The problem considered in this chapter is how to discover perceptual granules that are in some sense near each other. One approach to the solution to the problem of discovering perceptual granules close to each other comes from near set theory. This is made clear in this chapter by considering various nearness relations that define coverings of sets of perceptual objects that are near each other. A perceptual granule is something that is graspable by the senses or by the mind. Every perceptual granule is represented by a set of perceptual objects that have their origin in the physical world. This means that a perceptual granule does not include the empty set. Hence, each family of perceptual granules is a dual chopped lattice. Both perceptual near sets and tolerance near sets are presented in this chapter. Both the theory and applications of perceptually near granules are presented in this chapter.
Top1. Introduction
The better theory is the more precise description of the [object] it provides. –Ewa Orlowska, Studia Logica, 1990.
The main problem considered in this chapter is how to discover perceptual granules that are in some sense close to each other. An approach to the solution to this problem comes from near set theory (see, e.g., Peters (2007c, 2007d), Peters and Wasilewski (2009), Peters (2010)). Perceptual objects that have similar appearance are considered perceptually near each other, i.e., perceived objects that have matching or, at least, similar descriptions. A description is a tuple of values of functions representing perceptual object features.
Near set theory provides a basis for observation, comparison and classification of perceptual granules. That is, this article considers relations between perceptual granules that are near sets. A perceptual granule is a set of perceptual objects originating from observations of objects in the physical world. Near sets are disjoint sets that resemble each other. Sets of perceptual objects where two or more of the elements objects have similar descriptions are penultimate near sets. 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 physical objects carried out in cooperation with Zdzisław Pawlak in (Pawlak and Peters, 2002). This initial work led to the introduction of near sets (Peters, 2007b), elaborated in (Peters, 2007c), (Peters and Wasilewski, 2008), including tolerance near sets (Peters, 2009a), (Peters, 2010), (Pal & Peters (2010) having strong affinities with proximity space theory (Naimpally & Warrack (1970)), . Perceptual granules are markedly different from classical information granules. Basically, a perceptual granual is extracted from a physical continuum based on perception. By contrast, a classical information granule (at least in the rough set approach to information granulation) is extracted from an information table. The main idea for perceptual granules and perceptually near granules comes from Henri Poincaré (Poincare, 1905). The physical continuum (with elements that are sets of sensations such as light intensities in a visual space)are contrasted with the mathematical continuum (real numbers) where almost solutions are common and given equations 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 tolerance) (Sossinsky, 1986). The idea of an almost solution leads to the creation of tolerance perceptually near granules that are introduced in this article. A perception-based approach to discovering resemblances between disjoint sets of perceptual objects such as digital images leads to a tolerance space form of near sets that models human perception in a physical continuum (Peters, 2009a, 2010).