Leukocyte Detection Through an Evolutionary Method - Complex System Modelling and Control Through Intelligent Soft Computations - page 145

Fig. 2 Ellipse candidate
(individual) built from the
combination of points p 1 , p 2 ,
p 3 , p 4 and p 5

p

y

p

r

r

max

min

p

(, ) xy

p 00

p

x

represent the semi-major and semi-minor axis, respectively. However, MEA avoids
computing square root calculations by comparing the pixel separation distances.
A method for direct distance comparison is to test the halfway position between two
pixels (sub-pixel distance) to determine if this midpoint is inside or outside the
ellipse boundary. If the point is in the interior of the ellipse, the ellipse function is
negative. Thus, if the point is outside the ellipse, the ellipse function is positive.
Therefore, the error involved in locating pixel positions using the midpoint test is
limited to one-half the pixel separation (sub-pixel precision). To summarize, the
relative position of any point (x, y) can be determined by checking the sign of the
ellipse function:

8
<
:

0if ð x ; y Þ is inside the ellipse boundary

\

f ellipse ð x ; y Þ ¼ 0if ð x ; y Þ is on the ellipse boundary ð 12 Þ

if ð x ; y Þ is outside the ellipse boundary

[

The ellipse-function test in Eq. 12 is applied to mid-positions between pixels
nearby the ellipse path at each sampling step. Figure 3 a, b show the midpoint
between the two candidate pixels at sampling position. The ellipse is used to divide
the quadrants into two regions the limit of the two regions is the point at which the
curve has a slope of

1 as shown in Fig. 3 .
In MEA the computation time is reduced by considering the symmetry of
ellipses. Ellipses sections in adjacent octants within one quadrant are symmetric
with respect to the dy/dy =

−

1 line dividing the two octants. These symmetry
conditions are illustrated in Fig. 4 . The algorithm can be considered as the quickest
providing a sub-pixel precision (Van Aken 2005 ). However, in order to protect the
MEA operation, it is important to assure that points lying outside the image plane
must not be considered in S.
The objective function J(E) represents the matching error produced between the
pixels S of the ellipse candidate E and the pixels that actually exist in the edge
image, yielding:

−