IEEE Systems, Man and Cybernetics Magazine - April 2018 - 27

with centers (a 1, b 1) and (a 2, b 2), and radii R 1 and R 2,
respectively. That is,

and

f0 (x) = R 21 - (x - a 1) 2 + b 1

(7)

g 0 (x) = R 22 - (x - a 2) 2 + b 2 .

(8)

If the two circles intersect, taking an intersection point
of f0 (x) and g 0 (x) as the circle center and R 3 as the radius, the area of the new circle, which does not overlap with
f0 (x) and g 0 (x), is expressed as S [the shaded area in Figure 3(a)]. The value of S, therefore, is less than half of the
area of the circle.
According to the assumption whereby the graphical
area in a plane space is invariant to rotation, the graphic in
Figure 3(b) rotates and shifts its coordinates. Then, f0 (x)
and g 0 (x) can be changed to
(9)

x 2 + y 2 = R 21
x 2 + (y - m) 2 = R 22 .

and

(10)

To describe this relationship, a special case is used
to solve the problem. Assuming the existence of two
circles of equal area, the black circle is centered at (0, 0),
the center of the red circle has the same horizontal
coordinate as the black circle, and the radius of both is 1
(as shown in Figure 4). When the center of the red circle
shifts from (1, 0) to (2, 0), the area ratio decreases from
a reduction in the overlapping region and is always less
than 0.5 when there is a concave point. The larger the
degree of concavity, the smaller the area ratio; therefore, analyzing concave points using the method of area
ratio is reasonable from a theoretical aspect, which is
suitable for analyzing the situation of the deep adhesion
of the particles.
In practical applications, the area overlapping two particles in a circle mask is calculated as

A Framework for Particle Separation
Concave Point Detection
The method based on concave point analysis consists of
three steps. To ensure accuracy and efficiency, a detection
step based on a curvature scale space (CSS) corner detector and a refining step based on the area ratio are decomposed to determine the final position of the concave points.
The Screening of Concave Points
To overcome high computational costs, the CSS method is
used for rough screening. The key steps are as follows:
◆ step 1: apply a Canny edge detector to process the
input image, and obtain a binary edge-map
◆ step 2: extract the edge contours from the edge-map;
fill in the gaps in the contours and find the T-junctions
◆ step 3: compute the curvature at a high scale v high for
each edge contour
◆ step 4: consider those local maxima as initial corners
whose absolute curvatures satisfy the criteria
◆ step 5: track the corners from the highest scale to the
lowest scale to obtain a better position
◆ step 6: compare the T-junction to other corners, and
remove one of the two corners that are very close.
An example of image preprocessing is shown in Figure 5, where (a) is the gray image, (b) is the binary image,
(c) is the boundary contour image using a Canny edge
operator, and (d) shows the results of corner detection
using the CSS method, where the corners are marked with
red boxes.

y

y

g0(x )

R2

R3

R1

R1

f0(x )

f0(x )

x
g0(x )

x

Ratio = 1 -Ratiol = 1 - S .
SA

(11)

This ratio reflects the degree of concavity of the graphs.

f0(x )

g0(x )

f0(x )

(a)

g0(x )

(b)

Figure 3. the intersection of two circles (a) before

rotation and (b) after rotation.

f0(x )

g0(x )

f0(x )

g0(x )

Figure 4. a flowchart of the proposed method.

Ap ri l 2018

IEEE SyStEmS, man, & CybErnEtICS magazInE

27



Table of Contents for the Digital Edition of IEEE Systems, Man and Cybernetics Magazine - April 2018

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