Level Set Segmentation of Cellular Images Based on Topological Dependence.
ABSTRACT Segmentation of cellular images presents a challenging task for computer vision, especially when the cells of irregular shapes
clump together. Level set methods can segment cells with irregular shapes when signaltonoise ratio is low, however they
could not effectively segment cells that are clumping together. We perform topological analysis on the zero level sets to
enable effective segmentation of clumped cells. Geometrical shapes and intensities are important information for segmentation
of cells. We assimilated them in our approach and hence we are able to gain from the advantages of level sets while circumventing
its shortcoming. Validation on a data set of 4916 neural cells shows that our method is 93.3 ±0.6% accurate.

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ABSTRACT: Automatic and quantitative measurement of neurites is a challenging task, while it is critical in many neurological studies. We developed a fully automatic method to trace and quantitatively measure the neurites. Our measurements are validated by comparison with the semiautomatic NeuronJ and commercial software HCA Vision. The results demonstrate that the measurements of the three approaches have no significant difference. We also apply our approach for a biological study on neurite outgrowth and the measurements of four different conditions are presented.  [Show abstract] [Hide abstract]
ABSTRACT: Fluorescent microscopy imaging is a popular and wellestablished method for biomedical research. However, the large number of images created in each research trial quickly eliminates the possibility of a manual annotation; thus, the need for automatic image annotation is quickly becoming an urgent need. Furthermore, the high clustering indexes and noise observed in these images contribute to a complex issue, which has attracted the attention of the scientific community. In this paper, we present a fully automated method for annotating fluorescent confocal microscopy images in highly complex conditions. The proposed method relies on a multilayered segmentation and declustering process, which begins with an adaptive segmentation step using a twolevel Otsu’s Method. The second layer is comprised of two probabilistic classifiers, responsible for determining how many components may constitute each segmented region. The first of these employs rulebased reasoning grounded on the decreasing harmonic pattern observed in the region area density function, while the second one consists of a Support Vector Machine trained with features derived from the log likelihood ratio function of Gaussian mixture models of each region. Our results indicate that the proposed method is able to perform the identification and annotation process on par with an expert human subject, thus presenting itself a viable alternative to the traditional manual approach.Artificial Intelligence Review 10/2013; · 0.90 Impact Factor  SourceAvailable from: Simon Liao[Show abstract] [Hide abstract]
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Page 1
Level Set Segmentation of Cellular Images based
on Topological Dependence
Weimiao Yu1, Hwee Kuan Lee1, Srivats Hariharan2, Wenyu Bu2and Sohail
Ahmed2
1Bioinformatics Institute, #0701, Matrix, 30 Biopolis Street, Singapore 138671.
2Institute of Medical Biology, #0606, Immunos, 8A Biomedical Grove, Singapore
138648.
Abstract. Segmentation of cellular images presents a challenging task
for computer vision, especially when the cells of irregular shapes clump
together. Level set methods can segment cells with irregular shapes when
signaltonoise ratio is low, however they could not effectively segment
cells that are clumping together. We perform topological analysis on the
zero level sets to enable effective segmentation of clumped cells. Geomet
rical shapes and intensities are important information for segmentation
of cells. We assimilated them in our approach and hence we are able to
gain from the advantages of level sets while circumventing its shortcom
ing. Validation on a data set of 4916 neural cells shows that our method
is 93.3 ± 0.6% accurate.
1Introduction and Background
Biological science is in the midst of remarkable growth. Accompanying this
growth is the transformation of biology from qualitative observations into a
quantitative science. This transformation is driving the development of bio
imaging informatics. Computer vision techniques in bioimaging informatics have
already made significant impacts in many studies [1,2]. Cellular microscopy is an
important aspect of bioimaging informatics and they have their unique traits
and bring new challenges to the field of computer vision. Advances in digital
microscopy and robotic techniques in cell cultures have enabled thousands of
cellular images to be captured through High Throughput Screening and High
Content Screening. Manual measurement and analysis of those images are sub
jective, labor intensive and inaccurate. In this paper, we developed an efficient
algorithm for the segmentation of cells in highly cluttered environment, which
is a ubiquitous problem in the analysis of cellular images.
Accurate segmentation of the cellular images is vital to obtain qualitative in
formation on a cellbycell basis. Cellular images are usually captured by multi
channel fluorescent microscopes, in which one channel detects the nuclei. Since
nuclei contain important information, they generally serve as references for cellu
lar image segmentation. During the past 15 years, many efforts have been made
on automatic segmentation of nuclei from fluorescent cellular images, such as
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2 ISVC08 submission ID: 532
simple thresholding [3], watershed algorithm [4]∼[5], boundary based segmen
tation [6], flexible contour model for the segmentation of the overlapping and
closely packed nucleus [7]. Other related works on the automatic analysis of
cellular images can be found in [8,9].
Deformable models, also known as active contour, are popular and pow
erful tools for cell segmentation tasks. Among all the active contour models,
level set formalism has its superior properties, such as ease of implementation,
regionbased, robust to noise and no selfintersection, etc. Two concepts of level
set approach were discussed in O. Stanley’s original paper [10]. First, a level
set function in a higher dimensional space is defined to represent the regions,
which provided us with a nonparameterized model for segmentation. Second,
the curves are evolved according to their mean curvature. Thereafter, D. Mum
ford and J. Shah proposed their functional variation formulation to optimize the
segmentation of piecewise smooth images in [12]. Then ChanVese enhanced the
level set approach for region based image segmentation [13]. A comprehensive
review of level set approach for image processing are available in [14] and [15].
One longclaimed merit of level set methods is its ability to automatically
handle topological changes. However, this merit becomes a liability in many
cellular image segmentations, because nondividing cells can only contain one
nucleus. In a highly clustered image, as shown in Fig. 1, level set segmentation of
the cells (green channel) will results in many segments with multiple nuclei. The
contours of the snake model and geodesic active contours model can in principle
generate one cell segment per nucleus, but they need to be parameterized and the
node points may not be uniformly distributed along the length of the contours.
Thus they cannot capture the subtle details of irregular cell outlines. In this work,
we prefer the level set formulation since it is nonparameterized. We develop a
method to enforce the condition that one cell segment contains only one nucleus.
Watershed approach was first proposed in [16] and widely applied for cell
segmentation. Watershed approach was combined with the level set formulation
to segment cellular images and preserve the known topology based on sought
seeds in [17]. Similar seedsbased segmentation approach in [18] uses one level
set function for each individual corresponding cell to prevent the merging of
different cells. Simple point concept is applied to prevent the merging of the
cell segments during the evolution of level set function in [19] . However, a well
known problem of watershed approach is oversegmentation. Other methods have
been proposed to overcome this problem, such as rulebased merging [21] and
markercontrolled watershed correction based on Voronoi diagrams [20].
Generally, the cellular images are acquired by multichannel fluorescent mi
croscopes and nuclei are captured by one of the channels. Fig. 1 shows a few
examples of the cellular images captured by two channel fluorescent microscopy.
As shown in Fig. 1 , each cell consists of one nucleus and cells of irregular shapes
are crowded and touch each other. In this work, we first segment the nuclei,
since they are inside of the cell membrane and generally well separated. The
found nuclei serve as seeds for cell segmentation. We present a novel cell seg
mentation approach based on the concept of topological dependence, which will
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ISVC08 submission ID: 5323
be introduced shortly. In our approach, the level set curves propagate faster in
the regions of brighter image intensities. Hence the dynamics of the level set
curves incorporate essential information for cell segmentation. The utilization of
such dynamics in our approach is presented for the first time in literature. The
watershed lines are evolved dynamically based on the topological dependence at
each time step to segment the crowded cells with irregular morphology.
The remainder of the paper is structured as follows. Section 2 will provide
the definition of topological dependence. The level set formulation for two phase
segmentation will be presented in the length of Section 3. The dynamic watershed
transformation and the topological dependence preservation will be discussed in
Section 4. In Section 5, we will present our experimental results. The conclusion
in Section 6 will finalize this paper.
2Topological Dependence
In this paper, we use the images of two channels to illustrate our approach.
Generalization of our approach to the images of more than two channels is trivial
as long as we use one channel as reference. The nucleus is stained in blue and cell
cytoplasm is stained in green. We define the images on a finite subset in the two
dimensional Euclidean space Ω ⊂ R2. fn(x,y) : Ω → R and fc(x,y) : Ω → R
represent the intensities of nucleus and cytoplasm at (x,y) respectively. We call
fn(x,y) and fc(x,y) Nucleus Image and Cell Image. The superscripts ‘n’ and
‘c’ represent ‘nucleus’ and ‘cell’. Both of the functions are normalized to [0,1].
The segments of nucleus and cell images form connected regions in Ω. Due to
the limitation of space, we give a brief statement to define the connected region:
Connected region: A set of points π ⊆ Ω form a connected region if
for any two different points (x1,y1) ∈ π and (x2,y2) ∈ π, there exists a
path Γ connecting (x1,y1) and (x2,y2) such that Γ ⊆ π.
The segmentation of nucleus is relatively easy, since they are better sepa
rated. After the segmentation of nuclei, we obtain a set of connected regions,
e.g. segments of nuclei, denoted by ωn
nuclei is then determined. Each cell segment should contain exactly one nu
cleus segment. In order to describe this constraint in a rigorous mathematical
framework, we introduce the concept of topological dependence:
i,where i = 1,2,...L. The topology of the
Topological dependence: a set of connected regions πi, i = 1,2,...L
is said to be topologically dependent with another set of connected regions
θi, i = 1,2,...L if:
θi⊆ πi
i = 1,2,...L
(1)
Note that our definition of topological dependence is different from homeo
morphism[11]. Topological dependence is more relaxed. Due to the limitation of
space, we will not discuss in details here.
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4 ISVC08 submission ID: 532
3Level Set Segmentation
MumfordShah model of level set formalism is applied to obtain the segmentation
of nucleus images and the cell images, which is given by [12]:
E(φ,c1,c2) =µ · length{φ = 0} + ν · area{φ ≥ 0}
+ λ1
φ≥0
+ λ2
φ<0
?
?
u(x,y) − c1(φ)2dxdy
u(x,y) − c2(φ)2dxdy
(2)
where u(x,y) is the image intensity. µ, ν, λ1and λ2are parameters to regu
larize the contour length, area, foreground and background respectively. c1and
c2 are constants to be determined through the optimization. They are deter
mined by c1=
φ≥0dxdy
and area parameters µ and ν are set to zero in order to allow irregular con
tours and varying sizes of nuclei and cells. In general, we may set the parameters
accordingly when a priori knowledge on the length and area are available.
The optimal solution of MumfordShah model is given by the EulerLagrange
equation, which is an iterative procedure:
R
φ≥0u(x,y)dxdy
R
and c2=
R
φ<0u(x,y)dxdy
R
φ<0dxdy
. In our work, the length
φt+∆t= φt+ ∆t · δε[−λ1(u(x,y) − c1(φt))2+ λ2(u(x,y) − c2(φt))2]
t is the artificial time used for the evolution of the level set function and ∆t is
the time step. δ?is a regularized delta function defined in [13]. The selection of
the parameters is important to achieve a good segmentation. We will discuss the
parameter selection on λ1, λ2and ∆t in more details in Section 5.
In order to segment the nuclei, we initialize the level set function for nucleus
image as:
(3)
φn,t=0(x,y) = fn(x,y) −
?
Ωfn(x,y)dxdy
?
Ωdxdy
(4)
Substitute u(x,y) by Nucleus Image fn(x,y) and then evolve the level set
function φn,tusing Eq.(3). After the iterations converged, the set of points
{(x,y) ∈ Ωφn,t(x,y) ≥ 0} form L connected regions that define the nucleus
segments ωn
belonging to ωn
0 to represent the background. The detected nuclei will serve as seeds for the
cell segmentation.
After the nuclei are segmented, we need to include the information of the
nuclei into the cell segmentation. The level set function for the cell image seg
mentation is defined based on ωn
i, where i = 1,2,···L. L is the number of detected nuclei. The pixels
iare labeled by the integer i. All remaining pixels are labeled by
i:
φc,t=0=ˆfc(x,y) − 1(5)
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ISVC08 submission ID: 5325
where,
ˆfc(x,y) =
?
1
fc(x,y)
if (x,y) ∈?L
iωn
i
otherwise
(6)
Since fc∈ [0,1],ˆfc(x,y) ∈ [0,1]. Substitute u(x,y) byˆfc(x,y), then the level set
function for the cell segmentation φc,tis evolved according to Eq.(3). Unlike the
works in [18] where each individual cell has one corresponding level set function,
we use only one level set function to segment all cells in order to achieve better
computational efficiency.
In order to utilize the image intensity variation for cell segmentation, we
initialize the level set function for cell segmentation according to Eq. (5) instead
of traditional distance function. In addition, such initialization also ensures that
the zero level sets start from the nuclei and evolve outwards with a speed related
with the image intensity, e.g. brighter regions will be segmented as foreground
earlier.
4Preservation of Topological Dependence
The evolution of the level set function alone cannot ensure topological depen
dence between the cell segments and the nucleus segments. Dynamic watershed
lines is applied to preserve such topological dependence. Let’s define the seg
ments of cells at time t as ωc,t
According to the definition of φc,t=0, we know that ωc,t=0
This indicates ωc,t=0
i
is topologically dependent with ωn
that ωc,t
i
is topologically dependent with ωn
the watershed lines Wtby:
i. At t = 0, ωc,t=0
i
forms L connected regions.
= ωn
i. Under the condition
iat some time t, we may calculate
i
i, i = 1,2,···L.
Wt={(x,y) ∈ Ω : dmin[(x,y)ωc,t
for some i ?= j,where i,j = 1,2,...L,}
i] = dmin[(x,y)ωc,t
j],
(7)
where:
dmin[(x,y)ωc,ti] =min
(x?,y?)∈ωc,t
i
?
(x − x?)2+ (y − y?)2
(8)
The obtained watershed line at time t will be used to preserve the topological
dependence between ωc,t
i
and ωn
Preserving topological dependence and recovering the correct segmentation
consist of a series of relabeling steps. Firstly, the connected regions that do not
contain any nucleus segment are removed at each iteration, as shown by the gray
region in Fig. 2(b). If the remaining connected regions is topologically dependent
with ωn
the cell segments ωc,t+∆t
i
If the topological dependence is violated, relabel the connected regions as
“unknown” and the background as “0”. Then, obtain the intersection of the
connected regions and Wt, which forms a set of common boundaries
iat time t + ∆t.
i, then these regions will take the labels of ωn
iand we denoted them as
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6ISVC08 submission ID: 532
(a)(b)
(c)(d)
Fig.1. Captured cellular images with detected nuclei. Detected seeds are outlined in
blue and the geometric centers are marked by red dots.
(a) (b)
(c)(d)
Fig.2. Illustration of dynamic watershed lines and preservation of the topological
dependence. Nucleus segments are black. Labels of different regions at different time
are indicated by random colors. The dotted line represents the watershed line Wt.
The level set function evolves from t in (a) to t + ∆t in (b). Thereafter relabeling are
carried out in (c) and (d) to eliminate the residual regions and preserve the topological
dependence.
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ISVC08 submission ID: 5327
{st+∆t
and st+∆t
as different connected regions, then we obtain a new set of connected regions
βt+∆t
k
, k = 1,2,...K. (Note that K ≥ L). If ωn
we label βt+∆t
q
with the labels of ωn
to the above condition. Unlabeled regions are known as residual regions. An
iterative procedure is given to find the correct labels of residual regions:
1
,st+∆t
2
,···}. Two of such common boundaries are illustrated by st+∆t
in Fig. 2(b). Consider regions separated by the common boundaries
1
2
p⊆ βt+∆t
can be relabeled according
q
, for some p and q,
p. Not all βt+∆t
k
Relabeling of the residual regions: Any residual region must be
created by some common boundaries. One side of those common bound
aries must be adjacent to this given residual region and the other side
is adjacent to some other region that may or may not be successfully
relabeled previously. A given unlabeled residual region will take the label
of the adjacent region that shares the longest common boundary, which
is denoted by st+∆t
are unknown, then this residual region cannot be relabeled in the cur
rent iteration. Iterate this procedure until all unknown residual regions
are relabeled.
l,max. If all regions adjacent to this given residual region
We illustrate the preservation of topological dependence in Fig. 2, in which
the nucleus segments are indicated in black and the cell segments in red blue
and green. The dotted line represents the watershed line. Fig. 2(a) shows three
cell segments that are topologically dependent with the nucleus segments at
time t. Fig. 2(b) shows that when the level set function evolves to time t + ∆t,
the connected regions of the zero level set is no longer topologically dependent
with the nucleus segments. The gray region that does not contain any nucleus
is removed. In Fig. 2(c), the remaining connected regions are separated using
the watershed line Wtcalculated at the previous time step t. This produces
seven connected regions βt+∆t
1
,βt+∆t
2
,···βt+∆t
tain nucleus segments and are relabeled according to the corresponding nucleus
segments ωn
4
,βt+∆t
5
,βt+∆t
6
and βt+∆t
described in Relabeling of the residual regions. Note that βt+∆t
labeled with the same integer as βt+∆t
1
longer than st+∆t
2
. After the topological dependence is preserved, the watershed
lines are updated according to the new cell segments ωt+∆t
based on Eq. (7).
7
. βt+∆t
1
,βt+∆t
2
and βt+∆t
3
con
i. βt+∆t
7
are relabeled using the procedure
4
is re
because the common boundary st+∆t
1
is
1
,ωt+∆t
2
and ωt+∆t
3
5Experimental Results
We applied our segmentation approach to a neural cell study. In this study, we
want to automatically and quantitatively measure the length of the neurite. Ac
curate segmentation of the neural cell is a prerequisite to measure the length of
neurites and extract quantitative information on a cellbycell basis. As shown
in Fig.1, neurites are thin long structures that growth radially outwards from
the cells. More than 6000 images are acquired from fixed neural cells with DAPI
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8ISVC08 submission ID: 532
stain for nucleus and FITC stain for cell cytoplasm. Zeiss Axiovert 200M wide
field fluorescent microscope of two channels with Motorized XY Stage is applied
to capture the cellular images. The original images are captured at 20X magnifi
cation with 1366X1020 pixels of 12 bits accuracy. The camera is CoolSnap CCD
Camera and the resolution is 0.31 µm/pixel.
It is important to select the proper parameters for cell segmentation, such
as ∆t in Eq.(3), λ1 and λ2 in Eq.(2). We choose a big time step ∆t = 10 to
compromise accuracy and computation. To verify that using ∆t = 10 does not
introduce significant numerical errors, we performed our segmentations on eight
randomly selected images with three different time steps ∆t=1, 5 and 10. We use
Adjusted Rand Index [24] to compare the segmentations of ∆t=1 and ∆t=5 with
the segmentation of ∆t=10. Adjusted Rand Index is 0.9944± 0.0025 for ∆t=1
vs. ∆t=10 and 0.9952±0.0024 for ∆t=5 vs. ∆t=10. Results show that using
a big time step does not introduce significant numerical errors. Regarding the
regularization parameters, we set λ1= 1 and λ2= 50 for the cell segmentation
such that we may preserve the continuity of weakly connected neurites.
We choose the image in Fig. 1(a) and show ωc,t
ωn
by red dots. Different segments of the cells are shown by random colors. The
watershed line is illustrated by black solid lines. The segments of cells ωc,t
from the nucleus segments ωn
with the variation of image intensity. The watershed lines Wtalso evolve with
time t dynamically based on the constraint of topological dependence. The final
segmentation results of the cellular images in Fig. 1 are shown in Fig. 4. Although
the cells are irregular and clumpy, our approach can successfully separate them.
89% of 6000 cellular images can be segmented by our approach within 1 minute
on a desktop with 2.0GHz CPU and 1Gb RAM.
In order to testify and validate our approach, we compared our approach
with CellProfiler [25] and MetaMorph. CellProfiler is one of the popular cellu
lar image analysis freeware developed by the Broad Institute of Harvard and
MIT. MetaMorph is a commercial software specially developed for cellular im
age analysis by MDS Inc. The parameters for CellProfiler are suggested by the
software developers and the parameters for Metamorph are tuned by a service
engineer from MDS. 100 images containing a total of 4916 cells were randomly
selected from our database. They are segmented by CellProfiler, MetaMorph
and our algorithm to generate 300 segmented images. These segmentations were
then divided into 15 sets of 20 images each. They were randomly shuffled. Two
reviewers grade these segmented images without knowing which algorithm was
applied. They marked how many cells are segmented incorrectly. After the blind
evaluation, we count the number of incorrectly segmented cells for each approach.
The results are shown in Table 1.
Our approach achieved the best performance, which is about 2.5% better
than CellProfiler and much higher than MetaMorph. CellProfiler tends to over
segment the cells when the shapes of the nuclei and cells are irregular. Meta
Morph seems could not detect fine structures of the neurites.
i
at different time t in Fig. 3.
iare illustrated by the black regions and their geometrical centers indicated
i
start
iat t = 0 and evolve outwards with a speed related
Page 9
ISVC08 submission ID: 5329
(a) t = 0(b) t = 120
(c) t = 180(d) t = 200
Fig.3. Dynamical evolution of segments and watershed lines at different time t. Nuclei
segments are shown in black outlined by blue and their geometric centers are illustrated
by red dots. Watershed line are shown by the bold black line. Different cell segments
are indicated by random colors.
(a) (b)
(c)(d)
Fig.4. Segmentation results of our approach. The morphology of the cells in the cap
tured image is complicated. They are clumpy and touched with each other. Our ap
proach can successfully segment them .
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10ISVC08 submission ID: 532
Table 1. COMPARISON OF SEGMENTATION RESULTS
ApproachAccuracy
MetaMorph
CellProfiler
Our Approach
74.16%±1.02%
90.85% ±0.56%
93.25% ±0.57%
6Conclusion
The cell segmentation is nontrivial. It still remains a challenging problem in
many bioimaging informatics applications. Many segmentation algorithms could
not properly segment cells that are clumpy and touch each other, especially when
the intensity contrast at the boundaries is low and their geometrical shapes are
irregular. We proposed a novel segmentation approach for the cellular images
captured by twochannel microscope. The proposed approach combines the ad
vantages of level set and watershed method in a novel way based on the concept
of topological dependence. Utilization of the dynamics of level set curves in our
method is presented for the first time in the literature. Another novelty of our
method is that the watershed lines evolve dynamically at each time step t based
on the topological dependence, which is essential to prevent merging of cell seg
ments. This constraint also solved the oversegmentation problem of watershed
approach. We applied our approach on more 6000 cellular images of neural cells.
According to the validation of 100 randomly selected images including 4916
cells, our segmentation method achieved better performance than CellProfiler
and MetaMorph. We use only one level set function to segment all the cells in
an image, hence our algorithm is more efficient than the work in [18] where each
cell is associated with an individual level set function.1
Segmentation of the cells from the cellular images captured by multichannel
microscope is a common and important problem in many bioimaging applica
tions. Our approach is developed based on the assumption that the images are
captured by twochannel microscope, however, it can be easily generalized and
applied to cellular images of multichannel if the nuclei are captured by one of
the channels. Although our approach is not suitable to segment overlapped cells,
many biological assays seed and resuspend cells into monolayer such that the
cells do not overlap with each other. However, overlapped cells might happen in
some other applications and the problem itself is interesting and worth further
investigating.
1Video demonstrations and Matlab source codes are available at http://web.bii.a
star.edu.sg/∼yuwm/ISVC2008/.
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ISVC08 submission ID: 53211
7Acknowledgement
The authors would like to appreciate Dr. Anne Carpenter’s help to provide the
parameters for CellProfiler.
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