Multidimensional Data and Modelling

Transcription

1 Multidimensional Data and Modelling 1

2 Problems of multidimensional data structures l multidimensional (md-data or spatial) data and their implementation of operations between objects (spatial data practically cover just 1-, 2-, 3-dimensional data), l management of md-data splits into distinct types of processing: a) md-data or spatial data, b) conventional data, c) associations with spatial data, l formulation of queries was almost missing and too much programming was required, 2

3 Problems of multidimensional data structures l a new research effort was undertaken in the area of spatial databases, l efforts covered various sectors, such as design of efficient physical data structures and access methods, l approaches address spatial data modelling issues in: l an indirect way (spatial data modelling was not their primary objective), l direct way (research has been undertaken dedicated solely to the definition of data models). 3

4 Goals l Find an appropriate spatial data modelling approach, satisfying the following demands: 1. support of spatial data types: point, line & surface, 2. data structures should be simple, i.e. conform to the First Normal Form (1NF) of relational model, 3. spatial operations possible on any spatial data type. 4

5 Point -definition l A point is a location on a line (1-dimensional), in plane (2-d) or in space (3-d) with specific coordinates, i.e. (x), (x,y) or (x,y,z), corresponding to the axis in a coordination system. A set of points is a collection of more than one point. 5

6 Vector -definition l A vector is a straight line segment or movement defined by its end points or by the current position, i.e. (0,0), and one other point. Its length is magnitude and its orientation in space is direction. A vector exists in 1- d, 2-d and 3-d. 6

7 Line -definition l A line is defined by a set of points that have the same length (vector). It is also defined by a point and a vector, this line is called infinite line, whereas a segment of line has two (end-)points and is-called finite line. A line exists in 1-d, 2-d and 3-d. A set of lines is a collection of more than one line. 7

8 Surface -definition l A simple surface is defined by a point & two vector or by a point & one vector (parallel to axis) and exists in 2-d and 3-d. More complex surfaces need more vector. Any point of the surface is a linear combination of these vectors. A set of surfaces is a collection of more than one surface. 8

9 Hybrid surface -definition l An hybrid surface is a special surface. It consists of linked objects of at least 2 different dimensions, i.e. a line connected with a cube in space. An hybrid surface exists in 2-d and in 3-d. 9

10 Polyline -definition l A polyline or vector polyline consists of more than one point (so-called interpolation points), which are connected by lines/segments with each other depending on their coordinates. Polylines exist in 2-d and 3-d 10

11 Polygon -definition l A polygon or vector polygon is described by its vertices (points; min. three vertices!) which are connected by lines/segments with each other depending on their coordinates. However, the connecting lines are not allowed to intersect each other, they can only be tangent to their vertices. In fact, a polygon is a polyline which is closed, l Furthermore, there is special type of polygon called polygon with hole. Typical polygons are: triangle, rectangle, square, lozenge,.polygons exist in 2-d and 3-d. In 3-d, polygon is laying in some plane. 11

12 Polygon -definition 12

13 Overview: Definitions of spatial data operations A, B represents sets of data or geometric shapes, i.e. surfaces, lines in 1-d or x-dimension. 1. Spatial Union / Fusion: A union B: Result of union is one set or none, which contains all elements which are whether in A or B. 2. Spatial Difference: A difference B: Result of difference is one or more sets, containing all elements from A which are not in B. 3. Spatial Intersection: A intersects B: Result of intersection is one or more sets, containing all elements which are in A and in B. 13

14 Overview: Definitions of spatial data operations Overlay: 1. Inner Overlay 2. Left Overlay 3. Right Overlay 4. Full Overlay Unary operations: 1. Spatial Complementation 2. Spatial Boundary 3. Spatial Envelope 4. Spatial Buffer. 14

15 Definition - Spatial union / fusion A union / fusion operation on a set of spatial data objects which fully/partly intersects results in an aggregation of these sets to one set. Unless they haven t any intersections the union / fusion has no impact at all. Application is in 2-d and x-d possible on a set of two or more spatial objects. Application on two objects in 2-d 15

16 Definition Spatial difference l A difference operation on a set of spatial objects is comparable to the math. minus -operation. Applied to two different spatial objects it abstracts all common data of these objects from the first object. Application is possible in 2-d and x-d for a set of two spatial objects. Application on two objects in 2-d 16

17 Definition Spatial intersection l An intersection operation on a set of spatial objects extracts the common area of two or more spatial objects. Unless they haven t common area, the application of this operation results in no object. Application is possible in 2-d and x-d on a set of two or more spatial objects. Application on two objects in 2-d 17

18 Definition Overlay An overlay operation applied to a set of two spatial object extracts the visible front face of these objects. A requirement of successful overlay is intersection of these objects. Distinguish between inner (only intersecting area), left, right and full overlay. Application is possible in 2-d and x-d on a set of two spatial objects. l Inner Overlay: all elements that are in the intersection of A and B are shown. l l Left Overlay: all elements are in A and in the intersection of A and B are shown. Right Overlay: all elements that are in B and in the intersection of A and B are shown. 18

19 Definition Overlay l Full Overlay: all elements that are in A and B and in the intersection of A and B are shown. Application on two objects in 2-d 19

20 Definition Unary operations Unary operations on spatial data need no interaction between two or more spatial objects like the precedent operations. Each has a specific impact on the spatial objects applied on, needing no or only one input attribute to be set (only spatial buffer distance d). They can be applied on spatial objects in 2-d or x-d. l Spatial Complementation: all elements of the sets are converted to a different status (i.e., selected- >unselected & vice versa). l Spatial Boundary: all boundaries of the sets are selected. Application in 2-d 20

21 Definition Unary operations l l Spatial Envelope: the outer common boundary of the sets is determined and all sets inside regardless of their inner own boundaries are selected. Spatial Buffer: A buffer space size is defined in the first step. In the second all boundaries of the sets to each other are reduced by the buffer space size. Application in 2-d 21

22 Spatial operations in 3-d union / fusion 22

23 Spatial operations in 3-d union / fusion other cases l Point (0-d) lays inside cube (3-d): union includes a fixed pair of area of point + cube. l Point (0-d) lays outside cube (3-d): union results in area of point and area of cube. l Line (1-d) lays inside cube (3-d): union results in a fixed pair of area of line + cube. l Line (1-d) lays outside cube (3-d): union results in area of line and area of cube. l Line (1-d) intersects cube (3-d): union includes nonintersecting area of line + cube as one new object. 23

24 Spatial operations in 3-d union / fusion other cases l Polygon (2-d) lays inside cube (3-d): union results in a fixed pair of area of polygon + cube. l Polygon (2-d) lays outside cube (3-d): union has no effect on objects. l Polygon (2-d) intersects cube (3-d): union includes a fixed pair of non-intersecting area of polygon + cube. 24

25 Spatial operations in 3-d difference 25

26 Spatial operations in 3-d difference other cases l Point (0-d) lays inside cube (3-d): cube difference point results in area of cube as one new object. Point difference cube results in area of point as one new object. l Point (0-d) lays outside cube (3-d): difference results in no object. l Line (1-d) lays inside cube (3-d): cube difference line results in area of cube as one new object. Line difference cube results in area of line as one new object. l Line (1-d) lays outside cube (3-d): difference results in no object. 26

27 Spatial operations in 3-d difference other cases l Line (1-d) intersects cube (3-d): cube difference line results in non-intersecting area of cube with line as one new object. Line difference cube results in nonintersecting area of line with cube as one new object. l Polygon (2-d) lays inside cube (3-d): cube difference polygon results in area of cube as one new object. Polygon difference cube results in area of polygon as one new object. l Polygon (2-d) lays outside cube (3-d): difference results in no object. 27

28 Spatial operations in 3-d difference other cases l Polygon (2-d) intersects cube (3-d): cube difference polygon results in non-intersecting area of cube with polygon as one new object. Polygon difference cube results in non-intersecting area of polygon with cube as one new object. 28

29 Spatial operations in 3-d intersection 29

30 Spatial operations in 3-d intersection other cases l Point (0-d) lays inside cube (3-d): intersection results in area of point as new object. l Point (0-d) lays outside cube (3-d): intersection results in no object. l Line (1-d) lays inside cube (3-d): intersection results in area of line as one new object. l Line (1-d) lays outside cube (3-d): intersection results in no object. 30

31 Spatial operations in 3-d intersection other cases l Line (1-d) intersects cube (3-d): intersection results in intersecting area of cube with line as one new object. l Polygon (2-d) lays inside cube (3-d): intersection results in area of polygon as one new object. l Polygon (2-d) lays outside cube (3-d): intersection results in no object. l Polygon (2-d) intersects cube (3-d): intersection results in intersecting area of cube with polygon as one new object. 31

32 Spatial operations in 3-d overlay 32

33 Spatial operations in 3-d overlay other cases l Point (0-d) lays inside cube (3-d): LO results in area of point + cube. RO results in area of point + cube. IO results in area of point. FO results in area of point + cube. l Point (0-d) lays outside cube (3-d): overlay results in no object. l Line (1-d) lays inside cube (3-d): LO results in area of line + cube. RO results in area of line + cube. IO results in area of line. FO results in area of point + cube. l Line (1-d) lays outside cube (3-d): overlay results in no object. 33

34 Spatial operations in 3-d overlay other cases l Line (1-d) intersects cube (3-d): LO results in area of line + intersecting area with cube. RO results in area of cube + intersecting area with line. IO results in intersecting area of line with cube. FO results in area of line + cube. l Polygon (2-d) lays inside cube (3-d): LO results in area of polygon + cube. RO results in area of polygon + cube. IO results in area of polygon. FO overlay results in area of polygon + cube. l Polygon (2-d) lays outside cube (3-d): overlay results in no object. 34

35 Spatial operations in 3-d overlay other cases l Polygon (2-d) intersects cube (3-d): LO results in area of polygon + intersecting area with cube. RO results in area of cube + intersecting area with polygon. IO results in intersecting area of polygon with cube. FO results in area of polygon + cube. Note: Spatial overlay in 3-d is only applicable in combination with some transformation or rotation operations, whereas for geometric overlay in 2-d it is sufficient to use only the standard overlay operations. 35

36 GIS-centric approach The approach classifies operations into four categories: l (i) Local: The value of each location p depends on value of the same location p in input maps. l (ii) Zonal: The result value of each location p depends on values of the locations contained in zone of p in one or more input maps. l (iii) Focal: The result value of each location p depends on values of the locations contained in neighbourhood of p in one or more input maps. l (iv) Incremental: They extend set of Focal operations by taking into account type of zone at each location. One of the local operations resembles Full Overlay. 36

37 Example: Mapping SD to ORDMBS l Three maps with simple spatial data: 37

38 Result in ORDBMS - tables TABLE city_table TABLE highway_table id geom id geom 1 POINT(5,7) 1 LINE(0,7.2,5,7) 2 POINT(6,6) 2 LINE(5,7,8,6.5) 3 POINT(8,6.5) 3 LINE(8,6.5,6,6) 4 LINE(6,6,2.5,0) TABLE county_table id geom 1 POLYGON(points(POINT(0,2.1),POINT(8,6), POINT(0,6))) 2 POLYGON(points(POINT(0,6), POINT(7,9))) 3 POLYGON(points(POINT(11,8.5), POINT(7,6))) all variables in italics in the above displayed tables are references (addresses) of the actual geom-objects saved in different meta-tables in the db. In fact, POLYGON( ) with id 1 consists of references to three lower objects: First the POLYGON-object itself, the underlying POINTS-table and within these table there are POINTS objects saved. 38

39 Ex: Operations on raster grids Possible implementation: l A map is modelled as a 2-d raster grid data structure, which represents a partition of a given rectangular area into a matrix of a finite set of squares, called cells or pixels. l Each cell represents one of locations. l All these approaches consider only surfaces. Examples of operations on grids are shown below. Note that the functionality of operation Combine (Figure 3(f)) resembles that of Full Overlay on surfaces. 39

40 Ex: Operations on raster grids 40

41 Ex: Operations on raster grids The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. 41

42 Ex: Operations on raster grids of the spatio-temporal model (d Onofrio & Pourabbas, 2001). It considers 42 maps of surfaces or lines, but it does not achieve the functionality of all the operations

43 Ex: Functionality for maps l A map (called spatial partition) of a given area is defined as a set of non-overlapping, adjacent surfaces. l Each such surface is associated with a tuple of conventional data. l Surfaces associated with the same conventional data merge automatically into a single surface. l Point and Line types are not defined. l Three primitive operations are defined and, based on them, a representative functionality for map management is achieved (below). 43

44 Ex: Functionality for maps 44

45 Ex: Functionality for maps 45

46 Ex: relations and operations l Point, simple polyline and polygon data types (Figure 5(a-c)) are proposed as usually. l A map (called layer) M is defined as a mapping from a set of spatial values G to the Cartesian product of a set of conventional attributes (M: G C1,C2,...,Cn). Hence, a map can be seen as a relation with just one spatial attribute G. l Operations on maps also are defined. Operation Attribute derivation (Spatial computation) enables the application of conventional (spatial) functions and predicates. 46

47 Ex: relations and operations l Operation Reclassification merges into one all those tuples of a layer that have identical values in a given attribute and also are associated to adjacent spatial objects. l It can apply only to layers of type simple polyline or polygon. l Operation Overlay or Full Overlay applies to two maps L1 and L2 of any data type. l Its result is the union of three sets. 47

48 Ex: relations and operations l (i) I, consisting of the pieces of spatial objects l both in L1 and L2, l (ii) L, consisting of the pieces of spatial objects in L1 that are not inside the spatial objects in L2, l (iii) R, consisting of the pieces of spatial objects in L2 that are not inside the spatial objects in L1. 48

49 Representation of spatial objects in various approaches 49