Upper bounding overlaps by groupings

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ScienceDirect Fuzzy Sets and Systems 264 (2015) 76–99 www.elsevier.com/locate/fss

Upper bounding overlaps by groupings Nicolás Madrid a,∗ , Ana Burusco b , Humberto Bustince b , Javier Fernández b , Irina Perfilieva a a Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic b Dept. Computer Science and Artificial Intelligence, Public University of Navarra, C/ Arrosadía, s/n, Pamplona 31006, Spain

Received 4 March 2014; received in revised form 7 October 2014; accepted 21 October 2014 Available online 24 October 2014

Abstract The notions of overlapping and grouping arise from the problem how to assign an object to exactly one class among several available. In this respect, overlaps and groupings are connected with t-norms and t-conorms. However, there are some properties of t-norms and t-conorms that are not valid for overlaps and groupings, for example, that any t-norm is less than or equal than any t-conorm. This fact motivates the proposed research. Specifically, we determine conditions to ensure that an overlap is smaller than a certain grouping and methods to define operators satisfying such ordering. The relevance of this study is visible at the end of the paper by the inclusion of an illustrative example in the context of image processing. © 2014 Elsevier B.V. All rights reserved. Keywords: Overlaps; Groupings; Aggregators; Image processing

1. Introduction The notions of overlapping and grouping arise from a common problem in many fields: how to assign a given element or object to exactly one class among several available. In the course of carrying this task out, some complications can arise for two main reasons: either the separation between classes is unclear or there is no class to cover certain elements. In the former case, there is an overlap between classes, and in the latter case, the classes are unable to group the set of objects. The notion of overlap function was presented in [7] to address the former difficulty in the context of image processing. Consider the following classification task: given an image with an object on a background, separate those pixels that belong to the object from those others that belong to the background. The difficulty of such a task resides in those pixels that do not clearly belong to either one or to the other. A strategy can start by assigning to each pixel two membership degrees, one to represent the membership to the object and the other to the background. Then, overlap functions were introduced to measure to what extent a pixel belongs simultaneously to both classes by means of a degree of overlapping. Subsequently, the notion of grouping function was presented in [8] as a dual form of the * Corresponding author.

E-mail addresses: [email protected] (N. Madrid), [email protected] (A. Burusco), [email protected] (H. Bustince), [email protected] (J. Fernández), [email protected] (I. Perfilieva). http://dx.doi.org/10.1016/j.fss.2014.10.022 0165-0114/© 2014 Elsevier B.V. All rights reserved.

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notion of overlap function and, they were used to represent the degree of comparability and to measure the degree of indifference, respectively between two alternatives in decision-making problems. It is worth noting the relationship between overlaps and groupings with conjunctions and disjunctions, respectively. This fact is tangible simply by taking into account the underlying ideas behind both definitions: on the one hand, the membership to two classes simultaneously (the idea of conjunction) underlies overlaps; on the other hand, the membership to at least one class (the idea of disjunction) underlies groupings. Without any doubt, the most popular way to represent conjunctions in fuzzy logic is by means of t -norms and t-conorms. Thus researching theoretical properties of overlaps and groupings taking as references properties of t -norms and t -conorms looks to have sense. Actually, many results in the literature about overlaps and groupings are inspired in such a way [3,4]. Anyway, there are also some properties holding between t-norms and t -conorms that do not hold between overlaps and groupings. For instance, although any t-norm is lesser than any t -conorm, there are some overlaps that are not lesser than some groupings. This fact can open new applications to overlaps and groupings further the original motivational problem of classifying, which requires the strong assumption of an image of one object on a background (difficult to be achieved apart from medical images). This paper is divided in two main parts. The first one presents a purely theoretical study to determine under which conditions an overlap is smaller than or equal to a certain grouping, and viceversa. Moreover, other results establishing other relationships between overlaps and groupings are given. The second part is also theoretical but applicational oriented. In it, we pretend to present a potential application of the theory given in the first part of the paper. Specifically we focus on gradient operators in gray-scale images, which does not require the hard assumption of the original motivation in overlaps and groupings operators. We consider the gradient operator defined in [2,6,12] for edge detection and, assuming that a coherent gradient operator measures changes of intensity if and only if there is a change of intensity in the surrounding pixels we show that, although the approaches [2,6] look flexible due to the use of arbitrary t -norms and t -conorms, only one combination provides coherent results namely, that given by the maximum and the minimum operators. Therefore we show that the use of overlaps in the definition of gradients operators (in the sense of [2,6]) provides a real flexibility able to deal with variations under different luminosities. The structure of the paper is given as follows. The paper begin with the section of Preliminaries, where the notions of overlap and grouping functions are presented. Subsequently, the two main parts described above are given. Finally we include a section of conclusions and future work. 2. Preliminaries Let us begin by recalling the definition of overlap function [7], that was introduced to measure the possible overlapping between the pixels of an image when we try to classify them as belonging either to the background or to the object. Definition 1. An operator GO : [0, 1]2 → [0, 1] is an overlap if it is: • commutative, • monotonic, • continuous, and satisfies the marginal conditions: • GO (x, y) = 0 if and only if x = 0 or y = 0 and • GO (x, y) = 1 if and only if x = y = 1. Note that, from the image processing point of view, we say that the overlap function, applied to a given pixel, is equal to zero if and only if that pixel does not belong at all to either the object or the background. Analogously, we state that the overlap function is maximal (equal to one) if and only if we are absolutely certain that it belongs simultaneously to both the object and the background. Continuity is a natural requirement in image processing problems, as a slight variation on the considered values should not lead to dramatic changes in the outputs. Symmetry simply arises from taking into account that it is irrelevant what we consider to be the object and the background in order to measure

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the overlap. Finally, monotonicity simply follows from considering that when increasing the membership of a pixel to the object or background, the overlap between the object and the background for such a pixel cannot decrease. The definition of grouping function is motivated, also from the image processing point of view, as a tool to measure to what extent a pixel belongs to at least to one class. The restrictions of commutativity, monotonicity, continuity and marginal conditions can be motivated in a similar way as in the case of overlaps. Definition 2. An operator GG : [0, 1]2 → [0, 1] is a grouping if it is: • commutative, • monotonic, • continuous, and satisfies the marginal conditions: • GG (x, y) = 0 if and only if x = y = 0 and • GG (x, y) = 1 if and only if x = 1 or y = 1. It is worth noting the relationship between overlaps and groupings with conjunctions and disjunctions, respectively. This fact is tangible simply by taking into account the underlying ideas behind both definitions: on the one hand, the membership to two classes simultaneously (the idea of conjunction) underlies overlaps; on the other hand, the membership to at least one class (the idea of disjunction) underlies groupings. However, it is important to remark as well that overlap and grouping functions are different from conjunctive and disjunctive functions,1 as there exist overlap functions that are not smaller than or equal to the minimum (see Example 2) and there are grouping functions that are not greater than or equal to the maximum. Example 1. One of the most common ways of representing conjunctions and disjunctions is by means of t-norms (commutative, monotonic and associative mappings T : [0, 1]2 → [0, 1], with 1 as the identity element) and t -conorms (commutative, monotonic and associative mappings S: [0, 1]2 → [0, 1], with 0 as the identity element). Note that any continuous t -norm satisfying the marginal conditions of Definition 1 is an overlap as well. This fact gives us several examples of overlaps, such as the product t-norm TP (x, y) = x · y or the minimum (also called Gödel) t -norm TG (x, y) = min(x, y). Note also that there are t -norms that do not satisfy the marginal conditions of Definition 1, and therefore, they are not overlaps, as for instance, the Łukasiewicz t -norm TL(x, y) = max(0, x + y − 1). Reciprocally, any continuous t -conorm satisfying the marginal conditions of Definition 2 provides an example of grouping. Thus, we have that the product t -conorm SP (x, y) = x + y − x · y and the maximum (also called Gödel) t-conorm SG (x, y) = max(x, y) are groupings as well. Moreover, as in the case of t-norms, there are t -conorms that are not groupings, as for instance, the Łukasiewicz t -conorm SL (x, y) = min(1, x + y). Example 2. The geometric mean is an interesting example of an overlap function, as is not conjunctive (i.e., it is not smaller than or equal to the minimum). Overlaps and groupings are evidently related because, first, in both operators, commutativity, monotonicity and continuity are required, and second, the marginal conditions are very similar. Actually, there is a dual relationship between overlaps and groupings that recalls the duality between t-norms and t-conorms. We recall that a negation operator is any antitonic operator n: [0, 1] → [0, 1] satisfying n(0) = 1 and n(1) = 0. Definition 3. Let G be an overlap (resp. a grouping) and let n1 and n2 be two continuous negation operators such that: • n1 (x) = 0 (resp. n2 (x) = 0) if and only if x = 1 (i.e. non-vanish [1]), • n1 (x) = 1 (resp. n2 (x) = 1) if and only if x = 0 (i.e. non-filling [1]) 1 Let us recall that a conjunctive function is any function smaller than or equal to the minimum, and a disjunctive function is any function greater than or equal to the maximum.

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Then, the operator    Gn1 ,n2 (x, y) = n1 G n2 (x), n2 (y) is called the dual grouping (resp. overlap) of G with respect to n1 and n2 . The reader should note that the definition above needs a proof, i.e., it is necessary to prove that the operator Gn1 ,n2 is effectively a grouping (resp. an overlap). Such a result can be found in [11]. To end this section, we remark that the set W of all operators of arity two defined on the unit interval [0, 1] endowed with the point-wise ordering2 establishes a complete lattice. It is not difficult to check that the set of overlaps (resp. grouping) determines a sublattice of (W, ≤). However, this sublattice structure on the set of overlaps (resp. groupings) is not complete (see [7]). Therefore, in this paper, the operator supremum and infimum applied on an arbitrary set of overlaps (resp. groupings) is always considered in (W, ≤). 3. Upper bounding overlaps by groupings It is well known that, given a t -norm T and a t -conorm S, the inequality T ≤ S holds. In this section, we study conditions to ensure a similar relationship between overlaps and groupings. First, note that by the marginal conditions given in Definitions 1 and 2, a grouping can never be less than an overlap. Conversely, an overlap is not always less than a grouping, as the example below shows: √ √ Example 3. Consider the overlap given by GO (x, y) = min( x, y) and the grouping given by GG (x, y) = max(x 2 , y 2 ). Then: √ √   GO (0.5, 0.25) = min( 0.5, 0.25) > max (0.5)2 , (0.25)2 = GG (0.5, 0.25) Thus, in this case, GO  GG . The goal of this section can be formulated as follows: given an overlap GO (resp. a grouping GG ), determine a grouping GG (resp. an overlap GO ) such that GO ≤ GG . Keeping this idea in mind, we introduce below two notions: the notion of f -bound overlaps (resp. groupings) and f -diagonal overlaps (resp. groupings). 3.1. f -Bound groupings and overlaps We recall that one of the requirements imposed on t -norms and t -conorms is that the elements 1 and 0 be neutral elements, respectively. The following definition attaches a role to these two elements, 1 and 0, in overlaps and groupings, respectively. Definition 4. Let f be a mapping from [0, 1] to [0, 1]. An overlap GO (resp. a grouping GG ) is called f -bound if the equality GO (x, 1) = f (x) (resp. GG (x, 0) = f (x)) holds for all x ∈ [0, 1]. Some remarks about this definition are needed. First, note that every overlap GO is GO (x, 1)-bound, and each grouping GG is GG (x, 0)-bound. Second, fixing a mapping f , the families of f -bound overlaps (denoted hereafter by Of ) and f -bound groupings (denoted hereafter by Gf ) determine the disjoint classes of overlaps and groupings. Moreover, the set of overlaps (resp. groupings) coincides with the union of these classes Of (resp. Gf ). Third, note that the set of overlaps where 1 is the neutral element (resp. set of groupings where 0 is the neutral element) is exactly the class of id-bound overlaps (resp. id-bound groupings). Finally, not every mapping f from [0, 1] to [0, 1] specifies a non-empty class of f -bound overlaps or groupings. This is because the properties of overlaps and groupings place restrictions on such mappings. 2 Which means that given two operators A, B: [0, 1]2 → [0, 1], we say that A ≤ B if A(x, y) ≤ B(x, y) for all x, y ∈ [0, 1].

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Proposition 1. Let G be either an f -bound overlap or an f -bound grouping. Then: • • • •

f is continuous, f is monotonic, f (x) = 0 if and only if x = 0, f (x) = 1 if and only if x = 1.

Proof. Let us assume that G is an f -bound overlap. The continuity and monotonicity of f is a straightforward consequence of the continuity and monotonicity of G because of f (x) = G(x, 1). Moreover, as G(x, y) = 0 if and only if x = 0 or y = 0, we have that f (x) = G(x, 1) = 0 if and only if x = 0. Likewise, as G(x, y) = 1 if and only if x = y = 1, we have that f (x) = G(1, x) = 1 if and only if x = 1. The case for f -bound groupings is similar. 2 Let us denote by Ω the set of mappings satisfying the four properties given in the proposition above. Note that if f ∈ Ω is injective, then it is an automorphism on the unit interval. Note also that Ω can certainly be identified with the family of classes determined by f -bound conditions, because for each f ∈ Ω, the classes Of and Gf are not empty; consider the overlap min(f (x), f (y)) and the grouping max(f (x), f (y)). Furthermore, it is easy to check that (Ω, ≤) has a structure of lattice with the point-wise ordering induced by [0, 1]; however, such a lattice is not complete. Moreover, the lattice (Ω, ≤) has neither a minimal nor a maximal element. Therefore, for every mapping f ∈ Ω, there exist two mappings f1 , f2 ∈ Ω such that f1 < f < f2 . It is also interesting to note that for each f ∈ Ω, continuous t -norms can be used to find out elements in Of . For instance, if f ∈ Ω, then the mappings GO (x, y) = f (x)f (y) or the mapping GO (x, y) = min(f (x), f (y)) are overlap functions. More generally, we can state the following: Proposition 2. Let T and S be a continuous t -norm and a continuous t -conorm satisfying the marginal conditions given in Definitions 1 and 2, respectively, and let f ∈ Ω. Then: • f (T (x, y)) and T (f (x), f (y)) are f -bound overlaps. • f (S(x, y)) and S(f (x), f (y)) f -bound groupings. The following proposition justifies the names “f -bound” for overlaps and groupings. It shows how a certain upper (lower) bound can be constructed for an f -bound overlap (grouping) by means of the mapping f . Proposition 3. Let GO and GG be an overlap and a grouping, respectively, and f ∈ Ω. Then: • if GO ∈ Of , then GO (x, y) ≤ min(f (x), f (y)) for all x, y ∈ [0, 1]. • if GG ∈ Gf , then GG (x, y) ≥ max(f (x), f (y)) for all x, y ∈ [0, 1]. Proof. Let us prove only the case GO ∈ Of , because the case GG ∈ Gf is similar. By monotonicity of GO , we have:    GO (x, y) ≤ GO (x, 1) = f (x) ⇒ GO (x, y) ≤ min f (x), f (y) GO (x, y) ≤ GO (1, y) = f (y) for all x, y ∈ [0, 1].

2

The notion of f -bound overlaps and f -bound groupings allows us to give a sufficient condition to ensure that an overlap is less than or equal to a grouping. Proposition 4. Let GO and GG be an f1 -overlap and an f2 -grouping, respectively, with f1 , f2 ∈ Ω. If f1 ≤ f2 , then GO (x, y) ≤ GG (x, y) for all x, y ∈ [0, 1]. Proof. By monotonicity of GO and GG , we have the following chain of inequalities GO (x, y) ≤ GO (x, 1) = f1 (x) ≤ f2 (x) = GG (x, 0) ≤ GG (x, y) for all x, y ∈ [0, 1].

2

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The following result is a direct consequence of the above proposition. Corollary 1. Let GO be an overlap and GG a grouping. If both GO and GG are f -bound then GO ≤ GG . Note that Proposition 4 imposes a sufficient condition but not a necessary one, because we can find GO ∈ Of1 and GG ∈ Gf2 with f1 > f2 satisfying GO ≤ GG , as the following example shows: Example 4. Let us consider the Łukasiewicz t-norm TŁ (x, y) = max(x + y − 1, 0) and GG (x, y) = max(x 2 , y 2 ). Note that by Proposition 3, we can assert that TŁ (x, y) and GG (x, y) are an id-bound overlap and an x 2 -bound grouping, respectively. Let us show that TŁ ≤ GG despite x ≥ x 2 for all x ∈ [0, 1]. Let x, y ∈ [0, 1]. The case x + y − 1 ≤ 0 is straightforward. Thus, let us assume that x + y − 1 > 0 and, without loss of generality, that x ≤ y. Then: GG (x, y) = y 2 ≥ 2y − 1 ≥ x + y − 1 = max(x + y − 1, 0) = TŁ (x, y). Actually, the following result states that for any overlap GO and f ∈ Ω, there exists f -bound grouping Gf such that GO ≤ Gf . Note that the boundary condition of GO need not have any relationship with the f -boundary condition of Gf . Proposition 5. Let GO be an overlap (resp. let GG be a grouping). Then for each mapping f ∈ Ω there exists GG ∈ Gf (resp. GO ∈ Of ) such that GO ≤ GG . Proof. Let GO be an overlap and let f ∈ Ω. Let us show that the operator   GG (x, y) = max f (x), f (y), GO (x, y) is an f -bound grouping such that GO ≤ GG . It easy to see that the operator GG is a grouping because: • • • •

GG is commutative (straightforward) GG is monotonic (resp. continuous) because f, GO and max are monotonic (resp. continuous) operators. GG (x, y) = 0 if and only if f (x) = f (y) = GO (x, y) = 0, if and only if x = y = 0. GG (x, y) = 1 if and only if f (x) = 1 or f (y) = 1 or GO (x, y) = 1, which is equivalent to x = 1 or y = 1.

Moreover, GG is an f -bound grouping because     GG (0, x) = max f (0), f (x), GO (0, x) = max 0, f (x), 0 = f (x). Finally, the inequality GO ≤ GG holds by definition of GG .

2

By the lattice structure of Ω, it is obvious that there is neither a maximal nor a minimal overlap (resp. grouping).3 However, in each class Of (resp. Gf ), there is a greatest overlap (resp. least grouping). Corollary 2. Let f ∈ Ω, then • the operator GO (x, y) = f (min(x, y)) is the greatest f -bound overlap, and • the operator GG (x, y) = f (max(x, y)) is the least f -bound grouping. Proof. This is a straightforward consequence of Propositions 2 and 3.

2

The following result shows that there is neither a least element in Of nor a greatest element in Gf . Specifically, we will prove that the infimum of Of (resp. supremum of Gf ) is not even an overlap (resp. grouping) operator. 3 A formal result and proof of this assertion can be found in [11].

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Proposition 6. Let f ∈ Ω. Then the infimum of Of is: ⎧ ⎨ f (x) if y = 1  GO (x, y) = f (y) if x = 1 ⎩ 0 otherwise GO ∈Of and the supremum of Gf is: ⎧ ⎨ f (x) GG (x, y) = f (y) ⎩ 1 GG ∈Gf

if y = 0 if x = 0 otherwise.

Proof. Let us begin by proving that ⎧ ⎨ f (x) if y = 1 G(x, y) = f (y) if x = 1 ⎩ 0 otherwise is the infimum of Of . Obviously G(x, y) is a lower bound of Of . Thus, if we prove that there is a subset of Of whose infimum is G, then we obtain the proof. Let us take the subset of f -bound overlaps GkO , where k ∈ N  {0}, defined as follows:      GkO (x, y) = max f x k · f (y), f (x) · f y k Let us show that for all k ∈ N  {0}, each GkO (x, y) is an f -bound overlap. The commutativity, monotonicity and continuity are straightforward. To prove the marginal conditions, we proceed as follows: GkO (x, y) = 0 if and only if f (x k ) · f (y) = 0 and f (x) · f (y k ) = 0, if and only if (by properties of f ) x = 0 or y = 0. On the other hand, GkO (x, y) = 1 if and only if f (x k ) · f (y) = 1 or f (x) · f (y k ) = 1, and in both cases, it is equivalent to x = 1 and y = 1. Finally, by using that f is monotonic, f (1) = 1, and that x k ≤ x for all x ∈ [0, 1] and k ∈ N  {0}, we obtain          GkO (x, 1) = max f x k · f (1), f (x) · f 1k = max f x k , f (x) = f (x). Thus, GkO (x, y) is really an f -bound overlap for all k ∈ N  {0}.

Now let us show that k∈N (GkO ) coincides with G. Let us distinguish two cases. Firstly, if y = 1 (or x = 1) we have that GkO (x, 1) = f (x) for all k ∈ N  {0} So,     GkO (x, 1) = f (x) = f (x) = G(x, 1). k∈N0

k∈N0

Secondly, let us assume that x, y = 1. Note that by continuity of f and because f (x), f (y) = 1, we have that limk→∞ f (x k ) · f (y) = 0; thus k∈N (f (x k ) · f (y)) = k∈N (f (y k ) · f (x)) = 0 for all x, y ∈ [0, 1). Hence we obtain that for x, y = 1:         max f x k · f (y), f (x) · f y k = 0 = G(x, y). GkO (x, y) = k∈N0

k∈N0

To prove the other statement, proceed similarly to the above but consider the subset of f -bound groupings defined by           GkG (x, y) = 1 − min 1 − f x k · 1 − f (y) , 1 − f (x) · 1 − f y k where k ∈ N  {0}.

2

Note that the infimum of Of (resp. supremum of Gf ) can be written in terms of the drastic t-norm TD (resp. drastic

 t-conorm SD ) by GO ∈Of GO (x, y) = TD (f (x), f (y)) (resp. GG ∈Gf GG (x, y) = SD (f (f ), f (y))). Corollary 3. Let f ∈ Ω. Then, the set Of (resp. Gf ) does not have the least element (resp. greatest element).

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Proof. Simply note that the infimum of Of given by Proposition 6 is not continuous, so it cannot be an overlap. 2 The following consequence of Proposition 6 shows that there exist infinite strictly descending (resp. ascending) chains in Of (resp. Gf ). Corollary 4. Let f ∈ Ω. If G ∈ Of (resp. G ∈ Gf ), then there exists G ∈ Of (resp. G ∈ Gf ) such that G < G (resp. G < G). Proof. Consider G ∈ Of and the family of f -bound overlaps GkO defined in Proposition 6. Note that, as

k k k∈N (GO ) = TD (f (x), f (y)), by Proposition 6 there exists k ∈ N such that G  GO . Then, the f -bound overlap defined by G(x, y) = min{G(x, y), GkO (x, y)} satisfies G ∈ Of and G < G.

2

Another interesting property of Of and Gf is that they are dense as posets. Proposition 7. Let G1 and G2 be two overlaps in Of (resp. two groupings in Gf ) such that G1 < G2 . Then, there exists G ∈ Of (resp. G ∈ Gf ) such that G1 < G < G2 . Proof. Simply consider the f -bound overlap (resp. f -bound grouping) defined by G(x, y) =

G1 (x,y)+G2 (x,y) . 2

2

At the end of this section, we relate the notions of duality and f -bound overlaps. Primarily, however, it is convenient to show that an overlap need not be less than or equal to its respective dual groupings. 1/3 . Example 5. Consider the overlap GO (x, y) = (x · y)1/3 and its dual grouping G1−x GO (x, y) = 1 − ((1 − x) · (1 − y))

Then, GO (0.5, 0.5) = (0.5 · 0.5)1/3 ≈ 0.63  0.37 ≈ 1 − (0.5 · 0.5)1/3 = GG (0.5, 0.5). So GO  G1−x GO .

The following definition establishes a restriction on negations used in the dual construction in order to maintain the f -bound condition. Definition 5. Let G be an overlap (resp. grouping), f ∈ Ω an automorphism on the unit interval and n a bijective negation.4 Let us consider as well the negations defined by: • n1 (x) = f (n(x)) and • n2 (x) = f −1 (n−1 (x)). Then, the operator    Gn,f (x, y) = n1 G n2 (x), n2 (y) is called the f -dual grouping (resp. f -dual overlap) of G with respect to n. Note that both operators, n1 (x) = f (n(x)) and n2 (x) = f −1 (n−1 (x)), are negations satisfying the conditions given in Definition 3. In other words, the f -dual construction simply establishes a restriction on the negations chosen to make dual operators. Thus, the terminology f -dual groupings and f -dual overlaps makes sense. As we stated above, Definition 5 is motivated by the following result. Proposition 8. Let G ∈ Of (resp. G ∈ Gf ), f ∈ Ω an automorphism on the unit interval and n a bijective negation. Then the f -dual grouping (resp. overlap) of G w.r.t. n is f -bound. 4 In the Literature, this type of negation is called strict.

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Proof. Let G ∈ Of with f ∈ Ω be bijective and let n be a bijective negation. Let us show that Gn,f (x, 0) = f (x):       Gn,f (x, 0) = n1 G n2 (x), n2 (0) = f ◦ n ◦ G ◦ f −1 ◦ n−1 (x) , 1 for all x ∈ [0, 1]. Now, by using that G(x, 1) = f (x), we finally obtain: Gn,f (x, 0) = f ◦ n ◦ f ◦ f −1 ◦ n−1 (x) = f (x) for all x ∈ [0, 1].

2

Finally, as a consequence of Proposition 8 and Corollary 1, we can ensure that any f -bound overlap is less than or equal to any of its f -dual groupings. Corollary 5. Let f ∈ Ω, GO ∈ Of and let GO n,f be an f -dual grouping of GO . Then, the inequality GO ≤ GO n,f holds. And, reciprocally: Corollary 6. Let f ∈ Ω, GG ∈ Gf and let GG n,f be an f -dual overlap of GG . Then, the inequality GG n,f ≤ GG holds. Example 6. Let us consider the overlap given in Example 5, i.e., GO (x, y) = (x · y)1/3 . Because GO (x, 1) = x 1/3 , then GO is an x 1/3 -bound overlap. Let us determine the x 1/3 -dual grouping of GO with respect to the standard negation n(x) = 1 − x. First, let us determine the two negations used in the construction: • n1 (x) = (1 − x)1/3 and • n2 (x) = (1 − x)3 . Thus, GO 1−x,x

1/3

1/3 1/3    1/3 (x, y) = 1 − (1 − x)3 · (1 − y)3 = 1 − (1 − x) · (1 − y)

Moreover, because of Corollary 5, we have that  1/3 1/3 GO (x, y) = (x · y)1/3 ≤ 1 − (1 − x) · (1 − y) = GO 1−x,x (x, y) for all x, y ∈ [0, 1]. 3.2. f -Diagonal groupings and overlaps In this section, we study overlaps and groupings coinciding on the diagonal x = y, i.e., satisfying GO (x, x) = GG (x, x) for all x ∈ [0, 1]. To facilitate the description of this study, we introduce the notions of f -diagonal overlaps and f -diagonal groupings. Definition 6. Let f be a mapping from [0, 1] to [0, 1]. An overlap GO (resp. a grouping GG ) is called f -diagonal if the equality GO (x, x) = f (x) (resp. GG (x, x) = f (x)) holds for all x ∈ [0, 1]. The notion of f -diagonal overlap (resp. grouping) has some general similarities with the notion of f -bound overlap (resp. grouping). First, note that every overlap GO (resp. grouping) is GO (x, x)-diagonal. Second, for every suitable mapping f , we can define the class Ofd (resp. Gfd ) of f -diagonal overlaps (resp. groupings). Moreover, the set of overlaps (resp. groupings) coincides with the union of all such disjoint classes. Recall that a similar behavior occurs with the classes of f -bound overlaps (resp. groupings). Third, it is obvious that the properties of overlaps and groupings also induce properties on mappings f used to define f -diagonal overlaps and f -diagonal groupings. Curiously, these properties are exactly the same as those given in Proposition 1 under the context of f -bound overlaps and f -bound groupings. Namely, given an f -diagonal overlap (resp. grouping), the mapping f is:

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• • • •

85

continuous, monotonic, satisfies f (x) = 0 if and only if x = 0 and f (x) = 1 if and only if x = 1

The proof of the statement above is left to the reader because it is quite similar to the proof of Proposition 1. Let us recall that the set of mappings holding such properties was denoted by Ω. Furthermore, given f ∈ Ω, the classes Ofd and Gfd are non-empty because f (min(x, y)) and f (max(x, y)) are an f -diagonal overlap and an f -diagonal grouping, respectively. Therefore, as in the case of the set of classes Of and Gf , there is a bijection between Ω and the set of non-empty classes Ofd (resp. Gfd ). Finally, note that the notion of f -diagonal overlaps is more general than the notion of idempotent overlaps. Actually, the set of idempotent overlaps (resp. groupings) can be identified with the class Ofd (resp. Gfd ) with f (x) = x. The following result establishes another link between the notions of f -diagonal and f -bound overlap and grouping. Proposition 9. Let f, f1 , f2 ∈ Ω, GO ∈ Of1 and GG ∈ Gf2 . If both GO and GG are f -diagonal, then f1 ≥ f ≥ f2 . Proof. The proof consists in considering the following chain of inequalities: f1 (x) = GO (x, 1) ≥ GO (x, x) = f (x) = GG (x, x) ≥ GG (x, 0) = f2 (x) for all x ∈ [0, 1].

2

The following example focuses on the main goal of the section: determining conditions that ensure that an overlap is less than or equal to a grouping. Specifically, it shows that the same f -diagonal condition is insufficient to guarantee the desired inequality. In other words, from GO ∈ Ofd and GG ∈ Gfd , we cannot guarantee GO ≤ GG . Example 7. Consider the following idempotent overlap: 0 if x = y = 0 GO (x, y) = min(x,y)2 2 · min(x, y) − max(x,y) otherwise and the following idempotent grouping: 1 GG (x, y) = max(x,y)2 −2·x·y+min(x,y) 1−min(x,y)

ifx = y = 1 otherwise

Let us see first that they are an id-bound overlap and an id-bound grouping, respectively. The commutativity holds straightforwardly. The continuity comes simply from the following facts: lim

(x,y)→(0,0)

2 · min(x, y) −

min(x, y)2 =0 max(x, y)

and max(x, y)2 − 2 · x · y + min(x, y) =1 (x,y)→(1,1) 1 − min(x, y) lim

To see that GO and GG are also monotonic, consider the mappings g1 (x, y) = 2 · x − xy and g2 (x, y) = y −2·x·y+x . 1−x Note that g1 and g2 are both monotonic in the domains D1 = {(x, y) ∈ [0, 1]2 | x ≤ y and y = 0} and D2 = {(x, y) ∈ [0, 1]2 | x ≤ y and x = 1}, respectively. Thus, by composing g1 and g2 with max and min, we obtain that 2 max(x,y)2 −2·x·y+min(x,y) 2 · min(x, y) − min(x,y) are also monotonic in [0, 1]2  (0, 0) and [0, 1]2  (1, 1), respecmax(x,y) and 1−min(x,y) tively. Finally, note that the monotonicity on all the respective domains follows from the continuity 2 Let us verify the marginal conditions of GO . g1 (x, y) = 2 · x − xy = 0 if and only if 2 · x · y − x 2 = x · (2 · y − x) = 0 if and only if x = 0 or 2 · y = x. Note that the latter cannot holds in the case (x, y) ∈ D1 because, in such a case, 2

2

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x ≤ y and y = 0. Thus, g1 (x, y) = 0 in D1 if and only if x = 0. Because GO (x, y) = g1 (min(x, y), max(x, y)) if (x, y) = (0, 0), then G(x, y) = 0 if and only if min(x, y) = 0, if and only if x = 0 or y = 0. 2 To see the case GO (x, y) = 1, g1 (x, y) = 2 · x − xy = 1 (with (x, y) ∈ D1 ) if and only if y + x 2 − 2 · x · y = 0. Now, because 0 ≤ (x − y)2 ≤ y + x 2 − 2 · x · y, we have that g1 (x, y) = 1 implies x = y. However, if x = y, then g1 (x, x) = x; thus, g1 (x, y) = 1 if and only if x = y = 1. Because GO (x, y) = g1 (min(x, y), max(x, y)), then GO (x, y) if and only if x = y = 1. Let us verify the marginal conditions of GG . Let us consider first the case GG (x, y) = 0. Obviously, if x = y = 0, 2 then GG (x, y) = 0. To see the converse, note that g2 (x, y) = y −2·x·y+x = 0 (with x = 1) if and only if y 2 − 2 · x · 1−x y + x = 0. Because 0 ≤ (x − y)2 ≤ y 2 − 2 · x · y + x, we have that if y 2 − 2 · x · y + x = 0, then (x − y)2 = 0; thus, necessarily, x = y. In such a case, however, we have that x 2 − 2 · x · x + x = x(1 − x) = 0 if and only if x = y = 0 (because x = 1). Thus, because GG (x, y) = g2 (min(x, y), max(x, y)), we have that GG (x, y) = 0 implies x = y = 0. For the case GG (x, y) = 1, note first that x = 1 or y = 1 implies GG (x, y) = 1. Conversely, note that g2 (x, y) = y 2 −2·x·y+x = 1 if and only if (y − x)2 − (x − 1)2 = 0. Thus, if we consider (x, y) ∈ D2 , (y − x)2 − (x − 1)2 = 0 if and 1−x only if y = 1. From the fact that GG (x, y) = g2 (min(x, y), max(x, y)), we obtain that GG (x, y) = 1 implies x = 1 or y = 1. Finally, the proof of idempotency simply requires checking: GO (x, x) = 2 · x −

x2 =x x

and GG (x, x) =

x2 − 2 · x2 + x =x 1−x

Now, consider GO (0.5, 0.3) = 2 · 0.3 −

(0.3)2 = 0.42 0.5

and GG (0.5, 0.3) =

(0.5)2 − 2 · 0.5 · 0.3 + 0.3 ≈ 0.35. 1 − 0.3

So, GO  GG although GO (x, x) = x = GG (x, x). Let us recall that in Proposition 4, we showed that given GO ∈ Of1 and GG ∈ Gf2 with f1 ≤ f2 , then necessarily GO ≤ GG . Therefore, if we impose such a requirement to ensure that GO ≤ GG , and if we require that GO and GG are both f -diagonal, then by Proposition 9, we will obtain that f1 = f2 . However, we can be even more precise in such a case because, as a consequence of the following proposition, both operators, GO and GG , are unique. Actually, the result below states that the intersections of both classes Ofd1 ∩ Of2 and Gfd1 ∩ Gf2 are not empty if and only if f1 = f2 , and, in such a case, both intersections are singletons. Proposition 10. Let f1 , f2 ∈ Ω. GO ∈ Ofd1 ∩ Of2 and GG ∈ Gfd1 ∩ Gf2 if and only if: • f1 = f2 • GO (x, y) = f2 (min(x, y)) • GG (x, y) = f2 (max(x, y)). Proof. Let us assume that GO ∈ Ofd1 ∩ Of2 and GG ∈ Gfd1 ∩ Gf2 and let us begin by proving the second equality. By Corollary 2, we have that GO (x, y) ≤ f2 (min(x, y)) for all x, y ∈ [0, 1]. Let us prove that GO (x, y) ≥ f2 (min(x, y)) for all x, y ∈ [0, 1]:   GO (x, y) ≥ GO min(x, y), min(x, y) (by monotonicity of GO )   = f1 min(x, y) (GO if f1 -diagonal)   = GG min(x, y), min(x, y) (GG if f1 -diagonal)      ≥ f2 max min(x, y), min(x, y) = f2 min(x, y) (by Corollary 2)

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Thus, necessarily, GO (x, y) = f2 (min(x, y)). The equality GG (x, y) = f2 (max(x, y)) can be proved similarly. Finally, the proof of the first item is obtained straightforwardly from the other two because f1 (x) = GO (x, x) = f2 (min(x, x)) = f2 (x) for all x ∈ [0, 1]. The converse is straightforward. 2 Another interesting property of f -diagonal overlaps and f -diagonal groupings is that they can be characterized by boundary conditions. Proposition 11. Let f ∈ Ω and G be an overlap (resp. grouping). Then, G ∈ Ofd (resp. G ∈ Gfd ) if and only if:     f min(x, y) ≤ G(x, y) ≤ f max(x, y) for all x, y ∈ [0, 1]. Proof. Let us assume first that G is f -diagonal. Then, by monotonicity of G:     G(x, y) ≥ G min(x, y), min(x, y) = f min(x, y) and

    G(x, y) ≤ G max(x, y), max(x, y) = f max(x, y) .

Conversely, let us assume that f (min(x, y)) ≤ G(x, y) ≤ f (max(x, y)). Then:     f (x) = f min(x, x) ≤ G(x, x) ≤ f max(x, x) = f (x) Thus, every inequality above is indeed an equality, and therefore, G(x, x) = f (x). In other words, G is f -diagonal.

2

The following two results are important,5 although straightforward, consequences of the previous proposition. Corollary 7. Let f ∈ Ω, then: • f (min(x, y)) is the least element in Ofd • f (max(x, y)) is the greatest element in Gfd . Corollary 8. Let f ∈ Ω and GO ∈ Ofd (resp. GG ∈ Gfd ). Then, there exists a grouping GG ∈ Gfd (resp. an overlap GO ∈ Ofd ) such that GO ≤ GG . It is easy to see that, given f ∈ Ω, f (max(x, y)) is not an overlap because it does not satisfy the marginal conditions. Thus, by using Proposition 11, we can conclude that f (max(x, y)) is an upper bound of Ofd , but not its greatest element. The result below shows that f (max(x, y)) is something other than an upper bound because it coincides at almost every point with the supremum of Ofd . Proposition 12. Let f ∈ Ω. Then, the supremum of Ofd is

0 if x = 0 or y = 0 GO (x, y) = f (max(x, y)) otherwise GO ∈Ofd

and the infimum of Gfd is:

 1 if x = 1 or y = 1 GG (x, y) = f (min(x, y)) otherwise. GG ∈Gfd

5 The importance of these two results can be found in Section 4 after the requirements of coherence are introduced.

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Proof. Let us denote by G the operator:

0 if x = 0 or y = 0 G(x, y) = f (max(x, y)) otherwise Let us prove that G is the supremum of Ofd . We proceed as in the proof of Proposition 6. That is, by showing first that G is an upper bound of the set Ofd and, second, that G is the supremum of a specific subset of Ofd , G is the supremum of Ofd . Note that G is effectively an upper bound of Ofd thanks, on the one hand, to Proposition 11 and, on the other hand, to the fact that GO (x, y) = 0 if and only if x = 0 or y = 0 for all overlap GO . Now, consider the family of overlaps {GkO (x, y)}k defined by:    1 k−1  k ∈ N  {0} GkO (x, y) = f min(x, y) k · max(x, y) k It is easy to check that GkO ∈ Ofd for all k ∈ N  {0}. Let us show now that effectively supk {GkO } = G. Consider x, y ∈ [0, 1] and let us assume first that x = 0 (or equivalently y = 0). Then     supk GkO (x, y) = supk GkO (0, y) = supk {0} = 0 = G(x, y). Let us consider now, without loss of generality, that 0 = x ≤ y. Then, by using the continuity of f , we have:       1  1 k−1  k−1  = f supk x k · y k supk GkO (x, y) = supk f min(x, y) k · max(x, y) k 1

k1 −1

1

k2 −1

Now, by using that if k1 ≤ k2 , then x k1 · y k1 ≤ x k2 · y k2 , we have that:       1  1    k−1  k−1  = f lim x k · y k = f (y) = f max(x, y) supk GkO (x, y) = f supk x k · y k k→∞

as we want to prove. The other equality can be proved similarly as above, but in this case it is necessary to consider the family of groupings given by:  1 k−1  GkG (x, y) = f 1 − min(1 − x, 1 − y) k · max(1 − x, 1 − y) k for k ∈ N  {0}.

2

At the end of this section, we present a relationship between the notions of f -diagonal overlaps and duality. Take into account that duality is not enough to guarantee that an overlap and its dual grouping are both f -diagonal. Simply consider the t-norm product TP (x, y) = x · y and its dual t-conorm SP (x, y) = 1 − (1 − x) · (1 − y). In this case, TP is an x 2 -diagonal overlap, whereas SP is a (2x − x 2 )-diagonal grouping. The following result establishes a condition on negations in order to keep the same diagonal. Proposition 13. Let f ∈ Ω, G ∈ Ofd (resp. G ∈ Gfd ) with f bijective, and let n be a bijective negation. Then, the f -dual grouping (resp. f -dual overlap) Gn,f (x, y) with respect to n is also f -diagonal. Proof. Let us see that Gn,f (x, x) = f (x)    Gn,f (x, x) = f ◦ n GO f −1 ◦ n−1 (x), f −1 ◦ n−1 (x) = f ◦ n ◦ f ◦ f −1 ◦ n−1 (x) = f (x).

2

The following result relates the classes Ofd to the class Oxd (i.e., the set of idempotent overlaps). Specifically, it shows that there exists a bijection between both sets in the case that f is bijective. Proposition 14. Let GO ∈ Ofd (resp. GG ∈ Gfd ), where f ∈ Ω. If f is bijective, then there exists an idempotent overlap (resp. grouping) GO such that GO = f ◦ GO . Proof. The proof consists simply in checking that the operator GO (x, y) = f −1 (GO (x, y)) is an idempotent overlap. 2

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Fig. 1. Schema of the obtention of gradients from gray-scale images.

4. An illustrative example: Gradient operators from overlaps and groupings In this section, we present an illustrative example focused on edge detection for gray-scale images. We propose gradient operators that can be constructed with the help of overlaps and groupings. Our construction is similar to the one given in [6], but we allow here the use of overlaps and groupings instead of t -norms and t -conorms. Roughly speaking, the idea is to assign an interval of values to each pixel representing the variation of intensities in the neighborhood of this particular pixel. As a result, we can define an edge detector by associating “a pixel of an edge” with “a sufficiently large variation of intensities in the neighborhood of a pixel”.6 The resulting interval valued image represents the membership degree of a pixel in the set of edges of an image. Let us emphasize that what is described here is a pre-procedure, so further steps are needed in order to obtain the final edges of an image. In Fig. 1, we provide a possible method in three steps: Step 1 The creation of two images, one to represent how much darker the surroundings of each pixel are and the other to represent how much brighter the same surroundings are. Step 2 Combining the two images above to create an interval valued image. Step 3 A final procedure to measure the intensity variability represented by the interval valued image obtained in Step 2. It is worth mentioning that in this paper we only focus on Step 1 and we let the other steps for future research. Thus, in the examples used to illustrate the preprocessing, the second and third steps are obtained • by simply joining the two images obtained in Step 1 by considering them as the lower and upper bounds of the interval image and • by identifying the length of each interval with the variation of intensities associated to each pixel. To formalize the procedure, let us define an image as a mapping I: D ⊆ R2 → [0, 1]. The elements in D are called pixels, and the value I(x) ∈ [0, 1] represents the gray tone of the pixel x ∈ D, where 0 represents black and 1 white. 6 This association is made by the main approaches dealing with edge detection (as [9]).

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The neighborhood (or the set of surrounding pixels) of x ∈ D is defined as those pixels belonging to the window of radius δ ∈ N centered in x, i.e., Wδ (x) = {y ∈ D | x − y∞ ≤ δ}, where  · ∞ denotes the infinity norm. Finally, to determine the lower and upper bounds of an image I, we fix an overlap GO and a grouping GG and define the following two fuzzy sets on D:     LGO (I)(x) = min GO I(x), I(y)  y ∈ Wδ (x)     UG (I)(x) = max GG I(x), I(y)  y ∈ Wδ (x) G

Note that, thanks to the monotonicity of GO (resp. GG ), the darker (resp. brighter) the pixels in Wδ (x), the lower (resp. greater) the value LGO (I)(x) (resp. UGG (I)(x)). Thus, such values really are able to fulfill the goal of Step 1. Note also that in Step 2, these images are used later to create an interval image. Then, the fact that the lower construction is less than the upper construction is certainly welcome. However, this does not happen in general. The following result shows a necessary and sufficient condition to ensure LGO (I) ≤ UGG (I) for every image I. Actually, such a condition is closely related to the theoretic study presented in Section 3.1. Proposition 15. Let GO and GG be an overlap and a grouping, respectively. The inequality LGO (I)(x) ≤ UGG (I)(x) holds for every image I: D → [0, 1] and pixel x ∈ D if and only if GO ≤ GG . Proof. Let us assume first that GO ≤ GG and let I: D → [0, 1] be an image. Then, for each x ∈ D, we have:     LGO (I)(x) = min GO I(x), I(y)  y ∈ Wδ (x)     ≤ min GG I(x), I(y)  y ∈ Wδ (x)     ≤ max GG I(x), I(y)  y ∈ Wδ (x) = UG (I)(x) G

To prove the converse, let us assume that GO  GG , and let us show that there is an image I: D → [0, 1] such that UGG (I)(x) > LGO (I)(x) for some x ∈ D. Because GO  GG , there is (a, b) ∈ [0, 1]2 such that GO (a, b) > GG (a, b). Consider x ∈ D, and let us define the image I: D → [0, 1]:

a if y = x I(y) = b otherwise Then:

    LGO (I)(x) = min GO I(x), I(y)  y ∈ Wδ (x) = GO (a, b)     > GG (a, b) = max GG I(x), I(y)  y ∈ Wδ (x) = UGG (I)(x).

2

Another interesting point is that, if for every y ∈ Wδ (x) we have I(y) = I(x), then the interval value associated with x should have length zero. This is because, first, there is no variation of intensities in the surroundings of x, and second, the variation of intensities is given by the length of the interval. This is basically equivalent to saying that UGG (I)(x) = LGO (I)(x). Obviously, this is not guaranteed for the choice of arbitrary overlaps and groupings. Actually, the following example shows that it is neither guaranteed for arbitrary t -norms nor for t -conorms; so the freedom of [6] and [2] is just illusory. Example 8. Let us show that for any t -norm T and t -conorm S different from minimum and maximum, respectively, it is possible to obtain US (I)(x) = LT (I)(x) in a window Wδ (x) verifying I(y) = I(x) for all y ∈ Wδ (x). Because T is different from the minimum, there exists α ∈ [0, 1] such that T (α, α) < α. Note that S(α, α) ≥ α in any case. Consider now a window Wδ (x) assigning the value α to every pixel in Wδ (x). Then,         LT (I)(x) = min T I(x), I(y)  y ∈ Wδ (x) < α ≤ max S I(x), I(y)  y ∈ Wδ (x) = US (I)(x) Thus, in this case, in the center pixel of Wδ (x), a degree of variation of intensities is measured when obviously there is none. An extreme case is when the t -norm and t -conorm considered are the Łukasiewicz ones. In this case, if we consider an image assigning the degree 0.5 to every pixel, the values obtained for the Lower and Upper constructors are 0 and 1, respectively. In other words, the variation measured is the greatest one (the value 1), although there is no variation. Therefore, by using the Łukasiewicz t -norms and t -conorms, it is impossible to differentiate between a jump of intensities from 0 to 1 and an unvarying part of the image.

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The example above is enough to motivate the use of overlaps and groupings in the definition of the Lower and Upper constructors. The following result, which is closely related to Section 3.2, gives us the requirement we need to impose on overlaps and groupings in order to obtain the feature exposed above in our procedure. Proposition 16. Let GO and GG be an overlap and a grouping, respectively. For every image I: D → [0, 1] and pixel x ∈ D such that I(y) = I(x) for all y ∈ Wδ (x), we have UGG (I)(x) = LGO (I)(x) if and only if GO ∈ Ofd and GG ∈ Gfd for some f ∈ Ω. Proof. Let us begin by assuming that GO ∈ Ofd and GG ∈ Gfd for the same f ∈ Ω. Then, for any image I: D → [0, 1] and for each x ∈ D such that I(y) = I(x) for all y ∈ Wδ (x), we have:       LGO (I)(x) = min GO I(x), I(y)  y ∈ Wδ (x) = GO I(x), I(x)       = GG I(x), I(x) = max GG I(x), I(y)  y ∈ Wδ (x) = UGG (I)(x) To prove the converse, let us assume that GO ∈ Ofd1 and GG ∈ Gfd2 for f1 , f2 ∈ Ω with f1 = f2 . If we show that there is an image I: D → [0, 1] and x ∈ D such that for all y ∈ Wδ (x) we have I(y) = I(x), but UGG (I)(x) = LGO (I)(x), we will have completed the proof. Because f1 = f2 , then there is a ∈ [0, 1] such that f1 (a) = f2 (a). Let us define the image I(x) = a for all x ∈ D, i.e., the image assigning to every pixel the intensity a. Then:     LGO (I)(x) = min GO I(x), I(y)  y ∈ Wδ (x) = f1 (a)     = f2 (a) = max GG I(x), I(y)  y ∈ Wδ (x) = UGG (I)(x) for all x ∈ D.

2

Let us summarize the two results above. In order to obtain “coherent” results in the calculus of UGG (I)(x) and LGO (I)(x), it is necessary to choose one overlap GO and one grouping GG such that: • GO ≤ GG (property studied in Section 3.1), and • GO (x, x) = GG (x, x) for all x ∈ [0, 1] (property studied in Section 3.2). Hereafter, we will refer to the two properties above as requirements of coherence. Note that if we restrict the choice of our operators to t -norms and t -conorms (as is done in [2]), the first inequality always holds, but the second holds if and only if both operators are idempotent. Note that the only idempotent t -norm and t -conorm are the operators minimum and maximum, respectively. Thus, it is not surprising that in [2], the best results are obtained when these two operators are chosen. Note that due to Corollary 8 in Section 3.2 for a given overlap we can always find a grouping satisfying the requirements of coherence. So, as a specific case, allowing the use of overlaps and groupings provides more freedom at the time of choosing idempotent operators. This freedom is not trivial because it allows us to address the luminosity of an image in the following sense. Values in [0, 1] represent, somehow, intensities of gray. Thus, a subinterval [a, b] ⊆ [0, 1] determines a range of luminosity. That is, if every element y ∈ Wδ (x) satisfies I(y) ∈ [a, b], then the neighborhood of the pixel x has the luminosity given by the interval [a, b]. The following result shows how to choose overlaps and groupings that are more sensible at the time of measuring the variation of intensities under a certain range of luminosity. Proposition 17. Let I: D → [0, 1] be an image, and let G1O , G2O ∈ Ofd and G1G , G2G ∈ Gfd such that G1O (x, y) ≤ G2O (x, y) and G2G (x, y) ≤ G1G (x, y) for all x, y ∈ [a, b]. Then: • LG1 (I)(x) ≤ LG2 (I)(x) and O O • UG2 (I)(x) ≤ UG1 (I)(x) G

G

for all x ∈ D such that I(y) ∈ [a, b] for all y ∈ Wδ (x).

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Proof. Let x ∈ D such that I(y) ∈ [a, b] for all y ∈ Wδ (x), and let us show that LG1 (I)(x) ≤ LG2 (I)(x). O O  1    LG1 (I)(x) = min GO I(x), I(y) y ∈ Wδ (x) O     ≤ min G2 I(x), I(y)  y ∈ Wδ (x) = L 2 (I)(x) O

GO

The inequality UG2 (I)(x) ≤ UG1 (I)(x) is proved similarly. G

G

2

To interpret the result above, note that the images UGG (I)(x) and LGO (I)(x) will determine the bounds of the interval. Thus, the greater the distance between them, the greater the variation of intensity measured. In other words, the result above states that we can construct overlaps and groupings to detect more/less edges under a certain luminosity. Note also that the above result can be written in general terms, i.e., with [a, b] = [0, 1]. Corollary 9. Let G1O , G2O ∈ Ofd and G1G , G2G ∈ Gfd such that G1O ≤ G2O and G2G ≤ G1G . Then, LG1 (I(x)) ≤ O LG2 (I(x)) and UG2 (I)(x) ≤ UG1 (I)(x). O

G

G

Connecting the previous result with Corollary 7 in Section 3.2, we can ensure that the pair of t-norms maximum and minimum7 is the pair of overlap and grouping that provides • the greatest distance between UGG (I)(x) and LGO (I)(x) • among those pairs of idempotent operators holding the requirements of coherence. In other words, by choosing idempotent overlaps and groupings different from the minimum and maximum, respectively, we can only reduce the measure of variation of intensities, which implies a reduction in the number of pixels belonging to the edge of an image. Note that in some cases, for certain pairs of overlaps and groupings we can measure more variation of intensities than with minimum and maximum. In such a case, such overlaps and groupings cannot be idempotent. 4.1. Experimental results In order outline possible applications of the proposed research, we include here the results of some preliminary experiments that supports the idea of using overlaps and groupings for gradient operators. Let us begin by presenting an algorithm to construct the lower bound of the intervals associated with each pixel of an image by using an overlap function (GO ). Algorithm 1 Algorithm to construct the lower bound LGO (I) by using overlap functions. Input: An image, I, in greyscale. Output: The lower bound of intervals: a new gray-scale image J 1: Select a window size δ. 2: Select an overlap function GO . 3: for each pixel x of I do 4: Construct a matrix (2δ + 1) × (2δ + 1) so that all of the elements have the value of I(x). 5: Take the corresponding submatrix (2δ + 1) × (2δ + 1) of the image I centered on the pixel x. 6: Calculate the (2δ + 1) × (2δ + 1) values that are obtained by applying GO to each pair of pixels that occupy the same position in both matrices (2δ + 1) × (2δ + 1) considered in the two previous steps 4–5. 7: Calculate the min to the (2δ + 1) × (2δ + 1) values obtained in step 6. 8: Assign as the intensity of the pixel x of I the value obtained in step 7, LGO (I)(x). 9: end for

In an analogous way, an algorithm can be defined to construct the upper bound UGG (I). Simply substitute the overlap (GO ) by a grouping function (GG ) in step 2 and the minimum by the max in step 7. We will refer to this latter algorithm as Algorithm 2. 7 Note that the combination of these two operators gives the morphological gradient associated to a flat structuring element with the shape of a window.

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Fig. 2. Original images.

Subsequently, the interval valued image W : D → [0, 1]2 is constructed by assigning to each pixel x ∈ D the interval [LGO (I)(x), UGG (I)(x)], and the variation of intensity is measured by: GG ∇G (I)(x) = UGG (I)(x) − LGO (I)(x) O G

Note that ∇GOG (I)(x) is, in fact, a gray-scale image that represents, somehow, the edges of the original image I.

GG (I)(x) (for instance, the Rosin thresholding Thus, for the sake of the presentation we apply a threshold method to ∇G O method [15] with different thresholds of precision) and the result can be considered an edge detection method [5,10, 13,14]. In any case, note that in this work we do not focus on achieving an edge detector to procedure binary edges, rather, we focus on the potential use of overlaps and groupings in the definition of gradient operators to be used in other preprocessing like edge detection algorithms or image enhancement. To show the behavior of our algorithm, we use the following expressions of grouping and overlap functions:

• Idempotent overlap and grouping constructed by duality.  √ GO (x, y) = x · y and GG (x, y) = 1 − (1 − x) · (1 − y)

(1)

• Non-idempotent overlap and grouping constructed by x 2 -duality.   2 GO (x, y) = x · y and GG (x, y) = 1 − (1 − x) · (1 − y)

(2)

• A family of x 2n -diagonal overlaps and groupings without duality relationship. GO (x, y) = x n · y n and   GG (x, y) = max 2 · x 2n − x n − x n · y n + y n , 2 · y 2n − y n − x n · y n + x n • Another family of x 2n -diagonal overlaps and groupings without duality.   GO (x, y) = x n · y n and GG (x, y) = max x 2n , y 2n with n ∈ Q

with n ≤ 1/4

(3)

(4)

Next, we will apply our algorithm by means of Eqs. (1)–(4) to the original images shown in Fig. 2. For Eqs. (3), we have chosen n = 1/4 and for Eq. (4), n = 2 and n = 1/4. In every case, the length of windows considered is δ = 1, that is, windows of 3 × 3 pixels. The results are given in Appendix A (see Figs. A.1–A.4). We show, from left to right, the gray-scale images G obtained by Algorithm 1, Algorithm 2, and ∇GOG (after thresholding). Among the set of images obtained, we will focus on those obtained with respect to the fourth picture, labeled “Cameraman”, because we can find in it zones with different intensities of luminosity. Specifically, we will focus on three zones: the coat (a dark zone), the background buildings (a bright zone) and the grass (a zone with much noise after applying edge detection algorithms). Moreover, for the sake of the presentation, we have applied the Rosin thresholding method with precision 0.02 to the gray-image obtained by our procedure. The bi-valued images obtained after this thresholding are given in Fig. 3, where we can clearly see the behavior predicted by the theoretical results of this section. For instance, as Corollary 9 states, by using minimum–maximum, we obtain more edges than by using Eq. (1). It is also interesting to note that different pairs of overlaps and groupings can focus on different intensities in the image. For instance, by using Eq. (4) with n = 2 and n = 1/4, we focus differently on bright and dark zones, respectively, whereas (1) is more generally concerned with

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Fig. 3. Results of the algorithm on the image Cameraman w.r.t. different overlaps and groupings after applying the Rosin thresholding method with precision 0.02.

this aspect. Finally, note that with images obtained by using non-idempotent overlaps and groupings, we significantly reduce the noise produced by the grass with respect to the use of the operator minimum and maximum. 5. Conclusions and future work In this paper, we have presented results that ensure, first, that a certain overlap operator is less than or equal to a certain grouping, and second, that one overlap and one grouping coincide in the diagonal x = y. Specifically, the notions of f -bound grouping, f -bound overlap, f -diagonal grouping and f -diagonal overlap have been introduced to achieve both goals. We have shown that given an f -bound overlap GO , any f -bound grouping with f ≤ f is greater than or equal to GO . Additionally, we have proven that for the case f > f , although there exists some f -bound grouping GG such that GO  GG , we can always find an f -bound grouping greater than or equal to GO . Moreover, results concerning the structure of the set of f -bound groupings (f -bound overlaps, f -diagonal grouping and f -diagonal overlap) have been presented. Finally, we have related the construction of dual overlaps and groupings to the notions of f -bound and f -diagonal. Specifically, we have shown that the dual construction is insufficient to guarantee that an overlap is less than or equal to one of its dual groupings. Subsequently we have defined a new construction of dual operators induced by the original definition of duality but that preserves, somehow, the property of being f -bound (resp. f -diagonal). Finally, we have shown that, as a consequence of this feature, we can construct dual groupings GG from certain overlaps GO , verifying that GO ≤ GG (resp. GO (x, x) = GG (x, x)) and vice-versa. We have shown the potential importance of these theoretical results to measure variation of intensities in images. We have taken as a basis for such a goal the algorithm presented originally in [2]. We have shown that we can build the Lower and Upper constructors by using overlap functions instead of t -norms and grouping functions instead of t-conorms, respectively. We have also shown that we can obtain incoherent results in this preprocessing, even in the case where we use t-norms and t-conorms. Specifically, on the one hand, we can obtain lower bounds non-lesser than upper bounds, and on the other hand, we can obtain a positive measurement of variation in invariant windows. To avoid such shortcomings, we have required two conditions, called coherence requirements and introduced in Section 4. Such requirements are closely related to the theoretical study performed in the previous sections, and we have shown that the only t-norms and t-conorms satisfying them are the minimum and maximum. Summarizing, the theoretical study presented in this paper is motivated by the introduction of degrees of freedom in the operators used to define the Lower and Upper constructors in the algorithm given in [2]. Finally, we have shown in our experiments that such a freedom is not trivial because the use of overlaps and groupings can address different luminosities in images. As future work, other theoretical studies about overlaps and groupings should be developed. In the applied plane, it is interesting to seek for further applications of the theory presented in this paper. For instance, the gradient operators defined here can be further improved and used in various image preprocessing algorithms such as edge detection or enhancement.

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Acknowledgements This work was supported by the European Regional Development Fund projects CZ.1.05/1.1.00/02.0070 and CZ.1.07/2.3.00/30.0010 and by Spanish Ministry of Science funds through projects TIN2013-40765-P and TIN2012-39353-C04-04. Appendix A

Fig. A.1. First row: results of applying Lower and Upper constructors to the House. From the second to sixth row: results of applying Eqs. (1)–(4), respectively to the House.

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Fig. A.2. First row: results of applying Lower and Upper constructors to the Fireworks. From the second to sixth row: results of applying Eqs. (1)–(4), respectively to the Fireworks.

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Fig. A.3. First row: results of applying Lower and Upper constructors to the Plate. From the second to sixth row: results of applying Eqs. (1)–(4), respectively to the Plate.

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Fig. A.4. First row: results of applying Lower and Upper constructors to the Cameraman. From the second to sixth row: results of applying Eqs. (1)–(4), respectively to the Cameraman.

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