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Representation of spatial data
GIS architecture, raster and vector, conversion, administrative subdivisions: polygon-ring, topology, extended DCEL, continuous data: contours, DEMs, TINs
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Thematic map layers Separate storage of data according to theme: map 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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GIS data architectures
Structure/philosophy of how geometrical/ topological and attribute information should be stored and is accessible 1. Twofold or dual architecture 2. Layered architecture 3. Integrated architecture
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Pure database approach
Geometry and attributes in same relational data model + More concurrent users possible - Objects must be reduced to atomic parts, and partitioned over various tables - Retrieving original objects is expensive (join) - Query language doesn’t know spatial concepts (area, intersects, …)
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Twofold architecture Attributes in a DBMS
Geometry in separate files (by theme) Connection by unique identifier Two subsystems: the DBMS and one for the geometry E.g. ArcGIS
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Twofold: pros and cons + DB generally known (at organizations)
+ Geometry fast and easily accessible - More users difficult to incorporate concurrently - Maintaining consistency between 2 systems is tricky - Efficient transactions (optimizations) tricky because of two systems
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Layered architecture RDBMS with additional, geographically intelligent layer Layer contains extension with geographical data types, e.g. Point, pointcluster, polygon, line Layer offers extension to query language, and translates for the actual RDBMS
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Layered: pros and cons + More concurrent users possible
+ Spatial object types and concepts are present - Middle layer not extendible - Topological relations must be determined when they are needed - Practice: no object type for subdivision
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Integrated architecture
Spatial object types and functions in the database itself RDBMS or OO E.g. Postgres
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Integrated: pros and cons
+ No translation in middle layer necessary + Extendible with additional types and functions - Extension is rather complex - Practice: less GIS-functionality present by default
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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
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 C1 C2 C5 C4 C3 C6 C1: (..,..), (..,..), (..,..), ... C2: (..,..), (..,..), (..,..), ... C3: (..,..), (..,..), (..,..), ...
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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 P1 P2 P3 P1: (..,..), (..,..), (..,..), ... P2: (..,..), (..,..), (..,..), ... P3: (..,..), (..,..), (..,..), ...
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Subdivisions: topological structure
Nodes are objects with coordinates Edges are connections of nodes Sequences of edges along polygon boundaries are connected Polygons are objects of which the boundary is stored Doubly-connected edge list
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Subdivisions: topological chain structure
Splitting nodes are objects with coordinates Chains are connections of splitting nodes and contain zero or more nodes with coordinates Sequences of chains along polygon boundaries are connected Polygons are objects of which the boundary is stored Doubly-connected chain list
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Vector structures Memory Duplication Polygon Topology
retrieve retrieve Spaghetti Polygon ring DC edge list DC chain list
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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
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Thinning Raster-vector conversion Thinning
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Line simplification Douglas-Peucker algorithm from 1973
Input: chain p1, …, pn and error p1 pn
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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 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
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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 V = set of intersecting segments of Simp Until V is empty
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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)
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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)
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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
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 Non-linear interpolation ( ) / 4 =19.5
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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
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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
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Topological TIN structure
With explicit vertex and triangle representation w w t1 t2 t2 t1 t u v t t3 v u t3
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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 e3
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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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