Image Processing Reference
In-Depth Information
categories are the the fundamental building blocks of the knowledge and the family of all
categories in the knowledge base K = (U;R) are known asK-categories.
In the universe of discourseU, where X U andRis an equivalence relation,Xis
said to beR-denable orR-exact ifXis union of someR-basic categories; otherwiseX
isR-undenable orR-inexact orR-rough. TheR-exact sets are those sets of the universe
Uwhich can be exactly dened in the knowledge baseKandR-rough sets are those
subsets which cannot be dened in this knowledge base. However,R-rough sets can be
dened approximately by employing the two exact sets, referred to as a lower and an upper
approximation. The lower and upper approximations can be dened as follows:
R
X =
[
fY 2 U j IND (R) : Y Xg
(10.2)
RX =
[
fY 2 U j IND (R) : Y \ X 6= ;g
(10.3)
The set
R
X, also known asR-lower approximation ofX, is the set of all elements
of
U
which can be classied as elements ofXwith certainty in the knowledgeR. The set RX,
also known asR-upper approximation ofX, is the set of elements ofUwhich can possibly
be classied as the elements ofX, employing knowledgeR. Obviously, the dierence set
yields the set of elemen
ts
which lie around the boundary.
The set BN
R
(X) = RX
R
X is called theR-boundary ofXorR-borderline region of
X. This is the set of elements, which cannot be classied to X or to X using the knowledge
R. The borderline region actually represents the inexactness of the setXwith respect to
the knowledgeR. The greater the borderline region of the set more is the inexactness. This
idea can be expressed more precisely by the accuracy measure dened as:
R(X)
=
jRXj
jRXj
for X 6= ;
(10.4)
where j:j is the cardinality operator. The accuracy measure captures the degree of com-
pleteness of the knowledge about the setX. Here, we can also dene a measure to express
the degree of inexactness of the setX, calledroughnessmeasureorroughnessindexofX
orR-roughness ofX, given by
R
(X) = 1
R(X)
(10.5)
Obviously 0
R
(X) 1, for everyRand X U. If
R
(X) = 0, the borderline
region ofXis empty and the setXisR-denable i.e.Xis crisp or precise with respect to
the knowledgeR, and otherwise, the setXhas some non-emptyR-borderline region and
therefore isR-undenable i.e.Xis rough or vague with respect to the knowledgeR. Thus
any rough set has a non-empty boundary region (Pawlak, 2004), as depicted in Figure 10.1.
The histon (Mohabey and Ray, 2000) presents a new means for visualization of color infor-
mation for the evaluation of similar color regions in an image. It also presents a method
for the segregation of the elements at the boundary, which can be applied in the process of
image segmentation.
Histon is basically a contour plotted on the top of existing histograms of the primary
color components red, green, and blue in such a manner that the collection of all points
falling under the similar color sphere of the predened radius, called expanse, belong to one
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