Tolerance spaces: Origins, theoretical aspects and applications

Abstract

This article considers the origins, theoretical aspects and applications of tolerance spaces. In terms of the origin of tolerance spaces, this article calls attention to the seminal work by J.H. Poincaré (1854–1912) and E.C. Zeeman (1925–) on establishing the foundations for tolerance spaces. During the period from 1895 to 1912, Poincaré introduced sets of sensations and sequences of almost the same sensations as a means of characterizing the physical spectrum. The perception of physical objects that are almost the same leads to a tolerance space view of visual perception as well as other forms of perception such as touch and sound. Roughly 60 years later (in 1962), Zeeman formally introduced the notion of a tolerance space as a useful means of investigating a geometry of visual perception. In addition to the general theory of tolerance spaces, this article also carries forward earlier work on perceptual tolerance relations and considers the resemblance (nearness) between tolerance spaces. From an information systems point of view, it can be observed that tolerance spaces have proved to be fruitful in a number of research areas. Evidence of the utility of tolerance spaces in information systems can be seen in the introduction of tolerance rough sets, tolerance approximation spaces, and tolerance near sets. The contribution of this article is an overview of tolerance spaces considered in the context of visual perception and a presentation of a formal basis for the study of perceptual tolerance spaces.

Introduction

The idea of resemblance between objects is formalized with the notion of a tolerance space. In 1975, Shreider published a study of tolerance on a set in formalizing the notion of resemblance [45] (see, also, [58], [49], [46], [52], [54]). The exact idea of closeness or resemblance is universal enough to appear quite naturally in almost any mathematical setting [49, Section 2]. More precisely, a tolerance on a set is mathematical structure that formalises the idea of resemblance, i.e., the idea of being the same within some tolerance. Put another way, objects are considered near each other up to a small, allowable error. It is often the case that practical problems deal with approximate input data and solutions only require results with an acceptable level of error.

This article considers the origins, theoretical aspects and one of the principal applications of perceptual tolerance spaces. A perceptual tolerance space formalizes the study of resemblance between perceptual objects. The basic idea for tolerance spaces (also termed sensible spaces) was introduced by J.H. Poincaré in a series of articles and books published over a 10 year period, starting during the early 1890s, leading up to his view of physical continua [34] and his introduction of his analysis situs [33] (both works appearing in 1895). The Latin term analysis situs (analysis of place) coined by Poincaré provided a basis for what is now known as topology (from the Greek word τóπoς or topos, ‘place’, and logia, ‘study‘, i.e., study of (properties of) place). Poincaré’s notion of a sensible space and what he called a physical continuum derives from his interest in the interpretation of Fechner’s 1860 notion of sensory circles and measurement of sensitivity to changes in external stimuli [4], [3], mainly inspired by Weber’s experiments with human sensation [56].

During the period from the early 1880s, Poincaré introduced what he termed a representative space,¹ l’espace représentatif [34] of our daily experience that is a physical continuum with elements that are sets of sensations (see, e.g., [29], [32], [30], [31], [40]). For Poincaré, a representative space is a model for a physical continuum. The study of human sensation and sensitivity to stimuli led to the Weber-Fechner law: subjective sensation is proportional to the log of stimulus intensity.

Fechner’s experiments led to the formulation of a representative (sensed) space by Poincaré in a number of different books and articles (see, e.g., [29], [41], [42], [32], [30], [34], [40]). Poincaré’s representative space as a physical continuum (pc)² consists of linearly ordered sets of sensations, the elements of a pc. For example, assume a pc consists of two elements X, Y such that pc = {X, Y}. From linear ordering of pc elements X, Y that are sets of similar sensations, for all sensations x ∈ X and sensations y ∈ Y, it is the case that x < y. Implicit in this view of a pc is an underlying similarity relation that determines the members of a perceptual view of the world (sets of similar sensations). Such a similarity relation is perception-based and is characterized by term perceptually indistinguishable.

The notion of physical continuum and various tolerance spaces (tactile, visual, motor spaces) were introduced by Poincaré in an 1894 article on the mathematical continuum [42], an 1895 article on space and geometry [34] and a compendious 1902 book on science and hypothesis [29] followed by a number of elaborations, e.g., [40]. The 1893 and 1895 articles on continua (Pt. 1, ch. II) as well as sensed spaces and geometry (Pt. 2, ch IV) are included as chapters in [29].

A qualitative as opposed to a quantitative form of geometry called Analysis Situs (topology) was introduced by Poincaré in 1895 [33], followed by a succession of five supplements [35], [36], [37], [38], [39] (the 5th supplement was published in 1903). His Analysis situs was very much on Poincaré’s mind during the same timeframe when he introduced the distinction between representative space that are non-homogeneous, non-isotropic and geometric space that are homogeneous, isotropic. His 1902 monograph on science and hypothesis [29] (largely a compilation of previously published articles by Poincaré) contains the most detailed presentation of what have come to be known as tolerance spaces. Roughly 60 years after Poincaré’s initial work on representative spaces, Zeeman formalized the notion of a distance-based tolerance space as a means of characterizing visual perception [58], and, in particular, in [59].

The exposition of Poincaré’s sensed (aka tolerance) spaces in this article is mainly based on [29] and Zeeman’s formalization of the notion of tolerance spaces in [58]. A consideration of Poincaré view of the relation between physical and mathematical continua, the validity of Poincaré’s notion of a physical continuum, and Zeeman’s topology of the brain is outside the scope of this article. Instead, in pursuit of the origin of the notion of a tolerance space and its application, the focus in this article is on Poincaré sensed spaces and their formalization thanks to Zeeman’s work during the early 1960s.

From an information systems point of view, one can observe that tolerance spaces have proved to be fruitful in a number of research areas. Evidence of the utility of tolerance spaces in information systems can be seen in the introduction of tolerance rough sets [17], [14], [43], tolerance approximation spaces [46] (the results from 1996 have recently been extended in [50]), and tolerance near sets [27], [18] as well as recent studies of similarity relations (see, e.g., [52]) and their application (see, e.g., [9], [18], [25]) or applications of tolerance relations in granular computing (see, e.g., [47], [55]). In this article, the application of tolerance spaces is given in terms of near sets [24], [23], [28] and image analysis, e.g., [7], [18], [25], [26]. The contribution of this article is an overview of tolerance spaces considered in the context of visual perception.

This article has the following organization. The preliminaries concerning tolerance relations are given in Section 2. As a prelude to the remaining sections of this article, tolerance spaces in modelling visual spaces are briefly discussed in Section 3, where visual spaces are introduced in Section 3.2 and Zeeman’s view of visual spaces is presented in Section 3.3. This is followed by discussion of Zeeman’s tolerance between tolerance spaces in Section 4. Then an introduction to perceptual tolerance relations is presented in Section 5. Measurement of the resemblance between tolerance spaces is covered in Section 6.