Multidimensional Data and Modelling - Operations

Transcription

1 Multidimensional Data and Modelling - Operations 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 md-data or spatial data lacked an underlying formalism, l conventional data can only be achieved from within DBMS, l management of spatial data is not possible from within a conventional DBMS. 3

4 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). 4

5 Problems of multidimensional data structures The tasks: l surveys and evaluates spatial data modelling approaches, l reviewing restrictions of spatio-temporal models (spatial models with additional time coordinate) to the management of spatial data, l identifying data types, data structures used and operations supported by a data model for the management of cartography, topography, cadastral and relevant applications. 5

6 Problems of multidimensional data structures Background: l relevant applications require the processing of data that can geometrically be represented on a 2-d plane, a point, line or surface, but also including higher dimensions, l such piece of data, and any set of them as well, is termed spatial data or (spatial) object, l these data are distinguished from conventional data, such as a name, a number, a date, and so forth, l data modelling requires specifying at minimum data types, data structures and operations. 6

7 Problems of multidimensional data structures l to face individuality of spatial/multidimensional data and define closed spatial operations, many distinct spatial data modelling approaches have been proposed, l many of following characteristics: l adopt set-based data types, such as set of points, set of surfaces, etc. l use either complex data structures to record spatial data or two types of such structures, one to record spatial and another to record conventional data, l define operations that apply to spatial data of one specific type; for example, Overlay only between surfaces. 7

8 Problems of multidimensional data structures l other operations discard part of the result, for example, the point and line parts produced by the spatial intersection of two surfaces, l however, a data model should be simple, and enable a most accurate mapping of the real world. 8

9 Problems l Formal / expressive language for defining spatial data is missing, as... l too much effort has been put into geometric representation of spatial data, l underlying data management possibilities of spatial data and conventional data in one DBMS is not satisfying (not efficient, nor simple or nor optimized) 9

10 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. 10

11 Note: 1NF Definition l The term First Normal Form (INF) describes the tabular format in which l All the key attributes are defined. l There are no repeating groups in the table, id est each row/column intersection can contain one and only one value, rather than a set of values. l All attributes are dependent on the primary key. 11

12 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. 12

13 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. 13

14 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. 14

15 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. 15

16 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. 16

17 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 17

18 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. 18

19 Polygon -definition 19

20 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. 20

21 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. 21

22 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 22

23 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 23

24 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 24

25 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. 25

26 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 26

27 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 27

28 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 28

29 Spatial operations in 3-d union / fusion 29

30 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. 30

31 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. 31

32 Spatial operations in 3-d difference 32

33 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. 33

34 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. 34

35 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. 35

36 Spatial operations in 3-d intersection 36

37 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. 37

38 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. 38

39 Spatial operations in 3-d overlay 39

40 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. 40

41 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. 41

42 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. 42