Cells in a culture process transform with time and produce many overlapping cells in their vicinity. We are interested in a separation algorithm for images of overlapping cells taken using a fluorescence optical microscope system during a cell culture process. In this study, all cells are assumed to have an ellipselike shape. For an ellipse fittingbased method, an improved least squares method is used by decomposing the design matrix into quadratic and linear parts for the separation of overlapping cells. Through various experiments, the improved least squares method (numerically stable direct least squares fitting [NSDLSF]) is compared with the conventional least squares method (direct least squares fitting [DLSF]). The results reveal that NSDLSF has a successful separation ratio with an average accuracy of 95% for two overlapping cells, an average accuracy of 91% for three overlapping cells, and about 82% accuracy for four overlapping cells.
I. INTRODUCTION
Cells in a culture process transform with time. The analysis of the behavior of cells such as stem cells from the images taken during the early stage of a cell culture process is an important research topic in the fields of biology and medicine
[1
,
2]
. Recently, automatic cell tracking has been studied to observe cell characteristics such as apoptosis, aggregation, migration, and stretching in the cell differentiation process. In the cell culture process, two more cells may touch or overlap each other and be separated from one cell
[1

4]
.
A reliable cell tracking system should be capable of tracking well not only in individual cells but also in overlapping cells. The performance degradation of most existing cell tracking algorithms takes place when two and more different cells are overlapped or overlapped cells are divided. These errors are frequently compounded with the misclassification of the split and merge of the overlapping cell division or fusion events, resulting in erroneous cell lineages
[1

4]
.
However, cell objects in the cell images used in the published studies were very clear and large and had an ellipselike shape when viewed under a highmagnification optical microscope.
studies, researchers have used images similar to
Fig. 1
(a) whose background is clear and cell size is large.
Fig. 1
(b) contains many cells and the image quality is degraded.
Two different cell images: (a) one of cell image types which usually has been used in other researches [5] and (b) one of the images used in the present research
In this paper, we propose a separation algorithm for overlapping cells, which is based on two different ellipse fitting methods, namely the conventional direct least squares method and the improved direct least squares method, to split even multiple small overlapping cells. The proposed algorithm can be briefly described as follows:
We used a mixed threshold method to extract a cell from two or more overlapping cells even in an unclear image with background noise and poor illumination. First, an input cell image, which is a color or a grayscale image, was converted into a binary image by using the modified mixed threshold method. The background color of the image was white, and objects such as individual cells and overlapping cells were black in color. Second, we compared the conventional ellipse fitting with the improved ellipse fitting method based on an enhanced direct least squares method for tens of thousands of multiple small overlapping cells, as shown in
Fig. 1
(b).
II. MIXED THRESHOLD METHOD
To separate cell objects from the background, we need the binarization process for the given cell image. Upon binarization, pixels that have a higher intensity than the threshold
t
(
x, y
) are represented as objects and the other pixels are denoted as the background. The threshold function
g
(
x, y
) is defined as follows:
Automatic thresholding automatically selects an optimal threshold value for distinguishing objects from the background on the basis of their intensity distribution. There are two categories of automatic thresholding: global thresholding and local thresholding. Global thresholding selects a single threshold value from the histogram of the entire image. This method is simple, but it produces marginal noise for unevenly lighted images. In local thresholding, the threshold is determined for each pixel on the basis of its own gray value and the gray values of its neighborhood. This approach is called the adaptive thresholding method because it is adaptive to the local image characteristic. The adaptive threshold method has a high computational cost but gives good results even for a severely degraded image.
Most of the cell images obtained using optical microscopy are images with noise. In particular, we are interested in a degraded cell image because of the uneven microscope light. For our image, the adaptive threshold algorithm is good, but its computational cost is very high. Therefore, we used a mixed threshold method that modified the multistage adaptive threshold method proposed by Yan et al.
[6]
. The mixed threshold method chooses between a global threshold and an adaptive local threshold depending on the pixel intensity.
We use a global threshold
T
such that pixels whose gray values are less than
T
are classified as the background. If a pixel has a higher intensity than
T
, we compute the locally adaptive threshold
t
(
x, y
) for the pixel. Then, by using the adaptive threshold value
t
(
x, y
), we determine whether a pixel belongs to a cell object or the background. A mixed threshold function
t
(
x, y
) is defined as follows:
where
and
denote the mean and the standard deviation, respectively.
R
denotes a
b
×
b
neighborhood region centered at coordinates (
x, y
),
b
represents the mask size, and
k
indicates the constant coefficient. Because we consider very small overlapping cells, we used the constant 3 as the mask size. Further,
k
was set to a value of less than 1.
Sometimes, cells in a cell image can have holes, as shown in
Fig. 1
(b). This is observed when fluorescence does not stick close to the floor at the center of the cell. The next preprocessing step is to apply the flood fill algorithm to fill these holes. Finally, we extract the boundaries of the single cells and the overlapping cells. Before applying an ellipse fitting method, we split the boundaries into line segments. We use concave points on the boundaries to do this. This process involves splitting the touching cells into different ellipses, which represents separated cells. A fitted ellipse is used for representing a cell.
III. SEPARATION OF OVERLAPPING CELLS BASED ON ELLIPSE FITTING
 A. Direct Least Squares Fitting of Ellipses
We used two types of ellipse fitting algorithms: direct least squares fitting of an ellipse by Fitzgibbon et al.
[7]
(hereinafter Fitzgibbon’s method) and numerically stable direct least squares fitting of an ellipse by Halir and Flusser
[8]
(hereinafter Halir’s method). Both these methods are noniterative algorithms for fitting an ellipse to a set of data points. Let us briefly overview Fitzgibbon’s method.
An ellipse is a special case of a general conic, which can be described by an implicit secondorder polynomial as follows:
with an ellipsespecific constraint
where
a
= [
a, b, c, d, e, f
]
^{T}
denotes the coefficients of the ellipse and
x
= [
x
^{2}
,
xy
,
y
^{2}
,
x, y
, 1]
^{T}
represents the coordinates of the points of the line segment. Under appropriate scaling, the inequality constraint, shown in (6), can be changed into an equality constraint as follows:
The fitting of a general conic may be approached by minimizing the sum of the squared algebraic distance of points. Therefore, the ellipsespecific fitting problem can be described as follows:
Here,
D
denotes a matrix of the size
N
× 6 that presents the coordinates of all points and
C
represents the constraint matrix of the size 6 × 6.
By using the Lagrange multiplier
λ
and differentiating, we obtain the following system of simultaneous equations:
This may be rewritten as follows:
where
S
denotes the scatter matrix
DTD
. With the generalized eigenvectors of (12), we can get six real solutions (
λ_{J}, a_{j}
) for
S
. Further, the eigenvector of
S
corresponding to the minimal positive eigenvalue
λ
represents the bestfit ellipse for a given line segment.
All previously reported ellipse fittingbased methods that split overlapping cells are based on the direct least squares fitting of an ellipse
[9
,
10]
.
 B. Numerically Stable Direct Least Squares Fitting of Ellipses
According to Halir and Flusser
[8]
, the constraint matrix
C
is singular and the matrix
S
is also nearly singular. Therefore, the computation of the eigenvalues is numerically unstable and can produce incorrect results. In fact, when we used the generalized eigenvectors of (12), we could not obtain a minimal positive eigenvalue for a number of line segments. This implied that many of the overlapping cells could not be split into their separated cells.
To overcome these drawbacks of the original least squares method, we used Halir’s numerically stable direct least squares fitting of an ellipse. By using Halir’s method, we could rewrite (12) as the following set of equations:
where
M
denotes the reduced scatter matrix of the size 3×3.
where
Finally, we can find the eigenvector
a
_{1}
of matrix
M
, which yields a minimal nonnegative value. The values of
a
_{1}
represent the coefficients of the bestfit ellipse of the given line segment.
 C. Separation Algorithm of Overlapping Cells
The proposed algorithm is based on the ellipse fitting method to split the overlapping cells. The procedure is given below:
Algorithm (OverlappingCellPolygon CP)
The algorithm was implemented in Visual C++ and MATLAB.
IV. EXPERIMENTAL RESULTS
To verify the performance of the proposed algorithm, we used optical microscope images taken at intervals of 30 minutes during the cell culture process. The quality of data was degraded with severe noise and uneven microscope light. Further, there are tens of thousands of cells and overlapping cells in an image.
Fig. 2
shows the results of separating the overlapping cells. The second column shows the preprocessing results and the boundary information, and the third and fourth columns present the splitting results obtained by applying Fitzgibbon’s and Halir’s ellipse fitting method in the proposed algorithm, respectively. Both the methods gave good results for two overlapping cells. However, for three or more overlapping cells, the proposed algorithm using Fitzgibbon’s fitting method could not find the fitted ellipses in many overlapping cells. The third and fourth columns show the erroneous results obtained by an unstable computation of the eigenvalues such as infinite or complex numbers.
Separation results of overlapping cells (1st column: original images, 2nd column: its boundary, 3rd column: Fitzgibbon’s fitting method, and 4th column: Halir’s fitting method). DLSF: direct least squares fitting, NSDLSF: numerically stable direct least squares fitting.
Table 1
shows that the successful separation ratio was higher when the ellipse fitting method by Halir and Flusser
[8]
was used in the proposed algorithm. This is because of the fact that Fitzgibbon’s fitting method produced incorrect results for three or more overlapping cells among the many overlapping cells. In the proposed algorithm using Halir’s method, the overlapping cells were split with an average separation accuracy of 95%, 91%, and 82% for two overlapping cells, three overlapping cells, and four overlapping cells, respectively.
Performance of the proposed algorithms (unit: %)
Proportion = (current overlapping cells / all overlapping cells) × 100.
V. CONCLUSION
In this study, we proposed a separation algorithm for overlapping cells, which is based on two different ellipse fitting methods, namely the conventional direct least squares method and the improved direct least squares method. Through various experiments, the improved least squares method (numerically stable direct least squares fitting [NSDLSF]) is compared with the conventional least squares method (direct least squares fitting [DLSF]) and has proved to give a better result. The results reveal that NSDLSF has a successful separation ratio with an average accuracy of 95% for two overlapping cells, an average accuracy of 91% for three overlapping cells, and about 82% accuracy for four overlapping cells.
BIO
Migyung Cho
received her B.S. degree and Ph.D. in the area of design and analysis of computer algorithms from Department of Computer Science, Pusan National University, Pusan, in 1994 and 1998, respectively. She has been an associate professor at Tongmyong University in Pusan since September 2002. Her research interests include design and analysis of computer algorithms, image processing, and computer simulation of nanobio systems.
Fan J.
,
Zhang Y.
,
Wang R.
,
Li S.
2013
“A separating algorithm for overlapping cell images,”
Journal of Software Engineering
6
(4)
179 
183
Bise R.
,
Li K.
,
Eom S.
,
Kanade T.
“Reliably tracking partially overlapping neural stem cells in DIC microscopy image sequences,”
in Proceedings of MICCAI 2009 Workshop on Optical Tissue Image Analysis in Microscopy, Histopathology and Endoscopy (OPTIMHisE)
London
2009
67 
77
Mao K. Z.
,
Zhao P.
,
Tan P. H.
2006
“Supervised learningbased cell image segmentation for p53 immunohistochemistry,”
IEEE Transactions on Biomedical Engineering
53
(6)
1153 
1163
DOI : 10.1109/TBME.2006.873538
Di Ruberto C.
,
Dempster A.
,
Khan S.
,
Jarra B.
2002
“Analysis of infected blood cell images using morphological operators,”
Image and Vision Computing
20
(2)
133 
146
DOI : 10.1016/S02628856(01)000920
The Cell: an image library [Internet]
http://www.cellimageli brary.org/
Fitzgibbon A.
,
Pilu M.
,
Fisher R. B.
1999
“Direct least square fitting of ellipses,”
IEEE Transactions on Pattern Analysis and Machine Intelligence
21
(5)
476 
480
DOI : 10.1109/34.765658
Halir R.
,
Flusser J.
“Numerically stable direct least squares fitting of ellipses,”
in Proceedings of the 6th International Conference in Central Europe on Computer Graphics and Visualization (WSCG)
Plzen, Czech Republic
1998
125 
132
Cho M. G.
,
Shim J.
2012
“Ellipse based cell separation algorithm for tens of thousands of overlapping cells,”
International Conference on Convergence Technology
1
(2)
200 
204
Bai X.
,
Sun C.
,
Zhou F.
2009
“Splitting touching cells based on concave points and ellipse fitting,”
Pattern Recognition
42
(11)
2434 
2446
DOI : 10.1016/j.patcog.2009.04.003