Set coverings and tolerance relations

Gerasin, S. N.; Shlyakhov, V. V.; Yakovlev, Sergiy

doi:10.1007/s10559-008-9007-y

Cited by 18 publications

(5 citation statements)

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“…In general, this family does not coincide with the family A/ρ of "tolerance" classes of ρ : {(x] : x ∈ A}, where for any x ∈ A, (x] = {y ∈ A : (y, x) ∈ ρ} (in several papers, e.g., [3], [4], the elements from M ax(ρ) are just called tolerance classes while in the others, for example [2], these elements are said to be blocks of the tolerance ρ). The connections between both families are given in the following proposition.…”

Section: Corollary 12 the Complete Lattice (T ⊆) Of All Tolerance mentioning

confidence: 99%

Quasiorders, Tolerance Relations and Corresponding “Partitions”

Nowak

2016

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The paper deals with a generalization of the notion of partition for wider classes of binary relations than equivalences: for quasiorders and tolerance relations. The counterpart of partition for the quasiorders is based on a generalization of the notion of equivalence class while it is shown that such a generalization does not work in case of tolerances. Some results from [5] are proved in a much more simple way. The third kind of “partition” corresponding to tolerances, not occurring in [5], is introduced.

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Section: Corollary 12 the Complete Lattice (T ⊆) Of All Tolerance mentioning

confidence: 99%

Quasiorders, Tolerance Relations and Corresponding “Partitions”

Nowak

2016

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“…non‐empty intersections between TCs. The use of tolerance spaces in this work is directly related to recent work on tolerance spaces (Hassanien et al , 2009; Peters, 2009a, b, 2010; Peters and Ramanna, 2009; Gerasin et al , 2008; Zheng et al , 2005; Bartol et al , 2004; Skowron and Stepaniuk, 1996; Schroeder and Wright, 1992; Shreider, 1970; Pal and Peters, 2010).…”

Section: Definitionmentioning

confidence: 99%

Perception‐based image classification

Henry

Peters

2010

International Journal of Intelligent Computing and Cybernetics

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Pattern classification methodologies are present in many systems that we depend on daily. In these systems, classes are created based on human perception of the objects being classified. Thus, it is important to have systems that accurately model human perception. Near set theory provides a framework for measuring the similarity of objects based on features that describe them in much the same way that humans perceive objects. In this paper, we show that the near set approach can be used to classify images. Further, the results presented here suggest that the near set approach can be used in any image classification system. The contribution of this article is a perception based classification of images using near sets.

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“…Hence, the basic idea in a tolerance space view of images, for example, is to replace the indiscerniblity relation in rough sets [31] with a tolerance relation in partitioning images into homologous regions where there is a high likelihood of overlaps, i.e., non-empty intersections between image tolerance classes. The use of image tolerance spaces in this work is directly linked to recent work on tolerance spaces [9,11,13,38,39,43,[52][53][54]59]. The contribution of this article is the introduction of two new tolerance space-based image resemblance measures and a comparison of the new measures with the original Henry-Peters nearness measure.…”

Section: Tolerance Space For Image Recognitionmentioning

confidence: 99%

“…Let O, F be a perceptual system and let ε ∈ ℜ (set of all real numbers). For every B ⊆ F the perceptual tolerance relation B,ε is defined in (11).…”

Section: Definition 1 Perceptual Tolerance Relation [43]mentioning

confidence: 99%

Anisotropic Wavelet-Based Image Nearness Measure

Peters¹,

Puzio²

2009

IJCIS

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The problem considered in this article is how to solve the image correspondence problem in cases where it is important to measure changes in the contour, position, and spatial orientation of bounded regions. This article introduces a computational intelligence approach to the solution of this problem with anisotropic (direction dependent) wavelets and a tolerance near set approach to detecting similarities in pairs of images. Near sets are a recent generalization of rough sets introduced by Z. Pawlak during the early 1980s. Near sets resulted from a study of the perceptual basis for rough sets. Pairs of sets containing objects with similar descriptions are known as near sets. The proposed wavelet-based image nearness measure is compared with F. Hausdorff and P. Mahalanobis image distance measures. The results of three wavelet-based image resemblance measures for several well-known images, are given. A direct benefit of this research is an effective means of grouping together (classifying) images that correspond to each other relative to minuscule similarities in the contour, position, and spatial orientation of bounded regions in the images, especially in videos containing image sequences showing varied object movements. The contribution of this article is the introduction of an anisotropic wavelet-based measure of image resemblance using a near set approach.

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Set coverings and tolerance relations

Abstract: Properties of tolerance relations and tolerance classes are studied. Tolerance classes are demonstrated to form carrier set coverings. The conditions are proved under which tolerance and equivalence classes coincide.

Cited by 18 publications

References 1 publication

Quasiorders, Tolerance Relations and Corresponding “Partitions”

Quasiorders, Tolerance Relations and Corresponding “Partitions”

Perception‐based image classification

Anisotropic Wavelet-Based Image Nearness Measure

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