Journal of Universal Computer Science, vol. 6, no. 4 (2000), 447-459 submitted: 30/4/99, accepted: 28/1/00, appeared: 28/4/00 Springer Pub. Co. Region-based Discrete Geometry M. B. Smyth Department of Computing Imp erial College London SW7 2BZ mbs@do c.ic.ac.uk Abstract: This pap er is an essay in axiomatic foundations for discrete geometry in- tended, in principle, to b e suitable for digital image pro cessing and more sp eculatively for spatial reasoning and description as in AI and GIS. Only the geometry of convex- ity and linearity is treated here. A digital image is considered as a nite collection of regions; regions are primitiveentities they are not sets of p oints. The main re- sult Theorem 20 shows that nite spaces are sucient. The theory draws on b oth \region-based top ology" also known as mereotop ology and abstract convexity theory. Key Words: Discrete geometry, regions, mereotop ology, convexity. 1 Intro duction Digital top ology, as a study of the connectivity prop erties of digital images and the spaces in which they lie, is by now a fairly well-develop ed discipline. The approach is either graph-theoretic, as in [14], or else top ological in the strict sense, as in [8]. See also [9] and references given there. In our own work in this area, the emphasis has b een on an amalgamation of digital top ology with ordinary top ology, achieved by working with a single category in which all the spaces including the graphs exist as ob jects. In this category TopGr, for top ological graphs the usual \continuous" spaces arise as inverse limits of the digital spaces [15,16,17]. In the present pap er we take up the task of extending this approach to geometry prop er. We are not aware of any previous axiomatic theory of digital geometry, although it has o ccasionally b een prop osed that such a theory should b e develop ed for example by Zeeman [18]. Most often what is studied is some n version of the \grid" mo del: in e ect, Z taken with some graph adjacency structure, and with n restricted to b e 2 or 3. What we shall undertake here is an axiomatic approach to the geometry of convexity and linearity: no attempt will b e made to deal with parallels and con- gruence for the time b eing. Abstract convexity theory, esp ecially as develop ed by W. Prenowitz [12], is our exemplar; the problem is to adapt this material to \discrete" geometry. See also Copp el [4] for a recent treatment of abstract convexity theory. Both Prenowitz and Copp el provide non-trivial nite mo dels of parts of the theory. These, however, like the nite geometries of combinatorial theory [5], cannot b e said to resemble the spaces traditionally studied Euclidean or non-Euclidean, nor to b e capable of b eing viewed as discrete or approximate versions of these. In order to obtain discrete spaces that are suitable for digital image pro cess- ing, we shall adopt the region-based approach of what is known as mereotop ol- ogy [2, 6, 13, 7]. Pixels or voxels and other simple convex regions are not 448 Smyth M.B.: Region-based Discrete Geometry considered as sets of p oints, but as primitiveentities which are sub ject to a re- lation of partial order \part of " and a symmetric binary relation expressing closeness of two regions the \connection predicate". Our theory maythus b e considered as a sort of fusion of mereotop ology with abstract convexity theory. The primitives we adopt are not to o far removed from those whichhave b een adopted in some previous work on spatial reasoning [3] and references given there. The main di erences b etween what we are attempting here and what was done in those works are as follows. First, we aim for a mathematically adequate account of convexity. A simple criterion for this, often adopted in studies of abstract convexity, is that it p ermits the derivation of abstract versions of three famous theorems in convexity, namely those of Radon, Helly and Caratheo dory; see for example [1]. The axioms concerning convexityhavetobechosen so as to p ermit this. A second di erence is that we are willing to admit some p oints as regions. We are mainly interested in discrete geometries, and it do es not greatly matter whether the nitely many p oints which can b e found in a b ounded region of a discrete space are treated as primitive or as nite lters of regions. For the general theory, it would be desirable to resolve this question, however. A further imp ortant technical di erence from many region-based theories [13,10] is that we do not require our structures to b e Bo olean algebras; in particular, there is no op eration corresp onding to the intersection of regions. Despite these di erences, we think it p ossible that some development of the theory presented here will prove to b e useful for studies in spatial reasoning. n Note : As a reminder, Radon's Theorem for < says that any set of n + 2 or more p oints can b e partitioned into two disjoint subsets whose convex closures intersect. It is enough to lo ok at the case n = 2 to grasp the meaning of the theorem; while the case n = 1 is already of interest, as we shall see later. The Helly and Carathe o dory theorems are closely related to the Radon theorem, but they will not b e considered explicitly here. 2 Axiomatics Our basic structure is a triple Q,1,, where Q is a set of regions , 1 is a symmetric binary relation connection , and is a commutative asso ciative op eration product, satisfying the following Axioms: A 1 is \almost re exive": 9X:A 1 X A 1 A The e ect of this will b e that a non-null region is connected with itself. W B Fusion For any collection B Q, there exists a unique region B such that _ X 1 B , 9B 2B:X 1 B: W C Distr. distributes over . D Extension For any A; B 2 Q, there exists a region B=A such that X 1 B=A , AX 1 B Notice that o ccurrences of the op erator are generally omitted. Smyth M.B.: Region-based Discrete Geometry 449 E A 1 C & B 1 D _ B=A 1 D=C AD 1 BC E A stronger form of Axiom E A 1 C & B 1 D B=A 1 D=C AD 1 BC Some immediate consequences of these axioms: Prop osition 1. The relation , de nedonQ by * A B 8X:A 1 X B 1 X W isapartial order, with respect to which is the sup operation. Proof. Trivially, is a pre-order. It is a partial order since, if 8X:A 1 X , W B 1 X , then, by uniqueness of fusion, A = B = fAg.Itis then a trivial veri cation that fusion is the join for this partial order. Prop osition 2. Extension distributes over join. Proof. _ _ _ _ X 1 A= B,X B1 A _ _ , X B 1 A B 2B ,9A 2A; B 2B:X B 1 A ,9A 2A; B 2B:X 1 A=B: There follow some comments on the axioms: W { Taking Q together with and ,wehave a non-unital quantale W { Since wehave ,wehave a complete lattice. However, this is not required V to b e distributive, and has little signi cance compared with and 1. { It is easy to see that for any A 2 Q we have the \complement" W fX j:X 1 Ag, which satis es the usual join and meet conditions for Bo olean complement. By the preceding remark, however, we do not thereby obtain a Bo olean algebra. Moreover the complement so de ned is not in general an involution thus it is not an ortho complement. { An immediate consequence of E but not of E is that, if A; B are non- null regions, then A=B is non-null. A mo del based on a closed b ounded subset of Euclidean space will fail Axiom E see the interpretation 3 to be given in a moment, but may nevertheless be of interest for discrete geometry. In the pro ofs b elowwehave taken care to use E only when E is not sucient. Prenowitz works with p oint-based axioms corresp onding to Axiom E , whereas Bryant& Webster have, in e ect, only the weaker form E. { It may sometimes be convenient to drop the uniqueness requirement from Axiom B. The join is then a sp eci ed op eration satisfying the prop erty given in B, and the order Prop. 1 can only b e claimed to b e a pre-order. In the conclusions of one or two of the subsequent Prop ositions, equality should b e replaced by equivalence. 450 Smyth M.B.: Region-based Discrete Geometry { The main remaining axiom to b e considered is an Axiom of Order see Sec- tion 5. We next consider some of the main intended mo dels of the axiom system. n 1. Euclidean Q is: the subsets of < . A 1 B means: A meets B . The pro duct is de ned in the rst instance for p oints: x; y if x 6= y x y = x if x = y By distributivity, the de nition extends to arbitrary regions.