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Assessment of Line-GeneralizationAlgorithms Using Characteristic PointsEllen R. WhitePublished online: 14 Mar 2013.

To cite this article: Ellen R. White (1985) Assessment of Line-Generalization Algorithms UsingCharacteristic Points, The American Cartographer, 12:1, 17-28

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Assessment of Line-GeneralizationAlgorithms Using Characteristic Points

Ellen R. White

ABSTRACT. Line generalization, used to reduce scale in mapping, caricatures an original lineby retaining points essential to its shape-called characteristic, or critical, points-and discard-ing others. This study describes experiments utilizing base lines of perceptually selected criticalpoints to evaluate three common line-generalization algorithms: nth-point elimination, a perpen-dicular routine, and the Douglas algorithm. Base-line correspondence to computer generalizationswas measured by graphical overlay, areal offset, commonly held points, and mean number oftimes respondents judged particular points to be critical. Visual comparison of an original lineand its computer generalization followed. The Douglas algorithm's line generalizations provedmore faithful to original lines than those of perpendicular point selection and nth-point elimi-nation.KEY WORDS: line generalization, characteristic points, critical points, line perception, Douglasalgorithm

Generalization has always been a majorcomponent of the cartographic method.The Multilingual Dictionary of Tech-nical Terms in Cartography definesgeneralization as "the selection andsimplified representation of detail ap-propriate to the scale and/or purpose ofa map" (Brophy 1973, p. 301). Thoughthe concept is relatively simple, its im-plementation is complex, as the manystudies on problems of generalizationattest.

Cartographers use generalization toreduce scale: to simplify, symbolize, andselect the information that belongs on asmall-scale map. Researchers havedelved into the whats and whys of gen-eralization, producing elaborate modelsand formulas (Topfer and Pillewizer1966; Salichtchev 1976; Morrison 1974

Ellen R. White is Staff Cartographer and In-structor at the University of Oklahoma in Nor-man, OK 73019. This paper is based on herMaster's thesis, completed at Virginia PolytechnicInstitute and State University. She thanks Dr.Alan M. MacEachren, University of Colorado, andDr. J. Clark Archer, University of Oklahoma, fortheir comments on various drafts of this paper.

© 1985 American Congress on Surveying and Mapping0094-1689/85$2.50

and 1975), but not into the hows. Sinceno objective criteria define exactlywhich objects or points warrant selec-tion, individual cartographers, armedwith knowledge of theme and area to bemapped, must discriminate whichamong the many feature characteristicsand their relationships require inclusionon their maps (Wright 1942; Fahey1954; Lundquist 1959; Pannekoek 1962;Imhof 1963).

Line generalization is an area inwhich computer potential has been rec-ognized and applied. Many computerroutines have been developed for auto-mated line generalization "to reduce thenumber of points required to representa line and to produce abstractions, orcaricatures, of the line" (Douglas andPeucker 1973, p. 122). Point reductionconserves storage space, but its value islost if the resulting generalization doesnot retain the character of the originalline. The generalization must "lookright" to the map reader.

Psychologists, and others interestedin pattern recognition, have long beenconcerned with how people recognizelines and what line features facilitatethis recognition. Attneave determinedthat information is "concentrated at

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those points on a contour at which itsdirection changes most rapidly, i.e., atangles or peaks of curvature" (1954, p.184). Noton and Stark confirmed that"angles are the principal features thebrain employs to store and recognize"line drawings (1971, p. 37).

Linking these ideas to cartography,Arnheim (1976) stated that "generaliza-tion ... occurs spontaneously in all per-ception" and that, regardless of howcomplex the map, "simplified images arewhat remains in memory." In his studyof cartogram shapes, Dent (1972) notedthe importance of "major changes in di-rection" in determining perception ofshape. Freeman (1978) provided morespecific examples of these information-loaded points, including maxima, mini-ma, points of inflection, discontinuitiesin curvature, endpoints, intersections,and points of tangency. These points ofmaximum information are called char-acteristic, or critical, points.

Jenks identified two kinds of charac-teristic points: 1) those meeting thepsychological criteria suggested byAttneave, Noton, Stark, Dent, and Free-man "inherent in the linear configura-tion" and traditionally used to providecaricatures oflines (Jenks 1980, p. 214);and 2) those selected for their geograph-ical importance, for example, points oftangency between rivers and cities, in-tersections of political boundaries, andplaces of economic importance, like har-bors (Jenks 1981). Characteristic pointsof the first sort, because they provide theingredients essential for pattern recog-nition, are important regardless of mappurpose. Those of the second sort are de-termined by the purpose of an individualmap.

Marino (1978 and 1979) conducted anexperiment to test the usefulness of per-ceptually chosen characteristic pointsfor "establishing systematic criteria inline generalization" (1979, p. 70). Twogroups-one of cartographers, one ofnon-cartographers - selected character-istic points at three different levels ofgeneralization. Both groups consistentlyselected similar critical points on partic-ular lines; these points maintained their

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"hierarchy of importance" even with in-creasing degrees of generalization(1978). Her study confirmed earlier non-cartographic findings that angularchange is the factor principally deter-mining characteristic points.

So, in the spring and summer semes-ters of 1982, building on Marino's work,we created a baseline of perceptuallychosen critical points to be used in ex-periments to evaluate line-simplifica-tion algorithms. The closer an algorithmcame to approximating this standard-ized perceptual line, the better it wouldprovide map users with effective gener-alization. This study evaluated threeline-generalization algorithms' effec-tiveness in retaining rather than dis-carding information perceived as crucialby our map readers.

Faculty, staff, and students at theUniversity of Oklahoma comprised oneset of respondents. Another was associ-ated with the Oklahoma ArcheologicalSurvey and the Oklahoma GeologicalSurvey. Over half of a total of 90 re-spondents were undergraduates in in-troductory geography courses. Most ofthem (78 percent) had had no prior car-tographic experience.

To these two groups, we added a third,an independent control group consistingof 46 students enrolled in geographycourses at Virginia Polytechnic Insti-tute and State University.

THE ALGORITHMSFor evaluation, we selected three

poin t-selection/rejection algorithmswith the virtue of not relocating originalpoints: simple nth-point elimination; se-lection through a perpendicular calcu-lation between three consecutive points;and the Douglas algorithm.

Computationally, simple nth-pointelimination is the easiest of the three.The program reads and plots the firstpoint, then reads and eliminates succes-sive points up to a user-specified nthpoint. It continues through a line in thisfashion until it reaches and plots the fi-nal point (Figure 1A). Although this al-gorithm is quick and simple, its deletionprocedure is arbitrary: points selected

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Figure 1. Line-simplificationalgorithms: A)nth-point elimination, where n = 5; B) sim-ple tolerancing procedure using a perpendic-ular measure; C) Douglas (1973) algorithm.

thought to generalize linear features.Basically, the procedure defines astraight segment between a first point(the anchor) and a last point (the floater)on a line (Figure lC), and then calcu-lates perpendicular distances from thissegment to intervening points on theoriginal line. Where all distances lie be-Iowa specified-tolerance bandwidth, thestraight segment is presumed to repre-sent the line. If distances exceed thebandwidth, the point lying furthestaway becomes the new floater. "As thecycle is repeated the floating point ad-vances toward the anchor. When themaximum distance requirement is metthe anchor is moved to the floater"(Douglas and Peucker 1973, p. 117). Toavoid re-examination of the entire linewith each cycle, floating points arestacked in an array and processed in re-verse sequence to establish a new float-ing point after an anchor is stored. Thelist of stored anchors comprises the gen-eralized line at the end of the algorithm.

CREATION OF BASE LINESThree naturally occurring lines

formed the basis of our experiment. Be-cause angular change is an importantperceptual factor, we selected linear fea-tures from Rand McNally state-and-county base maps (Figure 2) to illustratecontrasting patterns of angular change(Table 2).

The least-complex line, LALINE, wascomposed of successive sections of theBlack, Red, and Mississippi Rivers inLouisiana. The next-most-complex line,ALLINE, consisted of portions of theSipsey and Tombigbee Rivers in Ala-bama. The most-complex line, CRLINE,from the northern Chesapeake Bay, ex-tended from Baltimore's northernmost

IBIA)

are unrelated to the specific character-istics of the line being generalized, anddistinctive features are arbitrarily elim-inated if they fall between consecutiventh points.

The perpendicular algorithm is a sim-ple tolerancing routine employing suc-cessive triads of points. It first calculatesa perpendicular from a point, B, to a seg-ment, AC, linking preceding and sub-sequent points (Figure IB), and it thencompares this length to a user-suppliedminimum distance, or tolerance band-width, below which it eliminates thepoint. In contrast, points exceeding thebandwidth are retained in the general-ization because they form distinctive de-partures from the general trend andrepresent points of rapid angularchange.

The Douglas algorithm also utilizes atolerance bandwidth, or corridor. Al-though the least efficient of the threeroutines (Table 1), it treats a line as anentity to mimic the way people are

Table 1. Comparisonof CPU times, in seconds,for three algorithms.

Seconds of CPU Time

Point eliminationPerpendicularDouglas

LALINE

1.211.602.99

ALLINE

1.341.793.87

CHLINE

1.411.994.71

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Figure 2. Naturally occurring lines used inthe study: A) ALLINE; B) LALINE; C)CHLINE.

city limits to Turkey Point, between theSusquehanna and Elk Rivers.

Marino's procedure for acquiring a setof perceived characteristic points was topresent subjects with a copy of a line anda specified number of straight pins thatthey were to insert into the line at pointsthey deemed "essential in preservingthe character of the line" (1979, p. 71).We adapted Marino's procedure to ourexperiment with modifications thatwould facilitate later comparison ofthese respondent-selected critical pointswith algorithm-selected critical points.

First, to clarify the degree of line gen-eralization, we generalized each line to

Table 2. Characteristics of base lines.

approximately 100 points. An nth-point-selection routine determined the rangeof points-from 91 to 107. A targetnumber, 100, seemed a readily identifi-able value that would represent a highdegree of generalization, yet be modestenough to prevent the task of identify-ing critical points from being tedious forrespondents.

Second, we required participants touse all of the pins given to them, en-abling later point-to-point comparisonbetween the perceptual and the algo-rithm-produced base lines.

Third, we asked respondents to in-clude end points in their generalizationsand to assume that successive pointswere connected by straight line seg-ments.

Each respondent selected character-istic points from only one line, which wethen plotted on a summary line to rankthe individual points selected. The mostfrequently chosen points created a com-mon perceptual base line for each algo-rithm (Figure 3).

To this first experiment of evaluatingalgorithms through comparison of criti-cal-point-selection results we added asecond, a line-comparison experimentoperating inversely to the first, in whichsubjects compared the computer-gener-alization results instead of creating ageneralization. Very simply, respon-dents viewed a sheet showing one of thenaturally occurring lines and the threecomputer-generalized versions and wereasked to select the generalized line bestrepresenting the original line. This ex-periment, using the same respondents,followed the first immediately.

With these two basic experiments we

LALINE ALLINE CHLINE

Total number of points 653 896 1,059Total segment length 15.1929 16.4217 18.6419Absolute line length 7.5754 7.9676 3.0500Average segment length 0.0233 0.0183 0.0176Total segment length

2.0057 2.0610 6.1121Absolute line lengthAverage angular deviation 169.6617 166.8158 164.7785

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tions of each naturally occurring line,employing a standardized error measurein which the total areal error was divid-ed by the length of the line being eval-uated.

With one exception, results for totalareal error were generally consistentacross all three lines: the Douglas algo-rithm, followed by the perpendicularand point-elimination algorithms, pro-duced generalized lines with the leasttotal error. The exception was the point-elimination generalization of LALINE,the least complex of all lines examined.The point-elimination and Douglas rou-

Figure 4. Graphic overlays of ALLINE forareal-offset analysis, where ELIMPTS =nth-point-elimination algorithm, PERPEN= perpendicular algorithm, and DOUGDouglas algorithm.

]'.. I

~

ran a third, a control, utilizing our 46-student control group. This final exper-iment was a simple repeat of the line-comparison experiment to determinewhether subject response to the inverseexperiment had been affected by expo-sure to the critical-point-selection pro-cedure, thereby skewing our results.

DATA ANALYSISData analysis of the quantitative com-

parison of perceptually created baselines and their computer-generatedcounterparts took two forms: measure-ment of the area of offset between theperceptual base lines and the corre-sponding algorithmic lines; and compar-ison of the number of points shared bybase and algorithmic lines.

Areal-Offset MeasurementOverlaying one line on another

creates a series of error polygons reflect-ing divergence between the two lines(Figures 4,5, and 6). We measured andsummed the data from these polygons tocompare the three computer generaliza-

Figure 3. Perceptually created base linesfrom characteristic points: A) ALLINE; B)LALINE; C) CHLINE.

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Figure 5. Graphic overlays of LALINE forareal-offset analysis, where ELIMPTS =nth-point-elimination algorithm, PER-PEN = perpendicular algorithm, and DOUG= Douglas algorithm.

tines yielded nearly identical areal-er-ror values for this line, perhaps becauseof the original, natural line's rather reg-ular sinuosity (Table 3). Total areal er-ror, standardized as error per centimeterof the computer-generalized line, showsall three generalizations for LALINEcontaining the least areal error of alllines examined (Table 4). The line pro-duced by the Douglas algorithm for bothCHLINE and ALLINE evinced the leastareal error of all lines examined, fol-lowed again by the generalizationscreated by the perpendicular and point-elimination routines.

Common-Point ComparisonThe number of points common to per-

ceptual base lines and their respectivecomputer-generalized lines can be used

22

Figure 6. Graphic overlays of CHLINE forareal-offset analysis, where ELIMPTS =nth-point-elimination algorithm, PERPEN= perpendicular algorithm, and DOUGDouglas algorithm.

as an alternative quantitative measureof line similarity. Because each line con-sists of a series of points connected bystraight segments, the higher the num-ber of shared points, the more alike thetwo lines.

We converted our results into per-centages of commonly shared points tocompare lines with varying numbers ofpoints. Percentages of common pointsranged from a low of 8.79 percent for thepoint-elimination generalization of AL-LINE at a tolerance of 0 to a high of

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A) tolerance,.-, zero B) toleronee" "0.025 em50 60---- 1\40 so

30 ~ 40

20 30

10 ~ 20

~10

lALINE AtuNE CHliNE lAUNE AlllNE CHliNE

• Douglas .•. Perpelldlculor .• POI"' EllmlllOllO"

C I loleranee ,'0.038 em D) toleronee ' ~0.051 em70 70

60

~

60 ~~

50 50

~40

~!

30

~

30

20 20lAUNE AlUNE CHUNE lAlINE AtUNE CHliNE

Figure 7. Plots of percentage of points heldin commonbetween a perceptual base lineand its corresponding computer-generalizedlines.

65.42 percent for the Douglas-derivedgeneralization ofCHLINE at a toleranceof 0.051 cm [0.02 in.] (Table 5). Unsur-prisingly, as the tolerance increased, so

Table 3. Areal-offsetvalues.

did all percentages. Still, only 19 of 48values were greater than 50 percent.

Rank orders of the generalization rou-tines over all tolerance levels were quiteconsistent, the exception being the per-pendicular generalization of ALLINE ata 0.025-cm [O.Ol-in.] tolerance, whichequaled the percentage of the Douglasgeneralization. The various generaliza-tion percentages are plotted in Figure 5.The point-elimination generalizationconsistently ranks far below the others,because of the relative arbitrariness ofits point-selection method .

Point commonality between percep-tual base lines and their computer gen-eralizations can also be measured by atechnique accommodating the relativeimportance of individual points. Be-cause individual points can be ranked bythe number of times they were chosen,certain points are deemed "more criti-cal" (more frequently agreed to be im-portant by respondents) than others.The computer generalizations are thenscored with the perceptually weightedcritical points. The more point numbersincluded considered critical by respon-dents, the more perceptually effectivethe generalization.

Our perceptual base lines were com-posed of characteristic points selected byat least 13 of the 30 subjects tested foreach line. Using the set of coordinatepairs common to each generalization (ato tolerance), we calculated a mean value

Total ArealError in Mean Standard

Square Centimeters Error Deviation

LALINEPoint elimination 18.722 0.290 0.310Perpendicular 21.658 0.258 0.258Douglas 18.710 0.290 0.323

ALLINEPoint elimination 122.374 1.342 1.890Perpendicular 113.045 1.316 2.090Douglas 89.722 1.181 1.806

CHLINEPoint elimination 121.058 1.458 1.548Perpendicular 94.516 1.258 2.032Douglas 67.135 0.690 0.987

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Table 4. Standardized areal-offset values.

Total Areal Mean StandardError Error Deviation

LALINEPoint elimination .299038 .036000 .003594Perpendicular .273122 .042000 .004561Douglas .256135 .040000 .004426

ALLINEPoint elimination 1.158920 .012735 .017923Perpendicular 1.005888 .011697 .018606Douglas .746934 .009826 .015039

CHLINEPoint elimination 1.071062 .014471 .031245Perpendicular .978953 .013052- .020464Douglas .570425 .005890 .008381

for each line based on average percep-tually weighted scores for algorithm-se-lected points (Table 6). Then we rankedthe computer generalizations by theircloseness to the perceptual base-linemean, evaluating differences betweenmean values with Student's t-values.

Based on these means, for ALLINEand CHLINE the Douglas generaliza-tion ranked highest in point common-ality, followed by the perpendicular andpoint-elimination generalizations. Allmeans for ALLINE and CHLINE weresignificantly different at the .10 level,whereas for LALINE the differences

were less distinct. Among the computer-generalized means, for LALINE Doug-las ranked first, perpendicular second,and point elimination third. However,though the Douglas and the perpendic-ular generalizations generally do notdiffer from each other significantly atthe .10 level, together at that level theydiffer significantly from the point-elim-ination generalization.

Line ComparisonThe line-comparison experiment was

simply another way to evaluate the com-puter generalizations, as well as to pro-

Table 5. Percentage of common points between perceptual base lines and computergeneralization.

LABASE ALBASE CHBASE Average

A) Tolerance = 0Point elimination 18.95 8.79 10.28 12.67Perpendicular 32.63 29.67 29.04 30.45Douglas 45.74 43.96 42.99 44.23

B) Tolerance = ± 0.025 cmPoint elimination 24.21 13.19 14.95 17.45Perpendicular 43.16 54.95 39.25 45.79Douglas 48.94 54.95 56.07 53.32

C) Tolerance = ± 0.038 cmPoint elimination 29.47 21.98 22.43 24.63Perpendicular 46.32 46.15 43.93 45.47Douglas 53.19 58.24 61.68 57.70

D) Tolerance = ±0.051 cmPoint elimination 36.84 27.47 28.97 31.09Perpendicular 51.58 51.65 55.14 52.79Douglas 57.45 62.64 65.42 61.84

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Table 6. Mean number of subjects to select points included in computer-generalizedlines.

Sample Standard SampleMean Deviation Size

LALINEPoint elimination 8.6 7.52 95Perpendicular 11.6 10.04 95Douglas 13.7 10.56 95Perceptual mean 22.9 5.51 95

ALLINEPoint elimination 6.1 5.57 91Perpendicular 10.0 8.99 91Douglas 13.2 10.31 91Perceptual mean 21.2 5.39 91

CHLINEPoint elimination 6.0 6.09 107Perpendicular 9.6 9.36 107Douglas 13.2 11.58 107Perceptual mean 23.9 5.10 107

vide a check on the use of critical pointsin line generalization. We asked all sub-jects to pick which of the three gener-alizations best represented the originalline, but we sampled two groups: one in-volved and the other (the control) unin-volved in the selection of characteristicpoints. We were trying to determine

whether subjects' involvement in choiceof characteristic points would influencetheir judgment of line similarity.

The generalizations produced by theDouglas algorithm were overwhelming-ly-by 86 percent of all sample sub-jects-deemed the best perceptualrepresentations of the original lines (Ta-

Table 7. Results of line comparisonexperiment.

University of Oklahoma-Population Tested Concurrently with Selection of Characteristic Points

Point eliminationPerpendicularDouglas

LALINE

o2

1517

ALLINE

o3

1821

CHLINE

oo

2121 59

Virginia Polytechnic Institute and State University-Independent Population

LALINE ALLINE

Point elimination 1 1Perpendicular 1 3Douglas 12 12

14 16

Combined Populations

LALINE ALLINE

Point elimination 1 1Perpendicular 3 6Douglas 27 30

31 37

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CHLINE

31

1216

CHLINE

31

3337

46

COMBINEDRESULTS

51090

105

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Table 8. Rank orders of computergeneralizations.

Standardized Common Critical Point LineAreal Offset Points* Comparison Comparison

LALINE DOUG DOUG DOUG DOUGPERPEN ELIMPTS PERPEN PERPENELIMPTS PERPEN ELIMPTS ELIMPTS

ALLINE DOUG DOUG DOUG DOUGPERPEN PERPEN PERPEN PERPENELIMPTS ELIMPTS ELIMPTS ELIMPTS

CHLINE DOUG DOUG DOUG DOUGPERPEN PERPEN PERPEN PERPENELIMPTS ELIMPTS ELIMPTS ELIMPTS

ELIMPTS = point elimination PERPEN = perpendicular DOUG = Douglas* zero tolerance

hIe 7). But the focus on characteristicpoints (or lack of it) did affect the twogroups' respective comparisons. Of theoriginal Oklahoma group 92 percent-but only 79 percent of the Virginia con-trol group-chose the lines generatedby the Douglas algorithm as the bestrepresentation. This difference is notstatistically significant (based on a Stu-dent's t-value), but it implies that, hav-ing taken part in both experiments, thesensitized Oklahoma subjects focusedsomewhat more on the individual char-acteristics of the generalized lines thanthe un sensitized Virginia group.

CONCLUSIONSComputer technology in cartography

is ubiquitous and its application to lin-ear generalization commonplace. Still,no matter how elegant an algorithm, ifits line generalization is not an easilyrecognizable caricature of the original,its purpose is not served.

One method of evaluating the percep-tual effectiveness of line-generalizationalgorithms is to assess their ability toidentify characteristic, or critical,points: those spots along a line whereinformation about its nature is concen-trated. These points must be retained forthe generalization to be recognized as alogical representation of the original.Critical points usually fall at sites ofmajor directional change such as maxi-ma, minima, and points of inflection.

Our research has sought, using baselines of perceptually selected character-

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istic points, to evaluate the line gener-alizations created by three algorithms.We used four separate measures to eval-uate the computer generalizations.Three of these compared computer-gen-eralized lines to a perceptual base line:areal offset; number of points commonto a perceptual base line and its re-spective computer generalization; andperceptually weighted critical-pointcomparison between a computer gener-alization and its corresponding baseline. The measure remaining was a vi-sual comparison by respondents of gen-eralized lines and the original lines theywere meant to represent. In all cases theDouglas routine produced the best gen-eralization (Table 8).

Characteristic points can be used tocreate a perceptual base line for theevaluation of computer-generalizedlines. Treating a line as this sort of en-tity thus appears to be a perceptuallyvalid approach, best approximated, inour experiments, by the Douglas line-generalization algorithm. But sinceeven the Douglas algorithm often failedto match points denoted as critical byrespondents, in the future, perhaps, al-gorithms should be generated with moreattention to map users' perceptions ofimportance than to the algorithm's meremechanical efficiency.

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28

Journal To Be Published QuarterlyAnd Distributed To ACA Members

On September 13, 1984 the ACSM Executive Committee ap-proved the following Publication Committee recommendations per-taining to The American Cartographer:

1) during 1985 The American Cartographer is to be publishedtwo times and is to be sent only to ACA members, except thatNSPS and AAGS members may receive copies at no chargeby so requesting on a form drafted by ACSM staff;

2) beginning in 1986, Surveying and Mapping will be distrib-uted only to NSPS and AAGS members;

3) The American Cartographer is to be published four times in1986 and sent only to ACA members. However, NSPS andAAGS members may request copies of the journal for a fee tobe determined.

These changes were made in an effort to reduce publication costsand to relieve the current backlog of accepted articles. In addition,a quarterly journal will make possible additional columns such asthe Software Reviews announced in this issue, as well as specialtopic issues focusing on areas of rapid development in cartography.The editorial staff welcomes suggestions for improving the rele-vancy of the journal and urges members to submit manuscripts,notes, or reviews describing advances in any aspect of cartography.

The American Cartographer

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