5. Spatial databases Characteristics: � The scene of spatial data is a geometric space , with some dimensionality (usually 2 or 3). � The shape and location are essential components of information � Dimension values are most often numeric, with some resolution, plus lower and upper bounds. � Application areas: Geographic information systems (GIS), computer-aided design (CAD), graphical user interfaces (GUI), virtual environments, computer games, robotics, animation, etc. Literature: � Güting, R.H.: ”An Introduction to Spatial Database Systems”, VLDB Journal , Vol. 3, 1994, pp. 357-399. � P. Rigaux, M. Scholl, A. Voisard: Spatial Databases, with Application to GIS , Morgan-Kaufmann, 2002 MMDB-5 J. Teuhola 2012 109
Abstract space modeling: Entity-based models Components of spatial objects: identity � description � spatial extent � Classification based on dimensionality: Choice depends on the application viewpoint. � (a) Zero-dimensional objects = points � Object does not have a shape , or it is not considered useful � Area quite small with respect to the embedding space, e.g. cities, buildings, road crossings on a map. � Depends e.g. on the scale of the map. MMDB-5 J. Teuhola 2012 110
Entity-based models (cont.) (b) One-dimensional = line objects � E.g. roads on a map � Main geometric type: polyline, consisting of a finite set of line segments ( edges ), so that each segment endpoint ( vertex ) is shared by exactly two segments, except the two endpoints (if any). � Simple polyline: no intersections. � Closed polyline: endpoints meet. � Any curve can be approximated arbitrarily closely with a polyline. MMDB-5 J. Teuhola 2012 111
Entity-based models (cont.) (c) Two-dimensional = surfacic objects � Represent entities with a non-zero area . � Main geometric type: Polygon = region bounded by a closed polyline. � Convex polygon: For any points A, B ∈ P , line segment AB is fully included in P . (d) Three-dimensional = volumetric objects ( polyhedrons ) (e) Four-dimensional = spatio-temporal objects MMDB-5 J. Teuhola 2012 112
Abstract space modeling: Field-based models � Called also space-based models � The spatial information is considered a continuous (though approximated) field, i.e. a function of coordinates (e.g. x and y ). � Each point of space is associated with one or more attributes. � Examples: � temperature, air pressure, height from sea level etc. at different points on maps, see e.g. http://ilmatieteenlaitos.fi/ � grey-level in a grey-scale digital image � red , green and blue components in a true-color (photographic) digital image MMDB-5 J. Teuhola 2012 113
Representation modes of spatial objects Tessellation mode Cellular decomposition (grid, mesh, tiling, etc.) � Fixed tessellation: regular grid (rastering) � Variable tessellation: different sizes of decomposition units � Regular/irregular tessellation � Default: N x M rectangular (usually square) cells, called pixels � Natural (discrete) representation for field-based data � Entity-based data: one pixel for points, a set of pixels for � polylines and polygons. A more precise representation requires more storage space, � and its processing takes more time. MMDB-5 J. Teuhola 2012 114
Representation modes of spatial objects (cont.) Vector mode Natural for the entity model. � Representation primitives: points and edges � Polygon and polyline are both represented as lists of points � 2 n representations for a polygon with n vertices (selection of starting vertex, clockwise/counterclockwise order) � A region is a set of polygons � Representation may be complemented by restrictions (e.g. to simple polygons) � Representing field-based data in vector mode; Digital Elevation Models (DEM): � Field values only for a subset of points � The rest of the values are interpolated . � Example: Triangulated Irregular Networks (TIN) MMDB-5 J. Teuhola 2012 115
Representation modes of spatial objects (cont.) Half-plane representation � Only a single primitive: half-plane (half-space generally) � Sound mathematical basis � Half-space definition in d -dimensional space: inequality a 1 x 1 + a 2 x 2 + ... + a d x d + a d +1 ≤ 0 � Convex polygon = intersection of a finite number of half-planes. � Polygon = union of a finite number of convex polygons. � Line segment = convex polygon of dimension 1 (intersection of two half lines or rays ) � Polyline = union of some line segments MMDB-5 J. Teuhola 2012 116
Compatibility of models and representations Space model Entity-based Field-based Possible Natural Tessellation Represen- tation Vectors Natural Possible Half-planes Possible Unlikely MMDB-5 J. Teuhola 2012 117
Computational geometry: typical problems � Is a point inside a polygon? � Intersection of line segments � Intersections of polylines � Intersection of polygons � Windowing and clipping with a rectangle � Polygon triangulation � Polygon trapezoidalization � Partitioning of a polygon into convex sub-polygons MMDB-5 J. Teuhola 2012 118
Computational geometry: algorithmic techniques for big problems (a) Incremental algorithms Solve the problem for a small subset of input and add the rest one � by one, maintaining the solution at each step. (b) Divide-and-conquer strategy Divide step : recursively split the task into subproblems, until those � can be solved easily. Conquer step : Merge the subproblem solutions bottom-up into a � global solution. (c) Sweep-line method Decompose the input into vertical strips , so that the information � related to the problem is located on lines separating the strips. MMDB-5 J. Teuhola 2012 119
Storage and retrieval of spatial objects Preliminary issues: � Arbitrary shapes difficult to handle ⇒ Restriction to axis-oriented Minimum Bounding Rectangles (MBR), called also Bounding Boxes (BB). � Dimensions are often transformed to the real interval [0, 1); the whole space is a hypercube, denoted E k . Performance factors: � Selected data structure � Dimensionality of the space � Distribution of objects in space: � density at point P = number of rectangles containing P � global density = maximum of local densities. MMDB-5 J. Teuhola 2012 120
121 MMDB-5 J. Teuhola 2012 Illustration of MBRs
Query types for spatial objects (1) Exact-match query : Not very common for spatial objects, except in the context of insert. (2) Point query : For a point P ∈ E k , find all rectangles R in the database such that P ∈ R . (3) Rectangle intersection : For a given rectangle S ⊆ E k , find all rectangles R with S ∩ R ≠ ∅ . (4) Rectangle enclosure : For a given rectangle S ⊆ E k , find all rectangles R with S ⊆ R . (5) Rectangle containment : For a given rectangle S ⊆ E k , find all rectangles R with R ⊆ S . (6) Volume query : Given v 1 , v 2 ∈ (0,1) and v 1 ≤ v 2 , find all rectangles with volume within [ v 1 , v 2 ]. (7) Spatial join : For two sets of k -dimensional rectangles, find all related pairs, satisfying a given join condition, such as intersection, enclosure, or containment. MMDB-5 J. Teuhola 2012 122
Illustration of spatial join S4 S2 R1 S1 R2 R3 S3 � Intersection-join of { R1, R2, R3 } and { S1, S2, S3, S4 } is { (R1, S2), (R2, S2), (R3, S3) } MMDB-5 J. Teuhola 2012 123
Transformation approach for organizing sets of spatial objects � k -dim. rectangle can be represented as a 2 k -dimensional point. � Alternatives e.g. in 2-dim. space: (a) ( c x , c y , e x , e y ), where ( c x , c y ) is the center point and e x and e y are the distances of the center from the sides. (b) ( l x , l y , u x , u y ), where ( l x , l y ) is the lower left, and ( u x , u y ) is the upper right corner of the rectangle. � Advantage of alternative (a): Location coordinates c x and c y are distinct from extension coordinates e x and e y . Special case: � 1-dimensional space [0, 1) � Rectangle = Line segment ⊆ [0, 1) � Alternative 2-dimensional representations: (a) ( c, e ) = (center, half of length) (b) ( l, u ) = (lower endpoint, upper endpoint) MMDB-5 J. Teuhola 2012 124
Example of the transformation approach S 3 1 • e L 4 • S 1 • L 3 u S 4 0.5 L 2 • P 1 • S 2 L 1 P 3 • • • P 2 P 4 0 1 0 1 0 1 c l Notes: � When PAMs are applied to transformed representation, they suffer from the empty triangles (so called dead regions ). � The (center, extension) approach can be improved, if we know an upper bound to the rectangle side; the ‘live space’ will look like a trapezoid, and the dead triangles are relatively small. MMDB-5 J. Teuhola 2012 125
Answering queries in the transformation approach � Successful areas for different types of queries can be derived by simple geometric calculations. � Example: 1-dim. rectangles (= line segments) represented as 2- dim. points using the (center, extension) approach; query rectangle S = ( c , e ): 0.5 R ⊇ S e R ∩ S ≠ ∅ R ∩ S ≠ ∅ R ⊆ S R ∩ S = ∅ R ∩ S = ∅ 1 c � Drawback: Close, but different-volume rectangles may be located quite far in 2 k -dimensional space. MMDB-5 J. Teuhola 2012 126
Recommend
More recommend
