Issues in Spatial Database and GIS

8/9/2019 Issues in Spatial Database and GIS

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Copyright 2007 Hanan Samet

These notes may not be reproduced by any means (mechanical or elec-tronic or any other) without express written permission of Hanan Samet

ISSUES IN SPATIAL DATABASES AND GEOGRAPHIC

INFORMATION SYSTEMS (GIS)

HANAN SAMET

COMPUTER

SCIENCE

DEPARTMENT AND

CENTER FOR AUTOMATION RESEARCH ANDINSTITUTE FOR ADVANCED COMPUTER STUDIES

UNIVERSITY OF MARYLAND

COLLEGE PARK, MARYLAND 20742-3411 USA


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PRINCE GEORGES COUNTYhi28

Copyright 2007 by Hanan Samet 2


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zk8

BACKGROUND (A PERSONAL VIEW!)

1. GIS originally focussed on paper map as output

anything is better than drawing by hand no great emphasis on execution time

2. Paper output supports high resolution

display screen is of limited resolution

can admit less precise algorithms

Ex: buffer zone computation (spatial range query)

a. usually use a Euclidean distance metric (L2)

takes a long time

b. can be sped up using a quadtree and aChessboard distance metric (L)

not as accurate as Euclidean but may notbe able to perceive the difference on a displayscreen!

as much as 3 orders of magnitude faster

3. Users accustomed to spreadsheets

GIS should work like a spreadsheet

fast response time

ability to ask what if questions and see theresults

incorporate a database for seamless integration ofspatial and nonspatial (i.e., attribute data)

Copyright 2007 by Hanan Samet


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zk9

GENERAL SPATIAL DATABASE ISSUES

1. Why do we want a database?

to store data so that it can be retrieved efficiently

should not lose sight of this purpose

2. How to integrate spatial data with nonspatial data

3. Long fields in relational database are not the answer

a stopgap solution as just a repository for data

does not aid in retrieving the data

if data is large in volume, then breaks down astuples get very large

4. A database is really a collection of records with fieldscorresponding to attributes of different types

records are like points in higher dimensional space

a. some adaptations take advantage of this analogy

b. however, can act like a straight jacket in case of

relational model5. Retrieval is facilitated by building an index

need to find a way to sort the data

index should be compatible with data being stored

choose an appropriate zero or reference point

need an implicit rather than an explicit index

a. impossible to foresee all possible queries inadvance

b. explicit would sort two-dimensional points onthe basis of distance from a particular point P impractical as sort is inapplicable to points

different from P



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zk10

6. Identify the possible queries and find their analogs inconventional databases

e.g., a map in a spatial database is like a relation ina conventional database (also known as spatialrelation)

a. difference is the presence of spatial attribute(s)

b. also presence of spatial output

7. How do we interact with the database?

SQLmay not be easy to adapt

graphical query language

output may be visual in which case a browsingcapability (e.g., an iterator) is useful

8. What strategy do we use in answering a query thatmixes traditional data with nontraditional data?

need query optimization rules

must define selectivity factors

a. dependent on whether index exists onnontraditional data

b. if no, then select on traditional data first

Ex: find all cities within 100 miles of the MississippiRiver with population in excess of 1 million

a. spatial selection first if region is small (implieshigh spatial selectivity)

b. relational selection first if very few cities with alarge population (implies high relationalselectivity)



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zk11

DATA IN SPATIAL DATABASES

1. Spatial information

locations of objects (are discrete, individual pointsin space)

space occupied by objects (are continuous; haveextent)

a. example objects

lines (e.g., roads, rivers)

regions (e.g., buildings, crop maps,polyhedra)

others ...

b. are objects disjoint or may they overlap?

e.g., several crop types may be grown on aplot of land

not concerned here with raster vs: vector issuesas these are data representationissues ratherthan data type issues

2. Non-spatial information

region names, postal codes, ...

city population, year founded, ...

road names, speed limits, ...



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hi27

EXAMPLE QUERIES ON LINE SEGMENT DATABASES

Queries about line segments

1. All segments that intersect a given point or set ofpoints

2. All segments that have a given set of endpoints

3. All segments that intersect a given line segment

4. All segments that are coincident with a given linesegment

Proximity queries

1. The nearest line segment to a given point

2. All segments within a given distance from a givenpoint (also known as a range or window query)

Queries involving attributes of line segments

1. Given a point, find the closest line segment of aparticular type

2. Given a point, find the minimum enclosing polygonwhose constituent line segments are all of a giventype

3. Given a point, find all the polygons that are incidenton it



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gs10

WHAT MAKES CONTINUOUS SPATIAL DATADIFFERENT

1. Spatial extent of the objects is the key to thedifference

2. A record in a DBMS may be considered as a point ina multidimensional space

a line can be transformed (i.e., represented) as apoint in 4-d space with (x1, y1, x2, y2)

(x2,y2)

(x1,y1)

good for queries about the line segments

not good for proximity queries since points outsidethe object are not mapped into the higherdimensional space

representative points of two objects that are

physically close to each other in the original space(e.g., 2-d for lines) may be very far from each otherin the higher dimensional space (e.g., 4-d)

A

B

Ex:

problem is that the transformationonly transforms the space occupiedby the objects and not the rest of thespace (e.g., the query point)

can overcome by projecting back to original space

3. Use an index that sorts based upon spatialoccupancy (i.e., extent of the objects)



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hi29.1

SPATIAL INDEXING REQUIREMENTS

1. Compatibility with the data being stored

2. Choose an appropriate zero or reference point

3. Need an implicit rather than an explicit index

impossible to foresee all possible queries inadvance

cannot have an attribute for every possible spatialrelationship

a. derive adjacency relations

b. 2-d strings capture a subset of adjacencies

all rows

all columns

implicit index is better as an explicit index which,for example, sorts two-dimensional data on the

basis of distance from a given point is impracticalas it is inapplicable to other points

implicit means that don't have to resort the data forqueries other than updates



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gs11

SORTING ON THE BASIS OF SPATIAL OCCUPANCY

Decompose the space from which the data is drawn intoregions called buckets(like hashing but preserves order)

Interested in methods that are designed specifically forthe spatial data type being stored

Basic approaches to decomposing space

1. minimum bounding rectangles

e.g., R-tree

good at distinguishing empty and non-empty

space drawbacks:

a. non-disjoint decomposition of space

may need to search entire space

b. inability to correlate occupied and unoccupiedspace in two maps

2. disjoint cells

drawback: objects may be reported more than once uniform grid

a. all cells the same size

b. drawback: possibility of many sparse cells

adaptive grid quadtree variants

a. regular decomposition

b. all cells of width power of 2

partitions at arbitrary positionsa. drawback: not a regular decomposition

b. e.g., R+-tree

Can use as approximations in filter/refine queryprocessing strategy



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MINIMUM BOUNDING RECTANGLEShi31

Objects grouped into hierarchies, stored in a structuresimilar to a B-tree

Object has single bounding rectangle, yet area that itspans may be included in several bounding rectangles

Drawback: not a disjoint decomposition of space

Examples include the R-tree and the R*-tree

a

b

c

d

e

f

g

h

i

1

b

Order (m,M) R-tree

1. between m M/2 and Mentries in each nodeexcept root

2. at least 2 entries in root unless a leaf node


2

r

R3

R4

R5

R6

ic feba hgd

hi31

R3: R4: R5: R6:Copyright 2007 by Hanan Samet 7

3

z

R4R3 R6R5

R1

R2

hi31

R2:R1:


4

g

R2R1

hi31

R0:

R0



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hi32SEARCHING FOR A POINT OR LINESEGMENT IN AN R-TREE

1

b

ba hgd ic fe

R2R1

R4R3 R6R5

ab

c

d

e

f

g

h

i

R3

R4

R5

R6

R2

R1

Q

Drawback is that may have to examine many nodes

since a line segment can be contained in the coveringrectangles of many nodes yet its record is contained inonly one leaf node (e.g., i in R2, R3, R4, and R5)

Ex: Search for a line segment containing point Q

R3: R4: R5: R6:

R1: R2:

R0:

R0


hi32

Q is in R0

2

v


hi32

Q can be in both R1 and R2

3

r


hi324z

Searching R1 first means that R4 is searched but thisleads to failure even though Q is part of i which is in R4


hi325g

Searching R2 finds that Q can only be in R5Copyright 2007 by Hanan Samet 8


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hi33DISJOINT CELLS

Objects decomposed into disjoint subobjects; eachsubobject in different cell

Drawback: in order to determine area covered byobject, must retrieve all cells that it occupies

Techniques differ in degree of regularity

R+-tree (also k-d-B-tree) and cell tree are examplesof this technique

a

b

c

d

e

f

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h

i

1

b

Q


hi332r

R3

R4

R6

R5

hgd ihc ifc ba e iR3: R4: R5: R6:


hi333z

R4R3 R6R5

R1

R2

R1: R2:


hi334g

R2R1R0:

R0



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hi33.1K-D-B-TREES

a

b

c

d

e

f

gh

i

1

b

Q

Rectangular embedding space is hierarchicallydecomposed into disjoint rectangular regions

No dead space in the sense that at any level of the tree,entire embedding space is covered by one of the nodes

Blocks of k-d tree partition of space are aggregated intonodes of a finite capacity

When a node overflows, it is split along one of the axes

Originally developed to store points but may be extendedto non-point objects represented by their minimumbounding boxes

Drawback: in order to determine area covered by object,must retrieve all cells that it occupies


hi33.12r

R3 R4 R6

R5

hgd ihc ifc ba e iR3: R4: R5: R6:Copyright 2007 by Hanan Samet

hi33.13z

R4R3 R6R5

R1 R2

R1: R2:


hi33.14g

R2R1R0:

R0



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UNIFORM GRID

Ideal for uniformly distributed data

Supports set-theoretic operations

Spatial data (e.g., line segment data) is rarely uniformlydistributed

hi34



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hi35

QUADTREES

Hierarchical variable resolution data structure based onregular decomposition

Many different decomposition schemes and applicableto different data types:

1. points2. lines3. regions4. rectangles5. surfaces6. volumes7. higher dimensions including time

changes meaning of nearesta. nearest in time, ORb. nearest in distance

Can handle both raster and vector data as just a spatialindex

Shape is usually independent of order of inserting data Ex: region quadtree

A decomposition into blocks not necessarily a tree!



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hi36

REGION QUADTREE

Repeatedly subdivide until obtain homogeneous region

For a binary image (BLACK 1 and WHITE 0) Can also use for multicolored data (e.g., a landuse

class map associating colors with crops)

Can also define data structure for grayscale images

A collection of maximal blocks of size power of twoand placed at predetermined positions

1. could implement as a list of blocks each of which

has a unique pair of numbers: concatenate sequence of 2 bit codes correspond-ing to the path from the root to the blocks node

the level of the blocks node

2. does not have to be implemented as a tree tree good for logarithmic access

A variable resolution data structure in contrast to apyramid (i.e., a complete quadtree) which is a

multiresolution data structure

A

B C D E

NW

NE SW

SE

F G H I J L M N O Q

K P

37 38 39 40 57 58 59 60

0

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B

60

37

L

J

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GF

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N

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M 5758

59

4039

38



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hi37

PYRAMID

Internal nodes contain summary of information innodes below them

Useful for avoiding inspecting nodes where there couldbe no relevant information

c1

c2

c3

c4

c5

c6

{c1,c2,c3,c4,c5,c6}

{c2,c3,c6} {c2,c3,c4,c5}

{c1,c2,c3,c4,c5,c6}

{c6}



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hi38

QUADTREES VS. PYRAMIDS

Quadtrees are good for location-based queries

1. e.g., what is at location x?2. not good if looking for a particular feature as have toexamine every block or location asking are you theone I am looking for?

Pyramid is good for feature-based queries e.g.,

1. does wheat exist in region x? if wheat does not appear at the root node, then

impossible to find it in the rest of the structure and

the search can cease2. report all crops in region x just look at the root

3. select all locations where wheat is grown only descend node if there is possibility that wheat is

in one of its four sons implies little wasted work

Ex: truncated pyramid where 4 identically-colored sonsare merged

c1

c2

c3

c4

c5

c6

{c1,c2,c3,c4,c5,c6}

{c2,c3,c6} {c2,c3,c4,c5}

{c1,c2,c3,c4,c5,c6}

{c6}

{c2,c3,c5} {c1,c2,c3,c5}

Can represent as a list of leaf and nonleaf blocks (e.g.,as a linear quadtree)



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PR QUADTREE (Orenstein)1 hp9b

Regular decomposition point representation

Decomposition occurs whenever a block contains more

than one pointUseful when the domain of data points is not discretebut finite

(0,100) (100,100)

(100,0)(0,0)

(35,42)

Chicago

Maximum level of decomposition depends on theminimum separation between two points

if two points are very close, then decomposition can bevery deep

can be overcome by viewing blocks as buckets withcapacity cand only decomposing the block when itcontains more than cpoints

1.

2.

3.

4.

Ex:c= 1


2

rhp9

(52,10)

Mobile


3

zhp9

(62,77)

Toronto


4

ghp9

(82,65)

Buffalo


5

vhp9

(5,45)

Denver


6

ghp9

(27,35)

Omaha


7

zhp9

(85,15)Atlanta


8

rhp9

(90,5)

Miami



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REGION SEARCH1 hp10b

Use of quadtree results in pruning the search space

Ex: Find all points within radius rof point A

A

r


hp10

If a quadrant subdivision point plies in a region l, thensearch the quadrants of pspecified by l

1. SE 6. NE 11. All but SW

2. SE, SW 7. NE, NW 12. All but SE3. SW 8. NW 13. All4. SE, NE 9. All but NW5. SW, NW 10. All but NE

1 2 39 10

13

1211

45

876

2

r


hp103z



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hp11FINDING THE NEAREST OBJECT

Ex: find the nearest object to P

1

b

P

12 8 7 6

13 9 1 4 5

2 3

10 11

D

E C

F

A

B

Assume PRquadtree for points (i.e., at most one point per block) Search neighbors of block 1 in counterclockwise order Points are sorted with respect to the space they occupy which enables pruning the search space

Algorithm:


hp112r

1. start at block 2 and compute distance to Pfrom A


hp113z

2. ignore block 3 whether or not it is empty as Ais closer to Pthan any point in 3


hp114g

3. examine block 4 as distance to SWcorner is shorter than the distance from Pto A; however, reject Bas it is further from Pthan A


hp115v

4. ignore blocks 6, 7, 8, 9, and 10 as the minimum distance to them from Pis greater than the distance

from Pto A


hp116z

5. examine block 11 as the distance from Pto the southern border of 1 is shorter than the distance from Pto A; however, reject Fas it is further from Pthan A


hp117r

If Fwas moved, a better order would have started withblock 11, the southern neighbor of 1, as it is closest

new F



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cd32

a

PM1 QUADTREE

1

DECOMPOSITION RULE:

Partitioning occurs when a block contains more thanone segment unless all the segments are incident atthe same vertex which is also in the same block

b

Vertex-based (one vertex per block)

Shape independent of order of insertion


cd32

b

2

r


cd323z

c


cd324g

d


cd325v

e


cd32

f

6

r


cd327z

g


cd32

h

8

g


cd32

i

9

v



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cd35

a

PMR QUADTREE 1

Split a block onceif upon insertion the number ofsegments intersecting a block exceeds N

b

Edge-based

Avoids having to split many times when two vertices orlines are very close as in PM1 quadtree

Probabilistic splitting and merging rules

Uses a splitting threshold value say N

DECOMPOSITION RULE:

Merge a block with its siblings if the total number of line

segments intersecting them is less than N

Merges can be performed more than once Does not guarantee that each block will contain at

mostNline segments

Splitting threshold is not the same as bucket capacity

Shape depends on order of insertion

Ex:N= 2


cd352r

b


cd353z

c


cd35

d

4

g


cd355v

e


cd35

f

6

r


cd35

g

7

z


cd35

h

8

g


cd35

i

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ADVANTAGES OF EDGE-BASED METHODS

The decomposition is focussed where the line segmentsare the densest

Handles the situation that several non-intersecting linesare very close to each other

cd46

Able to represent nonplanar line segment data

Good average behavior in terms of node occupancy

Example:

PM1 PMR (N=2)



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CONSISTENCY OF PM APPROACHcd47

Stores lines exactly

Each line segment is represented by a pointer to a

record containing its endpoints

Updates can be made in a consistent manner - i.e.,when a vector feature is deleted, the database can berestored to the state it would have been in had thedeleted feature never been inserted

Uses the decomposition induced by the quadtree tospecify what parts of the line segments (i.e., q-edges)are actually present

1. not a digitized representation2. no thickness associated with line segments

The line segment descriptor stored in a block onlyimplies the presence of the corresponding q-edge - itdoes not mean that the entire line segment is present

as a lineal feature

Useful for representing fragments of lines such asthose that result from the intersection of a line map withan area map

Ex:

1

b


2

rcd47


3

zcd47



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MAXIMAL SEARCH RADIUS

P

cd58

Properties of the PM quadtree family (PM1, PMR, etc.)greatly localize the search area for nearest line segment

Assume that the query point P falls in the SW corner ofthe small highlighted block

By virtue of the existence of a block of this size, we areguaranteed that at least one of the remainingsiblings contains a line segment

1

b


2

r

three

cd58


3

zcd58



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cd59NEAREST LINE SEGMENT ALGORITHM

A four stage intelligent search process

Maximal search radius equal to length of parent node'sdiagonal

Basic algorithm:

Search the block containing the query point1.

1

b


2

Search the three siblings2.

rcd59


3

Search the three regions of size equal to that of theparent that are incident to the block containing thequery point

3.

zcd59


4

Search the final four groups of two adjacent blocksto the previous step

4.

gcd59



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hp14MX-CIF QUADTREE (Kedem)

1

b

Collections of small rectangles for VLSI applications

Each rectangle is associated with its minimum

enclosing quadtree blockLike hashing: quadtree blocks serve as hash buckets

1.

2.

3.

1

2

3

45

6

9

87

10

11

12

D {11}

F {12}E {3,4,5}

B {1}

A {2,6,7,8,9,10}

C {}

A

B

C

E

D

F


hp142r

Collision = more than one rectangle in a block

resolve by using two one-dimensional MX-CIF trees tostore the rectangle intersecting the lines passingthrough each subdivision point

4.


hp14

one for y-axis

Binary tree for y-axis through A

Y1

Y2 10

Y4

2

Y5

Y3

6Y7

8

Y6

3

g


hp14

if a rectangle intersects both xand yaxes, thenassociate it with the yaxis

4

v


hp145z

one for x-axis

Binary tree for x-axis through A

X1

X39

X5

7

X4

X2

X6



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sf2

HIERARCHICAL RECTANGULAR DECOMPOSITION

Similar to triangular decomposition

Good when data points are the vertices of arectangular grid

Drawback is absence of continuity between adjacentpatches of unequal width (termed the alignmentproblem)

Overcoming the presence of cracks

1. use the interpolated point instead of the true point(Barrera and Hinjosa)

2. triangulate the squares (Von Herzen and Barr)

can split into 2, 4, or 8 triangles depending on howmany lines are drawn through the midpoint

if split into 2 triangles, then cracks still remain

no cracks if split into 4 or 8 triangles



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sf31b

RESTRICTED QUADTREE (VON HERZEN/BARR)

All 4-adjacent blocks are either of equal size or of ratio 2:1

Note: also used in finite element analysis to adptivelyrefine an element as well as to achieve elementcompatibility (termed h-refinementby Kela, Perucchio, andVoelcker)


sf32r


sf33z

8-triangle decomposition rule1. decompose each block into 8 triangles (i.e., 2 triangles

per edge)

2. unless the edge is shared by a larger block

3. in which case only 1 triangle is formed


sf34g

4-triangle decomposition rule

1. decompose each block into 4 triangles (i.e., 1 triangleper edge)

2. unless the edge is shared by a smaller block

3. in which case 2 triangles are formed along the edge


sf3

Prefer 8-triangle rule as it is better for display applications(shading)

5

v



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sf4

PROPERTY SPHERES (FEKETE)

Approximation of spherical data

Uses icosahedron which is a Platonic solid

1. 20 faceseach is a regular triangle

2. largest possible regular polyhedron



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sf5

ALTERNATIVE SPHERICAL APPROXIMATIONS

Could use other Platonic solids

1. all have faces that are regular polygons

tetrahedron: 4 equilateral triangular faces

hexahedron: 6 square faces

octahedron: 8 equilateral triangular faces

dodecahedron: 12 pentagonal faces

2. octahedron is nice for modeling the globe

it can be aligned so that the poles are at oppositevertices

the prime meridian and the equator intersect atanother vertex

one subdivision line of each face is parallel to theequator

Decompose on the basis of latitude and longitudevalues

1. not so good if want a partition into units of equalarea as great problems around the poles

2. project sphere onto plane using Lambertscylindrical projection which is locally area preserving

Instead of approximating sphere with the solids,project the faces of the solids on the sphere (Scott)

1. all edges become sub-arcs of a great circle

2. use regular decomposition on triangular, square, orpentagonal spherical surface patches



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hi60

OCTREES

1. Interior (voxels)

analogous to region quadtree approximate object by aggregating similar voxels

good for medical images but not for objects withplanar faces

Ex:

1 2 3 4 13 14 15

12111098765

B

A14 15

49 10

6

1 2

13

1211

5

2. Boundary

adaptation of PM quadtree to three-dimensionaldata

decompose until each block containsa. one face

b. more than one face but all meet at same edge

c. more than one edge but all meet at samevertex

impose a spatial index on a boundary model(BRep)



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hi39

EXAMPLE QUADTREE-BASED QUERY

Query: find all cities with population in excess of 5,000 inwheat growing regions within 10 miles of the Mississippi

River1. assume river is a linear feature

use a line map could be a region if asked for sandbars in the river

2. region map for the wheat3. assume cities are points

point map for cities

could be region is asked for high income areas

Combines spatial and non-spatial (i.e., attribute) data

Many possible execution plans - e.g.,

1. compute buffer or corridor around river2. extract wheat area3. intersect 1 with 2

4. intersect city map with 35. retrieve value of population attribute for cities in 4 fromthe nonspatial database (e.g., relational)

Regular decomposition hierarchical data structures suchas the quadtree

1. all maps are in registration

all blocks are in the same positions

not true for R+-trees and BSP trees disjoint decomposition of space - unlike R-tree

2. can perform set-theoretic operations on differentfeature types (e.g., 3 and 4)



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zk26

SAND BROWSER: A SPATIO-RELATIONALBROWSER

Assume a relational database

Relations have spatial and nonspatial attributes

Browse through tuples or objects (groups of tupleswith similar attribute values) of a relation one at a timeaccording to values within ranges of the

1. nonspatial attributes

2. underlying space in which the objects correspond-ing to the spatial attributes are embedded

Make use of indexes to facilitate viewing (termedranking) tuples in order of nearness to a referenceattribute value (e.g., zero, origin, etc.) and obtain tuplesin this order

Graphical user interface instead of SQLbut functionally

equivalent Graphical result of spatial and nonspatial queries

Output

1. display tuples satisfying the query one tuple or oneobject at a time

show the values of all of the attributes of the most

recently generated tuple cursor points at this tuple

2. cumulative display of spatial attributes

Can save the result of an operation as a relation forfuture operations (SAVE GROUP)



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zk36

SPECIFIC SPATIAL DATABASE ISSUES

1. Representation

bounding boxes versus disjoint decomposition

2. How are spatial integrity constraints captured andassured?

edges of a polygon link to form a complete object

line segments do not intersect except at vertices

contour lines should not cross

3. Interaction with the relational model

spatial operations dont fit into SQL

a. buffer

b. nearest to ...

c. others ...

difficult to capture hierarchy of complex objects

(e.g., nested definition)

4. Spatial input is visual

need a graphical query language



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zk37

5. Spatial output is visual

unlike conventional databases, once operation iscomplete, want to browse entire output togetherrather than one tuple at-a-time

dont want to wait for operation to complete beforeoutput

a. partial visual output is preferable

e.g., incremental spatial join and nearestneighbor

b. multiresolution output is attractive

6. Functionality

determining what people really want to do!

7. Performance

not enough to just measure the execution time ofan operation

time to load a spatial index and build a spatially-indexed output is important

sequence of spatial operations as in a spatialspreadsheet

a. output of one operation serves as input toanother

e.g., cascaded spatial join

b. spatial join yields locations of objects and notjust the object pairs



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zk38

CHALLENGES:

1. Incorporation of geometry into database querieswithout user being aware of it!

find geometric analogs of conventional databaseoperations (e.g., ranking semi-join yields discreteVoronoi diagram)

extension of browser concept to permit moregeneral browsing units based on connectivity (e.g.,shortest path), frequency, etc.

2. Spatial query optimization

different query execution plans use spatial selectivity factors to choose among them

3. Graphical query specification instead of SQL

4. Incorporation of time-varying data

how to represent rates?

5. Incorporation of imagery

6. Develop spatial indices that support both location-based (what is at X?) and feature-based queries(where is Y?)

7. Incorporate rendering attributes into databaseobjects or relations

queries based on the rendering attributes

Ex: find all red regions

query by content (e.g., image databases)

8. GIS on the Web and distributed data and algorithms

9. Knowledge discovery

10. Interoperability



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IEEE Internet Computing, 11(1):5259,world wide web.

Structures, MorganKaufmann, San Francisco, 2006.

AddisonWesley, Reading, MA, 1990.

[http://www.cs.umd.edu/~hjs/multidimensionalbookflyer.pdf]

Reading, MA, 1990. Graphics, Image Processing, and GIS, AddisonWesley,

2. H. Samet, Foundations of Multidimensional and Metric Data

3. H. Samet, Applications of Spatial Data Structures: Computer

4. H. Samet, Design and Analysis of Spatial Data Structures,

5. Spatial Data Applets at http://www.cs.umd.edu/~hjs/quadtree

FURTHER READING rf1


1. F. Brabec and H. Samet, Clientbased spatial browsing on the

Jan/Feb 2007.


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1 I n t r o d u c t i o n

S p a t i a l d a t a c o n s i s t s o f s p a t i a l o b j e c t s m a d e u p o f p o i n t s , l i n e s , r e g i o n s , r e c t a n g l e s , s u r f a c e s ,

v o l u m e s , a n d e v e n d a t a o f h i g h e r d i m e n s i o n w h i c h i n c l u d e s t i m e . E x a m p l e s o f s p a t i a l d a t a

i n c l u d e c i t i e s , r i v e r s , r o a d s , c o u n t i e s , s t a t e s , c r o p c o v e r a g e s , m o u n t a i n r a n g e s , p a r t s i n a

C A D s y s t e m , e t c . E x a m p l e s o f s p a t i a l p r o p e r t i e s i n c l u d e t h e e x t e n t o f a g i v e n r i v e r , o r t h e

b o u n d a r y o f a g i v e n c o u n t y , e t c . O f t e n i t i s a l s o d e s i r a b l e t o a t t a c h n o n - s p a t i a l a t t r i b u t e

i n f o r m a t i o n s u c h a s e l e v a t i o n h e i g h t s , c i t y n a m e s , e t c . t o t h e s p a t i a l d a t a . S p a t i a l d a t a b a s e s

f a c i l i t a t e t h e s t o r a g e a n d e c i e n t p r o c e s s i n g o f s p a t i a l a n d n o n - s p a t i a l i n f o r m a t i o n i d e a l l y

w i t h o u t f a v o r i n g o n e o v e r t h e o t h e r . S u c h d a t a b a s e s a r e n d i n g i n c r e a s i n g u s e i n a p p l i -

c a t i o n s i n e n v i r o n m e n t a l m o n i t o r i n g , s p a c e , u r b a n p l a n n i n g , r e s o u r c e m a n a g e m e n t , a n d

g e o g r a p h i c i n f o r m a t i o n s y s t e m s G I S B u c h m a n n e t a l . 1 9 9 0 ; G u n t h e r a n d S c h e k 1 9 9 1 .

A c o m m o n w a y t o d e a l w i t h s p a t i a l d a t a i s t o s t o r e i t e x p l i c i t l y b y p a r a m e t r i z i n g i t a n d

t h e r e b y o b t a i n i n g a r e d u c t i o n t o a p o i n t i n a p o s s i b l y h i g h e r d i m e n s i o n a l s p a c e . T h i s i s

u s u a l l y q u i t e e a s y t o d o i n a c o n v e n t i o n a l d a t a b a s e m a n a g e m e n t s y s t e m s i n c e t h e s y s t e m i s

j u s t a c o l l e c t i o n o f r e c o r d s , w h e r e e a c h r e c o r d h a s m a n y e l d s . I n p a r t i c u l a r , w e s i m p l y a d d

a e l d o r s e v e r a l e l d s t o t h e r e c o r d t h a t d e a l s w i t h t h e d e s i r e d i t e m o f s p a t i a l i n f o r m a t i o n .

T h i s a p p r o a c h i s n e i f w e j u s t w a n t t o p e r f o r m a s i m p l e r e t r i e v a l o f t h e d a t a .

H o w e v e r , i f o u r q u e r y i n v o l v e s t h e s p a c e o c c u p i e d b y t h e d a t a a n d h e n c e o t h e r r e c o r d s

b y v i r t u e o f t h e i r p r o x i m i t y , t h e n t h e s i t u a t i o n i s n o t s o s t r a i g h t f o r w a r d . I n s u c h a c a s e

w e n e e d t o b e a b l e t o r e t r i e v e r e c o r d s b a s e d o n s o m e s p a t i a l p r o p e r t i e s w h i c h a r e n o t

s t o r e d e x p l i c i t l y i n t h e d a t a b a s e . F o r e x a m p l e , i n a r o a d s d a t a b a s e , w e m a y n o t w i s h t o

f o r c e t h e u s e r t o s p e c i f y e x p l i c i t l y w h i c h r o a d s i n t e r s e c t w h i c h o t h e r r o a d s o r r e g i o n s . T h e

p r o b l e m i s t h a t t h e p o t e n t i a l v o l u m e o f s u c h i n f o r m a t i o n m a y b e v e r y l a r g e a n d t h e c o s t

o f p r e p r o c e s s i n g i t h i g h , w h i l e t h e c o s t o f c o m p u t i n g i t o n t h e y m a y b e q u i t e r e a s o n a b l e ,

e s p e c i a l l y i f t h e s p a t i a l d a t a i s s t o r e d i n a n a p p r o p r i a t e m a n n e r . T h u s w e p r e f e r t o s t o r e

t h e d a t a i m p l i c i t l y s o t h a t a w i d e c l a s s o f s p a t i a l q u e r i e s c a n b e h a n d l e d . I n p a r t i c u l a r , w e

n e e d n o t k n o w t h e t y p e s o f q u e r i e s a p r i o r i .

B e i n g a b l e t o r e s p o n d t o s p a t i a l q u e r i e s i n a e x i b l e m a n n e r p l a c e s a p r e m i u m o n t h e

a p p r o p r i a t e r e p r e s e n t a t i o n o f t h e s p a t i a l d a t a . I n o r d e r t o b e a b l e t o d e a l w i t h p r o x i m i t y

q u e r i e s t h e d a t a m u s t b e s o r t e d . O f c o u r s e , a l l d a t a b a s e m a n a g e m e n t s y s t e m s s o r t t h e

d a t a . T h e i s s u e i s w h i c h k e y s d o t h e y s o r t o n . I n t h e c a s e o f s p a t i a l d a t a , t h e s o r t s h o u l d b e

b a s e d o n a l l o f t h e s p a t i a l k e y s , w h i c h m e a n s t h a t , u n l i k e c o n v e n t i o n a l d a t a b a s e m a n a g e m e n t

s y s t e m s , t h e s o r t i s b a s e d o n t h e s p a c e o c c u p i e d b y t h e d a t a . S u c h t e c h n i q u e s a r e k n o w n

s p a t i a l i n d e x i n g m e t h o d s .

O n e a p p r o a c h t o t h e r e p r e s e n t a t i o n o f s p a t i a l d a t a i s t o s e p a r a t e i t s t r u c t u r a l l y f r o m

t h e n o n s p a t i a l d a t a w h i l e m a i n t a i n i n g a p p r o p r i a t e l i n k s b e t w e e n t h e t w o A r e f a n d S a m e t

1 9 9 1 a . T h i s l e a d s t o a m u c h h i g h e r b a n d w i d t h f o r t h e r e t r i e v a l o f t h e s p a t i a l d a t a . I n

s u c h a c a s e , t h e s p a t i a l o p e r a t i o n s a r e p e r f o r m e d d i r e c t l y o n t h e s p a t i a l d a t a s t r u c t u r e s .

T h i s p r o v i d e s t h e f r e e d o m t o c h o o s e a m o r e a p p r o p r i a t e s p a t i a l s t r u c t u r e t h a n t h e i m p o s e d

n o n - s p a t i a l s t r u c t u r e e . g . , a r e l a t i o n a l d a t a b a s e . I n s u c h a c a s e , a s p a t i a l p r o c e s s o r c a n

b e u s e d t h a t i s s p e c i c a l l y d e s i g n e d f o r e c i e n t l y d e a l i n g w i t h t h e p a r t o f t h e q u e r i e s t h a t

i n v o l v e p r o x i m i t y r e l a t i o n s a n d s e a r c h , a n d a r e l a t i o n a l d a t a b a s e m a n a g e m e n t s y s t e m f o r

t h e p a r t o f t h e q u e r i e s t h a t i n v o l v e n o n - s p a t i a l d a t a . I t s p r o p e r f u n c t i o n i n g d e p e n d s o n t h e

e x i s t e n c e o f a q u e r y o p t i m i z e r t o d e t e r m i n e t h e a p p r o p r i a t e p r o c e s s o r f o r e a c h p a r t o f t h e


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q u e r y A r e f a n d S a m e t 1 9 9 1 b .

A s a n e x a m p l e o f t h e t y p e o f q u e r y t o b e p o s e d t o a s p a t i a l d a t a b a s e s y s t e m , c o n s i d e r

a r e q u e s t t o n d t h e n a m e s o f t h e r o a d s t h a t p a s s t h r o u g h t h e U n i v e r s i t y o f M a r y l a n d

r e g i o n " . T h i s r e q u i r e s t h e e x t r a c t i o n o f t h e r e g i o n l o c a t i o n s o f a l l t h e d a t a b a s e r e c o r d s

w h o s e r e g i o n n a m e " e l d h a s t h e v a l u e U n i v e r s i t y o f M a r y l a n d " a n d b u i l d a m a p

N e x t , m a p i s i n t e r s e c t e d w i t h t h e r o a d m a p t o y i e l d a n e w m a p w i t h t h e s e l e c t e d

r o a d s . N o w , c r e a t e a n e w r e l a t i o n h a v i n g j u s t o n e a t t r i b u t e w h i c h i s t h e r e l e v a n t r o a d

n a m e s o f t h e r o a d s i n m a p . O f c o u r s e , t h e r e a r e o t h e r a p p r o a c h e s t o a n s w e r i n g t h e a b o v e

q u e r y . T h e i r e c i e n c y d e p e n d s o n t h e n a t u r e o f t h e d a t a a n d i t s v o l u m e .

I n t h e r e s t o f t h i s r e v i e w w e c o n c e n t r a t e o n t h e d a t a s t r u c t u r e s u s e d b y t h e s p a t i a l

p r o c e s s o r . I n p a r t i c u l a r , w e f o c u s o n h i e r a r c h i c a l d a t a s t r u c t u r e s . T h e y a r e b a s e d o n t h e

p r i n c i p l e o f r e c u r s i v e d e c o m p o s i t i o n s i m i l a r t o d i v i d e a n d c o n q u e r m e t h o d s . T h e t e r m

q u a d t r e e i s o f t e n u s e d t o d e s c r i b e m a n y e l e m e n t s o f t h i s c l a s s o f d a t a s t r u c t u r e s . W e

c o n c e n t r a t e p r i m a r i l y o n r e g i o n , p o i n t , r e c t a n g l e , a n d l i n e d a t a . F o r a m o r e e x t e n s i v e

t r e a t m e n t o f t h i s s u b j e c t , s e e S a m e t 1 9 9 0 a ; S a m e t 1 9 9 0 b .

O u r p r e s e n t a t i o n i s o r g a n i z e d a s f o l l o w s . S e c t i o n 2 d e s c r i b e s a n u m b e r o f d i e r e n t

m e t h o d s o f i n d e x i n g s p a t i a l d a t a . S e c t i o n 3 f o c u s s e s o n r e g i o n d a t a a n d a l s o b r i e y r e v i e w s

t h e h i s t o r i c a l b a c k g r o u n d o f t h e o r i g i n s o f h i e r a r c h i c a l s p a t i a l d a t a s t r u c t u r e s s u c h a s t h e

q u a d t r e e . S e c t i o n s 4 , 5 , a n d 6 d e s c r i b e h i e r a r c h i c a l r e p r e s e n t a t i o n s f o r p o i n t , r e c t a n g l e ,

a n d l i n e d a t a , r e s p e c t i v e l y , a s w e l l a s g i v e e x a m p l e s o f t h e i r u t i l i t y . S e c t i o n 7 c o n t a i n s

c o n c l u d i n g r e m a r k s i n t h e c o n t e x t o f a g e o g r a p h i c i n f o r m a t i o n s y s t e m t h a t m a k e s u s e o f

t h e s e c o n c e p t s .

2 S p a t i a l I n d e x i n g

E a c h r e c o r d i n a d a t a b a s e m a n a g e m e n t s y s t e m c a n b e c o n c e p t u a l i z e d a s a p o i n t i n a m u l t i -

d i m e n s i o n a l s p a c e . T h i s a n a l o g y i s u s e d b y m a n y r e s e a r c h e r s e . g . , H i n r i c h s a n d N i e v e r g e l t

1 9 8 3 ; J a g a d i s h 1 9 9 0 t o d e a l w i t h s p a t i a l d a t a a s w e l l b y u s e o f s u i t a b l e t r a n s f o r m a t i o n s

t h a t m a p t h e s p a t i a l o b j e c t h e n c e f o r t h w e j u s t u s e t h e t e r m o b j e c t i n t o a p o i n t t e r m e d a

r e p r e s e n t a t i v e p o i n t i n e i t h e r t h e s a m e e . g . , J a g a d i s h 1 9 9 0 , l o w e r e . g . , O r e n s t e i n a n d

M e r r e t t 1 9 8 4 , o r h i g h e r e . g . , H i n r i c h s a n d N i e v e r g e l t 1 9 8 3 d i m e n s i o n a l s p a c e s . T h i s

a n a l o g y i s n o t a l w a y s a p p r o p r i a t e f o r s p a t i a l d a t a . O n e p r o b l e m i s t h a t t h e d i m e n s i o n a l i t y

o f t h e r e p r e s e n t a t i v e p o i n t m a y b e t o o h i g h O r e n s t e i n 1 9 8 9 . O n e s o l u t i o n i s t o a p p r o x i -

m a t e t h e s p a t i a l o b j e c t b y r e d u c i n g t h e d i m e n s i o n a l i t y o f t h e r e p r e s e n t a t i v e p o i n t . A n o t h e r

m o r e s e r i o u s p r o b l e m i s t h a t u s e o f t h e s e t r a n s f o r m a t i o n s d o e s n o t p r e s e r v e p r o x i m i t y .

T o s e e t h e d r a w b a c k o f j u s t m a p p i n g s p a t i a l d a t a i n t o p o i n t s i n a n o t h e r s p a c e , c o n s i d e r

t h e r e p r e s e n t a t i o n o f a d a t a b a s e o f l i n e s e g m e n t s . W e u s e t h e t e r m p o l y g o n a l m a p t o r e f e r

t o s u c h a l i n e s e g m e n t d a t a b a s e , c o n s i s t i n g o f v e r t i c e s a n d e d g e s , r e g a r d l e s s o f w h e t h e r o r

n o t t h e l i n e s e g m e n t s a r e c o n n e c t e d t o e a c h o t h e r . S u c h a d a t a b a s e c a n a r i s e i n a n e t w o r k

o f r o a d s , p o w e r l i n e s , r a i l l i n e s , e t c . U s i n g a r e p r e s e n t a t i v e p o i n t e . g . , J a g a d i s h 1 9 9 0 ,

e a c h l i n e s e g m e n t c a n b e r e p r e s e n t e d b y i t s e n d p o i n t s . T h i s m e a n s t h a t e a c h l i n e s e g m e n t

i s r e p r e s e n t e d b y a t u p l e o f f o u r i t e m s i . e . , a p a i r o f c o o r d i n a t e v a l u e s a n d a p a i r o f

c o o r d i n a t e v a l u e s . T h u s , i n e e c t , w e h a v e c o n s t r u c t e d a m a p p i n g f r o m a t w o - d i m e n s i o n a l

O f c o u r s e , t h e r e a r e o t h e r m a p p i n g s b u t t h e y h a v e s i m i l a r d r a w b a c k s . W e s h a l l u s e t h i s e x a m p l e i n t h e

r e s t o f t h i s s e c t i o n .


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s p a c e i . e . , t h e s p a c e f r o m w h i c h t h e l i n e s a r e d r a w n t o a f o u r - d i m e n s i o n a l s p a c e i . e . , t h e

s p a c e c o n t a i n i n g t h e r e p r e s e n t a t i v e p o i n t c o r r e s p o n d i n g t o t h e l i n e .

T h i s m a p p i n g i s n e f o r s t o r a g e p u r p o s e s a n d f o r q u e r i e s t h a t o n l y i n v o l v e t h e p o i n t s

t h a t c o m p r i s e t h e l i n e s e g m e n t s i n c l u d i n g t h e i r e n d p o i n t s . F o r e x a m p l e , n d i n g a l l t h e

l i n e s e g m e n t s t h a t i n t e r s e c t a g i v e n p o i n t o r s e t o f p o i n t s o r a g i v e n l i n e s e g m e n t . H o w e v e r ,

i t i s n o t g o o d f o r q u e r i e s t h a t i n v o l v e p o i n t s o r s e t s o f p o i n t s t h a t a r e n o t p a r t o f t h e l i n e

s e g m e n t s a s t h e y a r e n o t t r a n s f o r m e d t o t h e h i g h e r d i m e n s i o n a l s p a c e b y t h e m a p p i n g .

A n s w e r i n g s u c h a q u e r y i n v o l v e s p e r f o r m i n g a s e a r c h i n t h e s p a c e f r o m w h i c h t h e l i n e s a r e

d r a w n r a t h e r t h a n i n t h e s p a c e i n t o w h i c h t h e y a r e m a p p e d .

A s a m o r e c o n c r e t e e x a m p l e o f t h e s h o r t c o m i n g o f t h e m a p p i n g a p p r o a c h s u p p o s e t h a t

w e w a n t t o d e t e c t i f t w o l i n e s a r e n e a r e a c h o t h e r , o r , a l t e r n a t i v e l y , t o n d t h e n e a r e s t l i n e t o

a g i v e n p o i n t o r l i n e . T h i s i s d i c u l t t o d o i n t h e f o u r - d i m e n s i o n a l s p a c e s i n c e p r o x i m i t y i n

t h e t w o - d i m e n s i o n a l s p a c e f r o m w h i c h t h e l i n e s a r e d r a w n i s n o t n e c e s s a r i l y p r e s e r v e d i n t h e

f o u r - d i m e n s i o n a l s p a c e i n t o w h i c h t h e l i n e s a r e m a p p e d . I n o t h e r w o r d s , a l t h o u g h t h e t w o

l i n e s m a y b e v e r y c l o s e t o e a c h o t h e r , t h e E u c l i d e a n d i s t a n c e b e t w e e n t h e i r r e p r e s e n t a t i v e

p o i n t s m a y b e q u i t e l a r g e .

T h u s w e n e e d d i e r e n t r e p r e s e n t a t i o n s f o r s p a t i a l d a t a . O n e w a y t o o v e r c o m e t h e s e

p r o b l e m s i s t o u s e d a t a s t r u c t u r e s t h a t a r e b a s e d o n s p a t i a l o c c u p a n c y . S p a t i a l o c c u p a n c y

m e t h o d s d e c o m p o s e t h e s p a c e f r o m w h i c h t h e d a t a i s d r a w n e . g . , t h e t w o - d i m e n s i o n a l

s p a c e c o n t a i n i n g t h e l i n e s i n t o r e g i o n s c a l l e d b u c k e t s . T h e y a r e a l s o c o m m o n l y k n o w n a s

b u c k e t i n g m e t h o d s . T r a d i t i o n a l l y , b u c k e t i n g m e t h o d s s u c h a s t h e g r i d l e N i e v e r g e l t e t a l .

1 9 8 4 , b a n g l e F r e e s t o n 1 9 8 7 , t r e e s H e n r i c h e t a l . 1 9 8 9 , b u d d y t r e e s S e e g e r a n d

K r i e g e l 1 9 9 0 , e t c . h a v e a l w a y s b e e n a p p l i e d t o t h e t r a n s f o r m e d d a t a i . e . , t h e r e p r e s e n t a t i v e

p o i n t s . I n c o n t r a s t , w e a r e a p p l y i n g t h e b u c k e t i n g m e t h o d s t o t h e s p a c e f r o m w h i c h t h e

d a t a i s d r a w n i . e . , t w o - d i m e n s i o n s i n t h e c a s e o f a c o l l e c t i o n o f l i n e s e g m e n t s .

T h e r e a r e f o u r p r i n c i p a l a p p r o a c h e s t o d e c o m p o s i n g t h e s p a c e f r o m w h i c h t h e d a t a

i s d r a w n . O n e a p p r o a c h b u c k e t s t h e d a t a b a s e d o n t h e c o n c e p t o f a m i n i m u m b o u n d i n g

o r e n c l o s i n g r e c t a n g l e . I n t h i s c a s e , o b j e c t s a r e g r o u p e d h o p e f u l l y b y p r o x i m i t y i n t o

h i e r a r c h i e s , a n d t h e n s t o r e d i n a n o t h e r s t r u c t u r e s u c h a s a - t r e e C o m e r 1 9 7 9 . T h e - t r e e

e . g . , B e c k m a n n e t a l . 1 9 9 0 ; G u t t m a n 1 9 8 4 i s a n e x a m p l e o f t h i s a p p r o a c h .

- t r e e a n d i t s v a r i a n t s a r e d e s i g n e d t o o r g a n i z e a c o l l e c t i o n o f a r b i t r a r y s p a t i a l

o b j e c t s m o s t n o t a b l y t w o - d i m e n s i o n a l r e c t a n g l e s b y r e p r e s e n t i n g t h e m a s - d i m e n s i o n a l

r e c t a n g l e s . E a c h n o d e i n t h e t r e e c o r r e s p o n d s t o t h e s m a l l e s t - d i m e n s i o n a l r e c t a n g l e t h a t

e n c l o s e s i t s s o n n o d e s . L e a f n o d e s c o n t a i n p o i n t e r s t o t h e a c t u a l o b j e c t s i n t h e d a t a b a s e ,

i n s t e a d o f s o n s . T h e o b j e c t s a r e r e p r e s e n t e d b y t h e s m a l l e s t a l i g n e d r e c t a n g l e c o n t a i n i n g

t h e m .

O f t e n t h e n o d e s c o r r e s p o n d t o d i s k p a g e s a n d , t h u s , t h e p a r a m e t e r s d e n i n g t h e t r e e

a r e c h o s e n s o t h a t a s m a l l n u m b e r o f n o d e s i s v i s i t e d d u r i n g a s p a t i a l q u e r y . N o t e t h a t t h e

b o u n d i n g r e c t a n g l e s c o r r e s p o n d i n g t o d i e r e n t n o d e s m a y o v e r l a p . A l s o , a n o b j e c t m a y b e

s p a t i a l l y c o n t a i n e d i n s e v e r a l n o d e s , y e t i t i s o n l y a s s o c i a t e d w i t h o n e n o d e . T h i s m e a n s

t h a t a s p a t i a l q u e r y m a y o f t e n r e q u i r e s e v e r a l n o d e s t o b e v i s i t e d b e f o r e a s c e r t a i n i n g t h e

p r e s e n c e o r a b s e n c e o f a p a r t i c u l a r o b j e c t .

T h e b a s i c r u l e s f o r t h e f o r m a t i o n o f a n - t r e e a r e v e r y s i m i l a r t o t h o s e f o r a - t r e e .

A l l l e a f n o d e s a p p e a r a t t h e s a m e l e v e l . E a c h e n t r y i n a l e a f n o d e i s a 2 - t u p l e o f t h e f o r m


46/66

s u c h t h a t i s t h e s m a l l e s t r e c t a n g l e t h a t s p a t i a l l y c o n t a i n s o b j e c t . E a c h e n t r y

i n a n o n - l e a f n o d e i s a 2 - t u p l e o f t h e f o r m s u c h t h a t i s t h e s m a l l e s t r e c t a n g l e t h a t

s p a t i a l l y c o n t a i n s t h e r e c t a n g l e s i n t h e c h i l d n o d e p o i n t e d a t b y - t r e e o f o r d e r

m e a n s t h a t e a c h n o d e i n t h e t r e e , w i t h t h e e x c e p t i o n o f t h e r o o t , c o n t a i n s b e t w e e n

e n t r i e s . T h e r o o t n o d e h a s a t l e a s t t w o e n t r i e s u n l e s s i t i s a l e a f n o d e .

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F i g u r e 1 : E x a m p l e c o l l e c t i o n o f l i n e s e g m e n t s e m b e d d e d i n a 4 4 g r i d .

F o r e x a m p l e , c o n s i d e r t h e c o l l e c t i o n o f l i n e s e g m e n t s g i v e n i n F i g u r e 1 s h o w n e m b e d d e d

i n a 4 4 g r i d . L e t = 3 a n d = 2 . O n e p o s s i b l e - t r e e f o r t h i s c o l l e c t i o n i s g i v e n i n

F i g u r e 2 a . F i g u r e 2 b s h o w s t h e s p a t i a l e x t e n t o f t h e b o u n d i n g r e c t a n g l e s o f t h e n o d e s i n

F i g u r e 2 a , w i t h b r o k e n l i n e s d e n o t i n g t h e r e c t a n g l e s c o r r e s p o n d i n g t o t h e s u b t r e e s r o o t e d

a t t h e n o n - l e a f n o d e s . N o t e t h a t t h e - t r e e i s n o t u n i q u e . I t s s t r u c t u r e d e p e n d s h e a v i l y

o n t h e o r d e r i n w h i c h t h e i n d i v i d u a l l i n e s e g m e n t s w e r e i n s e r t e d i n t o a n d p o s s i b l y d e l e t e d

f r o m t h e t r e e .

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F i g u r e 2 : a R - t r e e f o r t h e c o l l e c t i o n o f l i n e s e g m e n t s i n F i g u r e 1 , a n d b t h e s p a t i a l

e x t e n t s o f t h e b o u n d i n g r e c t a n g l e s .

T h e d r a w b a c k o f t h e s e m e t h o d s i s t h a t t h e y d o n o t r e s u l t i n a d i s j o i n t d e c o m p o s i t i o n o f

s p a c e . T h e p r o b l e m i s t h a t a n o b j e c t i s o n l y a s s o c i a t e d w i t h o n e b o u n d i n g r e c t a n g l e e . g . ,

l i n e s e g m e n t i n F i g u r e 2 i s a s s o c i a t e d w i t h r e c t a n g l e , y e t i t p a s s e s t h r o u g h

. I n t h e w o r s t c a s e , t h i s m e a n s t h a t w h e n w e w i s h t o d e t e r m i n e w h i c h o b j e c t i s

a s s o c i a t e d w i t h a p a r t i c u l a r p o i n t e . g . , t h e c o n t a i n i n g r e c t a n g l e i n a r e c t a n g l e d a t a b a s e ,

o r a n i n t e r s e c t i n g l i n e i n a l i n e s e g m e n t d a t a b a s e i n t h e t w o - d i m e n s i o n a l s p a c e f r o m w h i c h

t h e o b j e c t s a r e d r a w n , w e m a y h a v e t o s e a r c h t h e e n t i r e d a t a b a s e .

F o r e x a m p l e , s u p p o s e w e w i s h t o d e t e r m i n e t h e i d e n t i t y o f t h e l i n e s e g m e n t i n t h e

c o l l e c t i o n o f l i n e s e g m e n t s g i v e n i n F i g u r e 2 t h a t p a s s e s t h r o u g h p o i n t . S i n c e c a n b e i n

e i t h e r o f , w e m u s t s e a r c h b o t h o f t h e i r s u b t r e e s . S e a r c h i n g r s t , w e n d t h a t

c o u l d o n l y b e c o n t a i n e d i n . S e a r c h i n g d o e s n o t l e a d t o t h e l i n e s e g m e n t t h a t c o n t a i n s

e v e n t h o u g h i s i n a p o r t i o n o f b o u n d i n g r e c t a n g l e t h a t i s i n . T h u s , w e m u s t s e a r c h


47/66

a n d w e n d t h a t c a n o n l y b e c o n t a i n e d i n . S e a r c h i n g r e s u l t s i n l o c a t i n g , t h e

d e s i r e d l i n e s e g m e n t .

T h e o t h e r a p p r o a c h e s a r e b a s e d o n a d e c o m p o s i t i o n o f s p a c e i n t o d i s j o i n t c e l l s , w h i c h

a r e m a p p e d i n t o b u c k e t s . T h e i r c o m m o n p r o p e r t y i s t h a t t h e o b j e c t s a r e d e c o m p o s e d i n t o

d i s j o i n t s u b o b j e c t s s u c h t h a t e a c h o f t h e s u b o b j e c t s i s a s s o c i a t e d w i t h a d i e r e n t c e l l . T h e y

d i e r i n t h e d e g r e e o f r e g u l a r i t y i m p o s e d b y t h e i r u n d e r l y i n g d e c o m p o s i t i o n r u l e s a n d b y

t h e w a y i n w h i c h t h e c e l l s a r e a g g r e g a t e d . T h e p r i c e p a i d f o r t h e d i s j o i n t n e s s i s t h a t i n

o r d e r t o d e t e r m i n e t h e a r e a c o v e r e d b y a p a r t i c u l a r o b j e c t , w e h a v e t o r e t r i e v e a l l t h e c e l l s

t h a t i t o c c u p i e s . T h i s p r i c e i s a l s o p a i d w h e n w e w a n t t o d e l e t e a n o b j e c t . F o r t u n a t e l y ,

d e l e t i o n i s n o t s o c o m m o n i n t h e s e d a t a b a s e s . A r e l a t e d d r a w b a c k i s t h a t w h e n w e w i s h

t o d e t e r m i n e a l l t h e o b j e c t s t h a t o c c u r i n a p a r t i c u l a r r e g i o n w e o f t e n r e t r i e v e m a n y o f t h e

o b j e c t s m o r e t h a n o n c e . T h i s i s p a r t i c u l a r l y p r o b l e m a t i c w h e n t h e r e s u l t o f t h e o p e r a t i o n

s e r v e s a s i n p u t t o a n o t h e r o p e r a t i o n v i a c o m p o s i t i o n o f f u n c t i o n s . F o r e x a m p l e , s u p p o s e w e

w i s h t o c o m p u t e t h e p e r i m e t e r o f a l l t h e o b j e c t s i n a g i v e n r e g i o n . C l e a r l y , e a c h o b j e c t ' s

p e r i m e t e r s h o u l d o n l y b e c o m p u t e d o n c e . E l i m i n a t i n g t h e d u p l i c a t e s i s a s e r i o u s i s s u e

s e e A r e f a n d S a m e t 1 9 9 2 f o r a d i s c u s s i o n o f h o w t o d e a l w i t h t h i s p r o b l e m i n a d a t a b a s e

o f l i n e s e g m e n t s .

T h e r s t m e t h o d b a s e d o n d i s j o i n t n e s s p a r t i t i o n s t h e o b j e c t s i n t o a r b i t r a r y d i s j o i n t

s u b o b j e c t s a n d t h e n g r o u p s t h e s u b o b j e c t s i n a n o t h e r s t r u c t u r e s u c h a s a - t r e e . T h e

p a r t i t i o n a n d t h e s u b s e q u e n t g r o u p i n g s a r e s u c h t h a t t h e b o u n d i n g r e c t a n g l e s a r e d i s j o i n t

a t e a c h l e v e l o f t h e s t r u c t u r e . T h e - t r e e S e l l i s e t a l . 1 9 8 7 a n d t h e c e l l t r e e G u n t h e r

1 9 8 8 a r e e x a m p l e s o f t h i s a p p r o a c h . T h e y d i e r i n t h e d a t a w i t h w h i c h t h e y d e a l . T h e

- t r e e d e a l s w i t h c o l l e c t i o n s o f o b j e c t s t h a t a r e b o u n d e d b y r e c t a n g l e s , w h i l e t h e c e l l t r e e

d e a l s w i t h c o n v e x p o l y h e d r a .

- t r e e i s a n e x t e n s i o n o f t h e k - d - - t r e e R o b i n s o n 1 9 8 1 . T h e - t r e e i s m o t i v a t e d

b y a d e s i r e t o a v o i d o v e r l a p a m o n g t h e b o u n d i n g r e c t a n g l e s . E a c h o b j e c t i s a s s o c i a t e d w i t h

a l l t h e b o u n d i n g r e c t a n g l e s t h a t i t i n t e r s e c t s . A l l b o u n d i n g r e c t a n g l e s i n t h e t r e e w i t h t h e

e x c e p t i o n o f t h e b o u n d i n g r e c t a n g l e s f o r t h e o b j e c t s a t t h e l e a f n o d e s a r e n o n - o v e r l a p p i n g

. T h e r e s u l t i s t h a t t h e r e m a y b e s e v e r a l p a t h s s t a r t i n g a t t h e r o o t t o t h e s a m e o b j e c t .

T h i s m a y l e a d t o a n i n c r e a s e i n t h e h e i g h t o f t h e t r e e . H o w e v e r , r e t r i e v a l t i m e i s s p e d u p .

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F i g u r e 3 : a R - t r e e f o r t h e c o l l e c t i o n o f l i n e s e g m e n t s i n F i g u r e 1 a n d b t h e s p a t i a l

e x t e n t s o f t h e b o u n d i n g r e c t a n g l e s .

F i g u r e 3 i s a n e x a m p l e o f o n e p o s s i b l e - t r e e f o r t h e c o l l e c t i o n o f l i n e s e g m e n t s i n

F r o m a t h e o r e t i c a l v i e w p o i n t , t h e b o u n d i n g r e c t a n g l e s f o r t h e o b j e c t s a t t h e l e a f n o d e s s h o u l d a l s o b e

d i s j o i n t . H o w e v e r , t h i s m a y b e i m p o s s i b l e e . g . , w h e n t h e o b j e c t s a r e l i n e s e g m e n t s w h e r e m a n y l i n e s e g m e n t s

i n t e r s e c t a t a p o i n t .


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F i g u r e 1 . T h i s p a r t i c u l a r t r e e i s o f o r d e r 2 , 3 a l t h o u g h i n g e n e r a l i t i s n o t p o s s i b l e t o

g u a r a n t e e t h a t a l l n o d e s w i l l a l w a y s h a v e a m i n i m u m o f 2 e n t r i e s . I n p a r t i c u l a r , t h e e x p e c t e d

- t r e e p e r f o r m a n c e g u a r a n t e e s a r e n o t v a l i d i . e . , p a g e s a r e n o t g u a r a n t e e d t o b e f u l l

u n l e s s w e a r e w i l l i n g t o p e r f o r m v e r y c o m p l i c a t e d r e c o r d i n s e r t i o n a n d d e l e t i o n p r o c e d u r e s .

N o t i c e t h a t l i n e s e g m e n t s a p p e a r i n t w o d i e r e n t n o d e s , w h i l e l i n e s e g m e n t

a p p e a r s i n t h r e e d i e r e n t n o d e s . O f c o u r s e , o t h e r v a r i a n t s a r e p o s s i b l e s i n c e t h e - t r e e i s

n o t u n i q u e .

M e t h o d s s u c h a s t h e - t r e e a n d t h e c e l l t r e e a s w e l l a s t h e - t r e e B e c k m a n n e t a l .

1 9 9 0 h a v e t h e d r a w b a c k t h a t t h e d e c o m p o s i t i o n i s d a t a - d e p e n d e n t . T h i s m e a n s t h a t i t i s

d i c u l t t o p e r f o r m t a s k s t h a t r e q u i r e c o m p o s i t i o n o f d i e r e n t o p e r a t i o n s a n d d a t a s e t s e . g . ,

s e t - t h e o r e t i c o p e r a t i o n s s u c h a s o v e r l a y . I n c o n t r a s t , t h e r e m a i n i n g t w o m e t h o d s , w h i l e

a l s o y i e l d i n g a d i s j o i n t d e c o m p o s i t i o n , h a v e a g r e a t e r d e g r e e o f d a t a - i n d e p e n d e n c e . T h e y

a r e b a s e d o n a r e g u l a r d e c o m p o s i t i o n . T h e s p a c e c a n b e d e c o m p o s e d e i t h e r i n t o b l o c k s

o f u n i f o r m s i z e e . g . , t h e u n i f o r m g r i d F r a n k l i n 1 9 8 4 o r a d a p t t h e d e c o m p o s i t i o n t o t h e

d i s t r i b u t i o n o f t h e d a t a e . g . , a q u a d t r e e - b a s e d a p p r o a c h s u c h a s S a m e t a n d W e b b e r 1 9 8 5 .

I n t h e f o r m e r c a s e , a l l t h e b l o c k s a r e o f t h e s a m e s i z e e . g . , t h e 4 4 g r i d i n F i g u r e 1 .

I n t h e l a t t e r c a s e , t h e w i d t h s o f t h e b l o c k s a r e r e s t r i c t e d t o b e p o w e r s o f t w o , a n d t h e i r

p o s i t i o n s a r e a l s o r e s t r i c t e d .

T h e u n i f o r m g r i d i s i d e a l f o r u n i f o r m l y d i s t r i b u t e d d a t a , w h i l e q u a d t r e e - b a s e d a p -

p r o a c h e s a r e s u i t e d f o r a r b i t r a r i l y d i s t r i b u t e d d a t a . I n t h e c a s e o f u n i f o r m l y d i s t r i b u t e d

d a t a , q u a d t r e e - b a s e d a p p r o a c h e s d e g e n e r a t e t o a u n i f o r m g r i d , a l b e i t t h e y h a v e a h i g h e r

o v e r h e a d . B o t h t h e u n i f o r m g r i d a n d t h e q u a d t r e e - b a s e d a p p r o a c h e s l e n d t h e m s e l v e s t o

s e t - t h e o r e t i c o p e r a t i o n s a n d t h u s t h e y a r e i d e a l f o r t a s k s w h i c h r e q u i r e t h e c o m p o s i t i o n

o f d i e r e n t o p e r a t i o n s a n d d a t a s e t s . I n g e n e r a l , s i n c e s p a t i a l d a t a i s n o t u s u a l l y u n i -

f o r m l y d i s t r i b u t e d , t h e q u a d t r e e - b a s e d r e g u l a r d e c o m p o s i t i o n a p p r o a c h i s m o r e e x i b l e .

T h e d r a w b a c k o f q u a d t r e e - l i k e m e t h o d s i s t h e i r s e n s i t i v i t y t o p o s i t i o n i n g i n t h e s e n s e t h a t

t h e p l a c e m e n t o f t h e o b j e c t s r e l a t i v e t o t h e d e c o m p o s i t i o n l i n e s o f t h e s p a c e i n w h i c h t h e y

a r e e m b e d d e d e e c t s t h e i r s t o r a g e c o s t s a n d t h e a m o u n t o f d e c o m p o s i t i o n t h a t t a k e s p l a c e .

T h i s i s o v e r c o m e t o a l a r g e e x t e n t b y u s i n g a b u c k e t i n g a d a p t a t i o n t h a t d e c o m p o s e s a b l o c k

o n l y i f i t c o n t a i n s m o r e t h a n o b j e c t s .

A l l o f t h e s p a t i a l o c c u p a n c y m e t h o d s d i s c u s s e d a b o v e a r e c h a r a c t e r i z e d a s e m p l o y i n g

s p a t i a l i n d e x i n g b e c a u s e w i t h e a c h b l o c k t h e o n l y i n f o r m a t i o n t h a t i s s t o r e d i s w h e t h e r o r

n o t t h e b l o c k i s o c c u p i e d b y t h e o b j e c t o r p a r t o f t h e o b j e c t . T h i s i n f o r m a t i o n i s u s u a l l y i n

t h e f o r m o f a p o i n t e r t o a d e s c r i p t o r o f t h e o b j e c t . F o r e x a m p l e , i n t h e c a s e o f a c o l l e c t i o n

o f l i n e s e g m e n t s i n t h e u n i f o r m g r i d o f F i g u r e 1 , t h e s h a d e d b l o c k o n l y r e c o r d s t h e f a c t t h a t

a l i n e s e g m e n t c r o s s e s i t o r p a s s e s t h r o u g h i t . T h e p a r t o f t h e l i n e s e g m e n t t h a t p a s s e s

t h r o u g h t h e b l o c k o r t e r m i n a t e s w i t h i n i t i s t e r m e d a q - e d g e . E a c h q - e d g e i n t h e b l o c k

i s r e p r e s e n t e d b y a p o i n t e r t o a r e c o r d c o n t a i n i n g t h e e n d p o i n t s o f t h e l i n e s e g m e n t o f

w h i c h t h e q - e d g e i s a p a r t N e l s o n a n d S a m e t 1 9 8 6 . T h i s p o i n t e r i s r e a l l y n o t h i n g m o r e

t h a n a s p a t i a l i n d e x a n d h e n c e t h e u s e o f t h i s t e r m t o c h a r a c t e r i z e t h i s a p p r o a c h . T h u s n o

i n f o r m a t i o n i s a s s o c i a t e d w i t h t h e s h a d e d b l o c k a s t o w h a t p a r t o f t h e l i n e i . e . , q - e d g e

c r o s s e s i t . T h i s i n f o r m a t i o n c a n b e o b t a i n e d b y c l i p p i n g F o l e y e t a l . 1 9 9 0 t h e o r i g i n a l l i n e

s e g m e n t t o t h e b l o c k . T h i s i s i m p o r t a n t f o r o f t e n t h e p r e c i s i o n n e c e s s a r y t o c o m p u t e t h e s e

i n t e r s e c t i o n p o i n t s i s n o t a v a i l a b l e .


49/66

3 R e g i o n D a t a

A r e g i o n c a n b e r e p r e s e n t e d e i t h e r b y i t s i n t e r i o r o r b y i t s b o u n d a r y . I n t h i s s e c t i o n w e

f o c u s o n t h e r e p r e s e n t a t i o n s o f r e g i o n s b y t h e i r i n t e r i o r , w h i l e t h e u s e o f a b o u n d a r y i s

d i s c u s s e d i n S e c t i o n 6 i n t h e c o n t e x t o f c o l l e c t i o n s o f l i n e s e g m e n t s a s f o u n d , f o r e x a m p l e , i n

p o l y g o n a l m a p s . T h e m o s t c o m m o n r e g i o n r e p r e s e n t a t i o n i s t h e i m a g e a r r a y . I n t h i s c a s e ,

w e h a v e a c o l l e c t i o n o f p i c t u r e e l e m e n t s t e r m e d p i x e l s . S i n c e t h e n u m b e r o f e l e m e n t s i n

t h e a r r a y c a n b e q u i t e l a r g e , t h e r e i s i n t e r e s t i n r e d u c i n g i t s s i z e b y a g g r e g a t i n g s i m i l a r i . e . ,

h o m o g e n e o u s o r e q u a l - v a l u e d p i x e l s . T h e r e a r e t w o b a s i c a p p r o a c h e s . T h e r s t a p p r o a c h

b r e a k s u p t h e a r r a y i n t o 1 b l o c k s R u t o v i t z 1 9 6 8 . T h i s i s a r o w r e p r e s e n t a t i o n a n d

i s k n o w n a s a r u n l e n g t h c o d e . A m o r e g e n e r a l a p p r o a c h t r e a t s t h e r e g i o n a s a u n i o n o f

m a x i m a l s q u a r e b l o c k s o r b l o c k s o f a n y o t h e r d e s i r e d s h a p e t h a t m a y p o s s i b l y o v e r l a p .

U s u a l l y t h e b l o c k s a r e s p e c i e d b y t h e i r c e n t e r s a n d r a d i i . T h i s r e p r e s e n t a t i o n i s c a l l e d t h e

m e d i a l a x i s t r a n s f o r m a t i o n M A T B l u m 1 9 6 7 .

W h e n t h e m a x i m a l b l o c k s a r e r e q u i r e d t o b e d i s j o i n t , t o h a v e s t a n d a r d s i z e s s q u a r e s

w h o s e s i d e s a r e p o w e r s o f t w o , a n d t o b e a t s t a n d a r d l o c a t i o n s a s a r e s u l t o f a h a l v i n g

p r o c e s s i n b o t h t h e d i r e c t i o n s , t h e r e s u l t i s k n o w n a s a r e g i o n q u a d t r e e K l i n g e r

1 9 7 1 . I t i s b a s e d o n t h e s u c c e s s i v e s u b d i v i s i o n o f t h e i m a g e a r r a y i n t o f o u r e q u a l - s i z e

q u a d r a n t s . I f t h e a r r a y d o e s n o t c o n s i s t e n t i r e l y o f 1 s o r e n t i r e l y o f 0 s i . e . , t h e r e g i o n d o e s

n o t c o v e r t h e e n t i r e a r r a y , i t i s t h e n s u b d i v i d e d i n t o q u a d r a n t s , s u b q u a d r a n t s , e t c . , u n t i l

b l o c k s a r e o b t a i n e d p o s s i b l y 1 1 b l o c k s t h a t c o n s i s t e n t i r e l y o f 1 s o r e n t i r e l y o f 0 s . T h u s ,

t h e r e g i o n q u a d t r e e c a n b e c h a r a c t e r i z e d a s a v a r i a b l e r e s o l u t i o n d a t a s t r u c t u r e .

A s a n e x a m p l e o f t h e r e g i o n q u a d t r e e , c o n s i d e r t h e r e g i o n s h o w n i n F i g u r e 4 a w h i c h i s

r e p r e s e n t e d b y t h e 2 b i n a r y a r r a y i n F i g u r e 4 b . O b s e r v e t h a t t h e 1 s c o r r e s p o n d t o

p i x e l s t h a t a r e i n t h e r e g i o n a n d t h e 0 s c o r r e s p o n d t o p i x e l s t h a t a r e o u t s i d e t h e r e g i o n .

T h e r e s u l t i n g b l o c k s f o r t h e a r r a y o f F i g u r e 4 b a r e s h o w n i n F i g u r e 4 c . T h i s p r o c e s s i s

r e p r e s e n t e d b y a t r e e o f d e g r e e 4 .

0 0 0 0 00 0 0

0

0

0

0

0

0

0

0 0 0 0 0 0 0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

00

1

1

1 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1 1

1

1

1 1

1

1

11

2 3

4 5

6 7 8

9 1013

11 12

14

15 16

17 1819

A

NWNE SW

SE

1

2 3 4 5 6 11 12 13 14 19

7 8 9 10 15 16 17 18

B C

D F

E

(a) (b) (c)

(d)

F i g u r e 4 : a S a m p l e r e g i o n , b i t s b i n a r y a r r a y r e p r e s e n t a t i o n , c i t s m a x i m a l b l o c k s w i t h

t h e b l o c k s i n t h e r e g i o n b e i n g s h a d e d , a n d d t h e c o r r e s p o n d i n g q u a d t r e e .

I n t h e t r e e r e p r e s e n t a t i o n , t h e r o o t n o d e c o r r e s p o n d s t o t h e e n t i r e a r r a y . E a c h s o n o f

a n o d e r e p r e s e n t s a q u a d r a n t l a b e l e d i n o r d e r o f t h e r e g i o n r e p r e s e n t e d

b y t h a t n o d e . T h e l e a f n o d e s o f t h e t r e e c o r r e s p o n d t o t h o s e b l o c k s f o r w h i c h n o f u r t h e r

s u b d i v i s i o n i s n e c e s s a r y . A l e a f n o d e i s s a i d t o b e b l a c k w h i t e , d e p e n d i n g o n w h e t h e r

i t s c o r r e s p o n d i n g b l o c k i s e n t i r e l y i n s i d e o r e n t i r e l y o u t s i d e o f t h e r e p r e s e n t e d r e g i o n . A l l

n o n - l e a f n o d e s a r e s a i d t o b e g r a y . T h e q u a d t r e e r e p r e s e n t a t i o n f o r F i g u r e 4 c i s s h o w n i n


50/66

F i g u r e 4 d . O f c o u r s e , q u a d t r e e s c a n a l s o b e u s e d t o r e p r e s e n t n o n - b i n a r y i m a g e s . I n t h i s

c a s e , t h e s a m e m e r g i n g c r i t e r i a i s a p p l i e d t o e a c h c o l o r . F o r e x a m p l e , i n t h e c a s e o f a l a n d u s e

m a p , s i m p l y m e r g e a l l w h e a t g r o w i n g r e g i o n s , a n d l i k e w i s e f o r c o r n , r i c e , e t c . S a m e t e t a l .

1 9 8 4 .

T h e t e r m q u a d t r e e i s o f t e n u s e d i n a m o r e g e n e r a l s e n s e s t o d e s c r i b e a c l a s s o f h i e r a r c h i c a l

d a t a s t r u c t u r e s w h o s e c o m m o n p r o p e r t y i s t h a t t h e y a r e b a s e d o n t h e p r i n c i p l e o f r e c u r s i v e

d e c o m p o s i t i o n o f s p a c e . T h e y c a n b e d i e r e n t i a t e d o n t h e f o l l o w i n g b a s e s :

1 . t h e t y p e o f d a t a t h a t t h e y a r e u s e d t o r e p r e s e n t ,

2 . t h e p r i n c i p l e g u i d i n g t h e d e c o m p o s i t i o n p r o c e s s , a n d

3 . t h e r e s o l u t i o n v a r i a b l e o r n o t .

C u r r e n t l y , t h e y a r e u s e d f o r p o i n t s , r e c t a n g l e s , r e g i o n s , c u r v e s , s u r f a c e s , a n d v o l u m e s s e e

t h e r e m a i n i n g s e c t i o n s f o r f u r t h e r d e t a i l s o n t h e a d a p t a t i o n o f t h e q u a d t r e e t o t h e m . T h e

d e c o m p o s i t i o n m a y b e i n t o e q u a l p a r t s o n e a c h l e v e l t e r m e d a r e g u l a r d e c o m p o s i t i o n , o r

i t m a y b e g o v e r n e d b y t h e i n p u t . T h e r e s o l u t i o n o f t h e d e c o m p o s i t i o n i . e . , t h e n u m b e r

o f t i m e s t h a t t h e d e c o m p o s i t i o n p r o c e s s i s a p p l i e d m a y b e x e d b e f o r e h a n d o r i t m a y b e

g o v e r n e d b y p r o p e r t i e s o f t h e i n p u t d a t a .

U n f o r t u n a t e l y , t h e t e r m q u a d t r e e h a s t a k e n o n m o r e t h a n o n e m e a n i n g . T h e r e g i o n

q u a d t r e e , a s s h o w n a b o v e , i s a p a r t i t i o n o f s p a c e i n t o a s e t o f s q u a r e s w h o s e s i d e s a r e a l l

a p o w e r o f t w o l o n g . A s i m i l a r p a r t i t i o n o f s p a c e i n t o r e c t a n g u l a r q u a d r a n t s i s t e r m e d a

p o i n t q u a d t r e e F i n k e l a n d B e n t l e y 1 9 7 4 . I t i s a n a d a p t a t i o n o f t h e b i n a r y s e a r c h t r e e t o

t w o d i m e n s i o n s w h i c h c a n b e e a s i l y e x t e n d e d t o a n a r b i t r a r y n u m b e r o f d i m e n s i o n s . I t

i s p r i m a r i l y u s e d t o r e p r e s e n t m u l t i d i m e n s i o n a l p o i n t d a t a w h e r e t h e r e c t a n g u l a r r e g i o n s

n e e d n o t b e s q u a r e . T h e q u a d t r e e i s a l s o o f t e n c o n f u s e d w i t h t h e p y r a m i d T a n i m o t o a n d

P a v l i d i s 1 9 7 5 . T h e p y r a m i d i s a m u l t i r e s o l u t i o n r e p r e s e n t a t i o n w h i c h i s a n e x p o n e n t i a l l y

t a p e r i n g s t a c k o f a r r a y s , e a c h o n e - q u a r t e r t h e s i z e o f t h e p r e v i o u s a r r a y . I n c o n t r a s t , t h e

r e g i o n q u a d t r e e i s a v a r i a b l e r e s o l u t i o n d a t a s t r u c t u r e .

T h e d i s t i n c t i o n b e t w e e n a q u a d t r e e a n d a p y r a m i d i s i m p o r t a n t i n t h e d o m a i n o f s p a t i a l

d a t a b a s e s , a n d c a n b e e a s i l y s e e n b y c o n s i d e r i n g t h e t y p e s o f s p a t i a l q u e r i e s . T h e r e a r e

t w o p r i n c i p a l t y p e s A r e f a n d S a m e t 1 9 9 0 . T h e r s t i s l o c a t i o n - b a s e d . I n t h i s c a s e , w e

a r e s e a r c h i n g f o r t h e n a t u r e o f t h e f e a t u r e a s s o c i a t e d w i t h a p a r t i c u l a r l o c a t i o n o r i n i t s

p r o x i m i t y . F o r e x a m p l e , w h a t i s t h e f e a t u r e a t l o c a t i o n X ? " , w h a t i s t h e n e a r e s t c i t y

t o l o c a t i o n X ? " , o r w h a t i s t h e n e a r e s t r o a d t o l o c a t i o n X ? " T h e s e c o n d i s f e a t u r e - b a s e d .

I n t h i s c a s e , w e a r e p r o b i n g f o r t h e p r e s e n c e o r a b s e n c e o f a f e a t u r e , a s w e l l a s s e e k i n g

i t s a c t u a l l o c a t i o n . F o r e x a m p l e , d o e s w h e a t g r o w a n y w h e r e i n C a l i f o r n i a ? " , w h a t c r o p s

g r o w i n C a l i f o r n i a ? " , o r w h e r e i s w h e a t g r o w n i n C a l i f o r n i a ? "

L o c a t i o n - b a s e d q u e r i e s a r e e a s y t o a n s w e r w i t h a q u a d t r e e r e p r e s e n t a t i o n a s t h e y i n v o l v e

d e s c e n d i n g t h e t r e e u n t i l n d i n g t h e o b j e c t . I f a n e a r e s t n e i g h b o r i s d e s i r e d , t h e n t h e s e a r c h

i s c o n t i n u e d i n t h e n e i g h b o r h o o d o f t h e n o d e c o n t a i n i n g t h e o b j e c t . T h i s s e a r c h c a n a l s o

b e a c h i e v e d b y u n w i n d i n g t h e p r o c e s s u s e d t o a c c e s s t h e n o d e c o n t a i n i n g t h e o b j e c t . O n

t h e o t h e r h a n d , f e a t u r e - b a s e d q u e r i e s a r e m o r e d i c u l t . T h e p r o b l e m i s t h a t t h e r e i s n o

i n d e x i n g b y f e a t u r e s . T h e i n d e x i n g i s o n l y b a s e d o n s p a t i a l o c c u p a n c y . T h e g o a l i s t o

p r o c e s s t h e q u e r y w i t h o u t e x a m i n i n g e v e r y l o c a t i o n i n s p a c e . T h e p y r a m i d i s u s e f u l f o r


51/66

s u c h q u e r i e s s i n c e t h e n o d e s t h a t a r e n o t a t t h e m a x i m u m l e v e l o f r e s o l u t i o n i . e . , a t t h e

b o t t o m l e v e l c o n t a i n s u m m a r y i n f o r m a t i o n . T h u s w e c o u l d v i e w t h e s e n o d e s a s f e a t u r e

v e c t o r s w h i c h i n d i c a t e w h e t h e r o r n o t a f e a t u r e i s p r e s e n t a t a h i g h e r l e v e l o f r e s o l u t i o n .

T h e r e f o r e , b y e x a m i n i n g t h e r o o t o f t h e p y r a m i d i . e . , t h e n o d e t h a t r e p r e s e n t s t h e e n t i r e

i m a g e w e c a n q u i c k l y t e l l i f a f e a t u r e i s p r e s e n t w i t h o u t h a v i n g t o e x a m i n e e v e r y l o c a t i o n .

F o r e x a m p l e , c o n s i d e r t h e b l o c k d e c o m p o s i t i o n o f t h e n o n - b i n a r y i m a g e i n F i g u r e 5 a .

I t s t r u n c a t e d p y r a m i d i s g i v e n i n F i g u r e 5 b . T h e v a l u e s o f a n o n l e a f n o d e i n t h e t r u n c a t e d

p y r a m i d i n d i c a t e i f t h e f e a t u r e i s p r e s e n t i n t h e s u b t r e e s o f . I n t h e i n t e r e s t o f s a v i n g s p a c e ,

t h e p y r a m i d i s n o t s h o w n i n i t s e n t i r e t y h e r e .

A

B

C

D

E

F

{A,B,C,D,E,F}

{B,C,F} {B,C,D,E}

{A,B,C,D,E,F}

{B,C,E} {A,B,C,E}

(a) (b)

F i g u r e 5 : a S a m p l e n o n - b i n a r y i m a g e , a n d b i t s c o r r e s p o n d i n g t r u n c a t e d p y r a m i d .

Q u a d t r e e - l i k e d a t a s t r u c t u r e s c a n a l s o b e u s e d t o r e p r e s e n t i m a g e s i n t h r e e d i m e n s i o n s

a n d h i g h e r . T h e o c t r e e H u n t e r 1 9 7 8 ; M e a g h e r 1 9 8 2 d a t a s t r u c t u r e i s t h e t h r e e - d i m e n s i o n a l

a n a l o g o f t h e q u a d t r e e . I t i s c o n s t r u c t e d i n t h e f o l l o w i n g m a n n e r . W e s t a r t w i t h a n i m a g e

i n t h e f o r m o f a c u b i c a l v o l u m e a n d r e c u r s i v e l y s u b d i v i d e i t i n t o e i g h t c o n g r u e n t d i s j o i n t

c u b e s c a l l e d o c t a n t s u n t i l b l o c k s a r e o b t a i n e d o f a u n i f o r m c o l o r o r a p r e d e t e r m i n e d l e v e l

o f d e c o m p o s i t i o n i s r e a c h e d . F i g u r e 6 a i s a n e x a m p l e o f a s i m p l e t h r e e - d i m e n s i o n a l o b j e c t

w h o s e r a s t e r o c t r e e b l o c k d e c o m p o s i t i o n i s g i v e n i n F i g u r e 6 b a n d w h o s e t r e e r e p r e s e n t a t i o n

i s g i v e n i n F i g u r e 6 c .

1 2 3 4 13 14 15

12111098765

B

A14 15

49 10

6

1 2

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1211

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(a) (b) (c)

F i g u r e 6 : a E x a m p l e t h r e e - d i m e n s i o n a l o b j e c t ; b i t s o c t r e e b l o c k d e c o m p o s i t i o n ; a n d c

i t s t r e e r e p r e s e n t a t i o n .

T h e q u a d t r e e i s p a r t i c u l a r l y u s e f u l f o r p e r f o r m i n g s e t o p e r a t i o n s a s t h e y f o r m t h e b a s i s

o f m o s t c o m p l i c a t e d q u e r i e s . F o r e x a m p l e , t o n d t h e n a m e s o f t h e r o a d s t h a t p a s s t h r o u g h

t h e U n i v e r s i t y o f M a r y l a n d r e g i o n , " w e w i l l n e e d t o i n t e r s e c t a r e g i o n m a p w i t h a l i n e m a p .

F o r a b i n a r y i m a g e , s e t - t h e o r e t i c o p e r a t i o n s s u c h a s u n i o n a n d i n t e r s e c t i o n a r e q u i t e s i m p l e

t o i m p l e m e n t H u n t e r a n d S t e i g l i t z 1 9 7 9 .

I n p a r t i c u l a r , t h e i n t e r s e c t i o n o f t w o q u a d t r e e s y i e l d s a b l a c k n o d e o n l y w h e n t h e

c o r r e s p o n d i n g r e g i o n s i n b o t h q u a d t r e e s a r e b l a c k . T h i s o p e r a t i o n i s p e r f o r m e d b y s i -

m u l t a n e o u s l y t r a v e r s i n g t h r e e q u a d t r e e s . T h e r s t t w o t r e e s c o r r e s p o n d t o t h e t r e e s b

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