IEEE Systems, Man and Cybernetics Magazine - April 2018 - 29

This is shown in Figure 8, where the red rectangle is the
region containing a pair of concave points, and the upper
and right regions contain the gray projection of the rectangular region.
Segmentation Line Construction
Conventional methods for segmenting two touching
particles involve connecting paired concave points
directly through a line, as shown in [18], [20], and [22].
This method is simple, intuitive, and can directly separate adhesive particles, it does not, however, fit the cognitive law of human vision. Regardless of the touching
types, the touching border cannot be a straight line;
therefore, the geometric status of each particle prior to
touching must be estimated. Since the particles analyzed in this study are convex, they can be estimated
based on the concave point pair obtained and other
points on the given object.
According to geometrical principle, three noncollinear points can only form one circle. For individual
particles, the concave points and boundary points are
known; thus, making use of the two end points of the
dividing line and any other point along its edge, we can
achieve a three-point circle fitting. We adopt the perpendicular bisector intersection method, i.e., two straight
lines connecting three points, with the intersection of
their perpendicular bisectors being the center of a circle,
and the distance between the center and any of the three
points being the radius. Its geometric model is shown in
Figure 9, where A, B, and C are the three points in the
same plane, and their coordinates are (x a, y a), (x b, y b),
and (x c, y c), respectively.
Based on these principles, the processing steps in volved in arc fitting are as follows:
◆ step 1: collect the contour coordinates of the individual
particles split using a straight line between two concave
points, and mark the two separating points A and B
◆ step 2: obtain points C and D from the contour coordinates that are near points A and B, respectively, but do
not belong to the straight line connecting A and B
◆ step 3: fit circles with points A, B, and C, and A, B, and
D, respectively, and obtain the centers and radii of the
two circles
◆ step 4: get arcs Arc AB and ArclAB of the two circles
according to the spatial relationship between the circle
centers and the splitting points
◆ step 5: fill the regions encircled by arcs and segmentation lines with pixel value 1, and calculate areas S AB
and S lAB of the two regions
◆ step 6: compare S AB and S lAB, and choose the smaller
of the two as the compensation part of the particle.
Experimental Results
To evaluate the proposed method, several particle images from different conditions were employed for an experiment. The experimental results showed that the

proposed method is better than some traditional segmentation methods.
A Comparison of Methods
In this article, the ultimate erosion method [25] and marker
watershed method [27] were used for comparison with the
proposed method. Some experimental results are shown in
Figure 10, where the blue contours represent the markers
after separation. Figure 10(a) and (e) shows three types of
original particle images of different sizes on different rows,
and Figure 10(b) and (f) shows the results obtained by the
ultimate erosion method. Figure 10(b) and (f) indicates that
most touching particles could be split [only Figure 10(b)
exhibits undersegmentation], but restoring the original
shape before erosion is difficult. Figure 10(c) and (g) shows
the results of the marker watershed method, and although
it eased oversegmentation to some extent, it could not
avoid it entirely, and false separations exist in Figure 10(g).
For images in Figure 10(d) and (h), the shape of every particle was estimated based on the accurate detection of the
splitting points, which is consistent with the law of human
visual perception. To test the robustness of our proposed
method, the effect of noise on the concave points positioning and point-pair matching was analyzed.

δ
L

Figure 8. the gray projection of an image with two
concave points.

(x0, y0)
A (xa , ya)

LAB
LBC

C (xc , yc)
B (xb , yb)
Figure 9. the geometric model of a three-point

circle fitting.

Ap ri l 2018

IEEE SyStEmS, man, & CybErnEtICS magazInE

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