Leukocyte Detection Through an Evolutionary Method - Complex System Modelling and Control Through Intelligent Soft Computations - page 144

ax 2 by 2 þ 2hxy þ þ 2gx þ 2fy þ 1 ¼ 0 ð 5 Þ

guration of the edge points shown by Fig. 2 , the ellipse
center (x 0 , y 0 ), the radius maximum (r max ), the radius minimum (r min ) and the
ellipse orientation (

Considering the con

h ) can be calculated as follows:

hf bg

x 0 ¼ ; ð 6 Þ

C

gh af

y 0 ¼ ; ð 7 Þ

C

s

2

D

r max ¼ ; ð 8 Þ

ð þ Þ

C a b R

s

2

D

r min ¼ ; ð 9 Þ

C ð a þ b þ R Þ

1
2 arctan

2h

h ¼ ð 10 Þ

a b

where

0 1

ahg
hbf
gf 1

2

@ A :

R 2 4h 2 h 2 and

¼ð a b Þ þ ; C ¼ ab

D ¼

det ð 11 Þ

3.3 Objective Function

Optimization refers to choosing the best element from one set of available alter-
natives. In the simplest case, it means to minimize an objective function or error by
systematically choosing the values of variables from their valid ranges. In order to
calculate the error produced by a candidate solution E, the ellipse coordinates are
calculated as a virtual shape which, in turn, must also be validated, i.e. if it really
exists in the edge image. The test set is represented by S , where

N s are the number of points over which the existence of an edge point, corre-
sponding to E, should be tested.
The set S is generated by the Midpoint Ellipse Algorithm (MEA) (Bresenham
1987 ) which is a searching method that seeks required points for drawing an ellipse.
For any point (x, y) lying on the boundary of the ellipse with a, h, b, g and f, it does
satisfy the equation ffi ellipse ð

¼f s 1 ; s 2 ; ...; s N s g

r max x 2 r min y 2 r max r min , where r max and r min

x ; y Þ þ