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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
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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)
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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
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2
r
R3
R4
R5
R6
ic feba hgd
hi31
R3: R4: R5: R6:Copyright 2007 by Hanan Samet 7
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z
R4R3 R6R5
R1
R2
hi31
R2:R1:
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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
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hi32
Q is in R0
2
v
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hi32
Q can be in both R1 and R2
3
r
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hi324z
Searching R1 first means that R4 is searched but thisleads to failure even though Q is part of i which is in R4
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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
g
h
i
1
b
Q
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hi332r
R3
R4
R6
R5
hgd ihc ifc ba e iR3: R4: R5: R6:
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hi333z
R4R3 R6R5
R1
R2
R1: R2:
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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
Copyright 2007 by Hanan Samet
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:
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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
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
B
60
37
L
J
Q
GF
H
N
I
O
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
Copyright 2007 by Hanan Samet 15
2
rhp9
(52,10)
Mobile
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3
zhp9
(62,77)
Toronto
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4
ghp9
(82,65)
Buffalo
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5
vhp9
(5,45)
Denver
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6
ghp9
(27,35)
Omaha
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7
zhp9
(85,15)Atlanta
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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
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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
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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:
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hp112r
1. start at block 2 and compute distance to Pfrom A
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hp113z
2. ignore block 3 whether or not it is empty as Ais closer to Pthan any point in 3
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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
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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
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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
Copyright 2007 by Hanan Samet 17
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
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cd32
b
2
r
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cd323z
c
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cd324g
d
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cd325v
e
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cd32
f
6
r
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cd327z
g
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cd32
h
8
g
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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
Copyright 2007 by Hanan Samet 19
cd352r
b
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cd353z
c
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cd35
d
4
g
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cd355v
e
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cd35
f
6
r
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cd35
g
7
z
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cd35
h
8
g
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cd35
i
9
v
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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
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2
rcd47
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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
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r
three
cd58
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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
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2
Search the three siblings2.
rcd59
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3
Search the three regions of size equal to that of theparent that are incident to the block containing thequery point
3.
zcd59
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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
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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.
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hp14
one for y-axis
Binary tree for y-axis through A
Y1
Y2 10
Y4
2
Y5
Y3
6Y7
8
Y6
3
g
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hp14
if a rectangle intersects both xand yaxes, thenassociate it with the yaxis
4
v
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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)
Copyright 2007 by Hanan Samet 26
sf32r
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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
Copyright 2007 by Hanan Samet 26
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
Copyright 2007 by Hanan Samet 26
sf3
Prefer 8-triangle rule as it is better for display applications(shading)
5
v
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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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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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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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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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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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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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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
Copyright 2007 by Hanan Samet
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
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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
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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 .
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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
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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
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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
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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 .
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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 .
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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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