Compensation of Inhomogeneous Fluorescence Signal Distribution in 2D Images Acquired by Confocal Microscopy_preprint

Compensation of Inhomogeneous Fluorescence Signal
Distribution in 2D Images Acquired by Confocal Microscopy
JAN MICHA´ LEK,1
* MARTIN Cˇ APEK,1,2
AND LUCIE KUBIŃOVA´
AQ1
1
1
Department of Biomathematics, Institute of Physiology, Academy of Sciences of the Czech Republic, v.v.i.,
Vı´denˇska´ 1083, 14220 Prague 4, Czech Republic
2
Czech Technical University in Prague, Faculty of Biomedical Engineering, na´m. Sı´tna´ 3105, 272 01 Kladno, Czech Republic
KEY WORDS confocal laser scanning microscopy; image enhancement; morphology filters
ABSTRACT In images acquired by confocal laser scanning microscopy (CLSM), regions corre-
sponding to the same concentration of fluorophores in the specimen should be mapped to the same
grayscale levels. However, in practice, due to multiple distortion effects, CLSM images of even homo-
geneous specimen regions suffer from irregular brightness variations, e.g., darkening of image edges
and lightening of the center. The effects are yet more pronounced in images of real biological speci-
mens. A spatially varying grayscale map complicates image postprocessing, e.g., in alignment of
overlapping regions of two images and in 3D reconstructions, since measures of similarity usually
assume a spatially independent grayscale map. We present a fast correction method based on esti-
mating a spatially variable illumination gain, and multiplying acquired CLSM images by the inverse
of the estimated gain. The method does not require any special calibration of reference images since
the gain estimate is extracted from the CLSM image being corrected itself. The proposed approach
exploits two types of morphological filters: the median filter and the upper Lipschitz cover. The pre-
sented correction method, tested on images of both artificial (homogeneous fluorescent layer) and
real biological specimens, namely sections of a rat embryo and a rat brain, proved to be very fast and
yielded a significant visual improvement. Microsc. Res. Tech. 00:000–000, 2010. VVC 2010 Wiley-Liss, Inc.
INTRODUCTION
A confocal laser scanning microscope (CLSM) makes
it possible to acquire two-dimensional (2D) digital
images of thin optical sections within a thick specimen;
thus it enables us to obtain three-dimensional (3D)
image data composed from 2D images of perfectly reg-
istered serial optical sections (Pawley, 1995AQ2 ). For fur-
ther processing and analysis of images acquired by
CLSM, such as 3D volume reconstruction of selected
features in the specimen, it would be ideal if the image
brightness were proportional to the concentration of
the fluorescent dye in the specimen. However, in prac-
tice this is not the case, neither in the axial direction
where the light attenuation with increasing depth
inside the specimen can be observed, nor in the lateral
direction where irregularities in image brightness
across the field of view are encountered. In our previ-
ous study (Cˇ apek et al., 2006), we proposed methods
for compensation of light attenuation with depth, while
in the present study we focus on compensating inhomo-
geneous signal distribution in 2D images of focal
planes.
In the following, we assume according to Wilson
(2002) that the fluorescent field is proportional to the
intensity of the incident radiation. In an ideal case, a
2D digital image captured from specimen regions hav-
ing the same concentration of the fluorescent dye
should consist of equally bright pixels. This can be
expressed by the formula presented by Heintzmann
(2008 private communication):
Iem x; yð Þ ¼ Iex x; yð Þ Á Obj x; yð Þ ð1Þ
where Iem(x, y) denotes the intensity of the emitted
light, Iex(x, y) is the excitation intensity, and Obj x; yð Þ
represents the fluorescent dye concentration in the
pixel of the specimen image given by the coordinates
x; yð Þ:
In accordance with formula (1), after applying
constant excitation intensity across the field of view
the emitted light intensity for object regions having
the same fluorescent dye concentration should be the
same. However, in real CLSM images this is not the
case, and different pixels of the acquired images of
the specimen regions with the same fluorescent dye
concentration have different grayscale levels (Figs. F11
and F22). These irregularities in image brightness across
the field of view may be caused by (i) lateral chromatic
aberration of the microscope objective. In fluorescence
microscopy, the excitation wavelength is in general
different from the emission wavelength. Because of lat-
eral chromatic aberration, an off-axis ray at the excita-
tion wavelength will cross the intermediate image
plane at a different point than the longer wavelength
emission ray coming from the same point on the speci-
men. As a result, the emission ray may partially or
totally miss the pinhole, and the image will become
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*Correspondence to: Jan Micha´lek, Department of Biomathematics, Institute
of Physiology, Academy of Sciences of the Czech Republic, v.v.i., Vı´denˇska´ 1083,
14220 Prague 4, Czech Republic. E-mail: michalek@biomed.cas.cz
Received 6 September 2010; accepted in revised form 4 October 2010
Contract grant sponsor: Grant Agency of the Czech Republic; Contract grant
numbers: 102/08/0691, 304/09/0733; Contract grant sponsor: Academy of
Sciences of the Czech Republic (Institutional Research Concept); Contract grant
number: AV0Z50110509; Contract grant sponsor: Ministry of Education, Youth,
and Sports of the Czech Republic; Contract grant numbers: LC06063, ME09010,
and MSM6840770012
DOI 10.1002/jemt.20965
Published online in Wiley Online Library (wileyonlinelibrary.com).
VVC 2010 WILEY-LISS, INC.
MICROSCOPY RESEARCH AND TECHNIQUE 00:000–000 (2010)

darker away from the microscope optical axis (Pawley,
2006), (ii) curvature of field especially when using non-
plan objectives, when an image of a planar surface
loses intensity off optical axis (Pawley, 2006), (iii)
spherical aberrations of the point spread function
(PSF), which affect excitation efficiency in the sample
(iv) improper alignment of the optical parts of the con-
focal microscope system, etc.
The undesirable darkening of certain image regions
may shadow significant details of the specimen from
the observer’s eye. The effect may become even more
striking when a mosaic is composed of a number of ad-
jacent fields of view (Fig.F3 3a). Moreover, an automatic
mosaicking algorithm may fail to recognize overlap-
ping regions in two neighboring fields of view (Figs.
F5 5a–5c). The reason for the misalignment is that in simi-
larityAQ3 measures (Roche et al., 1999) used in image-
stitching software it is assumed that, between the
grayscale values of the two images to be stitched, there
exists either a functional relationship independent of
the pixel position or statistical relationship with sta-
tionary probability. However, in images with irregular-
ities in image brightness (Figs. 5a and 5b), these
assumptions are violated. As a result, the algorithm
may not identify the corresponding regions correctly
(Fig. 5c).
Spatially varying brightness mapping was dealt
with in various fields of biomedical imaging applying
different approaches (Hovhannisyan et al., 2008; Lee
and Bajcsy, 2006; Mangin, 2000). Mangin (2000) used
entropy minimization for automatic correction of in-
tensity nonuniformity in magnetic resonance (MR)
images. His method assumes that there is a narrow
intensity distribution for each tissue class; however,
this assumption is not necessarily satisfied in confocal
microscopy. Moreover, the algorithm is too slow (in the
order of minutes for a single image) to be of practical
interest for processing stacks of CLSM images. Lee
and Bajcsy (2006) proposed an intensity correction
technique in CLSM images, called mean-weight filter-
ing. Their method is based on searching for an opti-
mal, spatially adaptive, intensity transformation that
maximizes intensity contrast with respect to back-
ground, minimizes overall spatial intensity variation
for large area, and minimizes distortion of intensity
gradient for local features. The size and shape of
the filtering kernel has to be found experimentally,
thus the procedure is not fully automatic. Another
approach to image heterogeneity correction is based
on multiplication of the image acquired by CLSM or
multiphoton microscopy by a lateral correction factor
calculated from an image of a uniform fluorescent
sample (Hovhannisyan et al., 2008; Oldmixon and
Carlsson, 1993). However, a correction factor based on
a uniform sample makes it necessary to acquire
images of the calibration sample under conditions
very close to those applied to the acquisition of the
biological specimen, which can be tedious and difficult
to achieve, e.g., in optical sections deep inside a rela-
tively thick biological specimen.
In the present study, our aim was to find a fully auto-
matic, data driven (i.e., not relying on, e.g., a calibra-
tion set), method of compensation of lateral inhomoge-
neity in fluorescence signal distribution in CLSM
images, both for single frames and for mosaics com-
posed of a series of images acquired using, e.g., an
automatic motorized stage. Our direct motivation for
developing such a method was the need for fast
improvement of 3D volume reconstructions of biologi-
cal specimens larger than the field of view based on
composing stacks of CLSM images, both in lateral and
axial directions (Cˇ apek et al., 2009). In the recon-
structed volumes we were able to correct light attenua-
tion in axial direction (Cˇ apek et al., 2006), however, we
missed a convenient and efficient method for compen-
sation of image brightness irregularities in lateral
direction.
Fig. 1. CLSM image of homogeneous solution of DIOC3(3) fluores-
cent dye. HC PLAPO 320 water immersion objective (N.A. 5 0.70),
the excitation wavelength of 488 nm and emission wavelength range
from 531 to 627 nm were used. The image size is 550 lm 3 550 lm.
Fig. 2. CLSM image of rat embryo specimen, one field of view.
HCX PL FLUOTAR 35 dry objective (N.A. 5 0.15), the excitation
wavelength of 488 nm and emission wavelength range from 500 to
560 nm were used. The image size is 2.20 mm 3 2.20 mm.
Microscopy Research and Technique
2 J. MICHA´ LEK ET AL.

MATERIALS AND METHODS
Specimen Preparation
For testing of the proposed method we used images
of both artificial (homogeneous fluorescent layer) and
real biological specimens (rat embryo and brain).
Homogeneous Fluorescent Layer. A fluorescent
marker DIOC3(3) (Sigma-Aldrich Company) was dis-
solved in water to final concentration of 40 lM. A drop-
let of this solution was placed between a glass slide and
a cover glass and sealed with nail polish.
Rat Embryo Specimen. A 17-day-old embryo was
fixed and stained for 24 h in fixative (10% formalin
with 1% eosin). After dehydration the embryo was em-
bedded in paraffin. Then a series of 30-lm thick physi-
cal slices was cut by a rotary microtome HM 340 E
(MICROM Laborgera¨te, Walldorf, Germany) and
mounted on slides by poly-L-lysine (for details see
Cˇ apek et al., 2009; Jirkovska´ et al., 2005).
Rat Brain Specimen. The preparation of rat brain
specimen is described in detail by Mao et al. (2010).
Briefly, an anesthetized Sprague-Dawley rat received an
intravenous injection of biotin-labeled Lycopersicon
esculentum lectin (Vector Laboratories Burlingame, CA)
before vascular perfusion with 4% paraformaldehyde in
PBS (pH 7.4) via a 21-gauge cannula in the left ventri-
cle. After the perfusion, the brain was excised and
immersed in fixative, rinsed with PBS, infiltrated over-
night with 30% sucrose in PBS, and then embedded in
OCT (Sakura, Torrance, CA) and frozen. A 30-lm thick
brain coronal section was rinsed in PBS and incubated
in a solution containing fluorescently labeled streptavi-
din in 1% normal goat serum for 1 h for microvessel
staining. The sections were mounted with VectashieldTM
hard mount (Vector Laboratories, Burlingame, CA).
Acquisition of CLSM Images
Raw images of test specimens were acquired by a
Leica SPE CLSM using a solid state laser (15 mW)
yielding an excitation wavelength of 488 nm.
Homogeneous Fluorescent Layer. The images
(Fig. 1, 512 3 512 pixels, 550 lm 3 550 lm) were
acquired by a HC PLAPO 203 water immersion objec-
tive (N.A. 5 0.70) using the excitation wavelength of 488
nm and emission wavelength range from 531 to 627 nm.
Rat Embryo Specimen. The images (Fig. 2, 512 3
512 pixels, 2.20 mm 3 2.20 mm) were acquired by HCX
PL FLUOTAR 53 dry objective (N.A. 5 0.15), using the
excitation wavelength of 488 nm and emission wave-
length range from 500 to 560 nm. Series of images were
acquired from 64 successive physical sections of the rat
embryo. Each physical section was split into six to eleven
overlapping horizontal fields of view, and the correspond-
ing stacks of optical sections, 9.7-lm apart, were
acquired using a manual microscopic stage. Mosaics of
such overlapping stacks were composed semiautomati-
cally, using the algorithm implemented in the GlueMRC
software (Karen et al., 2003). Images used for our tests
were either single 2D images in one field of view (Figs. 2,
5a and 5b), or a mosaic of one optical section of the rat
embryo, composed of eight fields of view, shown in Figure
3a. For description of image acquisition, vertical linking
of 3D images of embryo physical sections and correspond-
ing volume reconstruction see Cˇ apek et al. (2009).
Rat Brain Specimen. Stacks of images (Fig. F44a, 512
3 512 pixels, 1.10 mm 3 1.10 mm) were acquired by HC
PLAPO 203 water immersion objective (N.A. 5 0.70),
using the excitation wavelength of 488 nm and emission
wavelength range from 500 to 600 nm.
Method for Compensation of Brightness
Inhomogeneities in CLSM Images
Our approach is based on estimating a spatially
varying gain which models the influence of effects such
as the lateral chromatic aberration, curvature of field,
and uneven excitation in the field of view on the signal
distribution, and correcting acquired images by invert-
ing the estimated gain. For gain estimation we
Fig. 3. Mosaic composed of CLSM images from eight fields of view of rat embryo specimen. (a) Origi-
nal image acquired by CLSM. (b) Lipschitz-cover estimated gain. (c) Image corrected after applying the
upper Lipschitz cover morphological operator without median filtering (d) Image corrected by using a
median filter and then applying the upper Lipschitz cover. The mosaic size is 4.3 3 7.2 mm2
.
3COMPENSATION OF CONFOCAL MICROSCOPY IMAGES

exploited a morphological operation applying the upper
Lipschitz cover and, moreover, in noisy images we
applied a median filter to reduce noise.
Estimation of the Space-Variable Gain Using the
Upper Lipschitz Cover Morphological Operator
In real CLSM images acquired using constant excita-
tion intensity Iex, the image brightness varies depend-
ing on the pixel location within the frame (Fig. 1), even
if the concentration of fluorescent dye is constant
across the specimen. To model this dependence we sup-
pose that the recorded light at different pixel positions
is a product of the dye concentration, the excitation
light intensity and a single function, gain x; yð Þ, that in
total accounts for any aberrations causing uneven sig-
nal distribution (Fig.F6 6):
Irec x; yð Þ ¼ gain x; yð Þ Á Obj x; yð Þ Á Iex ð2Þ
If an estimate of the gain function, gãin(x, y), were
known, one could easily correct the recorded image to
obtain the dye concentration in the specimen:
Obj x; yð Þ ¼
1
~gain x; yð Þ Á Iex
Á Irec x; yð Þ ð3Þ
To separate the gain from the object in the
recorded image, we need some qualitative features
that distinguish the function Obj(x, y) from
gain(x, y). For example, the gain changes slowly, is
continuous and has therefore only a small number of
minima or maxima, while the concentration of the
fluorescent dye in the microscope specimen changes
abruptly on boundaries of biological structures, which
in turn produces discontinuities and consequently
high frequencies. Such distinguishing features are
listed in Table T11.
It is obvious from formula (3) that in a CLSM
image of a specimen with uniform dye concentration
such as in Figure 1 the gray value distribution
assumes a shape identical to that of the gain func-
tion. To obtain a gain estimate for a real, nonuniform,
specimen, we assume that local maxima in the
acquired image correspond to specimen regions with
the highest fluorescent dye concentration, and try to
fill (pad) submaximal regions numerically with this
maximal concentration.
Because the form of the padded image should
reflect the form of the gain, it must satisfy the con-
dition that the rate of change is slow. This is guar-
anteed, if the padded function satisfies the Lipschitz
condition:
gain x1; y1ð Þ À gain x2; y2ð Þj j K Á x1; y1ð Þ À x2; y2ð Þj j ð4Þ
for any two pixels (x1, y1) and (x2, y2) of the image. K is a
constant factor called the Lipschitz constant, limiting
the maximum rate of change of gain(x, y). Padding the
acquired image numerically can be done very fast by
subjecting the image to a morphological operator called
‘‘the upper Lipschitz cover.’’ The upper Lipschitz cover of
an image I(x, y) is the infimum of functions L(x, y) satis-
fying the conditions:
Fig. 4. CLSM stack of images 7-lm apart from a rat brain speci-
men: (a) the original images, (b) the reciprocal of the Lipschitz-cover
estimated gain, (c) images corrected after applying the upper Lip-
schitz cover morphological operator. HC PLAPO 320 water immer-
sion objective (N.A. 5 0.70), the excitation wavelength of 488 nm and
emission wavelength range from 500 to 600 nm were used. The size of
each image is 550 lm 3 550 lm.

L x1; y1ð Þ À L x2; y2ð Þj j K Á x1; y1ð Þ À x2; y2ð Þj j
L x; yð Þ ! I x; yð Þ ð5Þ
The upper Lipschitz cover constructed in the process
is used as the gain estimate gãin(x, y). The Lipschitz
constant K which bounds the rate of change of the gain
estimate is a selectable parameter of the algorithm. A
fast algorithm for numerical computation of the Lipschitz
cover was presented by Sˇtencel and Janaćˇek (2006).
Noise Removal Using the Morphology Operator
Fast Median Filter
If the Lipschitz cover-based gain estimation is done
from noisy images, the upper Lipschitz cover creates
cones with vertices at noise peaks, which produce
undesirable artifacts. Therefore, in noisy images we
reduced the noise by using a median filter before apply-
ing the upper Lipschitz cover. The choice of the median
kernel size depends on the noise content of the particu-
lar CLSM image stack.
Computer Implementation of the Method
We used the algorithm of the upper Lipschitz cover,
described by Sˇtencel and Janaćˇek (2006). We were pro-
vided with the C-code of the algorithm by Dr. Janaćˇek.
Image handling like read/write, thresholding or multi-
plication by the correction factor were, for greater
convenience, programmed in MATLAB. The median fil-
ter implementation supplied with MATLAB’s Image
Processing Toolbox turned out to produce incorrect val-
ues at the image edges. Therefore, we used the elegant
median filter algorithm described by Perreault and
He´bert (2007), the C code of which is available from the
author’s homepage. Perreault’s algorithm runs in con-
stant time (the fastest run time possible), and is—
besides providing correct results—several times faster
on the images of interest than the MATLAB median fil-
ter. Both the C-coded Lipschitz filter and the median
filter were made callable from within MATLAB by
embedding them in respective MATLAB wrappers.
RESULTS
First, we applied our compensation algorithm to the
image in Figure 1 of a uniform sample represented by a
homogeneous fluorescent layer. Figure F77a shows the
upper Lipschitz cover of the uniform sample image in
Figure 1, Figure 7b the inverse gain, and Figure 7c the
corrected image. The gain estimate and its inverse
show circular artifacts which give rise to darker spots
centered at the noise peaks in the raw image. There-
fore, as described above, in noisy images we reduced
the noise by using a median filter before applying the
upper Lipschitz cover. Figure F88 shows the result when
the original image (Fig. 1) is median-filtered prior to
constructing the upper Lipschitz cover.
Further, we tested our approach for correction of a
highly nonhomogeneous image represented by a
mosaic composed of CLSM images from eight fields of
Fig. 5. Alignment of two fields of view of rat embryo. (a–c) Misalignment of two fields of view of rat
embryo specimen caused by space dependent grayscale mapping between the left and the right image:
(a) left field of view, (b) right field of view, (c) stitched image. (d–f) Proper alignment of the same two
fields of view after correction by the upper Lipschitz cover. (d) corrected left field of view, (e) corrected
right field of view, (f) stitched image. The size of each field of view is 2.20 mm 3 2.20 mm.

view of a rat embryo. After applying the upper Lip-
schitz cover morphological operator (Fig. 3c) the com-
pensation of inhomogeneous fluorescence signal distri-
bution was satisfactory, however the image appearance
was rather patch-like. This artifact was again compen-
sated using a median filter and then applying the
upper Lipschitz cover (Fig. 3d).
Finally, we checked our method for a fibrous struc-
ture with relatively homogeneous background. We cor-
rected a CLSM stack of images from a rat brain speci-
men (Fig. 4a) by first median filtering each image in
the stack and then applying the upper Lipschitz cover
morphological operator to estimate the inverted gains
(Fig. 4b). Multiplying the original images by the
inverted gains yielded the corrected stack (Fig. 4c).
The median filter is applied only for the gain esti-
mate, not on the corrected image. In Figure 3c, the
gain was estimated by applying the upper Lipschitz
cover on the raw image. Contrary to that, in Figures 3d
and 4c, the gain was estimated by applying the upper
Lipschitz cover on the median-filtered image. But in all
examples, the corrected images are obtained by multi-
plying the raw, noisy images, not median-filtered
images, by the inverse gain estimate. Thus, all fine
structures of the acquired images are preserved, and
no low-pass filtering takes place.
Figures 5d–5f demonstrates the improvement of
image stitching achieved when the spatial brightness
irregularity is corrected before stitching. In Figures
5a–5c, spatially variable grayscale mapping made a
stitching algorithm fail, because the measure of simi-
larity of the two images assumed spatially independent
grayscale mapping between the images. The two
arrows in Figure 5c indicate the structure which
is erroneously included twice in the stitched image.
Figure 5d–5f shows that the misalignment problem is
eliminated when Lipschitz-cover-based correction of
the spatially variable gain is applied before stitching.
The arrow in Figure 5f points to the location where the
previous alignment error has disappeared.
DISCUSSION
The method for lateral brightness correction pre-
sented in this article is based on the assumption that
the distortions are caused by a multiplicative gain
modeled by formula (2). The results shown in Figures
3, 4, and 8 suggest that the multiplicative assumption
captures well the image formation process. Median fil-
tering removing noise peaks in the raw images enables
the upper Lipschitz cover to reconstruct the slowly
varying gain precisely enough to yield corrected images
that—besides significant visual improvement—are
much better suited for further processing. The
improvement of image stitching achieved after the spa-
tial brightness irregularity is corrected is demon-
strated in Figures 5d–5f. In Figures 5a–5c, spatially
variable grayscale mapping made a stitching algorithm
fail, because the measure of similarity of the two
images assumed spatially independent grayscale map-
ping between the images. Figures 5d–5f show that the
misalignment problem is eliminated when Lipschitz-
cover based correction of the spatially variable gain is
applied before stitching.
Our approach proved to be advantageous when com-
pared with previous methods of lateral brightness cor-
rection in images. Unlike Mangin’s (2000) method
based on entropy minimization assuming a narrow in-
tensity distribution for each tissue class, we made
more realistic assumptions, tailored for confocal micro-
scopic images. Our technique is fully automatic, which
is not the case for the correction technique suggested
by Lee and Bajcsy (2006), requiring the experimental
assessment of the size and shape of the filtering kernel.
Moreover, our approach does not need any calibration,
which is necessary in some other correction methods,
usually applying a lateral correction factor calculated
from an image of a uniform fluorescent sample (Hov-
hannisyan et al., 2008; Oldmixon and Carlsson, 1993).
Thus, difficulties that may arise when attempting to
match exactly conditions of the test specimen acquisi-
tion and those applied to the acquisition of the biologi-
cal specimen are excluded.
Our new algorithm for lateral brightness correction
provides the user with high quality laterally corrected
images substantially faster than any of the aforemen-
tioned approaches. Fully automatic processing of a
whole stack of 60 CLSM images (512 3 512 3 8 bit)
takes typically about 11 s on a 3-GHz PC when the fast
C-coded algorithm for the upper Lipschitz cover
described by Sˇtencel and Janaćˇek (2006) and the fast
C-coded algorithm for the median filter running in O(1)
Fig. 6. Formation of the recorded image according to formula (2).
Iex—excitation intensity, Obj(x, y)—concentration of the fluorophore
at the point (x, y), gain(x, y)—gain at (x, y), Irec(x, y)—recorded light
intensity.
TABLE 1. Comparison of the features of the object function and the
gain function in CLSM images
Gain(x, y) Obj(x, y)
Continuity Continuous Discontinuous
Number of minima/maxima Small Large
Rate of change Slow Fast

time published by Perreault and He´bert (2007) are
used. Given its processing speed, our method is well
suited for routine lateral brightness correction of large
CLSM stacks, that can even be acquired using an auto-
matic motorized stage for composing mosaics of 3D
images of large portions of biological tissues. Thus, in
the reconstructed volumes, in addition to the light
attenuation in axial direction (Cˇ apek et al., 2006; Gopi-
nath et al., 2008; Wu and Ji, 2005), lateral brightness
variations can be compensated using the new method
outlined in this article.
ACKNOWLEDGMENTS
The authors thank Dr. Radomı´ra Va´gnerova´ (Insti-
tute of Histology and Embryology, 1st Faculty of Medi-
cine, Charles University, Prague, Czech Republic) for
preparing the rat embryo specimens used in Figures 2,
3, and 5, Dr. Xiao Wen Mao (Loma Linda University,
CA) for providing with the rat brain specimen (Fig. 4),
and Zuzana Burdı´kova´ for her help with the specimen
of homogeneous fluorescent layer (Fig. 1). They
thank Dr. Jirˇı´ Janaćˇek for providing the C-coded
Lipschitz-cover algorithm, as well as for proofreading
the manuscript.
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