Image Segmentation - Bezier and Splines in Image Processing and Machine Vision - page 48

inter-region boundaries. Merging should have very little effect on the over-
all contrast of the image. The following objective measures for quantitative
evaluation of segmentation are helpful.
a. Correlation
Correlation has already been used as a criterion for graylevel threshold-
ing and evaluation [31]. In the present context, it can be used to examine
the graylevel similarity between the segmented region/patches and the origi-
nal image. Consider the segmented image where all patches under respective
thresholds are replaced by their average value. The correlation between the
segmented and input images provides an idea about how a segmented patch
is nearer to the corresponding region in the original input image. For a good
segmentation, the correlation coecient between the two images should be
very high. However, if the segmented patches are not homogeneous, i.e., if
they have edges in them, the variance of the corresponding regions would be
high and as a result, the correlation coecient would be low. Thus correlation
between the two different images—input and segmented—can be an useful
measure to evaluate the quality of segmentation. The correlation coecient
can be calculated in the following way.
The coecient of correlation ρ xy for two sets of data X = { x 1 ,x 2 , ··· ,x N }

and Y = { y 1 ,y 2 , ··· ,y N } is given by

N

1
N

x i y i − xy

i =1

ρ xy =

x 2

, (2.29)

N N

1
N

x i −

1
N

y i − y 2

i =1 i =1

N N

1
N

1
N

where x = x i and y = y i . The correlation coecient, ρ xy takes on

i =1 i =1

values from +1 to -1, depending on the type and extent of correlation between
the sets of data.
b. Contrast
Another requirement for a good segmentation is that the contrast at inter-
region boundaries must be very high compared to that for the interior points.
This criterion immediately suggests that the average contrast, i.e., contrast
per pixel, say K b , of all inter-region boundary points in all subimages should
be high compared to that (say,

K Ω ) over all points enclosed within the bound-

aries. Therefore,

K b >> K Ω .
The contrast c ij , at the pixel position ( i, j ) can be computed as in equation
(2.21), which we repeat here as

c ij = |

B − B ij |

= |

B |

, (2.30)

B B