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Treasure Trove at Banacha. Set Patterns in Descriptive Proximity Spaces

Peters J.F., Ramanna S.

Fundamenta Informaticae

2013

Vol. 127, nr 1-4

357--367

This paper introduces descriptive set patterns that originated from our visits with Zdzisław Pawlak and Andrzej Skowron at Banacha and environs in Warsaw. This paper also celebrates the generosity and caring manner of Andrzej Skowron, who made our visits to Warsaw memorable events. The inspiration for the recent discovery of descriptive set patterns can be traced back to our meetings at Banacha. Descriptive set patterns are collections of near sets that arise rather naturally in the context of an extension of Solomon Leader's uniform topology, which serves as a base topology for compact Hausdorff spaces that are proximity spaces. The particular form of proximity space (called EF-proximity) reported here is an extension of the proximity space introduced by V. Efremovič during the first half of the 1930s. Proximally continuous functions introduced by Yu.V. Smirnov in 1952 lead to pattern generation of comparable set patterns. Set patterns themselves were first considered by T. Pavlidis in 1968 and led to U. Grenander's introduction of pattern generators during the 1990s. This article considers descriptive set patterns in EF-proximity spaces and their application in digital image classification. Images belong to the same class, provided each image in the class contains set patterns that resemble each other. Image classification then reduces to determining if a set pattern in a test image is near a set pattern in a query image.

Nearness of Sets in Local Admissible Covers : Theory and Application in Micropalaeontology

Peters J.F.

Fundamenta Informaticae

2013

Vol. 126, nr 4

433--444

This paper considers the nearness of sets in local descriptive admissible covers of nonempty sets and the problem of quantifying the nearness of such sets. A brief review of descriptive Efremovič spaces as well descriptive intersection and union provides a foundation for the study of descriptive admissible covers. Descriptively near sets in admissible covers contain sequences of points with members having similar descriptions. The motivation for this approach stems from the need to consider fine-grained neighbourhoods of points in admissible covers that facilitate highly accurate measures of nearness of tiny parts of sets of objects of interest. A practical application of local admissible covers is given in terms of micropalaeontology and the detection of minute similarities and differences in microfossils, useful in the study of climate change, mineral and fossil fuel exploration.