Perceptually near pawlak partitions

The problem considered in this paper is how to compare perceptually indiscernible partitions of disjoint, non-empty sets such as pairs of digital images viewed as sets of points. Such partitions are called perceptual Pawlak partitions, named after Z. Pawlak, who introduced a attribute-based equivalence relation in 1981 (the well-known indiscernibility relation from rough set theory). The solution to the problem stems from an approach to pairwise comparison reminiscent of the G. Fechner's 1860 approach to comparing perceptions in psychophysics experiments. For Fechner, one perception of an object is indistinguishable from the perception of a different object, if there is no perceptible difference in the particular sensed feature value of the objects, e.g., perceptions resulting from lifting small objects where the object feature is weight. In comparing visual perceptions, partitions of images determined by a particular form of indiscernibility relation ∼B are used. The L1 (Manhattan distance) norm form of what is known as a perceptual indiscernibility relation defined within the context of a perceptual system is used in this article to define what are known as perceptually indiscernible Pawlak partitions (PIPs). An application of PIPs and near sets is given in this article in terms of a new form of content-based image retrieval (CBIR). This article investigates the efficacy of perceptual CBIR using Hausdorff and Mahalanobis distance measures to determine the degree of correspondence between pairs of perceptual Pawlak partitions of digital images. The contribution of this article is the introduction of an approach to comparing perceptually indiscernible image partitions.