Near sets
Comparison of feature vectors provides a basis for measuring the closeness of descriptively near sets.
Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as engineering and science problems.
Near sets have a variety of applications in areas such as topology[37], pattern detection and classification[50], abstract algebra[51], mathematics in computer science[38], and solving a variety of problems based on human perception[42][82][47][52][56] that arise in areas such as image analysis[54][14][46][17][18], image processing[40], face recognition[13], ethology[64], as well as engineering and science problems[55][64][42][19][17][18].
From the beginning, descriptively near sets have proved to be useful in applications of topology[37], and visual pattern recognition [50], spanning a broad spectrum of applications that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analysis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, and topological psychology.
The notions of near and far[A] in mathematics can be traced back to works by Johann Benedict Listing and Felix Hausdorff.
Poincaré, who introduced sets of similar sensations (nascent tolerance classes) to represent the results of G.T.
Fechner's sensation sensitivity experiments[10] and a framework for the study of resemblance in representative spaces as models of what he termed physical continua[63][60][61].
Later, F. Riesz introduced the concept of proximity or nearness of pairs of sets at the International Congress of Mathematicians (ICM) in 1908[65].
Sossinsky observed in 1986[71] that the main idea underlying tolerance space theory comes from Poincaré, especially[60].
In 2006, a formal approach to the descriptive nearness of objects was considered by J. Peters, A. Skowron and J. Stepaniuk[C] in the context of proximity spaces[39][33][35][21].
The exact idea of closeness or 'resemblance' or of 'being within tolerance' is universal enough to appear, quite naturally, in almost any mathematical setting (see, e.g.,[66]).
It is especially natural in mathematical applications: practical problems, more often than not, deal with approximate input data and only require viable results with a tolerable level of error[71].
The words near and far are used in daily life and it was an incisive suggestion of F. Riesz[65] that these intuitive concepts be made rigorous.
1 are considered near each other, since these ovals contain pairs of classes that display similar (visually indistinguishable) colours.
The following EF-proximity[G] space axioms are given by Jurij Michailov Smirnov[67] based on what Vadim Arsenyevič Efremovič introduced during the first half of the 1930s[8].
For example, this leads to a proximal view of sets of picture points in digital images.
is directly related to the idea of closeness or resemblance (i.e., being within some tolerance) in comparing objects.
By way of application of Poincaré's approach in defining visual spaces and Zeeman's approach to tolerance relations, the basic idea is to compare objects such as image patches in the interior of digital images.
Finally, let the description of an object be given by the Green component in the RGB color model.
Using this information, tolerance classes can be formed containing objects that have similar (within some small
For example, the figure accompanying this example shows a subset of the tolerance classes obtained from two leaf images.
Each colour in the figures corresponds to a set where all the objects in the class share the same description.
is that the nearness of sets in a perceptual system is based on the cardinality of tolerance classes that they share.
It was motivated by a need for a freely available software tool that can provide results for research and to generate interest in near set theory.
The system implements a Multiple Document Interface (MDI) where each separate processing task is performed in its own child frame.
The system was written in C++ and was designed to facilitate the addition of new processing tasks and probe functions.
The Proximity System grew out of the work of S. Naimpally and J. Peters on Topological Spaces.
The Proximity System was written in Java and is intended to run in two different operating environments, namely on Android smartphones and tablets, as well as desktop platforms running the Java Virtual Machine.
With respect to the desktop environment, the Proximity System is a cross-platform Java application for Windows, OSX, and Linux systems, which has been tested on Windows 7 and Debian Linux using the Sun Java 6 Runtime.
In terms of the implementation of the theoretical approaches, both the Android and the desktop based applications use the same back-end libraries to perform the description-based calculations, where the only differences are the user interface and the Android version has less available features due to restrictions on system resources.
