Why Mathematical Morphology Needs Quantales

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Proceedings Paper: Stell, JG (2009) Why mathematical morphology needs quantales. In: Wilkinson, MHF and Roerdink, JBTM, (eds.) Extended abstracts of posters at the International Symposium on Mathematical Morphology, ISMM09. International Symposium on Mathematical Morphology, ISMM09, 24-27 August 2009, Groningen, The Netherlands. Institute for Mathematics and Computing Science, University of Groningen , 13 - 16.

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Paper: Stell, JG (2009) Why Mathematical Morphology Needs Quantales. In: Wilkinson, MHF and Roerdink, JBTM, (eds.) Proceedings of the International Symposium on Mathematical Morphology, ISMM09. International Symposium on Mathematical Morphology, ISMM09, 24-27 August 2009, Groningen, The Netherlands. Institute for Mathematics and Computing Science, University of Groningen , pp. 13-16.

White Rose Research Online [email protected] Why Mathematical Morphology Needs Quantales

John G. Stell∗ [email protected] School of Computing, University of Leeds, U.K.

Abstract The purpose of this extended abstract is to explain why another algebraic structure, called a quantale, The importance of complete lattices in mathemati- also deserves the attention of researchers interested cal morphology is well-known. There are some as- in the essence of mathematical morphology. The pects of the subject that can, however, be clarified title is a deliberate allusion to that of [7]. While by using complete lattices equipped with the ad- mathematical morphology does still need complete ditional structure of a binary operation subject to lattices, I will argue that for some purposes it also conditions making them quantales. This extended needs complete lattices with some extra structure, abstract shows how quantales, and more generally namely quantales. quantale modules, can be used to capture a num- Before giving the formal details, I will indicate ber of constructions for dilations in a uniform way. briefly why these structures are relevant to mor- The treatment is partly an exposition of material phology. Some further motivation appears in the that is not technically novel, but which provides a penultimate paragraph of Section 4. A quantale valuable conceptual perspective on mathematical is a complete lattice L where, besides the usual morphology. lattice operations, there is a binary operation (sat- Index Terms— Quantale, Complete Lattice, Dila- isfying certain conditions) on the elements of the tion, Relation lattice. This enables us to regard the elements of the lattice both as carrying the partial order structure, and simultaneously as being dilations on L . 1 Introduction (I will follow the practice of taking a dilation on L to mean a sup-preserving operation.) The bi- Complete lattices have been used in mathematical nary operation, in this view, models the composi- morphology as a framework which unifies several tion of dilations. This general situation is familiar important examples which originally appeared to to anyone with experience of mathematical mor- be distinct. Having such a framework not only al- phology. Taking E to be a Euclidean space, such lows a deeper understanding of these examples, it as R2, the subsets of E function both as binary im- has provided and continues to provide a founda- ages and as structuring elements which have asso- tion for advances in the subject. Additionally, as ciated dilations. In this case the powerset P(E) Heijmans and Ronse made clear [5, p253] is a quantale with the binary operation being the usual Minkowski addition, typically denoted ⊕. “an abstract approach provides a deeper For a full treatment of quantales, including the insight into the essence of the theory history of the development of the concept, the book (e.g., about what assumptions are min- by Rosenthal [8] may be consulted. The relevance imally required to have certain prop- of quantales to mathematical morphology has al- erties) and links it to other, sometimes ready been observed by Russo [9] in a paper that rather old, mathematical disciplines.” includes some of the examples I present below. ∗Supported by EPSRC project ‘Ontological Granularity for My aim here is partly to draw the attention of the Dynamic Geo-Networks’ morphology community to these known applications of quantales. However, the full range of ap- 3 Quantale-valued Functions plications of equation (1) below, including in particular relations, does not appear have been noted In mathematical morphology the lattice theoretic before. approach has been especially useful in situations where we study functions from a set X to a complete lattice. In general the set X carries some al- 2 Quantales gebraic structure, such as the group of translations of Euclidean space. L Definition 1. A quantale is a complete lattice However, a substantially weaker structure than equipped with a binary operation, &, which is as- an abelian group is adequate for the most basic L L sociative and for every A ⊆ and every x ∈ parts of the theory. We shall say that X is a partial satisfies (W A)& x = W(A & x) and x &(W A) = semigroup if it has a binary operation, , not nec- W(x & A) where A & x denotes {a & x | a ∈ A} essarily defined for all x, y ∈ X, such that when- and analogously for x & A. ever x, y, z ∈ X, the two expressions x (y z) The use of & to denote the binary operation and (x y) z are either (a) both undefined or (b) follows [8]. We have already mentioned one ex- both defined and equal to each other. For a lattice L L ample of a quantale, the set of subsets of a Eu- , the lattice of functions from X to with the L X clidean space E, with dilation as the binary opera- usual ‘pointwise’ ordering will be denoted . tion. Three further examples are as follows. Theorem 1. Let (L , &) be a quantale, and (X, ) 1. A Boolean algebra with the usual ordering a partial semigroup. Then L X is a quantale with and with & as ∧. a binary operation &X where for all F,G ∈ L X ,

2. The completed reals R = R ∪ {−∞, +∞} X (F & G)(x) = _ F (y)& G(z). (1) with + provided we require x + (−∞) = yz=x −∞ + x = −∞ for all x ∈ R. The theorem is proved by a straighforward ver- 3. The set of all relations on a set A, with com- ification of the necessary properties. Equation (1) position of relations. provides a unified treatment of several definitions Given a quantale (L , &) and x ∈ L we can of dilation, as the following examples demonstrate. define an operation δx by δx(y) = y & x. We see that δx is a dilation and δxδy(z) = δy&x(z). It 3.1 Binary morphology follows by well-known properties of adjunctions between complete lattices that each δx has an as- Here L is the two element Boolean algebra, and sociated upper adjoint (also called right adjoint in X is a Euclidean space with being the usual ad- the literature) denoted εx and that εxεy = εx&y. dition. The condition y z = x becomes y = We can define a second binary operation on L x − z. In this case &X is the usual dilation of bi- by x _ y = εy(x) and a short calculation will nary images by binary structuring elements. show that (x _ y) _ z = x _ (z & y). If we _ were to write & as ⊕ and as ⊖ we would ob- 3.2 Greyscale morphology tain the equations (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) and (x ⊖ y) ⊖ z = x ⊖ (z ⊕ y) where the unfamiliar We can take X to be Rn or Zn with as the order of y and z in the last equation is required as usual vector addition. By taking (L , &) to be the binary operation need not be commutative. (R, +), equation (1) gives the familiar operation It is also possible to define another family of of greyscale dilation, as found for example in [5, ′ dilations by δx(y) = x & y with associated ero- p250]. ′ sions εx. This situation has already been noted To use a bounded set of grey levels, whether in mathematical morphology by Roerdink [6] who an interval in R or in Z, requires specifying the explains the connection with residuated lattices, binary operation on the set of grey levels. It is now which are closely related to quantales. well-known, and was pointed out in [7], that using truncated addition with arbitrary functions in this cannot be formulated at the appropriate level of ab- context can lead to problems. The fact that since straction. Although the statement and derivation of truncated addition is not associative we fail to have equation (2) in the context of relations is entirely a quantale, is a useful way of understanding why straightforward, it provides a useful way of think- these problems arise. ing about the role of composition which is not al- ways made explicit in recent accounts such as [3, 3.3 Fuzzy morphology pp885–890]. There are several different approaches to fuzzy morphology [2] and not all the proposed definitions 3.5 Fuzzy relations of fuzzy dilations will fit into the quantale frame- There is a standard notion of a fuzzy relation tak- work. The central idea in fuzzy morphology is to ing values in L as a function A × A to L . If work with fuzzy sets which are functions from a we use a type of fuzzy logic with a conjunction, crisp set X to a complete lattice of truth values & such that (L , &) is a quantale then equation (1) which in many formulations is taken to be the in- gives the usual composition of such relations, tak- terval [0, 1] ⊆ R. ing the partial semigroup structure as in the crisp In several cases the lattice of truth values to- relation case. gether with the fuzzy conjunction will be a quan- Although the identification of fuzzy relations tale, in particular this happens with what Drossos with functions A × A → L is firmly established, called a “t-norm-like structure” [4]. When (X, ) it does mean that fuzzy relations no longer corre- is as in the binary or greyscale cases, and when the spond to dilations L A → L A. This is easily seen fuzzy conjunction makes the lattice of truth values when L is the 3-element chain {⊥, 0, ⊤} and A into a quantale, then equation (1) gives the appro- has one element. In this case L A×A has only three priate definition of fuzzy dilation. elements but there are six sup-preserving mappings L A → L A. 3.4 Relations Heijmans and Ronse showed [5, p270] that di- L A It is well-known that the lattice of all relations on a lations on are in a one-to-one correspondence set A provides a convenient description of the lat- with functions which assign to each element of L tice of all dilations on P(A). We can give the set A × A a dilation on . This suggests it would L A×A a partial semigroup structure where (a, b) be worth exploring the possibility that an -fuzzy (c, d) is defined iff b = c, and when this happens relation on A could be defined as an asssignment the value is (a, d). By taking the quantale (L , &) of this form. In this case the equation (1) provides to be the two-element Boolean algebra and X to us with the definition of composition of relations. be A × A with as just defined, we see that equation (1) gives the composition of relations. To observe that the lattice of relations on A is 4 Quantale Modules isomorphic to the lattice of dilations on P(A) re- veals only part of the picture. The fact that these We can think of a quantale as a complete lattice, L are isomorphic not merely as lattices but also as , where the elements of the lattice also serve as L quantales is what lies at the heart of many of the dilations on . This is a rather special case, be- manifestations of the equation cause we may well have a lattice of dilations which act on a structure other than this lattice itself. The δxδy = δy⊕x (2) quantale of relations has elements that we think of as being dilations on a powerset rather than on the in the mathematical morphology literature. set of relations. This is one of the key reasons why mathematical morphology does need quantales. The addi- Definition 2. A quantale module consists of a quan- tional structure provides a way of ‘indexing’ or pa- tale, (L , &), and a complete lattice, M, together rameterizing dilations without which equation (2) with a binary operation · : M ×L → M such that the following hold for all A ⊆ M, all X ⊆ L , all [2] I. Bloch. Duality vs. adjunction for fuzzy x, y ∈ L and all a ∈ M mathematical morphology and general form of fuzzy erosions and dilations. Fuzzy Sets 1. a · (x & y) = (a · x) · y, and Systems, 160:1858–1867, 2009. 2. ( A) · x = {a · x ∈ M | a ∈ A}, W W [3] I. Bloch, H. J. A. M Heijmans, and C. Ronse. 3. a · (W X) = W{a · x ∈ M | x ∈ X}. Mathematical morphology. In M. Aiello, The operation m 7→ m · x provides a dila- I. Pratt-Hartmann, and J. van Benthem, edi- tion δx : M → M, and we have δxδy = δy&x. tors, Handbook of Spatial Logics, chapter 14, Erosions can be obtained from these dilations and pages 857–944. Springer, 2007. many well-known properties of erosions and dila- [4] C. A. Drossos. Generalized t-norm struc- tions follow in this abstract setting. tures. Fuzzy Sets and Systems, 104:53–59, To treat the most essential features of dilations 1999. and erosions in an algebraic setting that uses only complete lattices means that we are unable to cap- [5] H. J. A. M Heijmans and C. Ronse. The al- ture the full structure. Just having the lattice M we gebraic basis of mathematical morphology. can consider dilations and erosions on M and their I. Dilations and erosions. Computer Vision composites and the order relation between them, Graphics and Image Processing, 50:245– but although we may form δxδy we cannot talk 295, 1990. about y & x or δy&x. Abramsky and Vickers [1] introduce quantale [6] J. B. T. M. Roerdink. Mathematical Mor- modules as models where the quantale L is a set phology with Non-Commutative Symmetry of observations of computational processes, and Groups. In E. R. Dougherty, editor, Math- the lattice M is a set of entities which change un- ematical Morphology in Image Processing, der these observations. This is closely related to chapter 7, pages 205–254. Marcel Dekker, the common description of mathematical morphol- 1993. ogy as a body of techniques in which an image is [7] C. Ronse. Why mathematical morphology ‘probed’ by a structuring element to obtain infor- needs complete lattices. Signal Processing, mation. We can view a dilation δ ∈ L as an ob- 21:129–154, 1990. servation which may be made of an image I ∈ M and the result of making the observation δ on I is [8] K. I. Rosenthal. Quantales and their applica- I · δ ∈ M. The · of the module models the action tions, volume 234 of Pitman Research Notes of observing, while the & in L models the ability in Mathematics. Longman, 1990. to build compound observations. Understanding morphology via quantales and [9] C. Russo. Quantale modules and their quantale modules can be expected to lead to use- operators with applications. to appear in ful links with other topics, especially in the area Journal of Logic and Computation. Preprint of logic, where quantales have important applica- at www.dmi.unisa.it/people/ tions to linear logic for example. In morphology, russo/www/publications.html, the graph morphology in [10] can usefully be stud- 2008. ied in this setting and other examples can be found [10] J. G. Stell. 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