Sup-Compact and Inf-Compact Representations of W ... - IOS Press

Fundamenta Informaticae 45 (2001) 283–294

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IOS Press

Sup-Compact and Inf-Compact Representations of W-Operators Junior Barrera and Ronaldo Fumio Hashimoto ∗ Departamento de Ciˆencia da Computac¸a˜ o Instituto de Matem´atica e Estat´ıstica. Universidade de S˜ao Paulo. S˜ao Paulo, Brazil. [email protected]; [email protected]

Abstract. It is well known that any W -operator can be represented as the supremum (respectively, infimum) of sup-generating and (respectively, inf-generating) operators, that is, the families of supgenerating and inf-generating operators constitute the building blocks for representing W -operators. Here, we present two new families of building blocks to represent W -operators: compositions of sup-generating operators with dilations and compositions of inf-generating operators with erosions. The representations based on these new families of operators are called, respectively, supcompact and inf-compact representations, since they may use less building blocks than the classical sup-generating and inf-generating representations. Considering the W -operators that are both antiextensive and idempotent –in a strict sense–, we have also gotten a simplification of the sup-compact representation. We have also shown how the inf-compact representation can be simplified for any W -operator such that it is extensive and its dual operator is idempotent –in a strict sense–. Furthermore, if the W -operators are openings (respectively, closings), we have shown that this simplified sup-compact (respectively, inf-compact) representation reduces to a minimal realization of the classical Matheron’s representations for translation invariant openings (respectively, closings).

Keywords: Mathematical Morphology, W -Operators, Compact Representation, Openings and Closings. ∗

Address for correspondence: Departamento de Ciˆencia da Computac¸a˜ o, Instituto de Matem´atica e Estat´ıstica., Universidade de S˜ao Paulo, S˜ao Paulo, Rua do Mat˜ao 1010 - Cidade Universit´aria, 05508-900 - S˜ao Paulo - SP, Brazil

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1. Introduction Operators in the family of W -operators (i.e., translation invariant and locally defined in a window W ) are often used to solve image analysis problems [11]. The desire of developing a formal approach to the automatic design of these operators has motivated the study of their representations [2]. Banon and Barrera [1] presented a general sup-representation (respectively, inf-representation) for translation invariant operators as the supremum (respectively, infimum) of sup-generating (respectively, inf-generating) operators, that is, the families of sup-generating and inf-generating operators constitute the building blocks for representing translation invariant operators. Barrera and Salas [4] extended these representations to W -operators. The sup-representation and inf-representation of W -operators have some nice properties as, for example, their simple structures allow to formalize prior knowledge about families of operators considered in automatic operator design. However, their parallel structures usually are not efficient for computation in conventional sequential machines. We find in the literature some works [3, 7, 6] that study the problem of transforming representation structures of W -operators. In this paper, we present two new families of building blocks to represent W -operators: compositions of sup-generating operators with dilations and compositions of inf-generating operators with erosions. The representations based on these new families of operators are called, respectively, supcompact and inf-compact representations, since they may use less building blocks than the classical sup-representations and inf-representations, respectively. A corollary to the results presented in this paper is that the sup-representation can be converted into the sup-compact representation by simple computational algorithms. This method is useful, for example, to optimize programs of Morphological Machines designed via learning algorithms, since the operators generated automatically by this approach are in the sup-representation and the sup-compact representation has better performance. Considering the W -operators that are both anti-extensive and idempotent –in a strict sense–, we have gotten a simplification of the sup-compact representation. We have also shown how the inf-compact representation can be simplified for any W -operator such that it is extensive and its dual operator is idempotent –in a strict sense–. Furthermore, in case the W -operators are openings (respectively, closings), we have gotten a minimal realization of the classical Matheron’s representation for translation invariant openings (respectively, closings) as a supremum (respectively, infimum) of morphological openings (respectively, closings) from the simplified sup-compact (respectively, inf-compact) representation. Following this introduction, Section 2 recalls the sup-representation and inf-representation of W operators. In Section 3, we present the sup-compact and inf-compact representations of W -operators. In Section 4, we show a non increasing specialization of the sup-compact and inf-compact representations. In Section 5, we deduce a minimal realization of Matheron’s representation for translation invariant openings (respectively, closings) from the sup-compact (respectively, inf-compact) representation. In Section 6, we present some examples of conversions of the sup-representation into the sup-compact representation. Finally, in Section 7, we give some conclusions and future directions of this research.

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2. Representation by Sup-Generating and Inf-Generating Operators Let E be a non empty set, which is an Abelian group with respect to a binary operation denoted by +. The zero element of (E, +) is denoted by o. Elements of E will be denoted by lower case letters a, b, c, ... Let P(E) denote the powerset of E. Elements of P(E) will be denoted by capital letters A, B, C, ... Collections of elements of P(E) will be denoted by capital script letters A, B, C, ... ˇ = {x ∈ E : −x ∈ X}. The reflection of a subset X ∈ P(E) is the subset X For any X ∈ P(E) and h ∈ E, Xh denotes the translation of X by h, that is, X h = {x ∈ E : x − h ∈ X}. A mapping from P(E) to P(E) is called an operator. The operators will be denoted by lower case Greek letters α, β, γ, ... The set of all operators will be denoted by Ψ. An operator ψ is called translation invariant (t.i.) if and only if (iff), for any h ∈ E and X ∈ P(E), ψ(Xh ) = ψ(X)h . Let W be a subset of E. An operator ψ is called locally defined within W iff, for any h ∈ E and for any X ∈ P(E), h ∈ ψ(X) ⇔ h ∈ ψ(X ∩ Wh ). An operator ψ is called a W -operator iff it is both t.i. and locally defined within W . The set of all W -operators will be denoted by Ψ W . The set ΨW with the partial order relation ≤, inherited from the inclusion relation on sets, constitutes a Complete Boolean Lattice [4]. The supremum and infimum of two operators ψ1 and ψ2 in ΨW verify, respectively, (ψ1 ∨ ψ2 )(X) = ψ1 (X) ∪ ψ2 (X) and (ψ1 ∧ ψ2 )(X) = ψ1 (X) ∩ ψ2 (X), for any X ∈ P(E). Let A, B ∈ P(W ) such that A ⊆ B. Two examples of operators in Ψ W are the sup-generating and W the inf-generating operators denoted, respectively, by λ W A,B and µA,B , or, simply, by λA,B and µA,B , and they are given by, for any X ∈ P(E), λW A,B (X) = {h ∈ E : A ⊆ X−h ∩ W ⊆ B}, W ˇ )∪B ˇ 6= W ˇ }. µA,B (X) = {h ∈ E : X−h ∩ Aˇ 6= ∅ or (X−h ∩ W When B = W the sup-generating (respectively, inf-generating) operator is called erosion (respectively, dilation) by A and it is denoted by εA (respectively, δA ), that is, εA = λA,W (respectively, δA = µA,W ). Let X, A ∈ P(E). The Minkowski addition and subtraction of X and A are the subsets given, respectively, by X ⊕ A = ∪{Xh : h ∈ A} and X A = ∩{X−h : h ∈ A}. One can easily prove that δA (X) = X ⊕ A and εA (X) = X A [1]. A fundamental property of dilation is that it commutes with supremum, that is, for any X 1 , X2 ∈ P(E), δA (X1 ∪ X2 ) = δA (X1 ) ∪ δA (X2 ) [8, p. 85]. The operator ι ∈ ΨW defined by ι(X) = X, for any X ∈ P(E), is called identity operator. The complement to a subset X ∈ P(E), denoted X c , is X c = {x ∈ E : x 6∈ X}. The operator ν defined by ν(X) = X c , for any X ∈ P(E), is called the negation operator. Note that νν = ι. The dual operator to the operator ψ, denoted by ψ ∗ , is ψ ∗ = νψν. Clearly, ψ ∈ ΨW iff ψ ∗ ∈ ΨW [4]. ˇ ˇ W W W ∗ An example of dual operators is λW ˇB ˇ , that is, µA, ˇB ˇ = (λA,B ) [4]. Consequently, if A,B and µA, B = W , we have that δA and εAˇ are dual operators, that is, εAˇ = (δA )∗ . The kernel of ψ ∈ ΨW is the set K(ψ) given by K(ψ) = {X ∈ P(W ) : o ∈ ψ(X)}. An important property of K(·) is that it is two-sides increasing [4], that is, for any ψ 1 , ψ2 ∈ ΨW , ψ1 ≤ ψ2 ⇔ K(ψ1 ) ⊆ K(ψ2 ).

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Let A, B ∈ P(W ) such that A ⊆ B. An interval with end-points A and B is the subset [A, B] of P(W ) given by [A, B] = {X ∈ P(W ) : A ⊆ X ⊆ B}. Collections of intervals contained in P(W ) will be denoted by capital bold face letters A, B, C, ... The cardinality of a collection of intervals X and of a subset X will be denoted, respectively, by |X| and |X|. An interval [A, B] in a collection of intervals X is called maximal if there does not exist an interval 0 [A , B 0 ] in X, distinct from [A, B], such that [A, B] ⊆ [A 0 , B 0 ]. The collection of all maximal intervals of X is denoted M ax(X). Note that when W is finite, for any [A, B] ∈ X there exists [A 0 , B 0 ] ∈ M ax(X) such that [A, B] ⊆ 0 [A , B 0 ]. Particularly, a collection of intervals X is called maximal if X = M ax(X). For any operator ψ ∈ ΨW , the basis of ψ is the collection B(ψ) of all maximal intervals contained in K(ψ), that is, B(ψ) = M ax({[A, B] ⊆ P(W ) : [A, B] ⊆ K(ψ)}). In particular, B(λ A,B ) = {[A, B]} [4]. Barrera and Salas [4] proved the following result. Proposition 2.1. Any operator ψ ∈ ΨW can be described in terms of sup-generating (respectively, infgenerating) operators by the sup-representation (respectively, inf-representation) given by W V ˇ ∗ ψ = {λA,B : [A, B] ∈ B(ψ)} and ψ = {µW ˇB ˇ : [A, B] ∈ B(ψ )}. A, An important property of the sup-representation (respectively, inf-representation) structure given in Proposition 2.1 is that it characterizes uniquely a W -operator ψ, in other words, if two sup-representations (respectively, inf-representations) of operators have the same basis, they, in fact, represent the same operator [4].

3. Sup-Compact and Inf-Compact Representations In this section, we will present two new decomposition structures for W -operators that simplify the suprepresentation and the inf-representation, in the sense that the number of building blocks used in these decompositions is smaller or, in the worst case, equal. Because of this property, we will call these two new decompositions, respectively, the sup-compact representation and inf-compact representation. Let us state an equivalence relation on a generic collection of maximal intervals X. Let [A, B] and [A0 , B 0 ] be two generic elements of X. We will say that [A, B] and [A 0 , B 0 ] are equivalent under translation iff one can be built by a translation of the other, that is, [A, B] ≡ [A 0 , B 0 ] iff there exists 0 ]. h ∈ E such that [A, B] = [A0−h , B−h As the equivalence under translation is an equivalence relation (i.e., reflexive, symmetric and transitive), the set of their equivalence classes (i.e., the sets composed exactly of all the equivalent elements in X) constitute a partition of X. We will denote by P(B(ψ)) the set of all the equivalence classes (under translation) on B(ψ). We will denote by E(ψ) a set composed by exactly one element of each equivalence class in P(B(ψ)), that is, E(ψ) is a set such that |E(ψ)| = |P(B(ψ))| and for each Y ∈ P(B(ψ)) there exists [A, B] ∈ E(ψ) such that [A, B] ∈ Y.

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Let [A, B] ∈ E(ψ), that is, [A, B] is an interval which represents an equivalence class. Let us denote by C[A,B] the equivalence class represented by [A, B]. Following this notation, we have P(B(ψ)) = {C[A,B] : [A, B] ∈ E(ψ)}. We will denote by C[A,B] the subset of E that is composed by all h ∈ E such that [A −h , B−h ] is in the equivalence class represented by [A, B], that is, for any [A, B] ∈ E(ψ), C [A,B] = {h ∈ E : [A−h , B−h ] ∈ B(ψ)}. Hence, the equivalence class represented by [A, B] is given by C[A,B] = {[A−h , B−h ] : h ∈ C[A,B] }. Note that, since C[A,B] ⊆ B(ψ) and B(ψ) is a collection of maximal intervals, then, clearly, so is C [A,B] . Let X be a collection of maximal intervals and C ∈ P(W ), we denote X ⊕ C = {[A c , Bc ] : c ∈ C, [A, B] ∈ X}. ˇ For any operator ψ ∈ ΨW 0 , the Let C and W 0 be two subsets of E such that W = W 0 ⊕ C. basis of the operator δC ψ ∈ ΨW (i.e., the operator built by the composition of ψ with δ C ) is given by ˇ ˇ = {X ∈ P(E) : B(δC ψ) = M ax({[A, B] ⊆ P(W ) : [A, B] ⊆ ∪(B(ψ) ⊕ C)}), where ∪(X ⊕ C) ˇ ˇ ∃[A, B] ∈ X ⊕ C, X ∈ [A, B]} [4]. Particularly, note that if B(ψ) ⊕ C is a collection of maximal ˇ intervals, then B(δC ψ) = B(ψ) ⊕ C. The following theorem gives the sup-compact representation for any W -operator. Theorem 3.1. If ψ ∈ ΨW , then ψ =

W {δC[A,B] λA,B : [A, B] ∈ E(ψ)}.

Proof: First, we show that, if [A, B] ∈ E(ψ), then {[A, B]}⊕ Cˇ[A,B] is a collection of maximal intervals. In fact, {[A, B]} ⊕ Cˇ[A,B] = {[Ah , Bh ] : h ∈ Cˇ[A,B] } = {[A−h , B−h ] : h ∈ C[A,B] } = C[A,B] . Consequently, since C[A,B] is a collection of maximal intervals, then so is {[A, B]} ⊕ Cˇ[A,B] . Hence, for any ψ ∈ ΨW , we have that P(B(ψ)) = {B(δC[A,B] λA,B ) : [A, B] ∈ E(ψ)}, since, for any [A, B] ∈ E(ψ), C[A,B] = {[A, B]} ⊕ Cˇ[A,B] = B(λA,B ) ⊕ Cˇ[A,B] (since B(λA,B ) = {[A, B]}) = B(δC[A,B] λA,B ). Therefore, the basis of the operator defined by this representation is B(ψ) and, consequently, it is a representation for the operator ψ. t u The next theorem gives the inf-compact representation for any W -operator. Theorem 3.2. If ψ ∈ ΨW , then ψ =

V ˇ ∗ {εCˇ[A,B] µW ˇB ˇ : [A, B] ∈ E(ψ )}. A,

Proof: By Theorem 3.1, we have, for any X ∈ P(E),

ψ ∗ (X) = ∪{δC[A,B] λA,B (X) : [A, B] ∈ E(ψ ∗ )}. By definition of ψ ∗ , we have that, ψ = νψ ∗ ν. Thus, for any X ∈ P(E),

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ψ(X)

= = = = = = =

νψ ∗ (ν(X)) ∗ ν(∪{δC[A,B] λW A,B (ν(X)) : [A, B] ∈ E(ψ )}) (since ψ ∗ ∈ ΨW ) ∗ ν(∪{δC[A,B] λW A,B ν(X) : [A, B] ∈ E(ψ )}) ∗ ∩{νδC[A,B] λW A,B ν(X) : [A, B] ∈ E(ψ )} ∗ ∩{νδC[A,B] ννλW A,B ν(X) : [A, B] ∈ E(ψ )} (since νν = ι) W ∩{(νδC[A,B] ν)(νλA,B ν)(X) : [A, B] ∈ E(ψ ∗ )} ˇ ∗ ∩{εCˇ[A,B] µW ˇB ˇ (X) : [A, B] ∈ E(ψ )}. A, ˇ

∗ Therefore, ψ = ∧{εCˇ[A,B] µW ˇB ˇ : [A, B] ∈ E(ψ )}. A,

t u

The sup-compact representation for W -operators, given in Theorem 3.1, may be still condensed. If with there exists X ⊆ E(ψ) W such that, for any [A, B] ∈ X, CW= C [A,B] , then, since δC commutes W supremum, we have that {δC[A,B] λ[A,B] : [A, B] ∈ X} = {δC λ[A,B] : [A, B] ∈ X} = δC ( {λ[A,B] : [A, B] ∈ X}). Note that with this factorization we economize |X| − 1 dilations.

4. Non Increasing Specialization In this section, we will show how the sup-compact representation is simplified for W -operators that are both anti-extensive and idempotent in a strict sense. We will also show how the inf-compact representation is simplified for extensive operators ψ ∈ Ψ W such that ψ ∗ is idempotent in a strict sense. An operator ψ ∈ ΨW is called anti-extensive (respectively, extensive) iff ψ ≤ ι (respectively, ι ≤ ψ). An operator ψ ∈ ΨW is called idempotent iff ψψ = ψ. A W -operator ψ is called increasing iff, for any X, Y ∈ P(E), X ⊆ Y ⇒ ψ(X) ⊆ ψ(Y ). The following result is an immediate consequence of the definition of dual operator. Proposition 4.1. Let ψ be an operator. (a) (b) (c)

ψ is increasing iff ψ ∗ is increasing; ψ is idempotent iff ψ ∗ is idempotent; ψ is extensive iff ψ ∗ is anti-extensive.

An interesting property of the anti-extensive operators is that, for any X ∈ K(ψ), o ∈ X, since o ∈ ψ(X) ⊆ X. Hence, o ∈ A, for any [A, B] ∈ B(ψ). Proposition 4.2. If an operator ψ ∈ ΨW is anti-extensive, then, for any [A, B] ∈ E(ψ), C [A,B] ⊆ A. Proof: For any [A, B] ∈ E(ψ), x ∈ C[A,B] is equivalent to [A−x , B−x ] ∈ B(ψ) and, since ψ is antiextensive, o ∈ A−x , consequently, x ∈ A. Hence, C[A,B] ⊆ A. t u

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Definition 4.3. An operator ψ ∈ ΨW satisfies the self bounding assumption if for any [A, B] ∈ B(ψ), X ∈ [A, B] implies that ψ(X) ∩ W ∈ [A, B]. On one hand, note that the self bounding assumption does not imply idempotence. For example, for any C ∈ P(W ) such that o ∈ C, δC satisfies the self bounding assumption, but it is not idempotent. On the other hand, an idempotent operator does not satisfy necessarily the self bounding assumption. For example, for any A, B ∈ P(W ) such that o ∈ A ⊂ B and |B| − |A| ≥ 2, the operator ψ = δ A λA,A ∨ δA λB,B is idempotent, but does not satisfy the self bounding assumption, since B ∈ [B, B] ∈ B(ψ) and ψ(B) = A ∈ / [B, B]. We say that an operator ψ ∈ ΨW is idempotent in a strict sense iff ψ is idempotent and satisfies the self bounding assumption. The next proposition gives a sufficient condition for the idempotence of non necessarily increasing operators. Proposition 4.4. If an operator ψ ∈ Ψ W is anti-extensive and satisfies the self bounding assumption, then ψ is an idempotent operator. Proof: On one hand, if x ∈ ψ(ψ(X)), then o ∈ ψ(ψ(X −x )), since ψ is t.i., which implies that o ∈ ψ(X−x ),since ψ is anti-extensive, and, consequently, x ∈ ψ(X), since ψ is t.i.. On the other hand, if x ∈ ψ(X), then X−x ∩ W ∈ K(ψ), since ψ is t.i. and locally defined in W , that implies that ∃[A, B] ∈ B(ψ) : X−x ∩ W ∈ [A, B], that implies that, ∃[A, B] ∈ B(ψ) : ψ(X −x ∩ W ) ∩ W ∈ [A, B], since ψ satisfies the self bounding assumption, that implies that ψ(X −x ∩ W ) ∩ W ∈ K(ψ), that implies that o ∈ ψ(ψ(X−x ∩W )∩W ), that implies that o ∈ ψ(ψ(X−x ∩W )) and, in the same way, o ∈ ψ(ψ(X−x )), since ψ is locally defined in W , and, consequently, x ∈ ψ(ψ(X)), since ψ is t.i.. t u In Theorem 4.6, we prove that operators of the type δ A λA,B , extensively studied by Ronse [10], constitute a building block family for the sup-compact representation of anti-extensive and idempotent –in a strict sense– operators. For that, let us first present the following proposition. Proposition 4.5. If an operator ψ ∈ ΨW satisfies the self bounding assumption, then, for any [A, B] ∈ E(ψ), ∀x ∈ A, [A−x , B−x ] ⊆ K(ψ). Proof: For any [A, B] ∈ E(ψ) and Y ∈ [A, B], if x ∈ A, then x ∈ ψ(Y ) ∩ W , since ψ satisfies the self bounding assumption and A is the least element of [A, B]. Thus, as x ∈ ψ(Y ), we have that o ∈ ψ(Y −x ), t u since ψ is t.i., and, consequently, [A −x , B−x ] ⊆ K(ψ). Theorem W 4.6. If an operator ψ ∈ ΨW is anti-extensive and satisfies the self bounding assumption, then ψ = {δA λA,B : [A, B] ∈ E(ψ)}.

Proof: This theorem is an immediate consequence of Propositions 4.2 and 4.5. Proposition 4.2 guarantees that B(ψ) is complete, while Proposition 4.5 guarantees that all other possible intervals generated by this representation are included in some interval of B(ψ). t u Theorem 4.7 proves that operators of the type ε A µA,B constitute a building block family for the inf-compact representation of the extensive operators ψ ∈ Ψ W such that ψ ∗ is idempotent in a strict sense.

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Theorem 4.7. If an operator ψ ∈ ΨW is such that ψ is extensive and ψ ∗ satisfies the self bounding assumption, then ˇ

∗ ψ = ∧{εAˇ µW ˇB ˇ : [A, B] ∈ E(ψ )}. A,

Proof: Since ψ is extensive, then, by Proposition 4.1, ψ ∗ is anti-extensive. Furthermore, as ψ ∗ also satisfies the self bounding assumption, by Theorem 4.6, we have that, for any X ∈ P(E), ψ ∗ = ∨{δA λA,B : [A, B] ∈ E(ψ ∗ )}. Since ψ = νψ ∗ ν, then, for any X ∈ P(E), ψ(X)

= = = = = = =

νψ ∗ (ν(X)) ∗ ν(∪{δA λW A,B (ν(X)) : [A, B] ∈ E(ψ )}) ∗ ν(∪{δA λW A,B ν(X) : [A, B] ∈ E(ψ )}) ∗ ∩{νδA λW A,B ν(X) : [A, B] ∈ E(ψ )} ∗ ∩{νδA ννλW A,B ν(X) : [A, B] ∈ E(ψ )} (since νν = ι) W ∩{(νδA ν)(νλA,B ν)(X) : [A, B] ∈ E(ψ ∗ )} ˇ ∗ ∩{εAˇ µW ˇB ˇ (X) : [A, B] ∈ E(ψ )} A,

ˇ

∗ Therefore, ψ = ∧{εAˇ µW ˇB ˇ : [A, B] ∈ E(ψ )}. A,

t u

We should remark that anti-extensive (respectively, extensive) and idempotent operators that do not satisfy the self bounding assumption may not be representable by the decompositions given by Theorem 4.6 (respectively, Theorem 4.7). Note that the computational implementation of anti-extensive operators that satisfy the self bounding assumption by the decomposition of Theorem 3.1 is more efficient than their implementation by the decomposition of Theorem 4.6, since, for any [A, B] ∈ E(ψ), C [A,B] ⊆ A. An invariant of an operator ψ is a subset X ∈ P(E) such that ψ(X) = X. The set of all invariants of ψ will be denoted by Inv(ψ). Note that as we are considering operators that are anti-extensive and idempotent, for any [A, B] ∈ B(ψ), A is an invariant of ψ, since, ψ(A) ⊆ A (by anti-extensivity), ψ(A) ∈ K(ψ) (by idempotence) and [A, B] is maximal.

5. Specialization for Translation Invariant Openings and Closings In this section, we will show the relation between the classical Matheron’s representation for t.i. openings (respectively, closings) and the sup-compact (respectively, inf-compact) representation. Related results to the ones presented in this section were obtained by Dougherty [5]. A W -operator ψ is called an opening (respectively, closing) if it is increasing, anti-extensive (respectively, extensive) and idempotent. For any A ∈ P(W ), the morphological opening γ A (respectively, morphological closing ϕ A ) is the operator built by the composition of the erosion ε A (respectively, dilation δA ) with the dilation δA (respectively, erosion εA ), that is, γA = δA εA (respectively, ϕA = εA δA ). The following proposition [8, p. 89, Eq. 4.56] gives a property of morphological openings.

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Proposition 5.1. Let A ∈ P(W ). For any X ∈ P(E), γA (X) = ∪{Ah : h ∈ E and Ah ⊆ X}. The kernel of a morphological opening is given in the next proposition. Proposition 5.2. If A ∈ P(W ), then K(γ A ) = {X ∈ P(W ) : A−h ⊆ X e h ∈ A}. Proof: For any Y ∈ P(W ), we have that, Y ∈ K(γA )

⇔ ⇔ ⇔ ⇔ ⇔ ⇔

o ∈ γA (Y ) o ∈ ∪{Ah : h ∈ E e Ah ⊆ Y } (by Proposition 5.1) ∃h ∈ E : o ∈ Ah e Ah ⊆ Y ∃h ∈ E : h ∈ A e A−h ⊆ Y ∃h ∈ A : A−h ⊆ Y Y ∈ {X ∈ P(W ) : A−h ⊆ X e h ∈ A}.

t u

A morphological opening (respectively, closing) is independent of translation, that is, for any h ∈ E, γA = γAh (respectively, ϕA = ϕAh ). Matheron [9, Prop. 7.1.3] proved the following decomposition theorem for t.i. openings and closings. Theorem 5.3. Let ψ ∈ Ψ. (a) (b)

W If ψ is a t.i. opening, then ψ = {γA : A ∈ Inv(ψ)} V If ψ is a t.i. closing, then ψ = {ϕAˇ : A ∈ Inv(ψ ∗ )}.

For any A, B ∈ P(W ), we say that B is A-open if there exists X ∈ P(E) such that B = δ A (X). If B is A-open, then γB ≤ γA and ϕA ≤ ϕB [11, p. 54]. This property permits the construction of granulometries and, by this reason, we called it granulometric absorption. Now, let us show that the sup-compact (respectively, inf-compact) representation of t.i. openings (respectively, closings) is an efficient realization of Matheron’s representation. Note that, as a consequence of Proposition 4.1, an operator ψ ∈ Ψ is an opening iff ψ ∗ is a closing. Let ψ ∈ ΨW be an opening or a closing. The increasing character ψ implies that its kernel has the inheritance property, that is, for any X, Y ∈ P(W ) such that X ⊆ Y , if X ∈ K(ψ), then Y ∈ K(ψ). Hence, for any [A, B] ∈ B(ψ), we have that B = W and the sup-generating (respectively, inf-generating) operators of the sup-representation (respectively, inf-representation) of ψ become erosions (respectively, dilations) [1]. Consequently, the sup-generating (respectively, inf-generating) operators of the sup-compact (respectively, inf-compact) representation also become erosions (respectively, dilations). Let ψ ∈ ΨW be an opening. As ψ is increasing, anti-extensive and idempotent, for any [A, B] ∈ B(ψ), A is an invariant of ψ (i.e., ψ(A) = A) and, consequently, ψ satisfies the self bounding assumption. As ψ is increasing, anti-extensive and satisfies the self bounding assumption, the decomposition structure of Theorem 4.6 reduces to an union of morphological openings. Taking W = E, this is a realization of

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the classical Matheron’s result for the representation of t.i. openings. Furthermore, the following theorem shows that it is a minimal realization, in the sense of the granulometric absorption, that is, morphological openings that are absorbed by other morphological openings are not considered in this decomposition. Theorem 5.4. Let ψ ∈ ΨW be an opening. If [A, W ] ∈ B(ψ), then there does not exist h ∈ E and A0 ∈ P(W ) such that A0 6= A, A is A0 -open and [A0h , W ] ∈ B(ψ). Proof: Let [A, W ] ∈ B(ψ). Suppose to the contrary that there exist h ∈ E and A 0 ∈ P(W ) such that A0 6= A, A is A0 -open and [A0h , W ] ∈ B(ψ). Since [A0h , W ] and [A, W ] are in B(ψ) and A0 6= A, then we can choose E(ψ) such that [A 0h , W ] and [A, W ] are in E(ψ). By the independence of translation of γ A0 and by simplification of Theorem 4.6 to t.i. openings, we have that γA0h = γA0 ≤ ψ. Since A is A0 -open and A0 6= A, then, by granulometric absorption, γA < γA0 . As K(·) is two-sides increasing and A is A 0 -open, we have that γA < γA0 = γA0h ≤ ψ ⇔ K(γA ) ⊂ K(γA0 ) = K(γA0h ) ⊆ K(ψ). Since [A, W ] ∈ E(ψ) and ψ is an opening (consequently, ψ is anti-extensive), then o ∈ A. By Proposition 5.2, we have that K(γA ) = {X ∈ P(W ) : A−z ⊆ X and z ∈ A} e K(γA0 ) = {X ∈ P(W ) : A0−z ⊆ X and z ∈ A0 }. Thus, since o ∈ A, then A ∈ K(γA ). Furthermore, as A ∈ K(γA ) ⊂ K(γA0 ) ⊆ K(ψ), we have that there exists z ∈ A 0 such that A0z ⊂ A e A0z ∈ K(ψ). Hence, [A, W ] ⊂ [A0z , W ] and [A0z , W ] ∈ B(ψ) and, consequently, [A, W ] 6∈ B(ψ), since [A, W ] is not a maximal interval in B(ψ). But it contradicts the hypothesis that [A, W ] ∈ B(ψ). Therefore, there does not exist such A0 and h. t u Let ψ ∈ ΨW be a closing. Since ψ is a closing (i.e., increasing, extensive and idempotent), then, by Proposition 4.1, ψ ∗ is increasing, anti-extensive and idempotent. Thus, for any [A, B] ∈ B(ψ ∗ ), A is an invariant of ψ ∗ (i.e., ψ ∗ (A) = A) and, consequently, ψ ∗ satisfies the self bounding assumption. As ψ ∗ is increasing, anti-extensive and satisfies the self bounding assumption, the decomposition structure of Theorem 4.7 reduces ψ to an intersection of morphological closings, since for any [A, B] ∈ B(ψ ∗ ), we have that B = W and the inf-generating operators are dilations [1]. Taking W = E, this is a realization of the classical Matheron’s result for the representation of t.i. closings. Furthermore, the following result shows that it is a minimal realization, in the sense of the granulometric absorption, that is, morphological closings that are absorbed by other morphological closings are not considered in this decomposition. Theorem 5.5. Let ψ ∈ ΨW be a closing. If [A, W ] ∈ B(ψ ∗ ), then there does not exist h ∈ E and A0 ∈ P(W ) such that A0 6= A, A is A0 -open and [A0h , W ] ∈ B(ψ ∗ ). Proof: Since ψ ∗ is an opening, by Theorem 5.4, there does not exist h ∈ E and A 0 ∈ P(W ) such that A0 6= A, A is A0 -open and [A0h , W ] ∈ B(ψ ∗ ). t u

6. Conversion of the Basis Representation to the Compact Representation We have implemented a software for the automatic transformation of the sup-representation (respectively, inf-representation) into the sup-compact (respectively, inf-compact) representation. The task of this software is first identify the equivalence classes in the basis and after identify the ones that can be factorized by having the same C[A,B] .

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Note that in the case of anti-extensive operators that satisfy the self bounding assumption, in the sup-compact representation we have a flexibility for chosen C [A,B] as any C such that C[A,B] ⊆ C ⊆ A. The idea here is to choose a C that maximizes the factorization effect. In the following, we give some examples of applications of this software in order to obtain supcompact representation from sup-representation. By duality, one can easily apply the same software to transform inf-representations into inf-compact representations. In the following examples, we will denote a subset of W by an array of ones (representing the points that are in the subset) and zeros (representing points that are not in the subset), with a bold" face element indicating the origin. For # 0 1 example, the subset {(−1, 0), (0, 0), (0, 1)} is represented by . In the following, consider A = 1 1       1 0 1 0 1 h i h i h i       1 1 1 , B =  1 , C =  1 1 1 , D =  0 , F = 1 1 0 and G = 0 1 1 .

1 0 1 0 1 Let ψ1 = δA εA ∨ δB εB . The basis of ψ1 is given by B(ψ1 ) = {[Aa , W ] : a ∈ A} + {[Bb , W ] : b ∈ B}, where + denotes the union of disjoint sets. Hence, in this case, the operator given is already in the sup-compact representation form. Now, let ψ 2 = δA εA ∨ δC εC . The basis of ψ2 is B(ψ2 ) = {[Aa , W ] : a ∈ A} + {[Cd , W ] : d ∈ D}. Hence, ψ2 can be rewritten as ψ2 = δA εA ∨ δD εC . Note that ψ1 and ψ2 are both increasing operators and that the sup-compact representation form of ψ 2 is better than the original one, since |D| < |C|. Let ψ3 = δA λA,A ∨ δC λC,C . The basis of ψ3 is given by B(ψ3 ) = {[Aa , Aa ] : a ∈ A} + {[Cc , Cc ] : c ∈ C}. Hence, the operator is already in the sup-compact representation form. Now, let ψ 4 = δA λA,C ∨ δC λC,C .The basis of ψ4 is B(ψ4 ) = {[Aa , Ca ] : a ∈ A} + {[Cd , Cd ] : d ∈ D}. Hence, ψ4 can be rewritten as ψ4 = δA λA,C ∨ δD λC,C . Note that ψ3 and ψ4 are both non-increasing operators and that the sup-compact representation form of ψ 4 is better than the original one, since |D| < |C|. ˇ Let ψ5 = δF λF,C ∨ δG λG,C . The basis of ψ5 is B(ψ5 ) = {[Ff , Cf ] : f ∈ Fˇ } + {[Gg , Cg ] : g ∈ G}. ˇ = Fˇ + h, where h = (−1, 0). Therefore B(ψ 5 ) = {[Ff , Cf ] : f ∈ Fˇ } + {[Ff , Ch+f ] : f ∈ Fˇ }. But G Hence, ψ5 can be rewritten as ψ5 = δF λF,C ∨ δF λF,Ch = δF (λF,C ∨ λF,Ch ). In this case, with the factorization we economize one dilation.

7. Conclusion We have introduced two new families of building blocks to represent W -operators: compositions of sup-generating operators with dilations and compositions of inf-generating operators with erosions. The representations based on these new families of operators are called, respectively, sup-compact and infcompact representations. An important property of the sup-compact (respectively, inf-compact) representation is that it may use a smaller number (or, in the worst case, the same number) of building blocks than the sup-representation (respectively, inf-representation). This property implies that the operators have better performance when implemented in the sup-compact or inf-compact representations. We have presented a simplification of the sup-compact representation for the family of idempotent, in a strict sense, and anti-extensive operators. We have also shown how the inf-compact representation can be simplified for any operator such that it is extensive and its dual operator is idempotent in a strict

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sense. Furthermore, from these simplified forms, we have gotten a minimal realization of Matheron’s representation for t.i. openings and closings. Finally, we have shown how the sup-representation (respectively, inf-representation) can be transformed into the sup-compact (respectively, inf-compact) representation. This method can constitute one of the first steps for approaching the quite complex problem of changing (when it is possible) automatically by computational algorithms the sup-representation into sequential representations.

Acknowledgements The authors thank to prof. Henk Heijmans for some interesting discussions about the representation of anti-extensive and idempotent operators. This work has been partially supported by “Olivetti do Brasil” and ProTeM-CC/CNPq through the AnIMoMat project, contract 680067/94-9.

References [1] G. J. F. Banon and J. Barrera. Minimal Representations for Translation-Invariant Set Mappings by Mathematical Morphology. SIAM J. Appl. Math., 51(6):1782–1798, December 1991. [2] J. Barrera, E. R. Dougherty, and N. S. Tomita. Automatic Programming of Binary Morphological Machines by Design of Statistically Optimal Operators in the Context of Computational Learning Theory. Journal of Electronic Imaging, 6(1):54–67, January 1997. [3] J. Barrera and R. F. Hashimoto. Compact Representation of W-Operators. In 10th Annual Symposium on Electronic Imaging, pages 84–94, San Jose, California, January 1998. SPIE Press. [4] J. Barrera and G. P. Salas. Set Operations on Collections of Closed Intervals and their Applications to the Automatic Programming of Morphological Machines. Journal of Eletronic Imaging, 5(3):335–352, July 1996. [5] E. R. Dougherty. Minimal Representation of τ -Openings via Pattern Bases. Pattern Recognition Letters, 14(3):1029–1033, 1994. [6] R. F. Hashimoto and J. Barrera. From the Sup-Decomposition to Sequential Decompositions. 2000. Submitted to Journal of Mathematical Imaging and Vision. [7] R. F. Hashimoto, J. Barrera, and C. E. Ferreira. A Combinatorial Optimization technique for the Sequential Decomposition of Erosions and Dilations. Journal of Mathematical Imaging and Vision, 13(1):17–33, August 2000. [8] H. J. A. M. Heijmans. Morphological Image Operators. Academic Press, 1994. [9] G. Matheron. Random Sets and Integral Geometry. John Wiley, 1975. [10] C. Ronse. A Lattice-Theoretical Morphological View on Template Extraction in Images. Journal of Visual Communication and Image Representation, 7(3):273–295, 1996. [11] J. Serra. Image Analysis and Mathematical Morphology. Academic Press, 1982.