Image Segmentation - Bezier and Splines in Image Processing and Machine Vision - page 49

where B is the immediate surrounding luminance of the ( i, j )th pixel with
intensity B ij .
Let SB be the set of all boundary points and SI be the set of all interior
points ( SB SI = null set). Contrast of all boundary points,

K b and that of interior points, K Ω are, therefore,
K b =

∪ SI = F , SB ∩

c ij and K Ω = c ij .

( i,j ) ∈ SB ( i,j ) ∈ SI

Note that K Ω is an indicant of homogeneity within regions—lower the
value of K Ω , higher is the homogeneity. The contrast per pixel, K b , of all
inter-region boundary points and that over all points enclosed within the
boundaries, K Ω can be obtained by dividing K b by the number of boundary
points and K Ω by the number of interior points.

2.6 Comparison with Multilevel Thresholding
Algorithms

Since the co-occurrence matrix contains information regarding the spatial dis-
tribution of graylevels in the image, several workers have used it for segmen-
tation. For thresholding at graylevel s , Weszka and Rosenfeld [174] defined
the busyness measure as follows:

s L − 1 L − 1 s

Busy ( s )= t ij + t ij . (2.31)

i =0 j = s +1 i = s +1 j =0

The co-occurrence matrix used in (2.31) is symmetric. For an image with only
two types of regions, say, object and background, the value of s which mini-
mizes Busy(s), gives the threshold. Similarly, for an image having more than
two regions, the busyness measure provides a set of minima corresponding to
different thresholds.
Deravi and Pal [58] gave a measure that they called “conditional probabil-
ity of transition” from one region to another as follows. If the threshold is at s ,
the conditional probability of transition from the region [0 ,s ]to[ s +1 ,L − 1]

is

s L− 1

t ij

i =0 j = s +1

P 1 = (2.32)

s s s L− 1

t ij + t ij

i =1 j =0 i =0 j = s +1

and the conditional probability of transition from the region [( s +1) , ( L − 1)]

to [0, s] is