1. Introduction
The focus of this paper is on groupoids in proximity spaces, which is an outgrowth of recent research [14]. A groupoid is a system consisting of a nonempty set together with a binary operation on . A binary operation maps into , where
is the set of all ordered pairs of elements of
.Groupoids can either be spatially near in a traditional Lodato proximity space [5, 6, 7] (see, also, [9, 8, 12]) or descriptively near in descriptive proximity space. A descriptive proximity space [15, 10]
is an extension of an Lodato proximity space. This extension is made possible by the introduction of feature vectors that describe each point in a proximity space. Sets
in an EFproximity space are near, provided . Sets in a descriptive proximity space are near, provided there is at least one pair of points with matching descriptions.In proximity spaces, groupoids provide a natural structuring of sets of points. For a proximity space , the structuring of results from a proximity relation on . Nonempty subsets of are further structured as a result of binary operations on , yielding a groupoid denoted by . These operations often lead to descriptions of the sets endowed for structures with potential applications in proximity spaces for pattern description, recognition and identification. In this work, the applications of descriptive proximitybased groupoids in digital image analysis is explored. A practical application of the proposed approach is in terms of the analysis of natural images for pattern identification and classification.
2. Preliminaries
In a Kovár discrete space, a nonabstract point has a location and features that can be measured [4, §3]. Let be a nonempty set of nonabstract points in a proximal relator space and let a set of probe functions that represent features of each . For example, leads to a proximal view of sets of picture points in digital images [10]. A probe function represents a feature of a sample point in a picture. Let denote a feature vector for , which provides a description of each . To obtain a descriptive proximity relation (denoted by ), one first chooses a set of probe functions. Let and denote sets of descriptions of points in , respectively. For example, .
Let be a nonempty set. A Lodato proximity is a relation on which satisfies the following axioms for all subsets of :
 (P0):

.
 (P1):

.
 (P2):

.
 (P3):

or .
 (P4):

and for each .
Descriptive Lodato proximity was introduced in [11]. Let be a nonempty set, . Recall that is a feature vector that describes , is the set of feature vectors that describe points in . The descriptive intersection between and (denoted by ) is defined by
A Descriptive Lodato proximity is a relation on which satisfies the following axioms for all subsets of , which is a descriptive Lodato proximity space:
 (dP0):

.
 (dP1):

.
 (dP2):

.
 (dP3):

or .
 (dP4):

and for each .
The expression reads . Similarly, reads . The descriptive proximity of and is defined by
That is, nonempty descriptive intersection of and implies the descriptive proximity of and . Similarly, the Lodato proximity of (denoted ) is defined by
In an ordinary metric closure space [17, §14A.1] , the closure of (denoted by ) is defined by
i.e., is the set of all points in that are close to ( is the Hausdorff distance [3, §22, p. 128] between and the set and (standard distance)). Subsets are spatially near (denoted by ), provided the intersection of closure of and the closure of is nonempty, i.e., . That is, nonempty sets are spatially near, provided the sets have at least one point in common.
The Lodato proximity relation (called a discrete proximity) is defined by
The pair is called an EFproximity space. In a proximity space , the closure of in coincides with the intersection of all closed sets that contain .
Theorem 1.
[16] The closure of any set in the proximity space is the set of points that are close to .
Corollary 1.
The closure of any set in the proximity space is the set of points such that .
Proof.
From Theorem 1, . ∎
The expression reads . The relation is called a descriptive proximity relation. Similarly, denotes that is descriptively far (remote) from . The descriptive proximity of and is defined by
The descriptive intersection of and is defined by
That is, is in , provided for some .
The descriptive proximity relation is defined by
The pair is called a descriptive Lodato proximity space. In a descriptive proximity space , the descriptive closure of in contains all points in that are descriptively close to the closure of . Let indicate that is descriptively close to . The descriptive closure of a set (denoted by ) is defined by
That is, is in the descriptive closure of , provided (description of ) matches for at least one .
Theorem 2.
[11] The descriptive closure of any set in the descriptive proximity space is the set of points that are descriptively close to .
Corollary 2.
The description closure of any set in the descriptive proximity space is the set of points such that .
Proof.
From Theorem 2, . ∎
3. Descriptive Proximal Groupoids
In the descriptively near case, we can consider either partial groupoids within the same set or partial groupoids that belong to disjoint sets. That is, the descriptive nearness of groupoids is not limited to partial groupoids that have elements in common. Implicitly, this suggests a new form of the connectionist paradigm, where sets are not spatially connected (sets with no elements in common) but are descriptively connected (sets have elements belong to the descriptive intersection of the sets). For more about this new form of connectedness, see, e.g., [13].
Recall that a partial binary operation on a set is a mapping of a nonempty subset of into . A partial groupoid is a system consisting of a nonempty set together with a partial binary operation “” on . Let
Next, to arrive at a partial descriptive groupoid, let “” be a partial descriptive binary operation defined by
4. Neighbourliness in Proximal Groupoids
In general, given a finite groupoid represented by an undirected graph, for let denote the fact that are neighbours. In particular, given a finite descriptive proximal groupoid , for , let denote the fact that are neighbours, i.e., points have matching descriptions.
Example 1.
Let be a descriptive proximity space. Then let denote an equivalence class represented by the feature vector that describes . Then , provided , i.e., have matching descriptions. Then, for example, define the descriptive groupoid such that
and
Then are neighbourly in groupoid , provided and belong to the same equivalence class, i.e., .
The notion of neighbourliness of elements in a groupoid extends to neighbourliness between disjoint groupoids. Let be disjoint descriptive proximity spaces and let be descriptive proximal groupoids such that . Then groupoids and are neighbourly, provided for some . This leads to the following result.
Theorem 3.
Let be descriptive proximity spaces, . If groupoids are neighbourly, then .
Proof.
Immediate from the definition of descriptive near sets [10]. ∎
5. Application: Digital Proximal Groupoids
A digital groupoid is a groupoid in a digital image. Let be a digital image and let be a nonempty subset in endowed with a binary operation . In that case, is a digital groupoid.
Example 2.
Let be a greyscale digital image and let be a subimage in . Let be pixels in the submimage in Fig. 1 and let be defined by greyscale intensity of . Further, define the set . Hence, is the set of feature vectors , where each feature vector contains a single number , which is the intensity of . Then define the binary operation by
From this, we obtain the digital groupoid .
5.1. Neighbourliness of digital groupoid elements
Let be a digital image shown in Fig. 1 and select the subset of pixel intensities . Let be a set of probe functions that represent features of pixels in and let be sets of feature vectors that describe . Let the binary operation to be that given in Example 2. Hence, is a digital groupoid. Next, define the pseudometric to be
Points are neighbourly, provided .
Example 3.
Let be the digital image shown in Fig. 1 and select the subset of pixel intensities (also shown in Fig. 1). Let , where equals the greyscale intensity of . Let be the pixel intensities labelled with a white in Fig. 1. Observe that . Hence, are neighbourly elements of this digital groupoid. There are other instances of neighbourly elements in the same groupoid.
5.2. Digital groupoid set pattern generators
A pattern generator is derived some form of regular structure [2]. In geometry, a regular polygon is a polygon with sides that have the same length and are symmetrically place around a common center. A geometric pattern contains a repetition of a regular polygon. An element in a groupoid is regular, provided , i.e., for some [1]. A groupoid is regular, provided every element is regular.
Example 4.
An algebraic pattern contains a repetition of regular structures such as a regular element or a regular groupoid. Again, for example, an algebraic pattern contains a repetition of neighbourly elements from groupoids such as the one in Example 3. In this work, the focus is proximal algebraic patterns. A proximal algebraic pattern in a descriptive proximity space (denoted by ) is derived from a repetition of structures such as groupoids that contain a mixture of neighbourly (regular) and nonneighbourly (nonregular) elements, i.e.,
Example 5.
From example 3, groupoid . From the digital image in Fig. 1, we can find additional groupoids containing neighbourly elements. Two such groupoids are given in Fig. 3. That is, the proximal groupoid given in Fig. 3.1 is derived from the digital image in Fig. 1 and has a number of neighbourly elements such as those pixels with intensities 0.36, 0.41, 0.44, 0.59. And and groupoid (in Fig. 1) are neighbourly (e.g., both and have pixels with intensity 0.83). Similarly, the proximal groupoid given in Fig. 3.2 is also derived from the digital image in Fig. 1 and has a number of neighbourly elements. Also notice that groupoid also a pixel with intensity 0.83. Then, from Theorem 3 and the definition of a proximal algebraic pattern, we obtain
Similarly, proximal groupoids from Fig. 4 are derived from the digital image in Fig. 2. In addition, and the proximal groupoid in Fig. 2 are neighbourly. Hence, we obtain a second proximal algebraic pattern, namely,
Patterns are examples of neighbourly patterns.
Theorem 4.
In a descriptive proximity space, a proximal groupoid containing regular elements generates a proximal groupoid pattern.
Proof.
Assume in a descriptive proximity space is the only proximal groupoid containing neighbourly elements. Then is a proximal groupoid pattern containing only the groupoid . Let be a pair of neighbourly proximal groupoids in . Then, from Theorem 3, and either or is a generator of a proximal groupoid pattern. ∎
Theorem 5.
In a descriptive proximity space, a proximal groupoid pattern is a collection of neighbourly groupoids.
Proof.
Immediate from Theorem 4 and the definition of a proximal algebraic pattern. ∎
The likelihood of finding proximal algebraic patterns in nature is high, since natural patterns seldom, if ever, contains regular structures such as regular polygons or regular algebraic structures. Proximal algebraic patterns are commonly found in digital images. Hence, proximal algebraic structures are useful in classifying digital images.
5.3. Neighbourly patterns in digital images
Let be descriptive proximity spaces and let be proximal groupoids that generate the proximal algebraic patterns in , respectively. A pair of patterns are neighbourly, provided , i.e., the description matches for some . In other words, neighbourly proximal algebraic patterns contain neighbourly sets.
Example 6.
In Example 5, patterns are neighbourly. To see this, consider, for instance, groupoid in Fig. 1 has a pixel with intensity 0.83 (labelled with a in Fig. 1). Similarly, groupoid in Fig. 2 has a pixel with intensity 0.83 (also labelled with a ). Consequently, and are neighbourly. Hence, groupoids generate neighbourly patterns.
In the search for neighbourly patterns in digital images , a proximal algebraic pattern in is salient (important) relative to a proximal algebraic pattern in , provided the number of neighbourly elements is high enough. A digital image belongs to the class of images represented by , if contains a salient pattern that is neighbourly with a pattern in . In that case, images are wellclassified.
Theorem 6.
Let be descriptive proximity spaces and let generate neighbourly patterns . If every is neighbourly with some , then is a salient proximal algebraic pattern.
Proof.
Immediate from the definition of a salient proximal algebraic pattern. ∎
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