Representation of spatial data GIS thematic layers, raster and vector, conversion, subdivision representation, continuous data: contours, DEMs, TINs

Thematic map layers Separate storage of data according to theme: map layers (or data layers) GIS typically use tens to hundreds of map layers For example: municipality borders, land use, cadastral boundaries, water pipes, churches, etc.

Example map layers Census data, 1995 (U.S.A.)

Geometry, topology and attributes Geometry: coordinates Topology: adjacency relations of objects Attributes: properties, values Example: Country map of South America Geometry: coordinates of the borders Topology: which countries border which Attributes: names of countries, population, etc.

Representation of geometry Two main approaches: raster and vector Can also be mixed in a GIS, any map layer Conversion raster-vector and vice versa possible Representation depends on type of data, way of acquisition, desired operations, etc.

Raster structure Division of space into equal-size cells (squares, pixels) Theme gives cells a value (nominal, ordinal, interval, ratio, vector, …) Cells should not contain any further spatial information (more detail)

Data in raster form Point object in raster form Line object in Plane object in raster form

Raster maps

Raster: pros and cons Simple structure Simple operations Obtained after scanning, remote sensing Less suitable for point and line objects: representation does not follow intuition Network analysis difficult Not adaptive: no difference in detail possible in different regions Either expensive in memory, or little precision Not obtained after digitizing

Raster: memory reduction Run-length encoding: no 2-dim array but coding start pixel with value and length of run Block encoding: 2-dim version Disadvantage: makes structure and operations much more complex (34,67) forest 9 (34,67) forest 4,6

Vector structure Objects stored as points, lines and areas Points have coordinates; lines connect points; areas are delimited by lines Attributes are stored with the objects (point, line or areal)

Vector: pros and cons Elegant structure; fits with both point, line and areal objects Small storage consumption Precise Adaptive: additional control points possible Network and cluster analysis possible Obtained after digitizing Relatively complex Map overlay and buffer computation complex

Vector representation of a region Not necessarily simply-connected: NL has islands NL has holes (Baarle-Nassau / Baarle-Hertog); there are even regions in these holes

Representation of subdivisions

Subdivisions: spaghetti model Every chain is represented by a list with coordinate pairs Split nodes are doubly stored Areas are not present explicitly C1 C2 C5 C4 C3 C6 C1: (..,..), (..,..), (..,..), ... C2: (..,..), (..,..), (..,..), ... C3: (..,..), (..,..), (..,..), ...

Subdivisions: polygon ring structure Every area is represented by a list with coordinate pairs Control points are doubly stored Neighbor areas are difficult to determine Consistency is difficult to maintain P1 P2 Consistency here refers to the property that a set of polygons should form a partition of a region. If you change the boundary of one polygon, you must change the boundaries of one or more other polygons as well to maintain the property that the polygons form a subdivision. Technically, consistency is not difficult to maintain, but it is an implementation hassle and causes inefficiency of updates, because these other polygons must be retrieved and changed as well. P3 P1: (..,..), (..,..), (..,..), ... P2: (..,..), (..,..), (..,..), ... P3: (..,..), (..,..), (..,..), ...

Subdivisions: topological structure (node-link structure) Nodes are objects with coordinates Edges are connections of nodes Sequences of edges along polygon boundaries form cycles Polygons are objects that can access their boundaries Doubly-connected edge list

Subdivisions: topological structure Edges are split into directed half-edges Half-edges have pointers to Twin half-edge Origin vertex Next and Prev half-edges of incident polygon Incident polygon Polygons have pointers to half-edges, one in each bounding cycle Origin polygon Twin Prev Next polygon

Subdivisions: topological chain structure Splitting nodes are objects with coordinates Chains are connections between splitting nodes and contain zero or more nodes with coordinates Sequences of chains along polygon boundaries form cycles Polygons are objects that can access their boundaries half-chains Doubly-connected chain list

Vector structures Memory Duplication Polygon Topology retrieve retrieve Spaghetti ++ + -- - Polygon ring - -- ++ - DC edge list -- ++ - + DC chain list ++ ++ + ++ Topology retrieve refers to the topology of the subdivision: region objects have a way to access adjacent regions efficiently. With a doubly-connected chain list, this comes down to following the chains and always looking at the other side of the chain for an adjacent region. This takes time linear in the number of adjacent regions, while for the doubly-connected edge list this takes time linear in the number of edges in the boundary of the region (which is much more). For the spaghetti and polygon ring structures, there is no easy access to adjacent regions and you have to traverse whole tables in the database.

Raster-vector conversion E.g. for data integration Vector-to-raster: Like in computer graphics: scan-conversion of lines, etc. Raster-to-vector: Consider pixel sides between pixels with different values as boundary and put in vector representation  Thinning, line simplification

Thinning Raster-vector conversion Thinning

Line simplification Douglas-Peucker algorithm from 1973 Input: chain p1, …, pn and error   p1 pn

DP-algorithm  Draw line segment between first and last point If all points in between are within error: ready Otherwise, determine farthest point and recursively continue on the part until farthest point and the part after farthest point

DP-algorithm DP-standard(i, j, ) Determine farthest point pk between pi and pj If distance(pk, pi pj) >  then DP-standard(i, k, ) DP-standard(k, j, ) Return the concatenation of the simplifications

  

  

Properties of the DP-algorithm DP-algorithm does not minimize the number of points in the simplification    DP-algorithm Optimal

Properties of the DP-algorithm Determining farthest point takes O(n) time Whole algorithm takes T(n) = T(m) + T(n-m+1) + O(n), T(2) = O(1) time, splitting in m and n-m+1 points “Fair” split gives O(n log n) time Worst case gives quadratic time

Properties of the DP-algorithm DP-algorithm may give self-intersections in the output  Solution: test output for self-intersections and continue adding control points if necessary

Improved DP-algorithm DP-improved(i, j, ) Simp = DP-standard(i, j, ) V = set of intersecting segments of Simp Repeat For all segments s  V: Refine(s) in Simp; do 1 refinement à la DP by adding the farthest point, giving a new Simp V = set of intersecting segments of Simp Until V is empty

Continuous data representation Digital Elevation Model (DEM) Data on interval or ratio measurement scale Data values of points near by will usually be not very different Representation is necessarily an approximation: finite representation of information with infinite detail Raster (1x) or vector (2x)

Elevation models Raster Vector Vector (Elevation) grid 21 20 21 20 15 20 19 25 10 10 (Elevation) grid Contour line model Triangulation (TIN; triangulated irregular network)

Grid elevation model

TIN elevation model

Elevation models Contour model well-suited for visualisation, not for representation or storage Interpretations grid: - elevation whole cel: not a continuous model - elevation middle cel: interpolation needed; how? Advantage grid: simple storage, operations simple too Advantage TIN: more efficient in storage, adaptive

Interpolation for grid 20 18 20 18 18 22 18 22 20 18 Linear interpolation; saddle point problem 18 22 20 18 20 18 18 22 18 22 Linear interpolation; additional point 20+18+18+22 4 = 19.5 Non-linear interpolation

Topological TIN structure With explicit vertex and triangle representation t2 w t3 t1 t1 t2 t t u v u w t3 v x, y-coordinates and elevation

Topological TIN structure With explicit vertex and triangle representation t2 w t3 t1 t1 t2 t t u v u w t3 v Because t1 has pointers to two the same vertices as t, we can determine their shared edge, even though it is not represented explicitly

Topological TIN structure With explicit vertex and triangle representation w w t1 t2 t2 t1 t u v t t3 v u t3

Topological TIN structure Alternatively, edges have an explicit representation too w t1 t2 w t1 t e1 e2 e1 e2 u e3 v t3 t u v e3

Summary representation Objects have geometry and attributes, at least the attributes are in a database Geometry can be stored in raster or vector form; each has advantages and disadvantages Important geometric types of representations are those for subdivisions and for elevation models For subdivisions, the doubly-connected chain list is the most suitable structure For elevation models, grids or TINs are most useful