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have similar descriptions.
Our classifications are often plainly influenced by
chains of a nities.
-Charles Darwin, 1859.
Related tolerance spaces have isomorphic set theories.
[Let ξ, η denote tolerance relations, then]
The right visual field (X, ξ) and the right visual lobe (X, η).
have isomorphic set theories.
-Sir E. Christopher Zeeman, 1962.
In general, the term a nity means close relationship based on a common origin or struc-
ture (Murray, Bradley, Craigie, and Onions, 1933). In this chapter, the term a nity means
close relationship between perceptual granules (particularly images) based on common de-
scription (Peters and Ramanna, 2009). In particular, this chapter considers a tolerance
near set solution to the image correspondence problem (Peters, 2009, 2010), i.e., where one
uses image matching strategies to establish a correspondence between one or more images.
Recently, it has been shown that near sets can be used in a perception-based approach to
discovering correspondences between images (see, e.g., (Henry and Peters, 2009c; Peters,
2009, 2010; Peters and Ramanna, 2009; Peters and Puzio, 2009; Henry and Peters, 2009a;
Meghdadi et al., 2009)). Sets of perceptual objects where two or more of the objects have
matching descriptions are called near sets. Detecting image resemblance and formulating
image description are part of the more general pattern recognition process enunciated by
K. Cyran and A. Mrozek in 2001 (Cyran and Mrozek, 2001). Work on a basis for near sets
began in 2002, motivated by image analysis and inspired by a study of the perception of the
nearness of perceptual objects carried out in cooperation with Z. Pawlak in (Pawlak and
Peters, 2002,2007) and is directly related to the more general setting of rough sets (Pawlak,
1981; Pawlak and Skowron, 2007c,b,a), especially if one considers approximation spaces (Pe-
ters, Skowron, and Stepaniuk, 2007, 2006). This initial work led to the introduction of near
sets (Peters, 2007b), elaborated in (Peters, 2007a, 2009, 2010; Peters and Wasilewski, 2009;
Peters and Puzio, 2009; Peters and Henry, 2009; Henry and Peters, 2009b). A perception-
based approach to discovering resemblances between images leads to a tolerance space form
of near sets that models human perception in a physical continuum.
The term tolerance space was coined by E.C. Zeeman in 1961 in modelling visual percep-
tion 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
(for all x, y∈X, x'y and x∼y) but transitivity of'is not required. For example,
it is possible to define a tolerance space relative to subimages of an image. This is made
possible by assuming that each image is a set of fixed points. Let O denote a set of percep-
tual objects (e.g., grey level subimages) and let gr(x) = average grey level of subimage x.
Then the tolerance relation' gr is defined as' gr ={(x, y)∈O×O||gr(x)−gr(y)|≤ε}
for some tolerance ε∈<(reals). Then (O,' gr ) is a sample tolerance space. A tolerance
threshold denoted by ε 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 (Sossinsky, 1986) observes that 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, Abraham, Peters, Schaefer, and Henry, 2009))
are contrasted with the mathematical continua (real numbers) where almost solutions are
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