The Polygon Overlay

The Polygon Overlay - http://www.ncgia.ucsb.edu/giscc/units/u186/u186.html

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Body of unit
Introduction
Traditions of polygon overlay use
Landscape planning
Set theoretic
Area interpolation
General concepts of polygon overlay operations
Operations requiring polygon overlay
Windowing
Buffering
Planar enforcement
Overlay algorithms
Example of the overlay operation
A more complex overlay example
Computational complexity
Intersection problems
Adjacent lines
Sliver polygons
Removing slivers during overlay
Removing slivers after overlay
Discussion and exam questions
References

Traditions of polygon overlay use
1.1.1. Landscape planning
superimposing layers of geographical data (e.g. environmental and social factors) so that their spatial relationships can be used in making land use decisions

a key reference is McHarg, 1969, Design with Nature
1.1.2. Set theoretic
a polygon can be thought of as representing a set

when two sets (polygons) A and B are overlaid, we have a graphic interpretation of the set concepts of intersection and union

it is possible to identify 16 such combinations, or Boolean expressions including e.g.
Figure 1 - Sixteen combinations of Boolean operations
A.AND.(NOT.B), i.e. all of the area of A which is not overlapped by B
NOT.(A.OR.B), i.e. the outside world (= (NOT.A).AND.(NOT.B)

in most polygon overlay operations it is the intersection which is of most interest, i.e. the area which is common to A and B
1.1.3. Area interpolation
Given: the population of area A and the fact that areas A and B overlap
Estimate: the population of area B

this problem can be solved by apportioning the population of A so that the amount in the area covered by B is proportional to the area of overlap

this is a simple way of estimating the population of areas from statistics based on other areal units
e.g. estimating populations of voting districts from populations of census tracts

this method assumes that density is uniform within A, but more realistic versions of the technique exist
see Unit 120 - Spatial Interpolation

2. General concepts of polygon overlay operations
in GIS, the normal case of polygon overlay takes two map layers and overlays them
each map layer is covered with non-overlapping polygons

if we think of one layer as "red" and the other as "blue", the task is to find all of the polygons on the combined "purple" layer
attributes of a "purple" polygon will contain the attributes of the "red" and "blue" polygons which formed it
can think of this process as "concatenating" attributes
usually a new attribute table is constructed that consists of the combined old attributes, or new
attributes formed by logical or mathematical operations on the old ones

number of polygons formed in an overlay is difficult to predict
there may be many polygons formed from a pair of "red" and "blue" polygons, with the same "purple" attributes

when two maps are overlaid, will result in a map with a mixture of 3 and 4 arc intersections
four arc intersections do not generally occur on simple polygon maps
2.1. Operations requiring polygon overlay
2.1.1. Windowing
the windowing operation, in which a window is superimposed on a map and everything outside the window is discarded, is a special case of polygon overlay
2.1.2. Buffering
buffering around points, lines and polygons is another case
buffers are generated around each point or straight line segment
the combined buffer is found by polygon overlay

2.1.3. Planar Enforcement
the process of building points, lines and areas from digitized "spaghetti"
wherever intersections occur between lines, the lines are broken and a point is inserted
the result is a set of points, lines and areas which obey specific rules:

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3. Overlay algorithms
to overlay polygons it is necessary first to find all intersections between "red" and "blue" boundary lines
arcs are more efficient than polygons for this operation because:
intersections need to be found only once rather than four times
more information is available about labels (see below)

3.1. Example of overlay operation
see Figure 2 - Example of overlay operation
Objective
overlay two maps of different themes and determine the combined attributes of the new polygons
e.g. overlay a soils map on a vegetation map and create a new set of polygons with a new set of attributes
Given
two overlapping polygons as follows:
red map: a polygon with attribute A
blue map: a polygon with attribute 1

the outside world labelled 0 on both maps
two intersecting arcs defining the boundaries of the polygons:
1. Red Map (light lines): (0,1) (0,3) (2,3) (2,1) (0,1)
Polygons - Right: A Left: 0
2. Blue Map (heavy lines): (1,0) (3,0) (3,2) (1,2) (1,0)
Polygons - Right: 0 Left: 1
Procedure
after intersections have been found, six new arcs are formed, three from arc 1 and three from arc 2:
1. (0,1) (0,3) (2,3) (2,2)
2. (2,2) (2,1) (1,1)
3. (1,1) (0,1)
4. (1,0) (3,0) (3,2) (2,2)
5. (2,2) (1,2) (1,1)
6. (1,1) (1,0)

because the right and left polygon labels are known for each input arc, we know the labels of the new polygons as soon as the intersections have been found
there are four new polygons
their attributes combine red and blue attributes: 00, A0, A1 and 01

the arc right and left labels, deduced from the geometry of the intersections, are:
Arc Right Left
1 A0 00
2 A1 01
3 A0 00
4 00 01
5 A0 A1
6 00 01

3.2. A more complex overlay example
Figure 3 - a more complex overlay example

in this case the right and left polygon labels for arcs 1, 2, 4, 5 and 7 would be known from the geometry of the intersections:
1R: A0 1L: 00
2R: A1 2L: 01
4R: A1 4L: 01
5R: 01 5L: 00
7R: A1 7L: A0

the labels of the remaining arcs must be determined
labels can be passed from one arc to another around a polygon:
3R: must be the same as 2R and 4R
6L: from 2L, 4L

arc 3 was part of the red network, so its soils labels are known, the remaining (blue) part of its left label must be the same as the blue part of its right label
3R is A1 - thus 3L is B1
thus 6R is B1

how to get the blue labels of arc 8?
use a point in polygon routine to find the enclosing blue polygon
use a data structure in which arcs on the inside of the polygon boundary "point" to arcs on the outside of any enclosed islands
e.g. 5R -> 4L -> 6L -> 8L -> 2L -> 5R
thus the labels of arc 8 are 8L: 01, 8R: C1

the final step in the algorithm is to identify all new polygons by following around each polygon from one arc to the next until every right and left side of every arc has been identified with a uniquely numbered polygon

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4. Computational complexity
polygon overlay is numerically intensive and time consuming, therefore it is the most complex operation of most vector-based GIS programs

notation: if computing time to process n objects is proportional to n, the computational complexity is "order n2, or O(n)
if it is proportional to n2, we way it is "order n squared" or O(n2)

it is important to know:
how long does it take to overlay a given number of polygons?

what affects the amount of time taken?
obviously, the number of arcs and polygons affects the number of computations required
it is usually possible to determine the number of polygons being overlaid
the number of arcs is roughly 3 times the number of polygons
other factors, such as the wiggliness of boundaries, affect the time, but it is difficult to measure these

if n1 polygons are overlaid on n2, how many polygons result? (assuming maps are different)
minimum is n1+n2, polygons on the two maps do not intersect at all
maximum is infinity, lines have infinite wiggliness

typical is 3 or 4 times (n1+n2), discounting spurious polygons

if every one of n1 "red" polygons has to be compared to every one of n2 "blue" polygons, the algorithm will be O(n1n2)

if we could immediately find all "blue" arcs likely to intersect with a given "red" arc, we could build an algorithm which would be O(n1), which means it would be much more efficient for a given size of problem

to find arcs in this way we would need an efficient method of spatial indexing
one of the most successful methods uses the moving band concept:
sort both "red" and "blue" arcs in ascending order of x
process the arcs beginning at the left, moving to the right, sweeping a "band" across the map
only those arcs falling in the band are examined
since the arcs are sorted, we can find those in the band on either red or blue maps quickly

some of the best polygon overlay routines now available in the GIS market operate in close to O(n1+n2) time
a map with tens of thousands of polygons can be overlaid on another map with a similar number in an hour of computing time on a moderate-sized machine

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5. Intersection problems
because of computer precision, lines will be represented in the computer with great precision even though the accuracy of the representation is low
5.1. Adjacent lines
the following diagram represents a case where two lines cross

5.2. Sliver polygons
overlay algorithms compute the exact intersections between lines

in any overlay operation, it is likely that there will be pairs of lines which should coincide, but do not because of differences in digitizing
these are called slivers, spurious polygons or "coastline weave"
Figure 4 - Overlay of two images

if an arc or polygon of n1 points is overlaid on one of n2 points, up to:
( 2n1n2/(n1+n2) - 3 ) slivers may be generated

two possible approaches:
a. delete during the overlay operation, or
b. delete afterwards
5.2.1. Removing slivers during overlay
this is the most common approach used in commercial GIS programs

this approach deals with the line as if it were fuzzy

define a tolerance limit for each line which indicates the amount of uncertainty that exists regarding the true position of the digitized line
provides a band of width "epsilon" around every line
for digitized lines this width may be 1 mm

using epsilon, can then conclude that lines which have a difference in position less than epsilon are the same line
these two lines can then be collapsed to represent a single line

it is easy to get into difficulty:
lines A and B within epsilon, therefore the same:
lines B and C within epsilon, therefore the same:
but lines A and C are not necessarily within epsilon of each other
polygon overlay routines which do sliver removal "on the fly" must deal with this problem
5.2.2. Removing slivers after overlay
need intelligent criteria to distinguish between slivers and real polygons

criteria for removal include:
area: slivers are small
shape: slivers are long and thin
number of arcs: slivers generally have only 2 bounding arcs while real polygons rarely have only 2
alternating attributes: if a "red" arc between polygons A and B is overlaid on a "blue" arc between polygons 1 and 2, the slivers will alternate between A2 and B1
junctions: slivers terminate in 4 arc junctions, but 3 arc junctions are more common in real polygons
chaining: slivers tend to occur in chains
it is likely that the confidence with which it can be concluded that two arcs are forming slivers will increase steadily as we work along the arc
i.e. the attribute "sliver" is strongly correlated

if a sliver is detected, it can be replaced by an arc along its center line

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6. Discussion and exam questions
McHarg described the overlay technique well before the advent of GIS and polygon overlay. Discuss the advantages and possible disadvantages of a computer implementation of the technique.

Write out and illustrate the 16 Boolean combinations of two polygons A and B.

Review and discuss the alternative forms of areal interpolation described by Goodchild and Lam, 1980.

Discuss the relative advantages of raster and vector approaches to polygon overlay. Identify the application areas likely to adopt each method given their advantages

The Polygon Overlay text