UNIT 48 - LINE GENERALIZATION
UNIT 48 - LINE GENERALIZATION
Compiled with assistance from Robert McMaster, Syracuse
University
A. INTRODUCTION
- generalization is a group of techniques that allow the
amount of information to be retained even when the amount
of data is reduced
- e.g. when the number of points on a line are
reduced, the points to be retained are chosen so
that the line does not change its appearance
- in some cases generalization actually causes an increase
in the amount of information
- e.g. generalization of a line representing a
coastline is done best when knowledge of what a
coastline should look like is used
- this unit looks at line generalization
- line generalization is only a small part of the
problem of generalization in cartography - the
larger problem includes e.g. generalization of areas
to points
- the focus of the unit is on line simplification
- simplification is only one approach to
generalization (see below)
B. ELEMENTS OF LINE GENERALIZATION
- generalization operators geometrically manipulate the
strings of x-y coordinate pairs
Simplification
- simplification algorithms weed from the line redundant or
unnecessary coordinate pairs based on some geometric
criterion, such as distance between points or
displacement from a centerline
Smoothing
- smoothing routines relocate or shift coordinate pairs in
an attempt to "plane" away small perturbations and
capture only the more significant trends of the line
Feature Displacement
- displacement involves the shifting of two features at a
reduced scale to prevent coalescence or overlap
- most computer algorithms for feature displacement in
vector mode concentrate on an interactive approach where
the cartographer positions displacement vectors in order
to initialize the direction for shifting
- another method uses a smaller-scale version of the
feature to drive the displacement process
Enhancement/Texturing
- enhancement allows detail to be regenerated into an
already simplified data set
- e.g. a smooth curve may not look like a coastline so
the line will be randomly textured to improve its
appearance
- one technique is to fractalize a line by adding points
and maintaining the self-similarity of the original
version
- this produces fake (random) detail
Merging
- merging blends two parallel features at a reduced scale
- e.g. the two banks of a river or edges of a highway
will merge at small scales, an island becomes a dot
- algorithms for merging fuse the two linear features
together
C. JUSTIFICATIONS FOR SIMPLIFYING LINEAR DATA
Reduced plotting time
- plotting time is often a bottleneck in many GISs
- as the number of coordinate pairs is reduced through the
simplification process, the plotting speed is increased
Reduced storage
- coordinate pairs are the bulk of data in many GISs
- simplification may reduce a data set by 70% without
changing the perceptual characteristics of the line
- this results in significant savings in memory
Problems with plotter resolution when scale is reduced
- as the scale of a digital map is reduced, the coordinate
pairs are shifted closer together
- with significant scale reduction, the computed
resolution could easily exceed the graphic
resolution of the output device
- e.g. a coordinate pair (0.1, 6.3) reduced by 50% to
(0.05, 3.15) could not be accurately displayed on a
device having an accuracy of 0.1. Simplification
would weed out such coordinate pairs before
reduction
Processing
- faster vector-to-raster conversion
- faster vector processing
- the time needed for many types of vector processing
including translation, rotation, rescaling,
cartometric analysis will be greatly reduced with a
simplified data set
- many types of symbol-generation techniques will also be
speeded up
- e.g. many shading algorithms calculate intersections
between shade lines and polygonal boundaries
- a simplified polygonal boundary will reduce
both the number of boundary segments and also
the number of intersection calculations
required
D. LINEAR SIMPLIFICATION ALGORITHMS
overhead - Linear Simplification Algorithms
Independent Point Routines
- these routines are very simple in nature and do not, in
any way, account for the topological relationship with
the neighboring coordinate pairs
1. nth point routine
- every nth coordinate pair (i.e, 3rd, 10th) is
retained
2. randomly select 1/nth of the coordinate set
Local processing routines
- these utilize the characteristics of the immediate
neighboring points in deciding whether to retain
coordinate pairs
1. Euclidean distance between points
2. Angular change between points
overhead - Perpendicular distance and angular change
3. Jenks's simplification algorithm
overhead - Jenk's simplification algorithm
diagram
- three input parameters:
MIN1 = minimum allowable distance from PT 1 to PT
2
MIN2 = minimum allowable distance from PT 1 to PT
3
ANG = maximum allowable angle of change between
two vectors connecting the three points
- algorithm:
IF distance from PT 1 to PT 2 < MIN1,
OR
distance from PT 1 to PT 3 < MIN2
THEN PT 2 is removed
ELSE
IF angle 123 < ANG
THEN PT 2 is removed
Unconstrained extended local processing routines
- these algorithms search beyond the immediate neighboring
coordinate pairs and evaluate sections of the line
- the extent of the search depends on a variety of
criteria, including:
- the complexity of the line
- the density of the coordinate set
- the beginning point for the sectional search
Reumann-Witkam simplification algorithm
overhead - Reumann-Witkam simplification algorithm
- the algorithm uses two parallel lines to define a
search region
- after calculating the initial slope of the search
region, the line is processed sequentially until one
of the edges of the search corridor intersects the
line
Constrained extended local processing routines
Global routines
E. MATHEMATICAL EVALUATION OF SIMPLIFICATION
F. LINEAR SMOOTHING
- smoothing is applied to digital line data in order to
improve the aesthetical qualities of the line and to
eliminate the effects of the digitizing device
- in general, it is felt that smoothing improves the
quality of these data
- smoothing increases the number of coodinates needed, so
is normally used only for output
REFERENCES
Buttenfield, B.P., 1985. "Treatment of the Cartographic
Line," Cartographica 22(2):1-26.
Douglas, D.H. and T.K. Peucker, 1973. "Algorithms for the
Reduction of the Number of Points Required to Represent a
Line or Its Character," The American Cartographer
10(2):112-123.
McMaster, R.B., 1987, "Automated Line Generalization,"
Cartographica 24(2):74-111.
McMaster, R.B., 1987. "The Geometric Properties of Numerical
Generalization," Geographical Analysis 19(4):330-346.
McMaster, R.B.,1989. "The Integration of Simplification and
Smoothing Algorithms," Cartographica 26(1).
Peucker, T.K., 1975. "A Theory of the Cartographic Line,"
Proceedings, Second International Symposium on Computer-
Assisted Cartography, AUTO-CARTO-II, September 21-25,
1975 U.S. Dept. of Commerce, Bureau of Census and ACSM,
pp. 508-518.
White, E., 1985. "Assessment of Line-Generalization
Algorithms Using Characteristic Points," The American
Cartographer 12(1):17-27.
DISCUSSION/EXAMINATION QUESTIONS
1. Discuss the differences between sequential and global
approaches to line simplification.
2. What are the five generalization operators for digital
line data? Discuss each one of these and give examples.
3. Using a series of diagrams, discuss the procedure used by
the Douglas algorithm.
4. Discuss the different approaches you might use to
evaluate the effectiveness of line simplification procedures
and the advantages and disadvantages in each case.
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Last Updated: August 30, 1997.