Applications of near sets (Q2883467)

This is an introductory survey paper in which the authors deal with nearness of sets in the sense of resemblance of features of the sets. It explains, in particular, the first author's approach [Appl. Math. Sci., Ruse 1, No. 53--56, 2609--2629 (2007; Zbl 1135.68570); Fundam. Inform. 75, No. 1--4, 407--433 (2007; Zbl 1108.68119)] to describe resemblance of objects in areas such as image analysis, image processing and pattern recognition.NEWLINENEWLINEAfter a brief history of concepts of nearness, from those given by Poincaré and Riesz around 1900 to current ones, the authors give a more conceptual account of nearness of sets leading straightforwardly to the notions of metric proximity (Čech) and proximity spaces in general. Near sets are then defined within proximity spaces. Furthermore, the authors describe how tolerance relations defined by real-valued feature functions may be added to proximity spaces in order to get a notion of nearness of objects relying on the numerical comparison of features of these objects: two sets (objects, images etc.) are near to each other if the numerical values of their features differ only within a given tolerance \(\varepsilon > 0\).NEWLINENEWLINEFinally, the authors discuss some examples that illustrate the great variety of possible features that may be assigned to pictures: not only those ones occurring naturally when dealing with digital images (such as colour, grey level etc.) but also quite different features, e.g. such ones describing what people depicted in a picture are doing. This way the reader gets a good impression of the wide applicability of the presented approach of near sets, e.g. in the areas mentioned at the start of this review.NEWLINENEWLINEThis is a nice (and nicely illustrated), short paper that introduces the reader in a pleasant way to the concept of near sets and their applications to, particularly, digital image representation and processing.