The article calls attention to the relationship between neighbourhoods and tolerance classes in the foundations of tolerance near sets. A particular form of tolerance relation is given by way of introduction to descriptively "-near sets. Neighbourhoods and tolerance classes have practical applications in digital image analysis.http://www.m-hikari.com/ams/ams-2011/ams-33-36-2011/henryAMS33-36-2011.pd

Henry, Christopher J.

Applied Mathematical Sciences

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Applied Mathematical Sciences, Vol. 5, 2011, no. 35, 1727 - 1732Neighbourhoods, Classes and Near SetsChristopher J. HenryDepartment of Applied Computer ScienceUniversity of WinnipegWinnipeg, Manitoba R3B 2E9, Canadach.henry@uwinnipeg.caAbstractThe article calls attention to the relationship between neighbour-hoods and tolerance classes in the foundations of tolerance near sets.A particular form of tolerance relation is given by way of introductionto descriptively ε-near sets. Neighbourhoods and tolerance classes havepractical applications in digital image analysis.Mathematics Subject Classification: 03E15, 11K55, 54E17, 54E40Keywords Description, neighbourhood, near sets, tolerance class1 IntroductionThe paper distinguishes between neighbourhoods and tolerance classes by wayof introduction to the foundations for tolerance near sets. This work is directlyrelated to recent work on near sets, in general, and tolerance near sets, inparticular (see, e.g., [1, 2, 4, 7, 12]) as well as recent work on distance functionscalled merotopies and approach spaces (see, e.g. [10]).2 PreliminariesTo introduce near set theory it is necessary to establish a basis for describingelements of sets. All sets in near set theory consist of perceptual objects.Definition 1. Perceptual Object. A perceptual object is something perceiv-able that has its origin in the physical world.A perceptual object is anything in the physical world with characteristics ob-servable to the senses such that they can be measured and are knowable to themind. Examples of perceptual objects include patients, components belonging1728 C. Henryto a manufacturing process, and camera images. Here, the term perception isconsidered relative to measurable characteristics called the object’s features.In keeping with the approach to pattern recognition suggested by M. Pavel[5], the features of an object are quantified by probe functions.Definition 2. Probe Function [6]. A probe function is a real-valued functionrepresenting a feature of a perceptual object.In this work, probe functions are defined in terms of digital images suchas: colour, texture, contour, spatial orientation, and length of line segmentsalong a bounded region. Specifically, objects in our visual field are viewedin terms of feature vectors that consist of extracted probe function values.Selected probe functions are used to measure characteristics of visual objectsand provide a basis for detecting similarities among perceived objects. Put~φ(x) = (φ1(x), ..., φn(x)) for a feature vector in Rn that describes an objectx. In sum, probe functions make it possible to determine if two objects areassociated with the same pattern without necessarily specifying which pattern(as is the case when performing classification).Next, a perceptual system is a set of perceptual objects, together with aset of probe functions.Definition 3. Perceptual System [7]. A perceptual system 〈O,F〉 consistsof a nonempty set O of sample perceptual objects and a nonempty set F ofreal-valued functions φ ∈ F such that φ : O → R.3 Neighbourhoods and Tolerance ClassesDefinition 4. Neighbourhood. Let 〈O,F〉 be a perceptual system, x ∈ Oand let ρ : O × O → R denote a distance function, e.g., ρ(x, y) = |x − y|(standard distance). For a set B ⊆ F and ε ∈ [0,∞], a closed neighbourhoodof x (denoted by N(x)) is defined byNρ(x, ε) = {y ∈ O : ρ(x, y) = |x− y| ≤ ε},with centre x and radius ε. For simplicity when the radius and distance func-tion ρ are understood, we write N(x). An example of a neighbourhood (alsoconcisely written nbd) in a 2D feature space is given in Fig. 1.1, where the po-sitions of the objects are given by the numbers 1 to 21, and the nbd is definedwith respect to the object labelled 1. That is, the nbd is the set of all theobjects within the circle. Notice that the distance between all the objects andobject 1 in Fig. 1.1 is less than or equal to ǫ = 0.1, and that the distances forall pairs of objects in the nbd of x = 1 are not necessarily less than or equalto ǫ = 0.1.Neighbourhoods, classes and near sets 17290.45 0.5 0.55 0.6 0.650.20.250.30.350.4Normalized featureNormalized featureε = 0.11234567891011121314151617181920211.1: Nbd & toleranceclass in 2D space1.2: Coverings with nocorrespondence1.3: Covering usingDef. 8Figure 1: Visualisation of Neighbourhoods & Tolerance ClassesDefinition 5. Perceptual Tolerance Relation [8, 9]. Let 〈O,F〉 be a per-ceptual system and let ε ∈ R. For every B ⊆ F, the perceptual tolerancerelation ∼=B,ǫ is defined as follows:∼=B,ε= {(x, y) ∈ O ×O :‖ ~φ(x)− ~φ(y) ‖≤ ε},Next, pre-classes are distinguished from tolerance classes.Definition 6. Pre-Class. Let 〈O,F〉 be a perceptual system. For B ⊆ F andε ∈ R, a set X ⊆ O is a pre-class iff x ∼=B,ǫ y for any pair x, y ∈ X.Definition 7. Tolerance Class. A maximal pre-class with respect to inclu-sion is called a tolerance class.The set coloured orange in Fig. 1.1 is a tolerance class, since no objects canbe added to the set and still satisfy the condition that any pair x, y ∈ Orange× Orange must be within ε of each other.Theorem 1. [1] All tolerance classes containing x ∈ O are subsets of theneighbourhood N(x).Proof. Given in [1].Next, observe that objects can belong to more than one tolerance class.Consequently, the following notation is required to differentiate between classesand facilitate discussions in subsequent sections. The set of all tolerance classesusing only the objects in O is given by H∼=B,ε(O) (also called the cover ofO), a single tolerance class is represented by C ∈ H∼=B,ε(O), and the set ofall tolerance classes containing an object x is denoted by Cx ⊂ H∼=B,ε(O).This section concludes with another tolerance relation similar to the weakindiscernibility relation [6].1730 C. HenryDefinition 8. Weak Perceptual Tolerance Relation [8] Let 〈O,F〉 be aperceptual system and let ε ∈ R. For every B ⊆ F the weak perceptual tolerancerelation ≅B,ǫ is defined as follows:≅B,ǫ= {(x, y) ∈ O ×O : ∃φi ∈ B  ‖ φi(x)− φi(y) ‖2≤ ε}.The weak tolerance relation can provide new information or relationships fora set of objects for a given application.4 Tolerance Near SetsSets of objects that have similar descriptions are called near sets, and a methodfor determining similarity is provided by way of the perceptual tolerance re-lation (and to a lesser degree with the weak perceptual tolerance relation).The following two definitions enunciate the fundamental notion of nearnessbetween two sets.Definition 9. Tolerance Nearness Relation [8, 9]. Let 〈O,F〉 be a percep-tual system and let X, Y ⊆ O, ε ∈ R. A set X is near to a set Y within theperceptual system 〈O,F〉 (X⊲⊳FY ) iff there exists x ∈ X and y ∈ Y and thereis B ⊆ F such that x ∼=B,ǫ y.Definition 10. Tolerance Near Sets [8, 9]. Let 〈O,F〉 be a perceptual sys-tem and let ε ∈ R,B ⊆ F. Further, let X, Y ⊆ O, denote disjoint sets with cov-erings determined by the tolerance relation ∼=B,ǫ, and let H∼=B,ε(X), H∼=B,ε(Y )denote the set of tolerance classes for X, Y , respectively. Sets X, Y are toler-ance near sets iff there are tolerance classes A ∈ H∼=B,ε(X), B ∈ H∼=B,ε(Y ) suchthat A ⊲⊳FB.Defs. 9 and 10 can be summarised in a single theorem.Theorem 2. The following assertions are equivalent.1. X, Y are tolerance near sets,2. There are A ∈ H∼=B,ε(X), B ∈ H∼=B,ε(Y ) such that A ⊲⊳F B,3. There are x ∈ X, y ∈ Y , B ⊆ F such that x ∼=B,ǫ y.Proof. Immediate from Def. 9 and Def. 10.Definition 9 fits nicely in applications that emphasise objects (see, e.g., [1,2]), while Definition 10 works best in applications that focus on toleranceclasses. Also, notice that tolerance near sets are a variation of the original nearsets using the indiscernibility relation [6]. Moreover, the original definition ofNeighbourhoods, classes and near sets 1731tolerance near sets given in [8, 9] defines nearness in terms of pre-classes (asopposed to tolerance classes as given in Definition 10), however the resultspresented in [1] are obtained using tolerance classes, and so the definition wasadjusted accordingly. Finally, an example of tolerance near sets is given inFig. 1.3, where the colours represent different tolerance classes, and classeswith the same colour represent the situation where A ⊲⊳FB. By contrast, theovals in Fig. 1.2 are not tolerance near sets.0 20 40 60 80 1000102030405060708090100110Query image 506.jpgRecall %Precision %  tNM ε = 0.7Figure 2: Precision vs. recall plot using tNM to perform CBIR on the SIM-PLcity image database with query image 506 [11]3.1: 3.2: 3.3: 3.4: 3.5:Figure 3: Top 5 query results (query image left).5 Application Using Near SystemA direct application of Defs. 7, 9, & 10, by way of the nearness measure tNM(see [1]), is in the area of Content-Based Image Retrieval (CBIR). In thisproblem domain, the goal is to retrieve all the images that are similar to aquery image based on the content of the image (rather than retrieval based onthe strings associated with the image). To perform CBIR, a nearness measureis used to quantify the number of tolerance classes shared between disjointsets obtained from two different images. This measure can then be used torank all the images in a databased based on the query image. Finally, if thedatabase is partitioned into categories, precision vs. recall plots can be used todemonstrate the ability of this approach to correctly retrieve images belongingto the same category as the query image. This approach was implemented in1732 C. Henrythe near system, introduced in [3], and the results are given in the followingFig. 2 & 3.References[1] C. Henry, Near Sets: Theory and Applications, Ph.D. Diss., supervisor:J.F. Peters, Dept. Elec. & Comp. Engg., U. of Manitoba, WPG, MB,Canada, 2010.[2] C. Henry, J.F. Peters, Perception-based image classification, Int. J. Intel.Comp. & Cyb., 3 no. 3 (2010), 410-430.[3] C. Henry, Near set Evaluation And Recognition (NEAR) System, in S.K. Pal and J. F. Peters, Eds., Rough Fuzzy Analysis Foundations andApplications, CRC Press, Taylor & Francis Group, 2010, 7-1 - 7-22.[4] S.A. Naimpally, Near and far. A centennial tribute to Frigyes Riesz,Siberian Electronic Mathematical Reports, 2 (2009), 144-153.[5] M. Pavel, Fundamentals of Pattern Recognition, Marcel Dekker, Inc., NY,1993.[6] J.F. Peters, Near sets. General theory about nearness of objects, AppliedMathematical Sciences, 1 no. 53 (2007), 2609-2629.[7] J. F. Peters and P. Wasilewski, Foundations of near sets, Inf. Sci., 179 no.18 (2009), 30913109.[8] J.F. Peters, Tolerance near sets and image correspondence, Int. J. Bio-Inspired Comp., 1 no. 4 (2009), 239-245.[9] J.F. Peters, Corrigenda and addenda: Tolerance near sets and image cor-respondence, Int. J. Bio-Inspired Comp., 2 no. 5 (2010), 310-318.[10] S. Tiwari, Some Aspects of General Topology and Applications. ApproachMerotopic Structures and Applications, Ph.D. thesis, Supervisor: M.Khare, Mathematics Dept., Allahabad Univ., 2010, vii + 112 pp.[11] J. Z. Wang, J. Li, and G. Wiederhold, SIMPLcity: Semantics-sensitiveintegrated matching for picture libraries, IEEE Transactions on PatternAnalysis and Machine Intelligence, 23 no. 9 (2010), 947-963.[12] M. Wolski, Perception and classification. A note on near sets and roughsets, Fundamenta Informaticae, 101 (2010), 143155.Received: December, 2010

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