Recently, methods that find approximate coordinates of cell region by
using Blob detector such as LoG or MSER, and find the cell region by using
local binarization and global binarization show good performance in cell region
detection 2021. However, in the case where the contrast between the
background and a cell region is low, the cell region can not be detected by the
global binarization and the local binarization complementing the problem can
not yield sufficiently good results. Therefore, a special image normalization
method capable of increasing the contrast difference according to the image is
required. In this paper, we propose a pixel-level cell region discriminator R
using statistical and histogram features and logistic regression to solve the
degradation of cell region detector performance due to low image contrast. The
statistical feature to be used in the logistic regression analysis is defined
as a five-dimensional feature vector composed of the median, mean, standard
deviation, maximum value, and the difference between the maximum value and the
minimum value for the pixel value of the local square image with one-side of

 pixels. In addition, the distribution feature
which directly expresses the distribution characteristics of pixel brightness
is defined as a brightness histogram that divides the range between the minimum
value and the maximum value in the image into n classes. To concentrate on the
low intensity value where the brightness values of the cell area and the
background area are densely distributed, the bin corresponding to the lower 25%
of the whole are used. The two kinds of features are categorized into the cell
region and the background, and the pixel of the feature is identified as the
cell region through the following logistic regression.

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 (1)

where

 is the sigmoid function,

 is the learned regression parameter vector,
and

 is the statistical feature or histogram
feature vector.

 is the label of

 and has 1 if it is a cell area or 0 if it is a
cell area.

The pixel-level
discriminator extracts features for every pixel of the image, so the training
data is very large. Therefore, the parameter vector

 is optimized by using the stochastic gradient
descent (SGD) defined as follows for fast convergence in learning.

                                (2)

where

 is the learning rate and

 is the mini batch sample number.

When identifying a
cell region through logistic regression analysis, one of the statistical and
distributional features has better discrimination power depending on the image.
Therefore, we define probability values of pixels for two features estimated
through logistic regression as a two-dimensional ensemble feature, and detect
the cell region stably using the second-order regression. The classification
threshold of each regressor is experimentally set to the value that best
classifies the training data.

3.    Multi-cell discriminator

3.1.  Convex surface transform

A cell region segment S detected in a pixel-level cell region
discrimination R actually contains one to dozens of cells. Therefore, S
consisting of a plurality of cells must be divided into individual cells for
precise cell segmentation. To this end, we adopt an existing study that assumes
each cell as a 2D GMM and clusters the pixel coordinates to each cell using the
Expectation-Maximization (EM) algorithm that find the parameters of the
probability model. The features to be used in EM are the pixel coordinates and
the coordinates of the local maximum point for them, and the initial cluster is
set through k-means clustering. The local maximum point coordinate is an
important factor that influences the performance of clustering, and the closer
the maximum point coordinates in a cell are, the more accurate the division is.
To reduce the variance of the local maximum point coordinates within a cell, we
combine the original cell image

 with the following distance image

 to convert the cell surface to a more convex
shape.

where,

 is a local image
containing only one region segment

, and

 is a Gaussian blur
kernel.

 is the set of edge
pixel coordinates

 of

, and

 is all pixel
coordinates of

.If the number of components k of the GMM is equal to
the number of actual cells in the region segment S, we can expect that a
meaningful division result is generated through the EM algorithm 2021. Depending
on various conditions such as brightness of

, smoothness of its surface, and the boundary
brightness between adjacent cells, the partitioning method 2021 of
selecting

 which minimize
the dissimilarity between the real cells and the virtual cells generated from
the GMM parameter may be not working well. Therefore, not a method using
dissimilarity, we propose a multi-cell discriminator M that divides a region

 into a binary
tree structure by determining whether it is a multi-cell. The feature of

 for multi-cell
identification is defined as least square error for surface fitting using 3rd
order polynomial for cell surface as follows

where,

 is the number of pixels belonging to

, and

 is a parameter for surface fitting. As shown
in the first row of Fig. 3, the least square error of a single cell by the
third-order polynomial surface fitting of region segment

 is smaller than that of multiple cells.

The
difference between

 consisted of a
single cell and consisted of multiple cells also appears in boundary sectional
area between cells divided by EM. If EM is divided into multiple cells of S,
the boundaries between the models will be formed at the boundary between the
two cells, as shown in Fig. 3(e) and (f). If S of the single cell is divided
into two cells, boundaries between the models will be formed at the vicinity of
the center of the cell because there are no obvious Gaussian mixture
distribution in S, as shown in Fig. 3(d). Therefore, the boundary sectional
area between cells divided by EM in

 of multiple
cells is smaller than the sectional area estimated in

 of single cell,
so that it is suitable for discriminating between a single cell and multiple
cells. Thus, we define the boundary sectional area feature of the cells divided
by EM as follows.

boundary area characteristics of divided cells for S of a single
cell and two cells, and discriminates whether any S is multiple cells that can
be divided. First, S is divided into two cells via EM in a learning stage of M.
is composed of a single cell in the ground truth, the class of the feature
vector extracted by Eqs. (4) and (5) in the divided region S is assigned as
single-cell class.

If S
is composed of multiple cells, the class of its feature vector is assigned as a
multi-cell class. With progressive partitioning on two divided regions

 and

 in S’ given a multi-cell label, a binary tree structure is
built, as shown in Algorithm 1. In
the learning of M, SVM is trained from extracted features for all nodes of the
tree. When testing the cell segmentation, if S is divided into 2 cells by EM
and it is discriminated as a multi-cell by using the learned M, S is divided