Applications of Near Sets

Near sets are disjoint sets that resemble each other. Resemblance is determined by considering set descriptions defined by feature vectors (n-dimensional vectors of numerical features that represent characteristics of objects such as digital image pixels). Near sets are useful in solving problems based on human perception [44, 76, 49, 51, 56] that arise in areas such as image analysis [52, 14, 41, 48, 17, 18], image processing [41], face recognition [13], ethology [63], as well as engineering and science problems [53, 63, 44, 19, 17, 18]. As an illustration of the degree of nearness between two sets, consider an example of the Henry color model for varying degrees of nearness between sets [17, §4.3]. The two pairs of ovals in Figures 1 and 2 contain colored segments. Each segment in the figures corresponds to an equivalence class where all pixels in the class have matching descriptions, i.e., pixels with matching colors. Thus, the ovals in Figure 1 are closer (more near) to each other in terms of their descriptions than the ovals in Figure 2. It is the purpose of this article to give a bird’s-eye view of recent developments in the study of the nearness of sets.