From Tolerance Near Sets to Perceptual Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 175

(9.1a) Original image
(Walnut)

(9.1b) Each subimage
is a perceptual object

(9.1c) Gray scale im-
age

(9.1d) Avg gray scale
used as a feature value

FIGURE 9.1: Images and perceptual objects

This approach to representation and comparison of feature values by probe functions
started with the introduction of near sets (Peters, 2007a, 2008b). Probe functions provide
a basis for describing and discerning a nities between sample objects in the context of a
perceptual information system. This approach is a generalization of the concept of attributes
in the approximation spaces exists in rough set theory (Peters, 2008d; Meghdadi, 2009).

Definition 9.2.1 Perceptual System
A perceptual systemhO, iis a real valued total deterministic information system where O

is a non-empty set of perceptual objects, while

F

F is a countable set of probe functions.

9.3 Perceptual Indiscernibility and Tolerance Relations

”The exact idea of closeness or of 'resembling', or of 'being within tolerance' is universal
enough to appear quite naturally in almost any mathematical setting. It is especially natural
in mathematical applications: practical problems, more often than not, deal with approx-
imate input data and only require viable results - results with a tolerable level of error.”
(Sossinsky, 1986)

In this section indiscernibility and tolerance relations are defined. Indiscernibility rela-
tion introduced by Z. Pawlak (Pawlak, 1981) is a key concept in approximation spaces
in rough set theory. Indiscernibility and tolerance relations are important and useful in
defining measures to compare a nities between pairs of perceptual objects in a perceptual
system, for example to compare perceptual images (Peters and Ramanna, 2008). The term
tolerance space was introduced by E.C. Zeeman in 1961 in modelling visual perception with
tolerances (Zeeman, 1961). A tolerance space is a set X supplied with a binary relation'
(i.e., a subset'⊂X×X) that is reflexive (for all x∈X, x'x) and symmetric (for all
x, y∈X, x'y and y'x) but transitivity of'is not required.

Definition 9.3.1 Perceptual Indiscernibility Relation
LethO, F ibe a perceptual system. For everyB⊆ F the indiscernibility relation∼ B is defined

as follows:

∼ B ={(x, y)∈O×O|∀φ∈Bkφ(x)−φ(y)k= 0}. (9.1)