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SPATIO-TEMPORAL
DATABASES
Spatial Databases
Spatial Data. Representing Spatial
Objects. Spatial Notions and
Semantics
 The workspace dimensionality: the space and the spatial objects are
usually represented as 1D, 2D, or 3D spatial information.
 Partition: Let S be a non-empty set. Then P = {P1, P2, …, Pp} is a
partition of S if and only if the following conditions are fulfilled:
  i:=1..p, Pi  
  i, j:=1..p, i  j, Pi  Pj = 
  Pi = S, i:=1..p
 In general, a partition is used in the decomposition of the workspace
[Sa90]. Eg. for a 2D region of space, a planar decomposition with
polygonal or non-polygonal shapes could be applied.
 One partition of the space is called regular if the polygonal
components are regular. Otherwise, it is called non-regular.
 If a partition allows the recursive decomposition of its elements, then
it is considered unlimited, otherwise – limited.
 Figure 1.1.– 3 types of partitions: regular non-limited,
regular limited, non-regular non-limited.
 Spatial objects: frequently encountered are point, line,
and region (simple, individual objects). Besides these,
there are special data types, such as segment, halfsegment (used in dual representation of a segment,
considering that a half-segment corresponds to a
segment end) [FG+00], circle, region with holes [FG+00,
GB+00] etc.
 In addition, some applications require modeling and
managing collections of spatial objects correlated in
space: partition (e.g. map of regions), network or graph
(e.g. transportation network, rivers, electricity network).
 Centroid: center of gravity / mass of a geometric figure.
The structure of space. There are two
techniques for modeling spatial data in a
computer system: raster (grid) and vector.
In a vector model, a spatial information
indicates where something is or where it
occurs, and in a raster model it indicates
something that exists or occurs
everywhere.
The structure of space
 Raster (grid) models are representing the space covered by a set of
cells and are treating information like temperature, pressure, altitude.
 Transforms a cell in a value of a given attribute domain.
 The neighbor cells that have associated the same value form a region,
and such regions form a partition of the workspace.
 This kind of modeling provides a continuous view of space [CZ00].
 It is generally materialized in the form of grid, within which each cell is
rectangular.
 The parameters that define a raster model are: grid size, grid
resolution, information about geography. The information about
geography associates a cell with a specific location in the shaped
reality.
 A cell may have associated values of certain attributes (thematic) (one
or more) to its central point. The raster representation of spatial
objects can capture continuous numeric values (qualitative
information, e.g. temperature) or continuous categories (qualitative
information, e.g. types of climates).
The structure of space
 The raster models are usually used in representing thematic maps
(e.g. a map of temperature levels, types of vegetation, etc.). Where
there are more than one such map defined on a spatial region for
various features, their superposition / overlap is performed.
 Also individual spatial objects can be represented in a space
partitioned as a grid (by the set of cells / pixels which intersect the
object’s shape => spatial data is not represented as continuous
geometrical shape, but is divided into discrete units of information).
 Advantages: uses simple data structures, simple procedures of
spatial analysis.
 Disadvantages: need a relatively large storage space (which
depends on the granularity of the grid cells and associated
information), lack of accuracy of visual result of data for a less fine
granularity.
The structure of space
 Vector-based models: any point is represented by coordinates
relative to reference point belonging to the workspace.
 The mentioned spatial data types are easily represented and
managed.
 Spatial data that is modeled using vector data represent discrete
features, and a vector model captures only the relevant information
(of represented objects), not of the whole workspace. Thus, it
provides a discrete view of space [CZ00].
 Advantages: less storage space than the raster models, topological
relations are easily determined, displaying data is much more
realistic and the resolution is not a parameter of the system, but is
simply given by the data received in the system.
 Disadvantages: need more complex data structures, more expensive
equipment and applications.
The discrete space domain
 The spatial domain of most of the application is seen (at least
theoretically) as being the Euclidian space. Yet, because a
computational system is limited in representing the infinite set of real
numbers, modeling spatial data uses:
 A discrete domain (pre-defined data types), or
 A discrete domain re-defined (a custom domain, UDT).
 Case to be discussed: the intersection of two line segments, if the
intersection point has coordinates that do not belong to chosen
domain. There are two strategies, depending on the spatial model
and the required accuracy:
 It is accepted that the final result is an approximation of the real result;
 Corrections are applied to the result by translating the real intersection
point to a point that is situated in the working space [GS93].
 (See intersection problems for realms.)
Spatial Databases
 Definition A spatial database is a database optimized for
storing, managing, and querying spatial data. It provides
spatial data types in its data model and query language,
support for spatial indexing and for spatial join [Gu94].
 Remark. The difference between image databases and
spatial databases: image databases manage data that is
introduced as digital images made with different
equipment (cameras, satellites and so on); these images
contain a set of objects, which then can be analyzed by
different applications; spatial databases do not record
images of objects, but values of spatial characteristics of
a set of objects.
Modeling Spatial Data
 Realms
 Uses a discrete domain for representing space and spatial objects
[GS93, GS95, Sc95]. The user can define a finite domain of
numerical values that is used as the base in defining a set of spatial
data types.
 Realm is a finite set of points and line segments defined over a finite
domain, of type grid, so that:




Each point is a point of the grid;
Each segment end is a grid point;
No point of the realm belongs to the interior of a segment;
Any two distinct segments do not intersect and do not overlap.
 The spatial objects considered in the design process using realms
are points, lines, and regions. These can be represented using only
points and segments of the realm. Basically, a spatial object is not
created on the realm, but there are construction elements associated
to it (points and segments).
Modeling Spatial Data
 Realms
 Updates are not performed on the object, but on the elements of the
realm, in this way they being propagated on the objects it contains.
 Advantages of modeling spatial data on a discrete domain using the
realms: the possibility of defining different types of spatial data in the
same area / domain, the property of closure is guaranteed (in spatial
operations), and forcing consistency of geometric objects in spatial
relations (e.g. adjacency of two objects of type region).
 Disadvantages: relatively difficult integration of realms in a DBMS,
the cost of restoring the spatial objects from realm elements.
 Figure 1.2. – discrete spatial domain of type grid and
spatial objects of type point (P), line (L), and region (R).
 Simplicial Complexes [Sc95]
 Considers the workspace as being continuous (theoretically) or discrete, uses
a collection of non-regular geometric shapes.
 The space and the spatial objects are modeled by joining such basic shapes,
called k-simplex.
 Definition Let k+1 points from Rn, v0, v1, ..., vk, such as the vectors v1 - v0,
..., vk - v0 are linearly independent. The set {v0, v1, ..., vk} is called
geometrically independent
and the set of points
k
k
x   i  vi ,  i  1, i  0
k = {x  Rn |
i: 0
i: 0
} Rn
is called simplex of dimension k (k-simplex), with vertexes v0, v1, ..., vk.
 A k-simplex represents the convex closure of the k +1 points in at least kdimensional space. Any k-simplex consists of k+1 simplexes of dimension k-1,
where they are called faces of the k-simplex object.
 Figure: simplexes of different dimensions
 Simplicial Complexes
 k-complex: finite set of simplexes; the greatest dimension of a
simplex is k.
 K-complex restriction: the intersection of two simplexes is the empty
set or a common face of them.
 The spatial objects are located in this space and any such object is
built by aggregation of objects of type simplex in the partition.
 Advantages: preservation of topological consistency between spatial
objects and easy implementation of data structures and algorithms
for management of simplicial complexes.
 Disadvantages: high cost of workspace triangulation and calculation
of numerical operations, such as distances.
 Figure
 Geo-Relational Algebra
 A data model defined on the Euclidian space (discretized)
[Gu88], proposed in order to be implemented on top of a
relational DBMS.
 Offers spatial data types and operators for spatial data
(=> geo-relational algebra)
 Spatial data types:
 Point
 Line – chained list of segments; simple lines (non self-intersecting
lines)
 Pgon – closed chained list of segments; simple polygons, convex
or concave
 Area – similar to Pgon; represents a region of a partition
 Geo-Relational Algebra
 One spatial object is represented by a tuple within a
table, and a table contains only objects of the same type
(set of points, set of lines, etc.).
 Does not use decomposition of objects and does not
borrow objects of the underlying space, but represents
them as they exist in reality.
 Simple data structures.
 Does not allow storing data of different types in the same
table within the database (the structure of the database
depends on the application’s characteristics).
 Spatial Model with Linear Constraints
 The database model with constrains [KK+90] was easily
used in representing spatial objects [BB+97, GR+98a].
Therefore, each geometric object is represented as
infinite set of points, by first-order formula.
 These formulas are given in the disjunctive normal form,
and their terms are linear constraints of the form
 p

  ai  xi  a0 
 i:1

where   {=, }, ai  Z, p  1.
 Spatial Model with Linear Constraints
 The geometric objects that can be represented using linear
constraints: point, line segment, semi-line, line, polygon, or any kind of
region (finite or infinite) of the space.
 Structure of space – this model corresponds to the vectorial one.
 Limit of the model: the possibility of representing only convex polygons
using a conjunction of linear constraints (two or more linear
constraints). In order to store a non-convex polygon, it is decomposed
into convex polygons (therefore – it is stored as a union of geometric
shapes = disjunction of conjunctions of linear constraints).
 Advantages: allows the representation of objects in an n-dimensional
space, where n  1, however large, even if physically it's hard to
imagine. In addition, it can represent infinite sub-spaces in a finite
(limited) manner.
 Figure 1.3. point (P), line segment (S), non-convex
polygon (Pg), infinite region (Ri)
Table 1.1. The linear constraints for example from figure 1.3.
Geometric object
Linear constraints
P
x=2y=5
S
3x - y = 2  -x  -1  x  2
Pg
-x  -5  x  7  -y  -2  y  6
Pg
-y  -1  y  2  -x  -5  x + y  8
R
-x + y  1  2x + y  18
 K-Spaghetti
 K-Spaghetti [LT92] – used for the representation of
spatial objects in a k-dimensional vectorial space. It is
frequently encountered in applications where the
dimension of the workspace is 2 or 3.
 The purpose of the k-Spaghetti model – to provide a
general way to represent geometric objects in a
relational tuple or a set of tuples.
 Each spatial object is triangulated and each such
obtained triangle is represented by a single tuple in the
relation that stores the spatial objects.
 Able to represent objects of type point, line segment or
polygon (possible, by degenerate rectangles)
 Figure 1.4. point (P), line segment (S), non-convex
polygon (Pg)
 Table 1.2. Records for example from figure 1.4.
OID
x1
y1
x2
y2
x3
y3
P
2
6
2
6
2
6
S
1
2
4
3
4
3
Pg
6
1
8
3
6
3
Pg
6
3
8
3
7
5
Pg
6
3
7
5
5
5