23

Fractal Enhancement ofCartographic Line Detail

Geoffrey H. Dutton

ABSTRACT. In plane geometry curves have a dimension of exactly 1 and no width. In nature, allcurvilinear features have width, and most have dimension greater than 1, but less than 2. Manyphenomena, such as coastlines, have the same "look," even when viewed at greatly varying scales. Theformer property is called "fractional dimensionality," and the latter is called "self similarity." Curvesdigitized from maps may be analyzed to obtain measures of these properties, and knowledge of themcan be used to manipulate the shape of cartographic objects. An algorithm is described which enhances thedetail of digitized curves by altering their dimensionality in parametrically controlled, self-similarfashion. Illustrations show boundaries processed by the algorithm.

Measuring and ModellingIrregularity in Nature

Only in the mind and works of man dostraight lines exist. Rarely does Naturerule with a straightedge, and even theselines are rough, seldom extending veryfar. But surrounded by rectilinear arti-facts, it is understandable why humanstry to measure and model the world withEuclidean precepts. Frustration in mak-ing certain measurements and in model-ling many natural forms can be attrib-uted to this view of space itself, in whichdistance between two given points isassumed to be Pythagorean.

Suppose one is surveying a section ofcoastline and wants to calculate its lengthaccurately and map it. A series of closely-spaced sightings must be made at thehigh-water mark. The cumulative dis-tance along these points can then beaccurately computed, and it is invariablygreater than the crow's-flight distancespanning the stretch of coast. Fig. IArepresents the profile of a fictitious coast-line. Its surveyed approximation is plot-ted in Fig. IB, and the crow's-flight version ofit is shown in Fig. 1C. A greater

Geoffrey Dutton is a programmer/analyst at theHarvard Laboratory for Computer Graphics andSpatial Analysis. His mailing address is GraduateSchool of Design, Harvard Univ., 520 Gund Hall,Cambridge, Mass. 02138.

number of sightings yields a closerapproximation to actual length andshape, even though the rate of increase oflength slows.

This lesson in approximation hasseveral morals. One is that surveyors runinto a real law of diminishing returnswhen trying for centimeter accuracies inthe lengths of complicated boundaries.Another is that the difficulty, hence theprobable error, in measuring coastlinesand the like varies from place to place. InFig. 1A, it is obvious that there is muchmore irregularity in the lower part of thecoast than in the upper part. This may bedue to the former being composed of rockoutcroppings and the latter being a sandybeach. But in trying to express this quali-tative difference quantitatively, one findsscientific vocabulary confusing and inad-equate. Literature in geography andimage processing abounds with indicesthat characterize the shapes of point setsand linear and areal features (Stoddard,1965; Boyce and Clark, 1964; Bunge,1961; Boots, 1972). However, to para-phrase Pavlidis (1978), these indices aredestructive of information and provideneither a general linguistic model nor ameasure suitable to allow manipulation ofcartographic detail. One wishes for ameasure of geometric complexity andirregularity that is as general as that of

The American Cartographer, Vol. 8, No. 1, 1981, pp. 23-40

The American Cartographer11

0

Fig. 1. A coastline and its approximations: (A) the original coastline, (B) segmentedapproximation. (C) original coastline with superimposed trend line.

entropy in thermodynamics. Fortunately,foundations for such a vocabulary and forsuch measures have been developed.Irregularity as FractionalDimensionality and Self Similarity

A suitably general approach to quan-tifying the complexity of irregular forms,and one that directly confronts the dilem-mas of Euclidean measurement, is that ofMandelbrot (1977). The phenomena thathe addresses-natural forms arising fromforces such as turbulence, curdling,Brownian motion, and erosion-have atall scales two related properties, self simi-larity and fractional dimensionality. Selfsimilarity means that a portion of anobject when isolated and enlarged exhib-its the same characteristic complexity asthe object as a whole. The shapes revealedmay be highly irregular, and none may beexactly alike, but they will have the samekind of irregularity over a wide range ofscales. Fractional dimensionality meansthat the Euclidean dimension thatnormally characterizes a form (1 forlines, 2 for areas, 3 for volumes)represents only the integer part of the truedimension of the form, which is a fraction.'

Mandelbrot treats dimension as acontinuum, in which the integer Euclid-ean dimensions merely represent limitingcases of topological genera, unlikely tooccur in nature. Thus the coastline in Fig.1A might have an approximate overalldimensionality of 1.2, but its two dissimi

lar subsections have different structureand dimensionality. The more irregularlower portion may have a dimension ofnearly 1.3, while the smoother upper partmay be of a lower dimension, less than1.1. There is only one version of the coastthat has a dimensrionality of exactly I (itsEuclidean dimensionality), and that isthe trendline shown in Fig. 1C.

This difference in dimensionality isquantified in Fig. 2. On this graph theabscissa symbolizes the number of sight-ings (or line segments) used to approxi-mate the entire coast (top curve) and itslower (middle curve) and upper (bottomcurve) portions. The values read from the

Fig. 2. Dependence of length on fractallimensionality and scale of measurement.

Vol. 8, No. 1, April

1981

3

ordinate represent the measured lengthyielded by a specified number of segmen-tations. The upper coast, being straighter(of lower dimension), requires fewerapproximations to minimize equally itsrate of increase. The lower coast, havinggreater dimensionality, cannot be asclosely approximated with as few sight-ings. Its asymptotic rise is slower, andcovers a greater range. The sum of thesetwo curves yields a curve for the entirecoast, the top one.

Fractal Forms in CartographyDigitized map data resemble fractals

much more than they resemble (continu-ous) functions which mathematiciansnormally study. Although certain carto-graphic objects, including both bounda-ries and terrain, can be approximatedusing real functions, e.g., trigonometricseries, the difficulty remains of repre-senting nonperiodic map features as well asones that are not single-valued, i.e.,places where curves reverse direction.Furthermore, fractal curves, such ascoastlines, can have islands, whose numberand size distribution varies with fractaldimensionality (Mandelbrot, 1977, p. 45).Given these similarities, perhaps it ispossible to subject strings of coordinatesdescribing lines on maps' to algorithmsthat modify them according to fractalcriteria. Why, for instance, cannot detailsbe inserted into a chain of coordinates toresemble the features already there? Whycannot digitized features be made moreprominent, as well as smoothed away?Seeing no reason why such methodsshould not be explored as cartographictechnique, the author has developedseveral algorithms in such an attempt.The procedure described below is thelatest of these, but it is by no means theonly possible approach to fractalizingmap data, nor the only use of fractaltheory in spatial analysis.

Physical and natural science deal withmany fractal phenomena, and the appli-cation of fractal concepts to geographyhas only just begun. Examples of suchanalyses abound in Mandelbrot's essay,

which is only incidentally concerned withcartography. What follows, then, de-scribes an experimental procedure fortransforming digitized curves fractally, orfractalizing them in self-similar fashion. Itis intended to enlarge the repertoire ofgeometric processing techniques for linereduction and enhancement, a repertoirethat includes coordinate weeding, culling,bandwidth filtering, and splining or arc-substitution methods (Jenks, 1980; Mor-rison, 1975; Douglas and Peucker, 1973;Rhind, 1973; Ramer, 1972).

Parametric Fractalization ofDigitized Curves

Like splining, in which smooth, mathe-matically-defined arcs are inserted inplace of one or more segments along achain, fractalizing can increase a chain'stotal length. Unlike it and other methodsfor coordinate reduction and chainsmoothing, fractalizing permits features tobe exaggerated and smaller-scale featuresto be introduced into digitized curves, aswell as allowing features to be eliminated.The exaggerations and additions are notarbitrary forms introduced to the chainbut are caricatures and recursions offorms already found there. Because theprocedure can be applied recursively, thereare geometric similarities between smallerfeatures introduced and larger featuresalready existing in cartographic lines.Such enhancements of chains may not haveas regular an appearance as a chainsmoothing via arc-substitution, but theymay preserve more concretely thequalities of the original chain. Thefollowing describes the most recentapproach to fractalization, whichreconfigures chains to desired dimen-sionality and detail.

As currently implemented, the proce-dure is given a list of coordinates for aninput chain and returns a fractalizedversion of it in a separate array, trans-formed according to four parameters:

(1) a Sinuosity Dimension (SD) (2) a Uniformity Coefficient (UC) (3) a Straightness Tolerance (ST) (4) a Smoothness Tolerance (SM)

The American Cartographer11

0

Fig. 3. Enhancement of a 13-point chain, showing the effects of varying SD.

The Sinuosity DimensionThe Sinuosity Dimension, SD, pre-

scribes the amount of waviness thatchains should possess after fractalization,but is tempered by the Uniformity Coef-ficient, discussed below. SD is a realnumber between 1 (minimum sinuosity)and 2 (maximum sinuosity), and specifiesa fractal dimension to characterize pro-cessed chains.

Fig. 3 demonstrates the effect of vary-ing SD from 1.1 to 1.9 in fractalizing a13-point chain. Three iterations of frac-talization were performed (second row),followed by one of smoothing (third row).The enhancements are overlaid with theoriginal chain in the fourth row. UC,discussed below, is fixed at 1.0 in each ofthe examples in Fig. 3.

Ignoring for the moment the effect ofUC, the algorithm works by moving alonga chain, relocating each vertex in thegeneral - direction of its angle bisector.The movement forms a new angle whichwill be the same at all vertices, as deter-mined by SD. Since all vertices are poten-tially subject to displacement, it isgeometrically necessary to introduce ad-ditional vertices, which are located at themidpoint of each segment along thechain. These serve as fixed points, andsuccessive ones are connected to formtriangle bases, where the apex is the vertexbetween these midpoints. To relocatesuch an apex, it is necessary andsufficient to know (a) the desired angle ofthe apex, which SD implies, and (b) thecurrent proportions of its two legs. In

Fig. 4. Effects of changing the number of iterations of fractalizing and smoothing.

order to maintain local self-similarity,this proportion is constrained to be thesame after fractalizing as before. Theactual computations require solving trig-onometric identities involving the law ofsines, law of cosines, and sums and differ-ences of sines and cosines. The result is tostandardize all junctions at the angledetermined by SD and to introduce inter-mediate vertices (at segment midpoints)having unstandardized included angles.This provides the degree of freedom neededto standardize the original junctions.Successive iterations of the procedurewill then standardize the angles at theformer midpoints, once again halving allsegments.

The effect of this, as the second row ofFig. 3 shows, is to create rather mechani-cal-looking chains. Such figures are notvery "cartographic," although they mayhave certain stylistic uses in thematicmapping. Their zigzags may be smoothed,as in the third row of Fig. 3, to soften

Vol.8, No. 1, April 1981

visual impact, and such post-processingshould be applied to chains thus fractal-ized. Usually, however, the goal is not tostandardize the geometry of a chain butto influence it in a self-similar fashion.Therein lies the utility of the secondparameter, UC.

The results of varying the number ofiterations of both fractalizing andsmoothing is demonstrated in Fig. 4. In itthe same chain used in Fig. 3 is subjected tocombinations of zero to three fractali-zations and zero to two smoothings. Ineach case, SD - 1.7, UC = 1.0, SM = 0.0,and ST = infinity. The effect of addi-tional iterations generating smaller-scaledetails is evident for fractalization, whilefor smoothing, additional i terationssoften detail without really eliminating it.The Uniformity Coefficient

Should extremely regular chains not bedesired, UC can he used to prevent their

27

The American Cartographer11

0

Fig. 5. Effects of uniformity (UC) and straightness (ST) on enhancement.

formation. This parameter specifies thedegree to which junctions may vary fromwhatever angle SD specifies. When UC is1 (its maximum), the dimensionality of afractalized chain is held constant at SDthroughout its length. When UC is zero,the appearance of chains will be unaf-fected by fractalizing them, althoughthey will gain extra coordinates atsegment midpoints. Given any value ofUC intermediate between 0 and 1, thealgorithm will displace vertices towardtheir "standardized" positions through adistance proportional to UC. That is, ifUC equals 0.5, a vertex will not bedisplaced to the location giving it a localdimensionality of SD but will be moved

only halfway towards that point. Thisallows SD to influence the dimension ofchains without fully standardizing them.

UC also may be set to less than zero,but not less than -1. When negative val-ues are used, their effect is to exaggeratejunctions; rather than being displacedtoward a specific angle, they are dis-placed in the opposite direction. Thedifferent effects of the sign of UC areillustrated in Figs. 5C and 5D. With nega-tive values of UC, the farther from thestandardized location a vertex initially is,the farther it will be displaced away fromit. Negative displacements can cause ajunction to reverse direction, should itsdimension initially be less than SD.

Vol. 8, No. 1, April1981

7

Another possibility, depending on localgeometry, is that of introducing nonpla-narities (crossing segments) into chains.This is a particular hazard in the vicinityof peninsulas, fjords, isthmuses, andother narrow features. Many such non-planarities, however, may be deleted bysubsequent application of smoothing,which will be discussed below.

Straightness andSmoothing Tolerances

Due to the fact that the geometriccharacter of a chain may vary considera-bly along its extent, as does the borderbetween Texas and Oklahoma, it is im-portant to be able to preserve "straight"features while modifying "curved" ones.Without such control, enhancementmight introduce major shape alterationswhere only minor ones were desired. Fig.5B illustrates the perils of foregoing suchcontrol in the case of Rhode Island. The "straightness" parameter (ST) specifies themaximum length of a segment allowedto be modified. It therefore has units oflength that are in the coordinate metric ofchains being processed. Long segmentswill be passed through without changewhen their extent is greater than the valuespecified for ST. This means that, forenhancement to affect a vertex, bothsegments that meet there must be less thanST units in length.

In a similar but inverse fashion, the "Smoothness Tolerance" specifies thesmallest segment allowed to be modified.SM determines the fineness of detailproduced by fractalizing anywhere along achain. Detail may also be limited bystopping the procedure after a fixednumber of iterations. Unlike ST, how-ever, SM can and should be altereddepending on the scale at which chainsare to be displayed. Together, then, STand SM define the upper and lower limitsfor the size of features subject to en-hancement and thus constrain the overallamount of added detail.

One application of the procedure willnominally double the number of seg-ments in each chain processed. Certain

vertices may not be transformed, how-ever, given the two tolerance criteria, STand SM. Should SM be set relativelyclose to ST, few vertices may be altered.As ST is increased, however, longersegments become candidates for altera-tion, yielding larger geometric changesand additional coordinates. As SM isreduced, more detail will be introduced,as shorter segments will be split.

Fig. 6 shows the sequence of operationsused to fractalize the coastline of New-foundland. The original digitization, 6A,having 158 points, is quite generalized,but when analyzed using Richardson'smethod (Mandelbrot, 1977, p. 32), it isfound to have a fractal dimensionality of1.22.3 This chain was first fractalizedusing SD = 1.3 and UC = 0.7. SM was setto roughly 1 percent of the width of themap window, and ST was set very large soas not to affect the results. This producedthe chain in 6B, which has 302 points.About 10 percent of these vertices wereeliminated by weeding out all segmentssmaller than SM, yielding the chainshown in 6C; this was then smoothedonce, raising the number of points to 534.The final result is shown in 6D and has ameasured dimensionality of 1.22.Smoothness

As stated earlier, the visual quality ofchains that have been fractalized may berather jagged, the degree of whichdepends on SD and SM. Before plottingchains, it is thus usually necessary tosmooth them, which does not appreciablyalter their dimensionality. The methodused is a form of splining, similar to thetechnique developed by Chaikin (1974,with comments by Riesenfeld, 1975).Both are fast methods, but the author's isa bit slower due to its capability to splineto different curvatures.

The smoothing procedure is illustratedin Fig. 7. During smoothing, each vertexis literally "snipped off" the chain some-where between the vertex and the mid-points of the two segments that join at thevertex. In Fig. 7, either one, two, orfour iterations of smoothing were used,each with Roundness set at .25, .50, and

The American Cartographer11

0

Fig. 6. Stages in enhancing a rough outline of Newfoundland.

.75. SM and ST were set so as not toaffect the results. Chaikin's algorithm wasdeveloped to rasterize smooth versions ofdigitized lines; the author's versionsimilarly can produce a series of adjacentraster coordinates if SM is set to a suitablysmall value and the procedure is calledrecursively.Illustrated Examples

To demonstrate fractalization in afamiliar context, a file of the contiguous U.S. state boundaries and its enhancementsare presented in Figs. 8 through 18. Thesefigures are reproduced at four graphicscales, roughly varying from 1:30,000,000to 1:3,000,000, with views zooming fromthe entire file to its north

west corner, the Puget Sound region.Source data for these figures consist of

state boundaries extracted from theCounty DIME File, distributed by theU.S. Bureau of the Census. This extractcontains 11,541 points, but its resolutionvaries due to inconsistencies in the sourcemaps originally used in digitizing. Tostandardize its resolution and to generate asuitably sparse representation withwhich to test enhancement, the coordi-nates were converted to miles on anAlber's equal-area projection, and theseconverted coordinates were then filtered,using the Douglas-Peucker Bandwidth-Tolerance Algorithm (Douglas andPeucker, 1973), at a bandwidth of 2 mi.This eliminated over 90 percent of the

Vol. 9, No. 1, April 1981 33

Fig. 7. Smoothing to different curvatures: the effect of varying Roundness for 1, 2, and 4smoothings.

original detail, resulting in a U.S. stateboundary file having 1,055 points. Asdrastic as this reduction is, when thefiltered file (Fig. 9) is compared to theoriginal (Fig. 8), only a few boundariesappear noticeably degraded.

Under magnification, however, theeffects of filtering are quite apparent, asFigs. 12, 14, and 16 reveal, especially incomparison with Fig. 18, a 10x -magnifica-tion of the unfiltered boundaries. Islands

disappear, peninsulas become triangles,and sinuous channels and coasts simplystraighten. It is these faint suggestions ofshape that fractal enhancement thenattempts to rejuvenate in the remainingfigures.

The enhancements are displayed inFigs. 10, 11, 13, 15, and 17, with increas-ing scale. Figs. 10 and 13 display oneiteration of fractalization; Figs. 11 and 15,two iterations; and Fig. 17, three itera

31

10 The American Cartographer

Fig. 8. State boundaries extracted from U.S. County DIME File: full detail of 11,541points.

Fig. 9. Filtered state boundaries: 1,055 points.

Vol. 11, No. 1, April1981

33

Fig. 10. One fractalization and smoothing of state boundaries: 3,193 points.

Fig. 11. Two fractalizations and one smoothing of state boundaries: 6,042 points.

12 The American Cartographer

Fig. 12. 2x zoom of Fig. 9 (filtered boundaries).tions. All are then given one iteration ofsmoothing. The parameters employedwere held constant: SD = 1.5; UC = 0.5;SM = 0.1 mi.; ST = 50 mi.; Roundness = 0.5. The value of 1.5 for SD (imposed fractaldimensionality) is probably excessive,even though UC at 0.5 tempers its effectsconsiderably. This tends to exag-gerate the gross features still preserved inthe filtered file but clearlydemonstrates the results ofenhancement. Fig. 18, which containsthe full DIME File detail, provides thestandard against which enhancementsmay be evaluated.Precedents, Precautions,Related Applications

The notion of reducing the number ofcoordinates in a chain is generally under-stood to be useful and appropriate instoring and plotting digital maps. Veryfew cartographers would argue that allcoordinates acquired through digitizationare necessarily accurate or should be usedin plotting. Fine detail often becomesredundant or distracting, given certain

map scales, purposes, and plotting resolu-tions, and may be eliminated by a variety ofline culling techniques. But whenmapmakers or map readers are presentedwith algorithms that add or displace coor-dinates, eyebrows rise. Somehow thenotion of creating detail seems arbitrary,inappropriate, or untruthful. In carelesshands, it may be argued, such algorithmscan yield maps conveying a false sense ofreality.

But any map is an abstraction in whichphenomena are selected, generalized,stylized, and emphasized by the map-maker. A map is not even an abstractionfrom reality, it is an abstraction ofideas about reality (Robinson and Sale,1969; Morrison, 1975). Both qualitativeand quantitative rules of thumb areemployed by cartographers in choosingwhich features to include, where to placethem in relation to one another, whatsymbols to employ, and how large eachshould be. While their choices may behighly informed, and a consistent set ofcriteria can be inferred from a well-conceived

Vol. 8, No. 1, April1981

Fig. 13. 2x zoom of Fig. 10 (first enhancement).map, rarely can a mapmaker or a mapreader formalize the criteria into rigoroussets of rules, i.e., algorithms.

Sometimes cartographers must guessintelligently where to place features on amap, for example, when interpolatingisolines. Elevation control points may betoo sparse to define features known toexist by the cartographer, such as ridgelines and course lines. Mapmakers nev-ertheless bend contours "uphill" whencrossing courses and "downhill" whencrossing ridges. Similarly, and even whencontrol points are. fairly dense, the "best"track for a contour line crossing an area ofvery low relief can become quite conjec-tural. Small errors in spot elevations cantranslate into quite visible horizontaldisplacements of contours, making theirinterpolation open to question.

Not only derivative map elements likecontour lines but visible features as wellmust sometimes be approximated. Forexample, in tracing a stream from anaerial photograph, one may encounterdifficulty at locations where vegetation

obscures the stream channel. In such casesthe cartographer must rely on clues, such aschanges in topography and vegetationtype, to estimate best the course line.Although the technique for linegeneralization presented in this paper doesnot attempt to infer shapes fromcontextual data, it should be evident thatsynthesis of detail is not foreign to car-tography. It is both held to be legitimateand practiced widely, but it should bedone with restraint and only when appro-priate.

The ability to manipulate the fractaldimensionality of cartographic objects isperhaps more useful for thematic map-ping than for other cartographic applica-tions. Certainly maps used for navigationor for displaying boundary surveys mustrespect actual measurements as closely aspossible or practicable. For such purposesthere is no substitute for careful survey-ing, drafting, and digitizing of features, andthe algorithmic addition and deletion ofcoordinate data is done at some peril.But thematic maps, in which boundary

35

Fig. 14. 4x zoom of Fig. 9 (filtered boundaries).

data are principally a matrix for commu-nicating other variables, can be enhancedwithout great risk of conveying falseinformation. In such maps it is not thelocation of boundaries that is of primeconcern but their appearance.

Although the author is not aware ofattempts elsewhere to enhance carto-graphic lines using fractal methods, therehave been some uses reported whichsynthesize and enhance digital terrainmodels (Carpenter, 1981; Fournier andFussell, 1981). Both of these approachesinterpolate cells in elevation grids usingstochastic neighborhood operators. Suchsynthesis of surfaces will no doubt beeagerly exploited in flight and other envi-ronmental simulation applications; not onlycan they drastically reduce the bulk of aterrain data base required for generatingrealistic images, but such enhancementalgorithms can be easily built intohardware, aiding real-time response.Fractal surface enhancement may alsoprove useful in generating specific tex-tures for analytical hill shading.

Concluding CommentsThere is no one "correct" or "best"

method of fractal enhancement. As statedabove, the approach presented here is butone of about half a dozen trial algorithms,and others have been outlined. Althoughits mechanics are inherently rigid, itsparameters provide a good measure ofcontrol over its results. In any case, itssensitivity to local conditions could beconsiderably improved upon. That is,rather than imposing an often inappro-priate fractal dimensionality at each pointalong each chain in an entire file, localdimensionality should be allowed to retainits variability.

To achieve this, SD must become a variable,although useful lower and upper limitsmight be imposed. Likewise, UC can alsobe locally determined. Local dimensionality,however, has no specific value; it will varyaccording to the extent of the localityused to calculate it and is dependent onthe consistency, resolution, andconditioning of coordinate data. But

The American Cartographer36

Vol. 8, No. 1, April1981

Fig. 15. 4x zoom of Fig. 11 (second enhancement).

Fig. 16. IOx zoom of Fig. 9 (filtered boundaries).

37

16

The American

Cartographer

Fig. 17. lOx zoom of third enhancement (3 fractalizations, l smoothing: c. 9,000 points).

Fig. 18, lOx zoom of original, unfiltered boundaries.

Vol- 8, No. 1, April1981

without sensitivity to changes in theirstructure, as Fig. 1 illustrates, enhance-ment of boundaries may homogenize aswell as elaborate their shape.

Computer-assisted interactive the-matic mapping is aided in two ways byfractalizing techniques. First, mapmakerscan have easily-specified ways to controlthe amount and kind of boundary detailto be displayed. This is particularlyuseful when a portion of a map is to beenlarged; the limited detail of the bound-ary file may be revealed, and withoutenhancement the boundaries would lookcrude, distracting from the thematicmessage of the map. Second, the tech-nique of generating self-similar detail letsthe enhancement be performed in aconsistent way and lessens the need for alarge, highly detailed data base (and theexpense of storing and processing it) justfor the few occasions when greateramounts of detail may be needed. Andgoing beyond cartographic display, frac-tal techniques may also enhance spatialanalysis methods; the abilities to measureand manipulate the dimensionality ofchains may prove to facilitate the analyticoverlay of different coverages and theautomatic recognition of features andclasses of features.ACKNOWLEDGMENTS

This paper is a revised version of one presented atthe CG'80 Conference, Brighton, U.K., in August1980 (sponsored by Online Conferences, Ltd.) andappearing in its Proceedings, pp. 371-381. The authoris indebted to Benoit Mandelbrot, NicholasChrisman, and Denis White for many usefulcomments on the manuscript of this paper, and toJames Dougenik for his assistance with some of thealgorithms and code for chain processing. Thanks isalso due to Mary Phelan, Robert Pierce, and Doro-thy Shamonsky for their assistance in preparing themanuscript and illustrations for publication. Copiesof the Fortran routines that generated the figures inthe text may be obtained from the author.

REFERENCES1. Fractional dimensions, or fractals for short,

represent an elegant way of dealing with manygeometric "monstrosities," such as Peano curves,Koch constructions, and Cantor sets. Such mathe-matical constructions have frustrated generations of

mathematicians who tried to subject them to realanalysis. Although continuous, such curves are notfunctions, and are locally-and in the limit globally-undifferentiable. In his essay Mandelbrotconstructs families of self-similar curves and derivestheir fractional dimensionality with extraordinarilysimple techniques. In his figures, objects with nearlythe same dimensionality exhibit similar profilecomplexity, even though their shapes may differ.

2. Throughout this paper, the term chainwill be used to denote such strings ofcoordinates describing cartographic lines. The morefamiliar word, polygon, could he used when thedigitized feature is a closed loop, but the term is lessgeneral. The only restriction on chains to beobserved are (a) a chain must contain at least twoline segments, i.e., be composed of at least threecoordinate pairs, and (b) chains should intersect oneanother or themselves only at their end-points. Whiletwo-point chains are perfectly legitimaterepresentations of boundaries, one must assume thattheir straightness is intentional, and in any case, twopoints provide insufficient data for any enhancementto be performed.

3. This approach to computing fractional dimen-sionality (D) is statistical. It is performed by parti-tioning a chain into segments of equal length. Theseare counted to derive the total length of the "regu-larized" chain. The longer the unit segments are, theshorter will be the distance along the regularizedchain. By repeating this procedure stepping through arange of segmentations, a regular, inverse relationshipbetween segment length and chain lengthemerges. The least-squares slope of log (chain length)as a function of log (segment length) isapproximately equal to €-D. This is a way toformalize analysis of the "measuring rod" phenome-non illustrated in Fig. 2.

Boots, B. (1972), Working Notes and Bibliographyon the Study of Shape in Human Geography andPlanning, Exch. Bibliog. 346, Council of PlanningLibraries, Monticello, ill.

Boyce and Clark (1964), "The Concept of Shape inGeography," Geographical Review, Vol. 54, pp.561-572.

Bunge, W. (1966), Theoretical Geography, Gleerup,Lund, Sweden.

Carpenter, L. C. (1981), "Computer Rendering ofFractal Curves and Surfaces," Communications,A.C.M. (forthcoming).

Chaikin, G. M. (1974), "An Algorithm for HighspeedCurve Generation," Computer Graphics andImage Processing, Vol. 3, No. 4, pp. 346-349.

Douglas, D. H. and T. K. Peucker (1973), "Algorithmsfor the Reduction of the Number of PointsRequired to Represent a Digitized Line or ItsCaricature," The Canadian Cartographer, Vol. 10,Na. 2, pp. 112-122.

Fournier, A. and D. Fussell (1981), "StochasticModelling in Computer Graphics," Communica-tions, A.C.M. (forthcoming).

Jenks, G. F. (1980), "Thoughts on Line Generaliza-tion," Proceedings of the International Sympo

39

18

The American

Cartographer

sium on Cartography and Computing (AUTO-CARTO IV), ACSM and ASP, pp. 209-220.

Mandelbrot, B. B. (1977), Fractals: Form, Chance, andDimension, W. H. Freeman, San Francisco.

Morrison, J. L. (1975), "Map Generalization: Theo-ry, Practice and Economics," Proceedings of theInternational Symposium on Computer-AssistedCartography (AUTO-CARTO II), ACSM and U.S.Bureau of the Census, pp. 99-112.

Pavlidis, T. (1978), "A Review of Algorithms forShape Analysis," Computer Graphics and ImageProcessing, Vol. 7, pp. 243-25S.

Ramer, U. (1972), "An Iterative Procedure for thePolygonal Approximation of Plane Curves,"

Computer Graphics and Image Processing, Vol. 1,pp. 244-256.

Riesenfeld, R. F. (1975), "On Chaikin's Algorithm,"Computer Graphics and Image Processing, Vol. 4,No. 3, pp. 304-310.

Rhind, D. W. (1973), "Generalization and Realismwithin Automated Cartographic Systems," TheCanadian Cartographer, Vol. 10, No. 1, pp. 51-52.

Robinson, Arthur H. and Randall D. Sale (1969),Elements of Cartography, 3rd ed., John Wiley &Sons, New York, p. 52.

Stoddart, D. R. (1965), "The Shape of Atolls,"Marine Geology, Vol. 3, pp. 369-372. ∎