Robust Representation and Analysis of Geo-information

One of the perennial difficulties in the representation of spatial data in database structures is that of the finite nature of the number representations. The vast majority of the mathematical analyses of the issue are based on the theory of real numbers, and metric space topology. The realization of this theory within a finite precision computer representation has been largely overlooked. Implementations often ignore this issue, resulting in unexpected errors, according to real number mathematics. In exceptional cases these errors will invalidate analysis computations such as map overlay, or cause data to become unusable after a coordinate transformation or projection, because the algorithms assume ‘correct’ real-number behaviour (which is not the case in a finite digital computer). Therefore alternative computer representations are being investigated: the “Constraint Spatial Database” approach (Kanellakis, Kuper et al. 1995), the “Rational Polygon” (Lemon and Pratt 1998), the “Realms” approach (Guting and Schneider 1993), the “Dual Grid” (Lema and Guting 2002), and the “Regular Polytope” (Thompson 2005a). These approaches share various difficulties which have, so far, inhibited their use in commercially available software, but have the major advantage that they provide support for a rigorous logic, with no complex “special case” programming being necessary to allow for finite arithmetic issues. In particular, the Regular Polytope approach implements the full Region Connection Calculus (RCC) (Randell, Cui et al. 1992) in a rigorous form such that the representation itself forms a topological space. Nevertheless, some practical issues remain with this approach. This paper will address the following issues with the Regular Polytope approach, and propose a possible alternative solution: • The Regular Polytope storage mechanism differs from the more familiar point/line/polygon paradigm commonly used in GIS, and requires non-trivial conversion routines. • The calculations require the use of very large precision integer arithmetic (as does the Dual Grid approach, and to an even larger extent, the Rational Polygon). • The storage requirements are significantly larger than required for simple polygon encoding. • It is not easy to map this storage form to/from the topological encoding form (Louwsma 2003). This proposed solution, the “Approximated Polytope”, while retaining the rigour of the Regular Polytope will address these issues, providing a mechanism which can use floating point arithmetic for the day-to-day calculations, uses a storage form more closely aligned to the point/line/polygon paradigm, and has space requirements somewhere between those of the Regular Polytope and those of polygon encoding. The Approximated Polytope is compatible with, and can utilise, the topological encoding method of storing spatial data. Therefore, the Approximated Polytope is potentially the first practical solution for robust representation and analysis of geo-information. 1. Outline of the Paper Section 2 consists of a discussion of the more conventional vertex-based representations. This is followed by a brief description of the regular polytope representation, in sections 3 to 6. Sections 7 to 9 describe the proposed “Approximated Polytope” model, based on a simplified database schema, including a discussion of conversion to and from other representations. Section 10 introduces the possibility of topological encoding within the model, while Section 11 an 12 discuss practical implementation issues, and further research possibilities respectively. 2. Vertex-based Representations In two-dimensional applications, the “Point/Line/Polygon” paradigm for the representation of spatial features is well entrenched, albeit with some significant variations (van Oosterom, Quak et al. 2003), and provides a degree of comfort in the user. This is spite of some serious difficulties in terms of rigorous definitions of concepts such as validity, and equality (Thompson 2005a). The available 3D structures take various forms (Arens, Stoter et al. 2005), with no one having proved to be the best in all circumstances (Zlatanova, Rahman et al. 2004). In this paper, we use the term “vertex based” representation to cover all ways to model spatial data in 2 or more dimensions based on point coordinates of vertices as the major determinants of the shape and position of the objects. For example, in the 3D FDS (Formal Data Structure) (Molenaar 1990), and the “Simplified Spatial Model” (SSM) (Zlatanova 2000), the node is defined as a point with coordinates (x,y,z), while all other geometric objects are defined in terms of sets of nodes or higher order constructive objects. This is true of virtually all 2 and 3 dimensional spatial data models, regardless of the level of topological encoding supported (Ellul, Haklay et al. 2005). One major challenge for 3D modelling is the fact that any definition of a face by more than three vertices runs the risk that that face may not be unambiguously planar. This could occur in two ways – the point values can be incorrectly calculated, or rounding errors can cause a small departure from planarity. Two different approaches may be taken 1) a tolerance value may be applied (provided that the departure of the face from planarity does not exceed a given tolerance, it is accepted); or 2) the faces may be triangulated (since any three points are always co-planar). The first of these approaches adds a certain level of extra complexity, and like all approaches that use a tolerance, raises issues of non-transitivity of operations (e.g. where A = B, B = C, but A ≠ C). The second strategy, of triangulation or tetrahedronisation of the objects, is quite acceptable for topographic applications, but in many applications, the loss of identity of the faces is significant. 3. The Regular Polytope A regular polytope as described by Thompson (2005a) represents spatial objects as the union of a finite set of (possibly overlapping) convex polytopes, which are in turn defined as the intersection of a finite set of half spaces (in 3D, half planes in 2D). These half spaces (planes) are defined by finite precision integer representations (3 values in 2D, 4 in 3D etc.). The concept of a domain-restricted rational number x, has been useful in the definition of continuity of regular polytopes. A dr-rational number x is defined as a pair of integers (I,J) of restricted value (not potentially infinite) –M1 ≤ I ≤ M1, 0 < J ≤ M2 (M1 and M2 being large positive integers), This is interpreted as x = I/J. In the following discussion, capital letters (such as X) will be used for to represent computational integers, or the integer values they represent. Lower case letters (such as x) will be used for rational or domain-restricted rational numbers, but occasionally lower-case will be used for small integer values (e.g. i=1..n). No notational distinction is made in this paper between computational operations +,-,., =, etc, and the mathematical operations they implement, since the integer and rational number arithmetic available in computers is exact. There is however, a distinction to be made. For example, it must be remembered that A+B as a computational operation may result in overflow. 1 By contrast, floating point is not exact, and it cannot be asserted that if a := b*c; (as a computation and assignment) then a = bc (as a mathematical equation). 4. Half Space Definition In 3D a half space H(A,B,C,D) is defined as the set of all dr-rational points P(x,y,z), -M ≤ x,y,z < M for which computational evaluation of the following inequalities yields these results: (A.x + B.y + C.z + D) > 0 or [(A.x + B.y + C.z + D) = 0 and A > 0] or 2 [(B.y + C.z + D) = 0 and A=0 and B>0] or [(C.z + D) = 0 and A=0, B=0 and C>0] Where M is the range of integer values allowed for point representations. The values of the integers A,B,C and D define the half space. In 3D applications, we place the restriction that –M < A,B,C < M, -3M < D <3M, H(0,0,0,0) is not a permitted half space. Two special half spaces are defined, Hφ = H(0,0,0,-1)(‘empty’ i.e. points for which –1 > 0). H∞ = H(0,0,0,1)(‘everything’ i.e. points for which 1 > 0). The complement of a half space is defined as: ) , , , ( D C B A H − − − − = , where ) , , , ( D C B A H = . Referring to the definition of a half space, it is readily apparent that: H p H p ∉ ⇔ ∈ , and that: H H = . 5. Convex Polytope Definition A convex polytope is defined as the intersection of any finite number of half spaces; see Figure 1 for a 2D and Figure 2 for a 3D example. Convex polytope representation C is defined as: } .. 1 , { n i H C i = = where Hi, i=1..n is a set of half spaces. This is interpreted as the intersection of the half-planes, i n i H C .. 1 = = I . 2 This form of the definition with four parts, rather than just (A.x + B.y + C.z + D) > 0, is chosen so as to ensure a clean definition of complement. This results in the regular polytope being a boundary-free representation. 3 In this paper, the term half space will be used generically to indicate half space or half plane depending on whether a 3D or 2D geometry is being considered. Most of the illustrations are in 2D for ease of visualisation. Convex region defined by half-planes Convex region defined not completely bounded Figure 1 Convex polytopes defined by half planes. Figure 2 A convex polytope in 3D defined by half spaces A special convex polytope is defined, C∞ = {} (no half spaces), with no constraints on allowed points, is used in the definition of O∞ (the infinite polytope). The intersection of two convex polytopes is defined as the intersection of the half planes that define each of them. It is clear that the intersection of two convex polytopes is itself a convex polytope. 6. Regular Polytope Definition Figure 3 Definition of Regular Polytope from convex polytopes. A regular polytope O is then defined as the union of a finite set of convex polytopes; see Figure 3 for example. U m i Ci O .. 1 = = where Ci, i =1..m are convex polytopes. Again, two special regular polytopes are defined, OΦ = {} (i.e. a set containing no convex polygons) O∞ = C∞. These sets are the empty and infinite sets required for the definition of a topological space. The union of a set of regular polytopes is simply the union of the sets of convex polytopes that define them: where . Note that this union is itself a regular polytope. U U m i n j m i ij i i C O , 1 . 1 , .. 1 = = = = . U i n j ij i C O .. 1 = = The intersection of two regular polytopes is the union of the pair-wise intersections of their component convex polytopes: . Again, note that this defines a regular polytope, since the intersections of convex polytopes are themselves convex polytopes. U U I I m j j n i i C C O' O .. 1 .. 1 ' = = = U I m j n i j i C C .. 1 , .. 1 ) ' (