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Near sets are disjoint sets that resemble each other. Resemblance between disjoint sets occurs whenever there are observable similarities between the objects in the sets. Similarity is determined by comparing lists of object feature values. Each list of feature values defines an object's description. Comparison of object descriptions provides a basis for determining the extent that disjoint sets resemble each other. Objects that are perceived as similar based on their descriptions are grouped together. These groups of similar objects can provide information and reveal patterns about objects of interest in the disjoint sets. For example, collections of digital images viewed as disjoint sets of points provide a rich hunting ground for near sets.
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Near set theory provides methods that can be used to extract resemblance information from objects contained in disjoint sets, i.e., it provides a formal basis for the observation, comparison, and classification of objects. The discovery of near sets begins with choosing the appropriate method to describe observed objects. This is accomplished by the selection of probe functions representing observable object features. A probe function is a mapping from an object to a real number representing a feature value. For example, when comparing fruit such as apples, the redness of an apple (observed object) can be described by a probe function representing colour, and the output of the probe function is a number representing the degree of redness (or whatever colour apple you prefer to eat). Probe functions provide a basis for describing and discerning affinities between objects as well as between groups of similar objects. Objects that have, in some degree, affinities are considered near each other. Similarly, groups of objects (i.e. sets) that have, in some degree, affinities are also considered near each other.
Near sets offer an ideal framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as engineering and science problems. In near set theory, perception is a combination of the view of perception in psychophysics with a view of perception found in Merleau-Ponty's work. In the context of psychophysics, perception of an object (i.e., in effect, our knowledge about an object) depends on signal values gathered by our senses. In this view of perception, our senses are likened to probe functions by considering them as mappings of stimuli to sensations that are a source of values assimilated by the mind. A human sense modelled as a probe measures observable physical characteristics of objects in our environment. The sensed physical characteristics of an object are identified with object features. In Merleau-Ponty's view, an object is perceived to the extent that it can be described. In other words, object description goes hand-in-hand with object perception. It is our mind that identifies relationships between object descriptions to form perceptions of sensed objects. It is also the case that near set theory has been proven to be quite successful in finding solutions to perceptual problems such as measuring image correspondence and segmentation evaluation.
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