Image Thresholding using Generalized Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 60

As described in Section 3.4, two values g d and g b are needed to be predefined in order
to use the proposed thresholding methodology. While using the proposed thresholding
methodology on gradient magnitude images, g d and g b represent two gradient magnitude
values and we consider the input parameters as g d = g min + max([10 g

]) and g b = g max −

3%

g 97% ]). The notation g ρ% denotes the ρ th

max([10 percentile of the gradient magnitude in

the distribution (gradient magnitude histogram).
Figures 3.10 and 3.11 give the edge extraction performance of the various thresholding
algorithms. In Figure 3.10, we find that the proposed technique does much better than the
others in determining the valid edges and eliminating those due to the inherent noise and
texture. In Figure 3.11, we find three regions in the gradient image. One (white) represents
the gradient values which surely correspond to valid edges, another (black) represents those
which surely do not correspond to valid edges and the third region (gray) represents the
gradient values which could possibly correspond to valid edges. Such multilevel threshold-
ing in gradient magnitude histograms could be used along with the hysteresis technique
suggested in (Canny, 1986) in order to determine the actual edges. We see from the figure
that the proposed techniques perform as good as or better than the others.
Note that, the gradient magnitude at every pixel in an image is obtained using the
operator given in (Canny, 1986), and edge thinning has not been done in the results shown
in Figures 3.10 and 3.11, as it is not of much significance with respect to the intended
comparisons.

3.5.2 Quantitative analysis
Here, we consider human labeled ground truth based quantitative evaluation of bilevel
thresholding based segmentation in order to carry out a rigorous quantitative analysis.

The Image Dataset Considered
We consider 100 grayscale images from the 'Berkeley Segmentation Dataset and Bench-
mark' (Martin, Fowlkes, Tal, and Malik, 2001). Each one of the 100 images considered are
associated with multiple segmentation results hand labeled by multiple human subjects and
hence we have multiple segmentation ground truths for every single image.

The Evaluation Measure Considered
We use the local consistency error (LCE) measure defined in (Martin et al., 2001) in
order to judge the appropriateness of segmentation results obtained by a bilevel thresholding
algorithm. Consider S H as a segmentation result hand labeled by a human subject and S A
as a segmentation result obtained applying an algorithm to be analyzed. The LCE measure
representing the appropriateness of S A with reference to the ground truth S H is given as

min n E(S H , S A , p i ), E(S A , S H , p i ) o

X

n

1
n

LCE(S H , S A ) = (3.46)

i =1

where

, p) = |R(S

, p)\R(S , p)|

1 2

E(S , S (3.47)

1 2

|R(S 1 , p)|

In the above,\represents set difference,|x|represents the cardinality of a set x, R(S, p)
represents the set of pixels corresponding to the region in segmentation S that contains
pixel p and n represents the number of pixels in the image under consideration. The LCE
take values in the range [0, 1], where a smaller value indicates more appropriateness of the
segmentation result S A (with reference to the ground truth S H ).