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

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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.

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

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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.

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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.

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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)

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

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Raster maps

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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

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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

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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)

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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

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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

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Representation of subdivisions

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Subdivisions: spaghetti model Every chain is represented by a list with coordinate pairs Split nodes are doubly stored Areas are not present explicitly C 1 : (..,..), (..,..), (..,..),... C 2 : (..,..), (..,..), (..,..),... C 3 : (..,..), (..,..), (..,..),... C1C1 C2C2 C3C3 C4C4 C5C5 C6C6

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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 P1P1 P2P2 P3P3 P 1 : (..,..), (..,..), (..,..),... P 2 : (..,..), (..,..), (..,..),... P 3 : (..,..), (..,..), (..,..),...

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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

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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 polygon Next Prev Twin Origin

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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 Doubly-connected chain list half-chains

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Vector structures Spaghetti Polygon ring DC edge list DC chain list Memory Duplication Polygon Topology retrieve

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Raster-vector conversion 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 E.g. for data integration

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Thinning Raster-vector conversion Thinning

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Line simplification Douglas-Peucker algorithm from 1973 Input: chain p 1, …, p n and error p1p1 pnpn

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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

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

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

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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

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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

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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

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Continuous data representation 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) Digital Elevation Model (DEM)

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Elevation models (Elevation) grid Contour line model Triangulation (TIN; triangulated irregular network) RasterVector

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Grid elevation model

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TIN elevation model

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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

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Interpolation for grid Linear interpolation; saddle point problem Linear interpolation; additional point Non-linear interpolation = 19.5

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Topological TIN structure t t1t1 t2t2 t3t3 u v w x, y-coordinates and elevation With explicit vertex and triangle representation t t1t1 t2t2 t3t3 u v w

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Topological TIN structure t t1t1 t2t2 t3t3 u v w Because t 1 has pointers to two the same vertices as t, we can determine their shared edge, even though it is not represented explicitly With explicit vertex and triangle representation t t1t1 t2t2 t3t3 u v w

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Topological TIN structure t t1t1 t2t2 t3t3 u v w With explicit vertex and triangle representation t t1t1 t2t2 t3t3 u v w

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Topological TIN structure Alternatively, edges have an explicit representation too e1e1 e2e2 e3e3 t e1e1 e2e2 e3e3 t1t1 w u t t1t1 t2t2 t3t3 u v w v

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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

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