IEEE Systems, Man and Cybernetics Magazine - April 2018 - 26

The concavity-based method searches for concave
points as potential separating points, and from a geometrical perspective, it considers particle shape and contour
information instead of gray information. For example,
Ohta et al. [21] extracted the contour of each particle and
used three points at intervals to form two vectors. They
then calculated the vector angles and selected the minimum angle as a potential splitting point relative to the
angle's proportion to concavity. This method ignores depth
information that can easily succumb to interference.
Kumar et al. proposed a rule-based approach for concave
point analysis [22] in which a support vector machine
(SVM) was used to screen the best separating line. This
method is suitable for the segmentation of particles with
varying shapes and sizes but needs a large number of
training samples and can easily lead to the loss of key
points in multipoint regions. Farhan et al. [23] improved
the approach involving the combination of segmentation
and key points but were unable to avoid excessive dependence on parameter settings and the oversegmentation
problem [23]. LaTorre et al. [24] proposed a concavity point
analysis-based clump splitting method by considering the
holes in the clumps, in which two-step binarization is used
for cell-particle extraction and the concave points are
detected. The separating lines between the two particles
are obtained by calculating the minimum distance. However, this method requires many parameters and its ro bustness is unsatisfied. Although these methods obtain
satisfactory results in different applications, methods
based on concave point analysis continue to face problems
as a result of the complexity of adhesion and overlapping.

This article proposes an automatic segmentation method
based on concave point analysis. The main contribution of
this study is the discovery of the location of concave points,
as well as the curve connection between concave points of
touching particles.
Methodology
Concavity and convexity are important features of geometric figures. In the particle image, most individuals appear
to be convex polygons, but when particles overlap, some
concave shapes appear in the touching area, which leads
to the particle mixing the convex and concave characteristics. The concave points are located in the concave region
of the contour, and if the concave region is extracted, the
concave points can be located easily and a separation line
obtained. If there is touching or overlapping among particles, the forms can be categorized into series touching,
parallel touching, and mixed touching, which are shown
in Figure 1(a)-(c).
Irrespective of the type of touching, the touching region
usually generates concave points. Touching involving
three or more particles may be decomposed into a situation involving the adhesion of two particles. For the adhesion of the two particles, the boundary can be expressed
as the intersection of functions f (x) and g (x), as shown
in Figure 2, where the circle with radius R and center
(x 0, y 0) is expressed as
(x - x 0) 2 + (y - y 0) 2 = R 2 .

(1)

In other words, y 0 = f (x 0) = g (x 0), and the intersection
of the circle and f (x) is
y 1 = f (x 1) = R 2 - (x 1 - x 0) 2 + y 0.

(2)

The intersection of the circle and g (x) is
y 2 = g (x 2) = R 2 - (x 2 - x 0) 2 + y 0.
(a)

(b)

(3)

(c)

Figure 1. the touching types of particles: (a) series

touching, (b) parallel touching, and (c) mixed
touching.

To obtain the area of the shaded part of the circle, the
integral function can be expressed as
S=

#

+
y

x0

x1

#

7 R 2 - (x - x 0) 2 + y 0 - g (x)A
x2

x0

7 R 2 - (x - x 0) 2 + y 0 - f (x)A .

(4)

Since the area of a circle is
(x2, y2)

(x1, y1)

S A = r R 2,

(x0, y0)

(5)

the area ratio of the shaded part and the circle is
g (x )

f (x )
x

Figure 2. the intersection of two functions.
26

IEEE SyStEmS, man, & CybErnEtICS magazInE A pr il 2 0 1 8

Ratiol = S .
SA

(6)

To describe the geometry of the two particles in the
image, f0 (x) and g 0 (x) are assumed to be two circles



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