Cantor, Fuzzy, Near, and Rough Sets in Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 11

1.5.2 Sample Rough Set

Any set X ⊂ U where

x / ∼ B = X .

x

/ ∼ B ⊆

U

/ ∼ B

In other words, if a set X does not equal its lower approximation, then the set X is rough, i.e. ,
roughly approximated by the equivalence classes in the quotient set U / ∼ B .

1.5.3 Rough Sets in Image Analysis

The essence of our approach consists in viewing
a digitized image as a universe of a certain
information system and synthesizing an indiscernibility
relation to identify objects and measure
some of their parameters.
- Adam Mrozek and Leszek Plonka, 1993.

In terms of rough sets and image analysis, it can be observed that A. Mr ozek and L. Plonka were
pioneers (Mr ozek and Plonka, 1993). For example, he was one of the first to introduce a rough
set approach to image analysis and to view a digital image as a universe viewed as a set of points.
The features of pixels (points) in a digital image are a source of knowledge discovery. Using Z.
Pawlak's indiscernibility relation, it is then a straightforward task to partition an image and to con-
sider set approximation relative to interesting objects contained in subsets of an image. This work
on digital images by A. Mr ozek and L. Plonka appeared six or more years before the publication
of papers on approximate mathematical morphology by Lech Polkowski (Polkowski, 1999) (see,
also, (Polkowski, 1993; Polkowski and Skowron, 1994)) and connections between mathematical
morphology and rough sets pointed to by Isabelle Bloch (Bloch, 2000). The early work on the use
of rough sets in image analysis has been followed by a number of articles by S.K. Pal and others
(see, e.g. , (Pal and Mitra, 2002; Pal, UmaShankar, and Mitra, 2005; Peters and Borkowski, 2004;
Borkowski and Peters, 2006; Borkowski, 2007; Maji and Pal, 2008; Mushrif and Ray, 2008; Sen
and Pal, 2009)).
From set composition Law 3, it can be observed that rough sets are Cantor sets.

1.6 Conclusion

In sum, fuzzy sets, near sets and rough sets are particular forms of Cantor sets. In addition, each of
these sets in the computational intelligence spectrum offer very useful approaches in image analysis,
especially in classifying objects.

Acknowledgements

This research by James Peters has been supported by the Natural Sciences and Engineering Re-
search Council of Canada (NSERC) grant 185986, Manitoba Centre of Excellence Fund (MCEF)
grant, Canadian Centre of Excellence (NCE) and Canadian Arthritis Network grant SRI-BIO-05,
and Manitoba Hydro grant T277 and that of Sankar Pal has been supported by the J.C. Bose Fel-
lowship of the Govt. of India.