Applied Mathematical Sciences, Vol. 5, 2011, no. 2, 73 - 78
Metric Spaces for Near Sets James F. Peters Department of Electrical and Computer Engineering University of Manitoba Winnipeg, Manitoba R3T 5V6, Canada
[email protected] Abstract The problem considered in this paper is the measurement of nearness between disjoint sets that resemble each other. Such sets are descriptively near sets. The solution to the problem results from the introduction of a normed feature space containing n-dimensional feature vectors of numbers representing descriptions of objects of interest and a metric that measures the distance between feature vectors. The nearness of disjoint sets is determined by measuring the distance between set element descriptions represented by feature vectors. An application of the proposed approach to measuring the nearness of sets is given in terms of the correspondence between digital images.
Mathematics Subject Classification: 03E15, 11K55, 54E17, 54E40 Keywords Descriptively near, feature space, metric space, near set, norm
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Introduction
The problem considered in this paper is the measurement of distances between disjoint sets that resemble each other. Such sets are descriptively near each other [10]. A metric space is introduced to define the gap distance [5] between feature vectors in a normed space representing descriptions of objects in disjoint sets. This makes it possible to quantise the nearness of disjoint sets. Such a metric space makes it possible to consider a nearness relation between disjoint sets that resemble each other (see, e.g., [7, 9]). The proposed approach to the nearness of sets is related to Alexandroff topological spaces [15] and recent work on approach merotopic spaces [11, 12] and nearness spaces [14].
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Preliminaries
In this article, the distance function ρ is defined in the context of a normed space. Let X be a linear space over the reals with origin 0. A norm on X is a function · : X → [0, ∞] satisfying several properties for a normed space [13]. Each norm on X induces a metric d on X defined by d(x, y) = x − y for x, y ∈ R [1]. For example, let a, b denote a pair of n-dimensional vectors of numbers that are positive real values representing perceived intensities of light reflected from objects in a visual field, i.e., a = (a1 , . . . , ai , . . . , an ), b = (b1 , . . . , bi , . . . , bn ) such that ai , bi ∈ R+ . Then, the distance function ρ· : Rn × Rn → [0, ∞] is defined by the · 1 norm called the taxicab distance, i.e., n ρ· (a, b) = a − b 1 = |ai − bi |. i=1
M. Fr´echet introduced the idea of a metric space in connection with a study of function spaces in 1906 [3]. Fr´echet observed that a distance function ρ : X × X → 0+ can be defined on any non-empty set X and called it a metric. Definition 1. Metric Space The pair X, ρ denotes a metric space that consists of a non-empty set X and real-valued function ρ : X × X → 0+ , assuming non-negative real values and satisfying the following conditions for all x, y, z ∈ X. (M.1) ρ(x, y) ≥ 0 for all x, y ∈ X, (M.2) ρ(x, y) = 0 if, and only if x = y ∈ X, (M.3) ρ(x, y) = ρ(y, x) for all x, y ∈ X, (M.4) ρ(x, z) ≤ ρ(x, y) + ρ(y, z) for all x, y, z ∈ X. Assuming that the function ρ has non-negative, real values, the pair X, ρ is a pseudometric space [2] if, and only if conditions M3 and M4 hold, and ρ(x, x) = 0 for every x ∈ X. The notion of a metric space gives rise to the notion of the nearness of points and sets. Put ρ(x, y) equal to the distance between points x, y ∈ X. For a point x ∈ X and a non-empty set B, define a Hausdorff lower distance [4] ρ(x, B) = inf{ρ(x, b) : b ∈ B}. In other words, ρ(x, B) is the greatest lower bound of the distances of x from points b ∈ B. A point x is close to a set B if, and only if the distance ρ(x, B) = 0. Notice that x does not necessarily belong to B. For example, let B = {y | 0 < y ≤ 1}, then ρ(0, B) = 0.
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Metric proximity (nearness) between sets is formalized with the gap functional Dρ (A, B) [6]. A is near B ⇐⇒ Dρ (A, B) = 0, where inf {ρ(a, b) : a ∈ A, b ∈ B}, if A and B are not empty, Dρ (A, B) = ∞, if A or B is empty. A descriptive view of nearness of sets based on resemblance (similar features of objects in sets) is given in [10, 9] and in [8]. Notation 1. Feature Vectors. Put ε ∈ [0, ∞). Let O denote a set of points with measurable features represented by probe functions φ : O → . Let (x) = B = {φ1 , φ2 , ..., φn } be a set of n probe functions. A feature vector φ B (φ1 (x), . . . , φn (x)) is an n-dimensional vector of numerical features of an object x ∈ O given by probe functions in B. Let F represent the set of feature vectors (x) ∈ F denotes a feature vector in corresponding to elements of O, where φ B F extracted from some x ∈ O. Put A, B ⊆ F equal to sets of feature vectors corresponding to elements of the sets X, Y ⊆ O, respectively. Let F = φB (x) | x ∈ O , A = φB (x) | x ∈ X , B = φB (y) | y ∈ Y . Definition 2. Feature Space-Based distance Let O is a set of points with measurable features represented by probe functions in B. A feature space-based distance function ρF : O → is defined by (x), φ (y)). ρF (x, y) = ρ· (φ B B Definition 3. Feature Space-Based Gap Functional Let O, ρF is a pseudometric space. Let X, Y ⊆ O with feature spaces A, B ⊂ F. The feature space-based gap functional DρF (X, Y ) is defined in terms of (x), φ (y)) = φ (x) − φ (y) = ρF (x, y) = ρ· (φ B B B B 1
n
|φi (x) − φi (y)|.
i=1
Remark 1. Normed Vector Space In a normed vector space X over or C, a function X → (written x → x ∈ ) is called a norm on X [13] such that (NVS.1) ∀x ∈ X, x ≥ 0 and x = 0 ⇐⇒ x = 0, (NVS.2) ∀x ∈ X and any scalar k, k · x = |k| x , (NVS.3) ∀x, y ∈ X, x + y ≤ x + y . Proposition 1. ρF is a pseudometric. Proof. (x), φ (y), φ (z) ∈ F. Then ρ defined over the normed Assume x, y, z ∈ O, φ B B B F space F satisfies the conditions for a pseudometric, i.e.,
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(PM.1) ∀x ∈ O ρF (x, x) = 0. (y) ∈ F, ρ (x, y) = φ (x)−φ (y) = −(φ (y)− (x), φ (PM.2) For any φ B B B B B F φB (x)) = −(φB (y) − φB (x)) = | − 1| φB (y) − φB (x) (from Def. NVS. 2) = ρF (y, x). (x), φ (y), φ (z) ∈ F, obtain ρ (x, z) = φ (x) − φ (z) (PM.3) For any φ B B B B B F B (y) + φ B (y) − φ B (z) ≤ φ B (x) − φ B (y) + B (x) − φ ≤φ (z) (from Def. NVS. 3) = ρ (x, y) + ρ (y, z) (y) − φ φ B B F F Remark 2. The focus of this work is on measuring the nearness of descriptions of objects in disjoint sets. For this reason, we consider the gap functional in terms of the greatest lower bound of the distances between feature vectors a, b for pairs of objects a, b ∈ A, B ∈ PX such that A ∩ B = ∅, i.e., A and B are disjoint. Let A ⊂ X. Let Φn (x) = (φ1 (x), . . . , φn (x)) for x ∈ A denote a feature vector, where φi : A → . In addition, let ΦA = {Φ1 (x), . . . , Φ|X| (x)} denote a set of feature vectors for objects x ∈ A. Similarly, let B ⊂ Y and let ΦB = {Φ1 (y), . . . , Φ|Y | (y)} denote a set of feature vectors for objects y ∈ B. In this article, a description-based norm gap functional DΦ,ρ· is defined in terms of the Hausdorff lower distance [4] relative to the norm on P(ΦB ) × P(ΦA ) for sets B, A, i.e., inf {ρ· (ΦA , ΦB )}, if ΦA and ΦB are not empty, DΦ,ρ· (A, B) = ∞, if ΦA or ΦB is empty. Definition 4. Descriptively Near Sets A nonempty set X is descriptively ε-near a nonempty set Y (denoted by X Y ) if, and only if there are A ⊂ X, B ⊂ Y such that DΦ,ρ· (A, B) ≤ ε. Φ,ε
Proposition 2. For nonempty sets X, Y , the following are equivalent. (NS.1) X is descriptively ε-near Y , (NS.2) X Y , Φ,ε
(NS.3) there are A ⊂ X, B ⊂ Y such that DΦ,ρ· (A, B) ≤ ε, (NS.4) 0 ≤ DΦ,ρ· (X, Y ) < ε. Example 5. Application of Descriptively Near Sets Given a pseudometric space O, DΦ,ρ· with a set of feature vectors Φ for objects in a set O. Let X, Y ⊂ O denote digital images viewed as non-empty, disjoint sets of points and let A, B ⊂ Φ represent sets of feature vectors for elements of X, Y , respectively. Let φ(x), φ(y) denote edge orientation for pixels x ∈ X, y ∈ Y . For simplicity, assume each feature vector in Φ consists of a
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Figure 1: Sample near eye images single number representing a single feature of a pixel x ∈ X. For example, (y) = φ(y) for y ∈ Y . Then X is (x) = φ(x) for x ∈ X and put φ put φ B B descriptively ε-near Y (i.e.,X Y ) if, and only if there exist A ⊂ X, B ⊂ Y Φ,ε
such that DΦ,ρ· (A, B) ≤ ε. Each tiny box in the companion images in Fig. 1 represents a subimage (collections of pixels), where the average edge orientations of the pixels in the subimage are similar (i.e., within ε of each other). Notice that there are many regions in each portrait, starting with the edge orientation of Mona Lisa’s right eyebrow and Lena’s left eyebrow, where the average edge orientations are similar. This indicates that X Y . Hence, from Prop. 2, the two portraits Φ,ε
are considered descriptively ε-near each other.
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[7] S.A. Naimpally, Near and far. A centennial tribute to Frigyes Riesz, Siberian Electronic Mathematical Reports 2 (2009), 144153. [8] S.A. Naimpally, Proximity Approach to Problems in Topology and Analysis, Oldenbourg Verlag, Munich, Germany, 2009, xiv+206 pp., ISBN 978-3-486-58917-7. [9] J. F. Peters and P. Wasilewski, Foundations of near sets, Inf. Sci. 179 (2009), no. 18, 30913109, http://dx.doi.org/10.1016/j.ins.2009.04.018. [10] J.F. Peters, Near sets. General theory about nearness of objects, Applied Mathematical Sciences 1 (2007), no. 53, 26092029. [11] J.F. Peters, S.A. Naimpally, Approach spaces and near families, General Mathematics Notes, 1 (1), 2010, 1-6. [12] J.F. Peters, Approach merotopies and near filters. Theory and Application, General Mathematics Notes, 2 (1), 2011, 1-14. [13] W.A. Sutherland, Introduction to Metric & Topological Spaces, OxfordUniversity Press, Oxford, UK, 1974, 2009, 2nd Ed., 2008. [14] S. Tiwari, Some Aspects of General Topology and Applications. Approach Merotopic Structures and Applications, Supervisor: M. Khare, Ph.D. thesis, Mathematics Dept., Allahabad Univ., 2010, vii + 112 pp. [15] M. Wolski, Perception and classification. A note on near sets and rough sets, Fundamenta Informaticae 101 (2010), 143155, doi: 10.3233/FI-2010281. Received: July, 2010
