Image Thresholding using Generalized Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 54

undefined region to the dark and the bright region are given as

Err d = (α + βe c ) + e p (3.43)

Err b = (α + βe c ) + e p

(3.44)

We calculate the association errors Err d and Err b for all graylevel bins corresponding
to g a ∈(g d , g b ), that is, the graylevel bins of the undefined region. We then compare the
corresponding association errors and assign these graylevel bins to one of the two defined
(dark and bright) regions that corresponds to the lower association error. In (3.43) and
(3.44), we consider α = β = 0.5 and hence force the range of contribution from the change
errors to [0, 1], same as that of the proximity errors. Thus the bilevel thresholding is achieved
by separating the bins of the graylevel histogram into two regions, namely, the dark and the
bright regions. As a region in the graylevel histogram of an image corresponds to a region
in the image, the aforesaid bilevel thresholding would divide the image into two regions.

3.4.2 Multilevel Thresholding

Here we extend above given novel bilevel image thresholding methodology to the multilevel
image thresholding problem. Note that, we do not posses the prior knowledge required
to assign more than two seed values for carrying out multilevel thresholding. Therefore,
we understand that the concept of thresholding based on association error can be used to
separate a histogram only into two regions and then these regions can further be separated
only into two regions each and so on. From this understanding, we find that the proposed
concept of thresholding using association error could be used in a binary tree structured
technique in order to carry out multilevel thresholding.
Now, let us consider that we require a multilevel image thresholding technique using
association error in order to separate a image into Θ regions. Let D be a non-negative
integer such that 2 D−1 < Θ≤2 D . In our approach to multilevel image thresholding
for obtaining Θ regions, we first separate the graylevel histogram of the image into 2 D
regions. The implementation of this approach can be achieved using a binary tree structured
algorithm (Breiman, Friedman, Olshen, and Stone, 1998). Note that in (Breiman et al.,
1998), the binary tree structure has been used for classification purposes, which is not
our concern. In our case, we use the binary tree structure to achieve multilevel image
thresholding using association error, which is an unsupervised technique. We list a few
characteristics of a binary tree below stating what they represent when used for association
error based multilevel image thresholding.

1. A node of the binary tree would represent a region in the histogram.
2. The root node of the binary tree represents the histogram of the whole image.
3. The depth of a node is given by D. At any depth D we always have 2 D

nodes

(regions).
4. Splitting at each node is performed using the bilevel image thresholding technique
using association error proposed in the previous section.
5. All the nodes at a depth D are terminal nodes when our goal is to obtain 2 D
regions in the histogram.

In order to get Θ regions from the 2 D regions, we need to declare certain bilevel thresholding
of histogram regions (node) at depth D−1 as invalid. In order to do so, we define a measure
(ι) of a histogram region based on the association errors Err d and Err b obtained for the