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In addition, the trial vector always inherits the mutant vector parameter according to
the randomly chosen index j rand , assuring that the trial vector differs by at least one
parameter from the vector to which it is compared (x i,t ).
Finally, a greedy selection is used to
find better solutions. Thus, if the computed
cost function value of the trial vector u i ; t is less or equal than the cost of the vector
x i,t , then such trial vector replaces x i,t in the next generation. Otherwise, x i,t remains
in the population for at least one more generation:
u i ; t ;
ð
u i ; t Þ
ð
x i ; t Þ;
if f
f
x i ; t þ 1 ¼
ð
4
Þ
x i ; t ;
otherwise
:
) represents the objective function. These processes are repeated until a
termination criterion is attained or a predetermined generation number is reached.
Here, f(
·
3 Ellipse Detection Using DE
Since WBC can be approximated by an ellipsoid form, an ellipse detector algorithm
may be successfully applied in order to recognize such elements. In this section, the
problem of ellipse detection is translated to an optimization task.
3.1 Data Preprocessing
In order to detect ellipse shapes, candidate images must be preprocessed
rst by an
edge detection algorithm which yields an edge map image. Then, the (xi, i , y i ) coor-
dinates for each edge pixel p i are stored inside the edge vector P
,
¼
p 1 ;
p 2 ; ...;
p N p
with N p being the total number of edge pixels.
3.2 Individual Representation
Just as a line requires two points to completely de
line its characteristics, an ellipse is
de
ned by
five points. Therefore, each candidate solution E (ellipse candidate)
considers
five edge points to represent an individual. Under such representation,
edge points are selected following a random positional index within the edge array
P. This procedure will encode a candidate solution as the ellipse that passes through
five points p 1 , p 2 , p 3 , p 4 and p 5 (E ={p 1 , p 2 , p 3 , p 4 , p 5 }). Thus, by substituting the
coordinates of each point of E into Eq. 5 , we gather a set of
five simultaneous
;
;
;
equations which are linear in the
ve unknown parameters a
b
f
g and h.
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