Edge Detection - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 181

For two norms S 1 and S 2 , the upper bound of the level is defined as

m
n

URxy SSRx iyjRxy
SS , [ (,)

= ( (( )( − − ), (,)))

12

12

i n
j m

=−
=−

Following the same procedure as the lower constructor, a new matrix
is formed. The upper constructor brightens the image. Max t -norm is the
smallest t -conorm. If the t -conorm is large, the intensity of the pixel is high
and the image is brighter.
A new fuzzy relation is constructed by taking the difference between the
lower and upper constructors:

(8.8)

WR xy URxy LRxy
SS

[ (,) = [ ]( ,) − [ (,)

, TT ,

12 12

W [ R ]( x , y ) is the fuzzy edge that denotes the intensity variation in its
neighbourhood. So, in the construction of a fuzzy edge, the length of the
interval represents the membership degree of each element in the new fuzzy
relation. Fuzzy edge is not the edge image as in Canny; rather, it is the change
in the intensity.

8.5 Construction of Enhanced Fuzzy Edge
Using Type II Fuzzy Set
The fuzzy edge image can also be obtained using Type II fuzzy set where
the membership function in an ordinary fuzzy set is considered as fuzzy [7].
The image is initially normalized to obtain the values in the range [0, 1]. For
each pixel, a 3 × 3 neighbourhood is selected and minimum and maximum
values are noted. This is done for all the pixels. This way, two image matri-
ces are obtained with maximum and minimum values of the pixel in a 3 ×
3 window. As the image is itself fuzzy, the maximum and minimum values
are also fuzzy, so for each image matrices, Type II levels are computed.
The upper and lower membership functions for the maximum value image
are written as

upper 075 .

μ = [ μ ( )] x

max max

lower 1075 /.

μ = [ μ ( )] x

max max

Likewise, the upper and lower membership functions of the minimum val-
ued image matrix are computed.