Image Processing Reference

In-Depth Information

When we have the means and variances of the different features the box classifier

should be replaced by a
statistical classifier
since this is a more accurate approach.

In the box classifier we have a binary decision; is a new feature vector (BLOB) in-

side or outside the rectangle? In a statistical classifier we instead measure a distance

between a new feature vector (BLOB) and the prototype. The smaller the distance

the more likely it is that the BLOB is the same type (here a large circle) as the proto-

type. To make this approach operational we need to threshold the distance and hence

end up with a binary decision region like the dashed box in Fig.
7.7
. The difference

is that the region is now a more precise ellipse and not a rectangle, see Fig.
7.7
.One

statistical classifier is the
weighted Euclidean distance
in our case defined as

(f
i
(
cir
)
−

mean
(
cir
))
2

variance
(
cir
)

(f
i
(
area
)
−

mean
(
area
))
2

variance
(
area
)

WED
(
f
i
,
prototype
)
=

+

(7.7)

where WED
(f
1
,
prototype
)
is the weighted Euclidean distance between feature vec-

tor
f
i
, i.e., the
i
th BLOB, and the prototype.
f
i
(
cir
)
and
f
i
(
area
)
are the circularity

and area of the
i
th BLOB, respectively. The rest of the parameters in the equation

are the means and variances of the two features of the prototype. In the general case

with
p
different features the weighted Euclidean distance measure is defined as

(f
i
(m
j
)

p

mean
(m
j
))
2

variance
(m
j
)

−

WED
(
f
i
,
prototype
)

=

(7.8)

j

=

1

where
m
j
is the
j
th feature. If the variances of all features are the same, then we

can ignore them and end up with the Euclidean distance measure (ED), where the

decision region is a circle in 2D (see Fig.
7.7
):

f
i
(m
j
)

p

mean
(m
j
)
2

ED
(
f
i
,
prototype
)

=

−

(7.9)

j

=

1

It should be noticed that the three equations above assume that the scale of the

features are the same. In our example the problem is that the area is measured in

1000 s and circularity is a value close to 1. This means that the area will dominate

the distance measure completely. The solution is to normalize the features so they

are scaled similarly and are in the same interval, e.g.

[

0
,
1

]

. This can be obtained,

for example, as

min
Area of BLOB

Area of Model
,
Area of Model

Area feature

=

(7.10)

Area of BLOB

min
Circularity
,

1

Circularity

Circularity feature

=

(7.11)