Vector Spatial Data Models and Structures

Transcription

1 Vector Spatial Data Models and Structures Fundamental Concepts of Vectors. Raster Comparison Geometric Representation Structured vs. Unstructured Data Topological Data Models From Map to Data Store Networks Example Polygon Models Topology in Commercial GIS Example algorithms

2 Simple Vectors A Euclidian view of space (coordinatised space) is assumed Mathematically, a vector expresses the notion of a quantity with both magnitude and direction In geometric terms we can think of it simply as a straight line segment defined by its end points The vector may or may not specify a direction between these end points The point locations can be in any co-ordinate system e.g. wind vectors

3 Initial Observations From this basic element we can digitally build up any more complex form we want The assembling of vector entities forms units which are apparently scaleless but are in fact controlled by: the level of generalisation used the ultimate discretisation capability of the underlying data storage These geometric objects can be associated with attributes so as to represent any realworld object we choose Spatial information is not so much contained in the location of the points but in the association between them It is the irregularity of spatial components and the need to associate them with nonspatial components that simultaneously brings many benefits but also presents many problems to GIS engineering

4 Rasters - just in case! Rasters are a structured array of data Usually represent some regular and continuous tessellation of space Each cell has an associated value They can be used to represent continuous fields or discrete objects Spatial relationship between cells is implicit What are the benefits of raster vs vector?

5 Raster/ Vector Comparisons Raster Vector Spatial location Implicit Explicit Attributes Explicit for each cell Implicit within data layer In either case the challenge is to: produce conceptual models of these data structures turn them into bytes of data in a computer system Efficiency considerations: - storage requirements - processing to structure data into the desired storage format - processing for simple retrieval and display of data - the suitability of the data model to allow the data to be manipulated for spatial operations or spatial modelling.

6 Interconversion VECTOR TO RASTER CONVERSION (RASTERISATION) RASTER TO VECTOR CONVERSION (VECTORISATION) Vectorisation is computationally more difficult Methods: Raster thinning Medial line finding Line fitting

7 Underlying Rationale for Vector-based Systems Rooted in the transition from analogue maps Good at conferring the meaning of complex information compactly. Our brains are wired up so as to rapidly make sense of vector images Naturalness - representing irregularly spatial objects with irregular spatial symbolism The human tendency to corral and classify A sense of precision, if not accuracy Nice lines make pleasing maps The typical sparseness helps us view multiple significant vectors with different attributes in a single view eg. contours, roads, rivers

8 Building Blocks Points Zero dimensions (but implicit vectors defining relative location) A set of geometric points can in themselves be associated with attributes to create meaningful entities eg. spot heights, post box locations, polygon centroids. May be: a standalone point a vertex defining the location of a link between one line and the next a node defining the location of a link between topologically structured lines

9 Building Blocks Lines, Chains, Arcs Line segment - a direct (but not necess. straight) line between two end points or vertices Link, a direct line between two nodes in a network Directed link, a link with a specified direction String, a sequence of line segments Chain, a directed sequence of nonintersecting line segments with nodes at each end (joining to other chains) - Called an arc by ESRI (Arc - a locus of points that forms a curve defined by a continuous mathematical function - eg. cubic splines)

10 Spatial Objects I

11 Spatial Objects II

12 Spatial Objects III One Polygon!

13 Building Larger Features Point, line, polygon model is hierarchical. At the base level it is defined by the set of points that define each location in the larger structures. How do we store these points to represent the line and polygon structures higher up the hierarchy? How do we link geometric features and attributes? How do we represent information about how the geometric units are spatially related? Historically, there are two major ways of storing vector data: Unstructured or Spaghetti data Structured or Topological data.

14 The Spaghetti Model Information stored as a list of co-ordinates, with some indication of point/line start/ finish Store attributes with an additional feature code at the start of each record or by lumping together all the similar features in a single coded block Road River Forestry block Roads 1 x 1 y 1 x 2 y 2 x 3 y Rivers 1 x 1 y 1 x 2 y 2 x 3 y Forests 1 x 1 y 1 x 2 y 2 x 3 y x n 2 x 1 y 1 y n x n 2 x 1 y 1 y n x n 2 y n x 1 y 1 x n y n x n y n x n y n

15 Spaghetti Model - Observations Separate feature classes (railways, rivers, roads) are not necessarily distinguished Lines can cross and the connectivity between lines is not described within the data structure Polygons are only represented by circumscribing boundaries there is no real way of referring to the areal unit encompassed by the boundary Boundaries (lines) shared by two polygons may be stored twice Storing as a sequential (possibly sorted) structure requires searching half the file on average Other operations such as finding intersections and identifying spatial relationships are tricky Often the form of raw digitised data May remain the primary data structure in cartographically-oriented GIS (as against GIS focussed towards spatial analysis)

16 Topology The study of form (different from topography or toponomy or indeed typology!) A mathematical concept which defines spatial relationships The recognition of spatial structure is selfevident from our ability to interpret spatial pattern Beyond basic location we want to represent the following features of spatial entities: connectivity (connectedness) adjacency (contiguity / nearness) containment Topological foundation of GIS Topology is necessary for: identifying contiguous areas routing through a network spatial analysis By reducing individual points or strings to lists of numbers we lose the topological information and the meaning of the spatial structure We thus have to find ways of representing these relationships in the data structure

17 Are these Similar?

18 Further Points on Topology Topological A point is at an end-point of an arc A point is on the boundary of an area A point is in the interior/exterior of an area An arc is simple An area is open/closed/simple An area is connected Non-topological Distance between two points Bearing of one point from another point Length of an arc Perimeter of an area Topology is often about neighbourhoods Used in the network data model Also basis for network spaces and measures Some GIS operations do not require coordinate locations, only topology eg. to find an optimal path between two points requires only the connecting links and the cost to traverse them in each direction Adjacency is implicit in a raster; connectivity is difficult to represent beyond the trivial

19 From Data Capture to Structured Vectors Vector Data Capture: Manual Digitising Scanning + Vectorisation Feature association Generalisation Interpolation/ Smoothing Cleaning Topology Construction Errors must be corrected using: automated methods (where possible) manual methods

20 Implications of Topology Digitised data is rarely in a consistent state, suitable for the creation of viable topology Lines do not cross and must connect Areas are explicitly defined Polygons must be closed Boundaries (lines) shared between two polygons are stored only once Lines / arcs are ordered, can be indexed and quickly searched There are different levels of topological representation (basic level, with opportunity to add sophistication if application so requires)

21 The Bridges of Königsberg A problem which puzzled the people of Königsberg (now Kaliningrad) for many years The Pregel River passes through the city Could someone tour the city crossing each of its bridges once and once only?

22 Topology and Graph Theory A mathematical proof which provides a solution to the Bridges of Königsberg problem was devised by the Swiss mathematician Euler in 1735 This involved the creation of what is now referred to as Graph Theory An Eulerian Path exists if: nodes are connected even number links at each node, excepting only start and end points Graph Theory is key to the understanding of networks Planar vs. non-planar graphs in planar graphs all lines intersect at nodes non-planar graphs are required to deal with underpasses, for example

23 The Most Famous Topological Model Real

24 Geometric vs. Logical Networks Which are we trying to model? Or both?

25 Linear Topology G 6 F 7 H B 1 2 C 5 3 D 4 Directed A E Link Connects to Link ? Link Connects to Node 1 C 2 C 3 D 4 D 5 F 6 F 7 H H 7 B G 2 C F 5 D 4 Undirected A E Link Connects to Link 1 2,3 2 1,3 3 1,2,4,5 4 3,5 5 3,4,6,7 6 5,7 7 5,6 Nodes Adjacent Links A 1 B 2 C 1,2,3.... Link Connects to Node 1 A,C 2 B,C 3 C,D 4 E,D 5 D,F 6 G,F 7 F,H

26 Google Routing Topology: Routing

27 Topology: Hydrological Model

28 Y Simplest Polygon Model B B 6 3 A C X A 6 C Only just one up from Spaghetti Stores all the points in the polygon. Overstorage, over representation One-to-one relationship between point, line and polygon features and the attributes they represent Potential problems: Knots and Weird Polygons 2 Polygon X Y ID Point A B Slivers and Gaps

29 Point-List Model B A C Points Co-ordinates X Y Nodes List of Points A 1,2,3,8,9 B 1,9,. Points are stored only once; one-to-many relationship between points and polygons Avoids double storage of points which are part of two polygons However, still does not record adjacency or connectivity directly

30 Line-Segment Encoding (DIME) 1 h 2 a g A 9 i 10 B f 8 b c 6 d e j 7 5 C k l 4 3 line Node Co-ordinates Polygon To From To From Left Right a ,14 1,10 B - B ,10 6,3 B Historically, the first topologically explicit data model was the line segment structure used by the US census Bureau in 1970 Known as the Dual Independent Map Encoding (DIME) format "Dual Independent" because, for each line segment, both the 'from node - 'to node' and 'left-poly' - 'right-poly' are stored Direction of segment important Simple line segments only with no intermediate vertices so major turning points have to be recognised as nodes as well as intersections Only line segments were directly addressed - polygons not explicitly stored

31 Chain model (POLYVRT) b B III I c 1 2 A IV f a e d II Chain Node or Point a III 6, 3 1 7,5 2 10,5 IV 11,8 b III 6,3 Chain Node Polygon From To Left Right a III IV B C b III I - B Peucker and Chrisman (1975) Extended DIME Added points along a chain can be stored Topology is explicit Topology and position are separately treated Used as the ArcInfo / ArcMap vector format Tom Waugh's GIMMS segment format extended POLYVRT

32 Doubly Connected Edge List (DCEL) next arc b left area end node arc begin node previous arc right area Explicit info on: sequence of arcs around nodes sequence of arcs around areas. a Arc Begin Node End Node Left Area Right Area Previous Arc Next Arc a 1 2 A X e d b 4 1 B X f a c 3 4 C X i b d 2 3 D X g c e 5 1 A B h b f 4 5 C B c e g 6 2 D A i a h 5 6 C A f g i 3 6 D C d h A D d g e B h i f C Single table c After Muller and Preparata, 1978

33 Winged Edge Representation p-vertex nc-edge n-face na-edge n-vertex edge pa-edge c=clockwise a=anticlockwise p-face pc-edge 1 a 2 e A g D d b B 5 h 6 4 f C 3 c edge n vertex p vertex n face p face nc edge na edge pc edge pa edge a 1 2 A X g e b d b 4 1 B X e f c a c 3 4 C X f i d b d 2 3 D X i g a c e 5 1 A B a h f b f 4 5 C B h c b e g 6 2 D A d i h a h 5 6 C A i f e g i 3 6 D C g d c h After Baumgart 1975, and Weiler 1985

34 Topological GIS GIS software tends to fall into one of three classes: Those systems where topology is build early in the process of creating a geodataset Those systems which require topology to be explicitly built when a particular operation requires it Those systems which build topology automatically on-the-fly as required This becomes a useful way of classifying systems Building and maintaining topology requires a more sophisticated data model to be used throughout (potentially a performance overhead) Spatial queries are much faster when pre-built Building topology on-the-fly is a compromise between minimising storage requirements and the time required to build it. Maintenance is not usually an issue; just rebuild as necessary

35 Topology in ESRI products There is no concept of topology in the ArcView shape file. Simple topology is built if needed but never stored (slow) ArcGIS has two simple concepts: planar topology, where polygons adjoin each other exactly, without gaps or overlaps geometric network, where network segments are unambiguously connected 'Real' topology has to be built on-the-fly SDE doesn't usually store topology, although can store coverages, which will Personal geodatabases (stored as MS Access MDB files) don't store topology MapObjects, used the ArcGIS concepts ArcInfo coverages implement three more sophisticated forms: polygon-arc topology (area definition) left-right topology (contiguity) arc-node (connectivity)

36 Polygon-Arc Topology Polygons are represented by the arcs which form their boundary, rather than a set of points ArcInfo therefore stores a list of arcs which make up each polygon (the polygon-arc list) The arcs must form a complete and closed boundary An arc may be stored in the list for two adjacent polygons However, the co-ordinates only need to be stored once This minimises the amount of data which needs to be stored, but perhaps more critically that the boundaries of adjacent polygons do not overlap

37 Left-Right Topology From the concept of arcs shared between polygons through the polygon-arc list, any polygon sharing an arc must be adjacent Because every arc has a direction, a list of polygons on the left and right side of each arc can be maintained A external (or 'out') polygon (polygon 1) is defined so that even arcs around the edge of s study area have a polygon on either side

38 Arc-Node Topology Arcs (lines) are shaped by points called vertices Arc must meet at their end points (nodes) Each arc has two nodes: a from-node and a to-node and therefore a direction ArcInfo records which arcs meet at which nodes This establishes which arcs are connected to which others, creating a network. Routing operations can therefore be carried out

39 Line Generalisation

40 Douglas-Peuker Line Generalisation Algorithm Frequently in GIS systems it is desirable to weed out unnecessary vertices from arcs These may have been generated by oversampling during digitisation. The Douglas-Peuker line generalisation algorithm works by reducing a point set by removal of vertices if they fall within a bandwidth tolerance. Further, it proceeds by progressive subdivision of the arc on either side of the vertex which lies furthest from the straight line between two end nodes of the sub-segment After David Douglas and Tom Peuker (Poiker)

41 Douglas-Peuker Example line which we wish to reduce

42 Draw a straight line between start and end nodes. Locate vertex which lies at the greatest perpendicular distance from the straight line. Examine to see if this lies within the linear tolerance set. if not, then repeat the process this time forming two new straight lines from the start node to the new node and from the new node to the end node. For each of these new lines now find the furthest point and examine to see if it lies within the set tolerance.

43 Repeat iteratively. If, as above, the points all fall within the tolerance set then proceed no further with this section of the arc. (If using a linked-list intervening points can be excluded. The remaining vertices are part of the desired generalisation.)

44

45 Original vs Final Result

46 Polygon Labelling Labelling is a method of attaching attribute data to a polygon Point-in-Polygon method used First step: Exclude points not in bounding rectangle

47 Point-in-Polygon Method Number of lines crossed from outward pointing ray: Odd: Within Polygon Even: Outside Polygon

48 Calculating Polygon Area Trapezoidal Rule Define a base and calculate area under upper edge of polygon

49 Subtract area under lower edge to gain area

50 Polygonisation In spatial analysis, the Thiessen Polygon attempts to define geometrically the region of influence of a point in terms of an area Alternatively VORONOI or DIRLICHET regions Point Network Right Angle Thiessen Poly Perpendicular bisectors are constructed on all axes radiating from the central node. These bisectors intersect at thiessen vertices to form the thiessen polygon This method is useful for generating polygons where only point data exists (eg. centroids of wards, EDs or parishes) Where the points are regularly spaced, hexagons result, which are less useful. Thus the technique is most useful for irregularlyspaced data Compare with Delaunay Triangulation

51 Leads to Triangulated Irregular Networks (TINs) But more of this later...

52 Other Vector Models There are various other models which could be used: agent simulation process Agents: A Vector-Object Model Sleep Flock Eat Drink Source: Conrad Rider

53 Agent Example II Weather parameters Aerial photograph Management choices Water sources Agent-based animals Field boundaries Pasture model Gates

54 Summary Vectors are basic building blocks These can be assembled into more complex structures The complexity is in the structuring, and the challenge of GIS is to find efficient ways of storing and manipulating this complexity and associating it with appropriate attribute information Benefits of vector data structures relate to the naturalness of the irregular features that may represented and the efficiency of not storing redundant data

55 Some References Baumgart B 1975 A polyhedron represenation for computer vision. Proceedings AFIPS National Conference, 44, Cova, T. J. and Goodchild, M. F., 2002, Extending geographical representation to include fields of spatial objects. International Journal of Geographical Information Science, 16(6), pp Dutton, G. (1972) Harvard papers on topological data structures. Harvard Lab for Computer Graphics & Spatial Analysis, Vols Egenhofer, M. J. and Herring, J. R. (1991) High level spatial data structures for GIS. In Maguire, D. J., Goodchild, M. F. and Rhind, D. W. (Eds.) Geographical Information Systems: Principles and Applications. Longman, Harlow. Chapter 16. Franklin, W. R. (1991) Computer systems and low-level data structures for GIS. In Maguire, D. J., Goodchild, M. F. and Rhind, D. W. (Eds.) Geographical Information Systems: Principles and Applications. Longman, Harlow. Chapter 15. Laurini, R. and Thompson D. (1992) Fundamentals of Spatial Information Systems. Academic Press, London. Muller D.E and Preparata F.P Finding the intersection of two complex polyhedra, Theoretical Computer Science, 7, (2), Peuquet, D.J. (1984) A conceptual framework and comparison of spatial data models. Cartographica, 21 (4). Piwowar, J. M., Ledrew, E. F. and Dudyeha, D. J. (1990) Integration of spatial data in vector and raster formats in a GIS environment. International Journal of Geographical Information Systems, 4 (4), Weiler 1985 Edge-based data structures for solid modelling in curved-surface environments, Computer graphics Applications, 5(1), Wise, S., (2002). GIS Basics. CRC Press, New York. Worboys, M. and M. Duckham, (2004). GIS: A Computing Perspective. CRC Press, New York.