Image Processing Reference
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O /∼ B
X
Y
FIGURE 7.2: Example of Defn. 2.
space-based approach to perceiving image resemblances hearkens back to the observation
about perception made by Ewa Orlowska in 1982 (Orlowska, 1982) (see, also, (Orlowska,
1985)), i.e., classes defined in an approximation space serve as a formal counterpart of
perception.
The term tolerance space was coined by E.C. Zeeman in 1961 in modeling visual per-
ception with tolerances (Zeeman, 1962). 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 (i.e., for all x,y∈X, x = y implies y = x) but transitivity of = is not
required. The tolerance is directly related to the exact idea of closeness or resemblance
(i.e., being within some tolerance) in comparing objects. The basic idea is to find objects
such as images that resemble each other with a tolerable level of error. Sossinsky (Sossin-
sky1986, 1986) observes that the main idea underlying tolerance theory comes from Henri
Poincare (Poincare, 1913). Physical continua (e.g., measurable magnitudes in the physical
world of medical imaging (Hassanien et al., 2009)) are contrasted with the mathematical
continua (real numbers) where almost solutions are common and a given equation have no
exact solutions. An almost solution of an equation (or a system of equations) is an object
which, when substituted into the equation, transforms it into a numerical 'almost identity',
i.e., a relation between numbers which is true only approximately (within a prescribed tol-
erance) (Sossinsky1986, 1986). Equality in the physical world is meaningless, since it can
never be verified either in practice or in theory. Hence, the basic idea in a tolerance space
view of images is to replace the indiscerniblity relation in rough sets (Pawlak, 1982) with a
tolerance relation. The use of image tolerance spaces in this work is directly related to re-
cent work on tolerance spaces (see, e.g., (Hassanien et al., 2009; Peters, 2009c,b; Peters and
Ramanna, 2009; Bartol, Miro, Pioro, and Rossello, 2004; Gerasin, Shlyakhov, and Yakovlev,
2008; Schroeder and Wright, 1992; Shreider, 1970; Skowron and Stepaniuk, 1996; Zheng,
Hu, and Shi, 2005)).
Formally, the Perceptual Tolerance Nearness Relation is defined in Defn. 3.
Definition 3 Perceptual Tolerance Nearness Relation (Peters, 2009c)
LethO,
the tolerance relation = B
F
ibe a perceptual system and let
R
. For everyB⊆
F
is defined as follows:
= B,ε ={(x,y)∈O×O :kφ B (x)−φ B (y)k≤ε}.
, instead of = {φ} we write = φ . Further, for notational conve-
nience, we will write = B instead of = B, with the understanding that ε is inherent to the
definition of the tolerance relation.
IfB={φ}for some φ∈
F
As in the case with the perceptual indiscernibility relation, a tolerance class can be defined
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