active contours (Dagher
and El Tom, 2008), snake-lets (Charłampowicz et al.,
2014), level sets (Zhou, 2007), and wavelets (Khan
et al., 2007). The data mining and rough sets theory
found an application in the solution designed by Poletti
and Ruggeri (2014); while Ruggeri et al. (2010) used
an artificial neural network to classify whether a
pixel belongs to a cell body or boundary oriented at
different angles; the genetic algorithm was exploited
by Ruggeri and Scarpa (2015); Scarpa and Ruggeri
(2015), who supported it with information about pixel
intensities and regularity of cell shapes. Finally, the
third stage is thinning, which is used to locate cell
borders precisely. Saeed et al. (2010) introduced the
K3M algorithm for image skeletonization. Recently,
a very interesting comparison that paid attention to
cell border determination accuracy and the influence
on parameter calculation of a repeatability framework
was presented by Piorkowskietal. (2016).
EXISTINGMEDICALPARAMETERS
Images of the corneal endothelium with marked
cell borders are insufficient for a physician. An
automatic method for comparison and endothelial
layer state description is necessary in order to support
tracking of endothelial structure changes due to ageing
or damage caused by disease or surgical procedures.
Straightforward comparison of a grid which
represents cell borders is impossible, because available
distance metrics do not correspond to the way humans
interpret images, as described by Gavet and Pinoli
(2013). Therefore, in medicine, for each cell, a shape
parameter value is calculated and the image content is
described using a statistical approach that exploits this
data.
Historically, Rao et al. (1982) introduced
calculation of the size of cells recorded in the image
first. Next, Doughty (1990) described a coefficient
of variation that describes changes in cell area
size in images, and later Doughty (1992) discussed
the hexagonality measure, which is understood as
the probability of finding a hexagonal cell in the
image. Ollivier et al. (2003) proved that these
parameters are likely to describe endothelial layer
function, because the researchers noticed that size
distribution varies between healthy and pathological
tissues. Gronkowska-Serafin and Piorkowski (2014)
introduced a novel parameter that describes the
average coefficient of variation of cell side lengths
and Piorkowski et al. (2015) proved its high stability
compared to previous cell shape parameters.
Another approach to cell shape description
presents a granulometric measure that assumes that
cell shape is related to a disk, and therefore includes
in calculations cells that are only partially visible in
the image. Thanks to this assumption, it is possible
to estimate the radius of each cell, which part was
detected in the image (usually on its boundary) and
consider its area in calculations. In this case, Ayala
et al. (2001) suggested using a probability density
function, while granulometric moments were exploited
by Zapteretal. (2005) for data description. According
to D´ ıazetal. (2007), this manner of endothelial image
description is better because it combines information
about size, shape, and spatial distribution. There are
also approaches which exploit fractal dimensions for
nuclei description (Oszutowska-Mazurek et al., 2012;
2013; 2015), but they have not find an application in
endothelial image analysis yet.
RESEARCHOBJECTIVES
Researchers who design medical parameters
concentrate on features that are considered during the
visual inspection of data. This approach appears to be
valid; however, in reality it is unknown how shapes
are compared by humans. Moreover, adapting human
thinking when preparing an algorithm for a computer
may be misleading.
This work addresses the problem of shape
description for endothelial cells. Since we are working
with digital data, a plethora of object shape features
known from the image processing domain may find
an application and create a perfect image content
descriptor. Such parameters should make it possible
to differentiate between healthy and pathological
endothelial layer structures, and should give similar
results irrespective of the cell border determination
method applied. Therefore, this work compares its
descriptive qualities, searches for possible correlation
in order to chose such parameters, which application
gives new and broad data description removing
unnecessary redundancy. Additionally, the overview of
the broad range of existing parameters should give a
better insight into endothelial cell shape characteristics
and in the future may lead to a better understanding of
shape changes.
This paper is structured as follows. The
section Material and Methods introduces parameters
used for endothelial image description widely applied
in medical practice (subsection Medical parameters)
and the standard shape parameters known from the
image processing domain, which are presented in
subsection Shape parameters. The description of
preprocessing of collected medical data is given
in subsection Experimental data, while results are
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Image Anal Stereol 2016;35:149-158
presented in section Results section Discussion
discusses the results and draws the conclusions.
MATERIALANDMETHODS
MEDICALPARAMETERS
As was stated in section Existing Medical
Parameters, two groups of parameters are considered
for describing the shape of endothelial cells. The first
examines cell shape characteristics and is already used
in medicine (Rao et al., 1982; Doughty, 1990; 1992).
The second is a granulometry measure (Ayala et al.,
2001; Zapter et al., 2005; D´ ıaz et al., 2007) which, to
the best our knowledge, did not find an application.
Therefore, the second approach is not included in this
research.
The most common endothelial cell descriptor
used in medicine is the cell density CD measure. It
correlates the average cell size to the total investigated
area. This parameter is automatically calculated by
the TOPCON software, therefore became widely
exploited.
A coefficient of variation in the area size of
different cells was described by Doughty (1990).
It measures cell size distribution over the whole
population and usually has a value below 30% for
healthy structures. It is defined as follows:
CV =
1
m
c
s
1
N
N
 i=1
(c
i
 m
c
)
2
 100%; (1)
wherec
i
is the area ofith cell,m
c
is the average area of
cells in the image, and N is the number of cells. This
formula is a general statistical method, which enables
describing the data standard distribution in population
and can be applied to any other measure.
The hexagonality coefficient H was discussed
by Doughty (1992) and calculates the proportion of
hexagonal cells (N
6
) to those of other shapes (N
T
)
following the equation:
H =
N
6
N
T
 100%: (2)
This measure states the proportion of cells with
hexagonal shape, has a value above 50% for healthy
structures, and decreases with age.
The coefficient of cell side length variation
CVSL was introduced by Gronkowska-Serafin and
Piorkowski (2014) and is given as:
CVSL=
1
N
N
 j=1
1
¯
SL
j
v
u
u
t
1
NL
j
NL
j
 i=1
(l
i
 ¯
SL
j
)
2
; (3)
wherel
i
is the length of theith side of the jth cell,
¯
SL
j
is the average length of all sides of jth cell and NL
j
is the number of sides and N is the number of cells.
This parameter is sensitive to deformation of cells,
especially for stretching.
SHAPEPARAMETERS
Several techniques have been developed in order
to describe the shape of objects presented in binary
images. Some of them exploit topological information
about pixel distribution within the object; others
describe its statistical qualities. In this work, both
approaches have been used and their accuracy for
the description of endothelial image content has been
evaluated.
Spatialmoments
A probability theory concerning spatial moments
was adopted by Gonzalez and Woods (2001) to
describe binary images, especially the shape of objects.
In this approach, an imageI is understood as a discrete
function I(x;y), where pixels describing the object
have ones as values and the background is marked
with zeros. The summation over such defined function
enables to define the spatial momentSM of the(m;n)th
order as follows:
SM
m;n
=
1
W
m
 H
n
W
 x=1
H
 y=1
x
m
 y
n
 I(x;y); (4)
where x and y are coordinates of pixels in image,
and W and H are the width and height of the image,
respectively. Usually, moments of the following orders
are considered: (0;0); (1;0); (0;1); (2;0); (1;1);
(0;2); (3;0); (2;1); (1;2); (0;3). This formulation
assumes that the origin of coordinates is in the
lower left corner of an image, and the first factor
in Eq. 4 removes scaling dependency from the
spatial moment calculation. The derived moments have
several properties, for example the SM
0;0
describes
object area and the relation between SM
1;0
and SM
0;0
,
andSM
0;1
andSM
0;0
makes it possible to calculate the
objects center of gravity ( ¯ x; ¯ y).
If the centroid of the object is known, it is possible
to derive spatial central momentsCM:
CM
m;n
=
1
W
m
 H
n
W
 x=1
H
 y=1
(x ¯ x)
m
 (y ¯ y)
n
 I(x;y) (5)
and spatial normalized central moments NM,
additionally prone to scale variations:
NM
m;n
=
CM
m;n
SM
m+n
2
+1
0;0
: (6)
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The last moments were introduced by Hu (1962), who
used them to derive seven invariant moments IM,
whose values do not change under translation, rotation,
and scaling of the object.
Topologicalattributes
Shape parameters based on topology are not
affected by translation and rotation. Several definitions
are given by Gonzalez and Woods (2001); Russ
(1998); Jahne (2002). In a binary image, the area
and perimeter are the basic shape characteristics of
the object. The area A counts pixels which belong to
the object, whereas the perimeter P is defined as the
number of pixels which belong to the object, but are
placed on its boundary.
With information about area and perimeter of
an object, several geometric measures are defined
which are not affected by rotation, translation, and
changes of scale. Several of them are the circularity
measurements, which are given by different formulas:
C
1
=
4 p A
P
2
; (7)
C
2
=
P
2
A
; (8)
C
3
=
2 p A
P
; (9)
Malinowska=
P
2
p
p A
 1; (10)
Shapeless=
P
2
4 p A
: (11)
The C
1
, C
2
, C
3
, and “Shapeless” result in unity value
for circular shapes, while “Malinowska” has a zero
value. The greater the variation from unity (or zero),
the more elongated and complex a shape is. There are
also other parameters concerning circularity:
C
4
=
P
p
; (12)
C
5
=
A
MAX
P
; (13)
Roughness= 2 r
A
p
; (14)
where A
MAX
is the maximal area calculated as the
multiplication of maximal width and height of object.
Other definitions calculate the elongation of object
as the ratio between the maximum diameters D
orthogonal to each other:
Feret=
D
I
D
II
: (15)
There are also several complex parameters that
except object’s area use information about distribution
of the shortest distancessd between each of the objects
pixels and the contour:
Blair-Bliss=
A
q
2 p  i
sd
2
i
; (16)
Danielsson=
A
3
( i
sd
i
)
2
: (17)
Other parameters focus the calculation on the distances
d between the points on the contour and the center of
gravity of the object:
Haralick=
s
( i
d
i
)
2
P  i
d
2
i
 1
: (18)
Another parameter expresses the ratio between the
largest and smallest distance of contour points of the
object to its center of gravity (PD):
W7=
minPD
maxPD
: (19)
For each shape parameter, the value describing
the whole image was calculated following the Eq. 1,
where p stands for chosen parameter value and m
p
for
its average:
Param=
1
m
p
s
1
N
N
 i=1
(p
i
 m
p
)
2
: (20)
This approach, which calculates general standard
deviation of each parameter, has been chosen as
an objective mean for descriptive parameter values
comparison.
EXPERIMENTALDATA
The experimental data set was acquired with an
inverse phase contrast microscope (CK 40, Olympus)
at 200 magnification and an analogue camera (SSC-
DC50AP, Sony) by Ruggeri et al. (2010). The set
consists of 30 images of a corneal endothelium taken
from 30 porcine eyes stained with alizarine. The
images are monochromatic in JPEG format at 768 576 resolution. The number of cells detected within
images ranges from 188 to 388 with average of 232,
whereas the cell average size is 272.76 pixels. Fig. 1a
presents an exemplary image from this set. This dataset
is publicly available at http://bioimlab.dei.unipd.it/. It
is worth to point out that the shape of cells in human
corneal endothelial layer and the ground truth data are
corresponding.
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Image Anal Stereol 2016;35:149-158
(a) Example of input data
(b) Original cell borders (c) Thinned cell borders
Fig. 1. Exemplary image of endothelial layer with
marked cell borders: (b) existing in the data set; (c)
thinnedforcalculation.
Table 1. Masks set used for thinning. X denotes pixels
notincludedinthesquaremask.
2
4
X 1 1
0 1 1
0 0 X
3
5
2
4
1 1 1
0 1 0
0 0 0
3
5
2
4
X 1 X
X 1 X
0 0 0
3
5
2
4
X X 0
0 1 0
0 0 0
3
5
2
4
0 X X
0 1 0
0 0 0
3
5
Except for original endothelial layer images, the
dataset contains manually prepared segmentations in
seed form. This means that cell borders are thicker
than 1 pixel. Therefore, before information about cell
shapes can be acquired, thinning was necessary. The
skeletonization procedure was performed using all
masks from the set masks shown in Table 1 with
orientations: 0
 , 90
 , 180
 , and 270
 .
Fig. 2. Original cell borders for Fig. 1a with
additionally marked in red thinned locations of
cells considered for calculation in order to assure
informationaboutcellsides.
The thinning procedure was sufficient in order
to determine the shape parameters, but in the case
of H and CVSL metrics, some additional processing
was necessary. The computation of this cell feature is
strongly related to its neighbors, due to the information
about number and length of cell sides; therefore, using
the data prepared by the authors of the dataset would
result in unreliable results for all cells on the region
border. In order to meet the demands and assure that
for each cell there is information about how many
sides it has, we decided to remove from the whole
computations the first envelop of cells, which lacks
such information, as is depicted in Fig. 2. Moreover,
the black lines correspond to the skeletonized cell
borders calculated from original data, whereas the red
lines approximate straight connections between nodes
(tripple points) calculated in the points where three
sides were crossing. This approximation is necessary
for correct computation ofH andCVSL parameters.
In this work, manual segmentation was chosen
for better cell border determination, however the
authors are aware of two methods (Piorkowski and
Gronkowska-Serafin, 2015; Selig et al., 2015a;b)
that enable automatic and precise cell border
location. In (Latała and Wojnar, 2001), for a similar
grain detection problem, the authors suggest that
using computer-aided methods involves determining
software parameters, which is time consuming
and difficult. However, some processing to assure
repeatability in thinning was exploited.
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NURZYNSKA K ET AL: Thecorrelationanalysisoftheshapeparameters
Table 2. Absolute correlation value range for a group
name.
Name Min Max
Perfect 0.99 1.00
Very good 0.95 0.98
Good 0.90 0.94
Possible 0.80 0.89
Satisfactory 0.70 0.79
Weak 0.50 0.69
None 0.00 0.49
RESULTS
The goal of this experiment was to find which of
the presented parameters are correlated and therefore
should not be used together to describe the endothelial
layer image content. Using such parameters in sets to
describe the data should be avoided, as they do not
convey any additional information.
In order to calculate the correlation measure for
each testing image, the cell boundaries were located,
then a chosen parameter was calculated for each
cell in the image and the final value was obtained
following Eq. 20. These values were used to calculate
the correlation.
When plotting all calculated correlation values on
one graph it turned out that there are three aggregations
of correlated parameters between each other only and
several correlated pairs of features. Since presenting
a plot with 55 features is impossible, the authors
decided to present each aggregations on separate
plots. Therefore, the following figures depict groups
correlated together. However, for interested reader the
matrix presenting all correlation values is accessible as
a supplementary file.
DISCUSSION
Fig. 3 shows Group I of features, which are
characterized in most cases by very good or perfect
correlation. This group consists of 17 features. It
is not surprising to see that A, SM
0;0
, and CM
0;0
are very strongly correlated, as all these measures
define the area of an object. Next, good or better
correlation was recorded between all possible orders
of SM. Then, several shape parameters (A, P,
C
3
, C
4
, and “Roughness”) reported satisfactory or
better correlation. Finally, a weak correlation with
“Danielsson” feature was noticed.
Fig. 3. Group I of features with almost perfect
correlation.
Group II presented in Fig. 4 only consists of
shape parameters whose correlation is in most cases
above the possible range. A good correlation between
C
1
, C
2
, “Malinowska”, “Shapeless” was suspected
as all of these features are a scaled or inverse of
relation P
2
=A. However, it is interesting to see that
they do not show correlation with others circularity
measures, which belong to Group I. Next, the perfect
correlation between “Blair-Bliss” and “Haralick” is
also interesting as these measures are calculated using
different shape properties. The “Feret” feature behaves
similarly to the aforementioned features.
Fig. 5 groups features for which the correlation
from weak to possible range was calculated: Group III.
Except for one very good correlation between NM
0;1
andIM
1
, the other features proved to be less related to
each other. It is interesting to find that such a simple
measure asC
5
is correlated with the statistical features
presented in this group. The correlation between others
probably results from common parts in the formula
definition.
Finally, Group IV consists of pairs of features:
– IM
5
andIM
6
which are possibly correlated,
– CM
1;0
andNM
1;0
which are satisfactory correlated,
– CM
0;2
andNM
0;2
which are satisfactory correlated,
– CM
2;1
andNM
2;1
which are satisfactory correlated.
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Image Anal Stereol 2016;35:149-158
Fig. 4.GroupIIofshapefeatureswithatleastpossible
correlation.
Fig. 5. Group III of features with at least weak
correlation.
The correlation value presented and discussed
above is not sufficient to verify whether correlation
between two datasets exists. It is necessary to plot the
data and see whether the trend is visible. For each
pair of features, such a scatter-plot was prepared and
checked. The examples of data distribution showed
that the calculated correlation corresponds well to the
plots. Moreover, when the results of correlation plots
for different groups were compared, it was seen how
the distribution varies more in two dimensions when
the correlation value lowers. In almost all cases, a
positive correlation took place, only for pairs with
“Malinowska” was a negative correlation recorded.
Additionally, for each pair of features the p-value
was calculated to verify a null hypothesis that the
data describing two features comes from independent
random samples. Values lower than the significance
level set to 0.05 allow to reject this hypothesis for all
elements in Group II and most in Group III (except
three pairs consisting of Shapeless,C
1
, andC
2
). In case
of Group I, the hypothesis is rejected for pairs created
from “Danielsson”, “Roughness” (except pair with
C
3
),C
4
(except pair with P),C
3
, and P. In other cases
there are no bases to reject the hypothesis. Considering
Group IV the hypothesis can be rejected for IM
5
and
IM
6
pair as well as forCM
2
1
andNM
2;1
.
From the presented experiment and the gathered
data, one can conclude that it is possible to distinguish
several endothelial layer descriptive features that are
not correlated to each other. Moreover, it is possible
to choose one representative from those correlated
to be used to describe the data. Table 3 names the
non-correlated and chosen representatives, which are
characterized by at least possible correlation value.
The next experiment assumed calculation of
correlation between parameters used in medicine (CD,
CV ,CVSL, andH) and those discussed in the previous
experiment. It was found out that, since the method for
parameter calculation was derived fromCV definition,
it behaves asA because its calculation is similar. Next,
the CVSL parameter proved to have weak correlation
with features considered in Group II. Only theCD and
H features are not correlated to any other features.
Table 3.Notcorrelatedfeatures.Featuresmarkedby
 have very low correlation which exists only between
two measures, hence it was decided to join them to
non-correlatedgroup.
Features
Group I A
Group II C
2
Group III CM
0;1
Group IV IM
5
Non-correlated IM
7
NM
 1;0
,NM
2;0
,NM
1;1
,NM
 0;2
,
NM
3;0
,NM
 2;1
,NM
1;2
,NM
0;3
CM
 1;0
,CM
2;0
,CM
1;1
,CM
 0;2
,
CM
3;0
,CM
 2;1
,CM
1;2
,CM
0;3
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NURZYNSKA K ET AL: Thecorrelationanalysisoftheshapeparameters
This article investigates the shape parameters used
in the image processing domain as a means for
endothelial image description. The main objective
of the presented experiments is to verify which
parameters should be considered for such data
descriptions, and which could improve understanding
and support automatic analysis of endothelial images.
It was very important to remove the information
redundancy that exists in correlated measures.
According to the performed experiments that
addressed the spatial moments and topological
attributes, there are four groups of features correlated
to each other, but also 17 non-correlated metrics were
distinguished. From each group, a representative with
the lowest computation overload was suggested for
further consideration. Additionally, it was found that
the medical parameterCV and CVSL are correlated to
a group of features, whileCD and H does not exhibit
such properties.
In further research, application of chosen shape
features for differentiation between healthy and
damage tissue will be considered. Moreover, the
accuracy of endothelial image quality description with
these features will be investigated.
ACKNOWLEDGEMENTS
K. Nurzynska was supported by statutory funds
for young researchers (BKM/507/RAU2/2016) of
the Institute of Informatics, Silesian University of
Technology, Poland. A. Piorkowski was financed by
the AGH – University of Science and Technology,
Faculty of Geology, Geophysics and Environmental
Protection as a part of statutory project.
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