Image Segmentation: A Rough-set Theoretic Approach - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 193

The pixels in the neighborhood fall in the sphere of the similar color if the distance
d T (m;n) is less than expanse. We dene a matrixXof the size M N such that an

element X (m;n) is given by

( 1; d T (m;n) < expanse
0; otherwise

X (m;n) = (10.9)

The mathematical formulation of histon can now be given as (Mushrif and Ray, 2008)

M X N X

h i (g) = (1 + X (m;n)) (I (m;n;i) g)

m=1 n=1

for 0 g L 1 and i = fR;G;Bg (10.10)

10.3.2Roughnessmeasure
The histogram and the histon can be correlated with the concept of approximation space
in the rough set theory. The histogram value of the g th intensity is the set of pixels which
denitely belong to the class of intensitygand therefore, can be considered as the lower
approximation and the histon value of the g th intensity represents all the pixels which
possibly belong to the same segment or a class of similar color value and therefore, may be
considered as the upper approximation.
The roughness index of the set at each intensity may be dened as

i (g) = 1 jh i (g)j
jH i (g)j
for 0 g L 1 and i = fR;G;Bg

(10.11)

The roughness measure is always in the range [0; 1]. The object regions in the image are
more or less homogeneous so that there is a very little variation in the pixel intensities in this
region. The number of pixels belonging to the similar color sphere is large and therefore the
roughness is more in the homogeneous object region. Whereas, near the object boundaries
the homogeneity is less and variation in pixel intensities more. The number of pixels in the
similar color sphere is small resulting into a smaller value of roughness measure. Roughness
index, when evaluated at every intensity value, thus develops crests and troughs just as in
case of a conventional histogram. It exploits the correlation among the neighboring pixels
better than that in case of the conventional histograms as well as the histon and therefore
it is better suited for segmentation of natural color images.
In a histogram based segmentation scheme, the peaks in the histogram represent the dif-
ferent regions and the valleys represent the boundaries between those regions. In a similar
fashion the peaks and valleys of the graph of roughness index versus intensity can also be
used to segregate dierent regions in the image. The roughness index based segmentation
scheme has some distinct advantages over the histogram based and histon based segmenta-
tion schemes such as:

In conventional histogram as well as in histon, the small but important seg-
ments may not be detected due to insignicant peak and valley points. Since the
roughness index, dened by eq. (10.11), is a ratio, the peak and valley points are
signicant even if the segment is small. Due to this property, roughness index
based segmentation technique detects such small but signicant regions which
results in a better segmentation performance.