an optimized rubber-sheet algorithm for continuous area...

An Optimized Rubber-Sheet Algorithm for Continuous Area

Cartograms∗

Shipeng SunUniversity of Waterloo

This article optimizes a continuous area cartogram algorithm published in The Professional Geographer byDougenik, Chrisman, and Niemeyer (DCN) in 1985. The DCN algorithm simulates a rubber sheet andis an iterative and approximate solution of cartogram construction. Although it remains popular because ofits conceptual simplicity and cartographic quality, the DCN algorithm cannot completely preserve topologyand its mathematical properties are inadequately explained. This article presents an optimization to theDCN algorithm, named Opti-DCN, with three improvements. First, it provides a mathematical conditionfor topology preservation. Second, new transformation equations that meet this condition are deduced frommathematics, which simultaneously optimize the global elasticity coefficient, a key parameter that greatlyimpacts the convergence rate of the rubber-sheet algorithm and the topological integrity of its generatedcartograms. Last, the new algorithm simplifies the way of generating transforming forces in DCN andimproves its efficiency of geometric transformation. Comparison shows that Opti-DCN is significantly fasterto converge to equal-density cartograms and can mathematically and practically eliminate topological errors.Key Words: algorithm optimization, continuous area cartogram, rubber-sheet algorithm.

En este artıculo se optimiza un algoritmo de cartograma de area continua publicado en The ProfessionalGeographer por Dougenik, Chrisman y Niemeyer (DCN) en 1985. El algoritmo de DCN simula una laminade caucho y es una solucion iterativa y aproximada para la construccion de cartogramas. Aunque sigue siendopopular debido a su simplicidad conceptual y calidad cartografica, el algoritmo de DCN no puede preservartotalmente la topologıa y ademas sus propiedades matematicas se explican de manera inadecuada. El presenteartıculo presenta una optimizacion del algoritmo DCN, identificada con el nombre Opti-DCN, al cual sele introducen tres mejoras. Primero, suministra una condicion matematica para la preservacion topologica.Segundo, las nuevas ecuaciones de transformacion que satisfacen esa condicion se deducen de la matematica, locual simultaneamente optimiza el coeficiente de elasticidad global, parametro clave que impacta fuertementela tasa de convergencia del algoritmo lamina de caucho y la integridad topologica de los cartogramas quegenere. Por ultimo, el nuevo algoritmo simplifica la manera de generar fuerzas transformadoras en el DCNy mejora su eficiencia de transformacion geometrica. La comparacion permite ver que el Opti-DCN essignificativamente mas rapido para converger en cartogramas de igual densidad y puede eliminar errores

∗This work is supported in part by the National Aeronautics and Space Administration New Investigator Program in Earth-Sun System Science(NNX06AE85G) and the University of Minnesota. The author gratefully acknowledges the efforts of the editor and three anonymous reviewersfor their valuable comments and constructive suggestions. As always, the author takes full responsibility for any remaining errors or omissions.

The Professional Geographer, 65(1) 2013, pages 16–30 C© Copyright 2013 by Association of American Geographers.Initial submission, January 2011; revised submissions, April and June 2011; final acceptance, June 2011.

Published by Taylor & Francis Group, LLC.

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Optimized Rubber-Sheet Algorithm for Continuous Area Cartograms 17

topologicos de manera matematica y practica. Palabras clave: optimizacion algorıtmica, cartograma dearea continua, algoritmo de lamina de caucho.

C artograms are transformed maps in whichareas or distances represent values having

little or partial relevance to the original map ge-ometries. After more than forty years of devel-opment, geographers, along with researchers inother fields, have achieved remarkable break-throughs in computer cartograms (Tobler2004). Not only have new forms of cartograms,such as circular cartograms and value-by-alphamaps, been introduced but, more notice-ably, efficient cartogram algorithms havebeen developed and numerous cartogramshave been widely disseminated to the public(Dorling 1994, 1996; Roth, Woodruff, andJohnson 2002). Cartograms produced withthe Gastner–Newman algorithm, for instance,drew much attention in academia and in the2004 and 2008 U.S. presidential elections(Gastner and Newman 2004; Gastner, Shalizi,and Newman 2005; Webb 2006). A dedicatedWeb site (www.worldmapper.org) publishesnearly 700 high-quality cartograms using thisalgorithm (Dorling, Barford, and Newman2006; Dorling 2007).

Continuous area cartogram algorithms canbe technically classified into three categories:cartographic and computational geometrymethods, math-physical process simulation,and artificial intelligence. Despite advance-ments evidenced by a variety of cartogramalgorithms, they have various limitationsin efficiency, accuracy, and conceptual andcomputational simplicity, as well as carto-graphic and aesthetic qualities. For instance,the ultimate, absolute equal-density mapproduced by the Gastner–Newman diffusionalgorithm would resemble circles in theory.This “ballooning” effect, however, negativelyimpacts map readers perceiving cartograms(Kocmoud 1997; House and Kocmoud 1998;H. Sun and Li 2010). It is therefore necessaryto pursue new cartogram algorithms and toimprove existing ones to create cartogramshaving different and better features.

This article reexamines one of the earliestcomputer algorithms for continuous areacartograms developed by Dougenik in the late1970s and published by Dougenik, Chrisman,and Niemeyer (DCN) later in The Profes-sional Geographer (Dougenik, Chrisman, and

Niemeyer 1985). This DCN algorithm simu-lates a rubber sheet and gradually transforms amap to equal-density status. It calculates forcevectors pointing from polygon vertices to theirdesired positions and generates a vector fieldto transform space. Although the DCN hasdifficulty in transforming complex maps com-posed of polygons with diverse shapes, sizes,and statistical values, it retains much popularitydue to its simplicity and ability to preserveshapes (Tobler 2004; Henriques, BaCao, andLobo 2009). Two of the three continuous areacartogram programs written in ArcscriptsTM

and publicly available on the Internet use theDCN algorithm, the other being Gastner andNewman’s diffusion algorithm. That said, theDCN algorithm cannot guarantee topologicalintegrity and its mathematical propertiesare not well explained and thus not wellutilized to optimize the algorithm (Gastnerand Newman 2004; Tobler 2004). This articleoptimizes the DCN algorithm by providinga more formal mathematical foundation,adopting a new and uniform force generationequation, and proposing a new mathematicallydeduced parameter—the global elasticitycoefficient—that rescales the transformingforces in a way that ensures topologicalintegrity and optimizes convergence rate.The purpose of the optimized rubber-sheetalgorithm (named Opti-DCN) is to make theDCN mathematically rigorous, algorithmicallymore efficient, and possibly open for furtherimprovement. To simplify the description, inthis article, cartograms are continuous area car-tograms unless otherwise noted, and the statis-tical values used to determine the desired sizesof areas in cartograms are called the population.

The rest of the article is organized asfollows. The next section briefly reviewsmajor cartogram construction algorithmsdeveloped in the last forty years. I then lay outa mathematical foundation for the Opti-DCNalgorithm and propose new equations andendogenous parameter-setting mechanisms.After that, I present the U.S. continental popu-lation cartograms produced by the Opti-DCNalgorithm and compare them with the originalrubber-sheet algorithm. The article concludeswith discussion of the value of the optimized

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18 Volume 65, Number 1, February 2013

algorithm and its mathematical founda-tion, as well as some directions for furtherimprovement.

Continuous Area Cartogram

Algorithms

From a technical viewpoint, the nearly fiftyyears of continuous area cartogram algo-rithm development roughly diverged intothree methodological families: cartographicand computational geometry, math-physicalprocess simulation, and artificial intelligence.Although each family of methods has its uniquecharacteristics and advantages, all algorithmscome with certain limitations. This article onlybriefly summarizes some features of these threecategories. For more general and detailed re-views on cartogram and related algorithms,see Dorling (1996), Kocmoud (1997), Skupinand Fabrikant (2003), Tobler (2004), and Hen-riques, BaCao, and Lobo (2009).

As its name implies, a cartogram is initiallyconceptualized and created by cartographers(Friis 1974). One of the earliest examples isa rectangular cartogram developed by Raisz(1934, 1936). In the early stage of cartogram de-velopment, cartographers and artists manuallyplotted “equal-density” maps or “perceptual”maps based on cartographic principles. Withthe availability of new information technology,especially geographic information systems,computerized programs captured the coreprocedure of these manual cartogram creationmethods and made new improvements basedon computational geometry (to continue thecase of rectangular cartograms, see Heilmannet al. 2004; Keim, North, and Panse 2004;Florisson, van Kreveld, and Speckmann 2005;van Kreveld and Speckmann 2007). In recentyears, researchers have utilized more generaland more advanced computational geometryconcepts and measures, such as medial axis andtriangulation, to advance cartogram algorithms(House and Kocmoud 1998; Keim, Panse, andNorth 2005; Inoue and Shimizu 2006).

Simulating mathematical and physical pro-cesses facilitated developing the earliest andlately some of the best computer cartograms.Preserving topology or, in mathematical terms,maintaining topological equivalence betweenthe original and transformed spaces, is one fun-

damental requirement for a cartogram algo-rithm. In mathematical studies of topology, arubber sheet is most often used to illustrate ab-stract concepts and the discipline of topologyis therefore informally referred to as “rubber-sheet geometry” (Johnson and Glenn 1960;Sauvy and Sauvy 1974). This metaphor alsohelped develop cartogram algorithms. As a pi-oneer on computer cartograms, Tobler devel-oped rubber-map and pseudo-cartograms bysimulating rubber sheet on a regular grid (To-bler 1973, 1986). The DCN algorithm alsosimulates similar systems, where each vertexgenerates forces with distance decay. Theseforces form a vector field and transform thespace. This type of process usually cannotachieve the ideal shapes with single iteration;instead, it requires multiple steps to approach anear-equilibrium status, in which forces at anylocations are no longer large enough to mobi-lize any points.

In addition to these approximate force-fieldmethods, more complicated yet mathemat-ically accurate algorithms are developed forcartograms. Gusein-Zade and Tikunov (1993),for example, used line integral to deduce themathematical equation for cartogram trans-formation. Their method divides the spaceinto small surface elements and assigns themnew sizes according to the population. Afterintegrating these changes from one side to theother, the equation can calculate all coordinatesin one step. Besides pure mathematics, intu-itive physical processes can also help producecartograms. Diffusion is one such physicalprocess. In an ideal diffusion, fluid movesfrom high-density to low-density areas until itreaches an equal-density status, which is exactlythe goal of cartograms. Diffusion theory hasbeen well studied in physics and its applicationto cartograms turns out to be a success in termsof achieving desired areas and preserving topol-ogy (Gastner and Newman 2004; Gastner,Shalizi, and Newman 2005; Dorling 2007).

Artificial intelligence is another direction ofcartogram algorithm development. Cellular au-tomata (CA) and self-organizing maps (SOMs)have been applied to generate cartograms. Dor-ling (1996) used a CA machine to make car-tograms. The space is divided into a regular gridand each grid cell adjusts itself to approach thedesired size without violating topological re-lationships with neighbors. Accompanying the

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advancement of general artificial intelligencetechnologies, new methods are developed forcartogram applications, of which SOMs, a neu-ral network technology, are the most notice-able (Henriques, BaCao, and Lobo 2009). SOMessentially maps an input space to an outputspace and preserves “topology” because closeentities are kept close throughout the mappingprocess. For cartograms, SOM uses points in aregular grid as input. The training set is com-posed of points that are randomly generatedwithin each polygon according to its density.SOM trains the input points so that they mimicthe density-controlled points. Through a re-verse labeling process, these input points buildthe skeleton of the cartogram, with which anypoints in the space can be interpolated.

As many researchers in this field haveobserved, there is no “silver bullet” solutionfor cartograms to date. It seems that no onealgorithm is universally better than others(Henriques, BaCao, and Lobo 2009). Thecartogram algorithms already discussed haveadvantages and disadvantages in simplicity,efficiency, effectiveness, cartographic quality,topology preservation, shape preservation,and many other aspects (Kocmoud 1997;Henriques, BaCao, and Lobo 2009). Althoughdeveloping new algorithms is critically impor-tant, improving existing ones is also beneficial,not only to cartograms but to relevant spatio-temporal transformation and geovisualizationresearch as well (e.g., Ahmed and Miller 2007).The DCN algorithm is easy to understand,simple to implement, and can produce qualityand readable cartograms. Improving the effi-ciency of the DCN algorithm and eliminatingits problem of topological errors would greatlyexpand its applicability to cartograms and togeovisualization as well.

Opti-DCN Algorithm

Mathematical definitions of the cartogramproblem have been provided by previousstudies (for recent ones, see Henriques,BaCao, and Lobo 2009; Lekien and Leonard2009; H. Sun and Li 2010). Unlike thosedefinitions where the basic transformationunits are polygons, the DCN algorithm canactually be defined by point displacements.Define a point set of vertices of all originalpolygons So

pnt = {Po

i (xoi , yo

i ), i = 1, 2, . . . , n}.

According to the relationship between thearea of a polygon and its population, definea point set for the transformed polygonsSt

pnt = {Pt

i (xti , yt

i ), i = 1, 2, . . . , n}, where Pt

i

is the new position of the original point Poi ,

so that Area(Polygontm) = Populationm

Density . The DCNalgorithm is essential to find a mathematicaltransformation T : So

pnt → Stpnt.

Like many multidimensional data visualiza-tion algorithms, the DCN algorithm simulatesa system of springs that exert forces generatedby the vertices of polygons to transform a rub-ber sheet. In the original DCN algorithm, theforces are calculated using the coordinates ofpolygon vertices and their new coordinates inan assumed situation where polygons are in-dependently and linearly transformed accord-ing to their population. The magnitudes of theforces are specified using a continuous, differ-entiable function with distance decay and theirdirections are parallel to the line from the poly-gon centroids to the vertices. In each step, theforces mobilize points in the space toward de-sired locations. As points gradually approachtheir new position, the forces become weakerand will finally reach an insignificant level. Inthat status, the transformed areas will be veryclose to desired sizes. In the whole process, aforce reduction factor is calculated using thesize error to make vertices move toward theright direction and to avoid polygon overlap-ping possibly caused by “too strong forces”(Dougenik, Chrisman, and Niemeyer 1985).

The optimization presented here keeps theiterative logic of DCN but extends this rubber-sheet algorithm in three aspects. First, withsuch a mathematical definition of the problem,a condition that can preserve topology, namely,avoid any line intersecting and polygon over-lapping, is proposed. Second, a set of force gen-eration functions that satisfy the condition isdesigned. Finally, using these functions, a sim-plification to the original DCN force genera-tion mechanism is proposed, which improvesthe efficiency of the DCN algorithm and re-tains its effectiveness.

Preserving TopologyOne of the major concerns in cartograms ispreserving topology, which is also an unre-solved problem of the DCN algorithm, es-pecially in the mathematical sense (Kocmoud

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1997; Tobler 2004). A good cartogram algo-rithm should be able to prevent line segmentsfrom intersecting and make polygons free fromoverlapping. In the viewpoint of topology,the key to preserving topological relation-ships is keeping vertices in order. Math-ematically, bicontinuous transformation canguarantee topological equivalence; that is, bi-continuity is a sufficient condition for topo-logical equivalence (Sauvy and Sauvy 1974;Binmore 1980). It is easy to design a contin-uous transformation that smoothly maps onespace to another as the original DCN algo-rithm does; however, it is difficult to guaran-tee that the reverse transformation exists andis continuous, too; that is, make the continu-ous transformation also bicontinuous. The coreof cartogram transformation, therefore, is tochoose or design a continuous transformationfunction with an inverse function that existsand is also continuous. Monotonically increas-ing or decreasing continuous functions musthave inverse functions. A continuous compos-ite function composed of differentiable ele-mentary functions like arithmetic, exponential,logarithm, power, and four elementary oper-ations must be continuous also if the inversefunction does exist.

Based on this mathematical reasoning, wecould design a continuous, differentiable trans-formation function using arithmetic, exponen-tial, logarithm, power, and other elementaryfunctions. As long as the inverse of this func-tion exists, it should be continuous, too. Thistransformation, therefore, would be bicontin-uous and could preserve topology. Because amonotonically increasing continuous functionmust have inverse function, a transformationfunction that is continuous and monotonicallyincreasing would certainly lead to bicontinuityand then to topological equivalence. As a result,the essential function of designing cartogramalgorithms is to find such continuous, differen-tiable, and invertible transformation functions.

To simplify the discussion, a one-dimensional case is used to explain theproblem (Figure 1). ∀xo

i , xoj ∈ So , So ⊆ �1, and

xoi < xo

j define a continuous, differentiabletransformation T : So → St, St ∈ �1 andT(xo

i ) = xti , T(xo

j ) = xtj , where o and t denote

the original and transformed spaces, respec-tively. Then xt

i < xtj ⇒T is monotonically

increasing ⇔ T′ > 0 ⇒ T−1 exists ⇒T is

bi-continuous, and ⇒ So and St are topolog-ically equivalent (for a formal definition andproof of this proposition, see Binmore 1980).This implies that, in a one-dimensional space,topology will be preserved if the order of anypair of the transformed points is the sameas that of the original points (Figure 1A). Inother words, a continuous and monotonicallyincreasing function must have an inverse func-tion. If it is a composite function composedof elementary functions, its inverse functionmust be continuous, too. This bicontinuousfunction could then guarantee topologicalequivalence. If the transformation functionis not monotonically increasing, at least twopoints in the original space would be mappedto the same point in the transformed space,which implies that the space “collapses” at thatpoint, thus violating topological equivalence(Figure 1B).

Extend this formula to a two-dimensionalspace. Define a continuous, differentiabletransformation T : So → St, So ∈ �2, St ∈ �2.It can be written in matrix format as( xt

y t

)= T

( xy

)=(Tx(x, y)

Ty (x, y)

). Its Jacobian ma-

trix is JT =(

∂Tx∂x

∂Tx∂y

∂Ty∂x

∂Ty∂y

). It has been pointed out

by many researchers that the determinant of theJacobian matrix, det JT = ∂Tx

∂x · ∂Ty∂y − ∂Tx

∂y · ∂Ty∂x ,

should be equal to the ratio of global averagedensity to local density for the purpose ofproducing ideal cartograms (Gusein-Zade andTikunov 1993; Gastner and Newman 2004;Lekien and Leonard 2009). The relationshipbetween the Jacobian and topological equiva-lence, however, has never been discussed in de-tail for cartogram algorithms. According to theinverse function theorem, if the determinant ofa Jacobian matrix, det JT , is nonzero, then thetransformation T has a local inverse functionand therefore is locally invertible. If the deter-minant is greater than zero, the transformedspace also has the same orientation as the orig-inal one (Nijenhuis 1974; Binmore 1980). So,for such a continuous, differentiable transfor-mation T, det JT > 0 ⇒ T−1exists locally ⇒ Sand T(S) are topological equivalents. This suffi-cient condition literally means, to any small re-gion around point Po (x, y) in the original spaceS, after such a transformation, the region will becontinuously projected to another region that

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Figure 1 Monotonic increase, inverse function, and topological equivalence in one-dimensional space.(A) Topological equivalence. (B) Point overlapping.

has the same orientation. Because the projec-tion is invertible, any point in the transformedregion can also be continuously projected backto the original region. This implies a bicon-tinuous transformation and thus guaranteestopological equivalence. Although this con-dition only applies to local regions, it is stilleffective when points are evenly distributed inthe space.

It must be noted, however, that even withsuch a sufficient condition, the discrete space ofvector polygons cannot completely preserve themathematical property of topological equiva-lence as in a continuous space. For example, ina continuous space, the direct line of DE would

become a curve to avoid overlapping with ABand BC; in the discrete space, however, DE willintersect with AB and BC (Figure 2A). Appro-priate densification of vertices on polygon arcs,therefore, is necessary to preserve topology in adiscrete space. If there was a point at F, for in-stance, the topology would be preserved underthe same transformation condition (Figure 2B).

Previous implementation of the DCNalgorithm mentioned the value of densificationfor avoiding polygon overlapping, althoughwithout much explanation (Du and Liu 1999).In fact, much topological error producedby the DCN can be effectively eliminatedafter inserting more points on those long

Figure 2 The value of point densification in topology preservation. (A) Broken topology. (B) Preservedtopology.

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polygons arcs. Point densification in DCNand Opti-DCN can be done by imposing anupper limit on the length of the line segmentbetween any adjacent polygon vertices. Ifthe distance between two adjacent vertices isgreater than a maximum length, a new pointwould be inserted between them. In general,one two-hundredth of the map width or height,whichever is lower, is small enough for sucha maximum length. In some rare cases whereneighboring polygon vertices experience ex-ceptionally strong forces in opposite directions,this maximum value might need to be set evensmaller to produce a smoother transformation.

Force Generation FunctionWith this condition for topology preservation,the key optimization to the DCN algorithm isto design and calibrate a force generation equa-tion that can meet such a condition. Essentially,it is to find a continuous function with a differ-ential or determinant of its Jacobian matrix thatis greater than zero. Before doing this, I first layout the input and output of the force generationfunction.

To generalize the problem, a set of force-generating point pairs is taken as algorithm in-puts. On a two-dimensional Cartesian surface,a set of points Po

i (xoi , yo

i ) is to be transformed toPt

i (xti , yt

i ), i = 1, 2, 3, . . . . Each pair of points

(Poi , Pt

i ) forms a vector⇀

Pi = Pti − Po

i and ex-erts forces on the surface, thus forming a vec-tor field. For a cartogram application, thesepairs of points could be the vertex points inthe polygons to be transformed; however, theycould also be other points as long as the vectorforce field generated by them can transform thespace. The optimized algorithm proposed hereessentially specifies the forces to achieve thefollowing: (1) T(Po

i ) is as close to Pti as possible,

and ideally T(Poi ) = Pt

i ; and (2) the T trans-formed polygons retain their topology; that is,T meets the condition specified in the previoussection.

I use two steps to specify a mathematical formfor the forces generated by such a point pair.First, using a one-dimensional case, I give somebasic properties and constraints of such forces.Then, I give some adjustment and extend it totwo-dimensional situation.

In a one-dimensional space, let −−−→Fk·i (x) bethe force generated by vector −→Pi = Pt

i − Poi at

Pk(x), where x is the coordinate of the point

k. It is obvious that the magnitude of the forceshould be proportional to the distance betweenpoints Po

i and Pti . So, define the magnitude of

the force generated by the i th point pair at pointk as

∥∥∥−−−→Fk·i (x)∥∥∥ =

∥∥∥−→Pi

∥∥∥ · f

⎛⎝∥∥Px − Po

i

∥∥∥∥∥−→Pi

∥∥∥⎞⎠

= dot · f(

do x

dot

)= dot · f (d ) (1)

where do x is the distance from the original pointPo

i to Pk(x), dot is the distance from Poi to Pt

i,and d is a standardized distance. The directionof the force is obviously the same as that of −→Pi .There are four requirements for the propertiesof f (d), the basic transforming force function(S. Sun and Manson 2007). First, f (0)=1; thatis, Po

i should be displaced to Pti after applying

this force. Second, f ′(d ) exists on [0, +∞); thatis, the force is differentiable and thus continu-ous. Third, f ′(d ) < 0 when d > 0; that is, f (d) ismonotonically decreasing. This means distancedecay of the forces. Finally, T(d ) = d + f (d ) ismonotonically increasing; that is, T′(d ) > 0 andf ′(d ) > −1 when d > 0. This requirement isthe equivalent of the condition specified earlier,which ensures topological integrity because ev-ery point being transformed will be kept in theoriginal order. Examples of such functions in-clude f (d ) = e−d and f (d ) = 2

ed +e−d . These twofunctions reveal little difference. For conve-nience of discussion and calculation, the simpleexponential form is used for optimization.

Now, extend the situation to two-dimensional with a single pair of originand destination points. The force exerted bythe ith point pair {Po

i (xi ·o , yi ·o ) → Pti (xi ·t, yi ·t)}

at any point Pk(x, y) in the original space is

−−−−−→Fk·i (x, y) = Fk·i ·x(x, y) · −→μx + Fk·i ·y (x, y) · −→μy ,

where Fk·i ·x and Fk·i ·y are forces on the X and Yaxis, respectively, and −→μx and −→μy are unit vectors(Figure 3). Then

Fk·i ·x(x, y) = di ·o t · f(

di ·o x

di ·o t

)· cos

(−−−→Po

i P ti

)

= di ·o t · f(

di ·o x

di ·o t

)·(

xi ·t − xi ·odi ·o t

)

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Figure 3 Force field generated by onepoint pair Po → pt. (Color figure availableonline.)

= (xi ·t − xi ·o ) · f(

di ·o x

di ·o t

)

= (xi ·t − xi ·o ) · e− di ·o xdi ·o t . (2)

The summation of all forces at point Pk(x,y)along the X axis is

Fk·x(x, y) =n∑

i=1

Fk·i ·x(x, y)

=n∑

i=1

(xi ·t − xi ·o ) · e− di ·o xdi ·o t . (3)

Similarly,

Fk·y (x, y) =n∑

i=1

(yi ·t − yi ·o ) · e− di ·o xdi ·o t (4)

Then, define( xt

y t

)= T

(xy

)=(

Tx(x, y)Ty (x, y)

)=(

x + c k · Fk·x(x, y)y + c k · Fk·y (x, y)

), 0 < c k < 1, where c k is a

local elasticity coefficient at the point Pk(x, y).To meet the condition of topological equiva-lence specified earlier, we must find a value forc k that makes the determinant of the Jacobiangreater than zero; that is, det JT = ∂Tx

∂x · ∂Ty∂y −

∂Tx∂y · ∂Ty

∂x > 0. The four partial derivatives of T

are not difficult to calculate. For instance,

∂Tx

∂x= 1 + c k · ∂ Fk·x(x, y)

∂x

= 1 + c k ·n∑

i=1

(xi ·t − xi ·o ) · e− di ·o xdi ·o t · ∂di ·o x

∂x

= 1 + c k ·n∑

i=1

(xi ·t − xi ·o ) · e− di ·o xdi ·o t · x − xi ·o

di ·o x

(5)

So, det JT = ∂Tx∂x · ∂Ty

∂y − ∂Tx∂y · ∂Ty

∂x = (1 + c k ·∂ Fk·x∂x ) · (1 + c k · ∂ Fk·y

∂y ) − c k · ∂ Fk·x∂y · c k · ∂ Fk·y

∂x .Then this can be rewritten as a quadraticequation:

det JT =(

∂ Fk·x∂x

· ∂ Fk·y∂y

− ∂ Fk·x∂y

· ∂ Fk·y∂x

)· c 2

k

+(

∂ Fk·x∂x

+ ∂ Fk·y∂y

)· c k + 1 (6)

Simply solving equation det JT = ε > 0 witha constraint of 0 < c k < 1 can give a value ofc k at point Pk(x, y). If there is no solution forthe equation or c k is negative, make c k one.Experiments show that 0.05 is large enough forε to preserve topology.

Alternatively, the elasticity coefficient c k canbe calculated in a simpler but mathematicallyrelaxed way. Topologically, if the order of

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points along the curve with maximum distor-tion remains the same after cartogram transfor-mation, the transformed space would preservetopology. Similar to the one-dimensionalcases, as long as the transformation function Tis monotonically increasing along the steepestdirection (i.e., the gradient), T−1 will existlocally, and S and T(S) will be topologicallyequivalent. Although this condition is not asstrict as the Jacobian determinant, it also largelymaintains the topological integrity of areacartograms in most cases. To make sure T−1

exists along X, Y, and the direction of gradient,

c k =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1.

if∂ Fk·x(x, y)

∂x≥ 0 and

∂ Fk·y (x, y)∂y

≥ 0

(ε − 1)/min(

∂ Fk·x(x, y)∂x

,∂ Fk·y (x, y)

∂y

).

if∂ Fk·x(x, y)

∂x· ∂ Fk·y (x, y)

∂y< 0

(1−ε)/

(∂ Fk·x(x, y)

∂x

2

+ ∂ Fk·y (x, y)∂y

2)0.5

.

if∂ Fk·x(x, y)

∂x< 0 and

∂ Fk·y (x, y)∂y

< 0

(7)For any point to be transformed Pk(x, y), al-

though c k is not, in the strictest sense, the valuerequired for the existence of an inverse for theglobal transformation function, it is easy to cal-culate and actually reduces enough forces toretain topological integrity of cartograms whileallowing a near optimal convergence rate.

With such mathematical deductions, theoptimized force generation equation can belaid out as follows. Calculate the local elas-ticity coefficient c k at Pk(x, y) as in Equa-tion (6) or (7) and the global elasticity coef-ficient as c = min(c k), k = 1, 2, 3, . . . . Thenthe new coordinates of point Pk(x, y) inone single iteration are Pd

k (x, y) = Pk(x, y) +−−−−→Sk(x, y). −−−−→Sk(x, y) = −→Sk·x + −→Sk·y = c · Fk·x · −→μx +c · Fk·y · −→μy , where Fk·x , Fk·y are forces alongthe X, Y dimension at the point Pk(x, y). Or in

a matrix form,( xt

y t

)=(

x + c · Fk·x(x, y)y + c · Fk·y (x, y)

).

Simplified Force Generation MechanismIn the original DCN algorithm, the force-generating point pairs are generated usingpolygon centroids and vertices. When using

Figure 4 A simplified force generation mecha-nism. (Color figure available online.)

all vertices of polygons, levels of generaliza-tion can impact the forces calculated from thesame polygons. For example, without any otherchanges, adding more points to a specific arcin a polygon increases the forces around thatarc and causes different cartograms. Moreover,calculating forces generated by all vertices iscomputationally more costly than using sim-plified shapes. The application of Opti-DCNshows that using simplified shapes could be asefficient as using all vertices.

In the Opti-DCN algorithm, only circles andsquares are used to approximate polygons (Fig-ure 4). For each polygon, Opti-DCN creates acircle with the polygon centroid being its cen-ter and the polygon size being its area. A squareis then created around the circle. On the circle,sixteen points are evenly sampled. Accordingto the population of the polygon, the intendedcoordinates of these origin points on the circleand the square can be easily calculated. It isproved that such extreme simplification workswell for geometric shapes of U.S. continentalstates. Of course, such simplification wouldreduce the transforming forces because,compared with all polygon vertices, the simplecircles and squares generally have fewerforce-generating point pairs. An exaggerationmechanism therefore is applied to enlarge the

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Figure 5 Force exaggeration.

transformation and to recover the lost forces.Suppose the ratio between the desired and theoriginal radii of a circle is ρ = rd /ro . The ex-aggeration mechanism should, in a continuousmanner, make the exaggerated ratio ρe smallerthan ρ when ρ is less than one but make ρebigger when ρ is greater than one. An exampleof such exaggeration is ρe = ρ · [1 + 8·ln(ρ)

Ns tep :]

(Figure 5). The new radius of the desired circleis rd

e = ρe · ro . This mechanism essentially fur-ther shrinks the desired circles and squares thatshould shrink according to the population ofthe polygon being approximated; accordingly,it also further enlarges the circles and squaresthat should be enlarged. The exaggeratedcircles and squares therefore generate strongertransforming forces, recover some lost forcesdue to using fewer transforming point pairs,and help prevent Opti-DCN from losingalgorithmic convergence efficiency.

It should be noted that the number of sam-pled points on the circle and the specifica-tion of the exaggeration function are flexible.Changing the number of points sampled or theform of the force exaggeration formula wouldslightly change the convergence rate and theshape of the resultant cartogram, which couldhelp develop variants of the Opti-DCN algo-rithm. Nevertheless, one advantage of Opti-DCN is its ability to preserve topological re-lationships under various specifications as long

as the determinant of the Jacobian matrix ofthe transformation function is greater thanzero.

In summary, Opti-DCN keeps the algo-rithmic logic and main steps of the originalrubber-sheet algorithm and it optimizes DCNin three critical ways. First, the points used togenerate forces are changed from all vertices topoints on circles and squares. Second, the forcegeneration equation in Opti-DCN is a scalednegative exponential function. Last, a globalelasticity coefficient is calculated from theJacobian matrix of the transformation functionand can largely guarantee topological integrityand optimal convergence rate. Because thedifferential of an exponential function is itself,calculating the new, optimized global elasticitycoefficient adds little computational burden toOpti-DCN.

Results and Comparison

Visually comparing the 2000 U.S. populationcartograms produced by the original and opti-mized DCN algorithms further illustrates thevalue of the optimization. Because Opti-DCNis essentially a variant of the DCN algorithm,conclusions from previous comparisons be-tween the DCN algorithm and other cartogramalgorithms similarly apply to Opti-DCN aswell. The main improvement to the DCNalgorithm is that Opti–DCN is able to preservetopology with point densification (i.e., addingnew vertices if the distance between two neigh-boring vertices is greater than a predefinedcriterion; in the case of the continental UnitedStates, for example, one two-hundredthof the minimum dimension of the map).In addition, Opti-DCN is algorithmicallyfaster converging to the ideal equal-densityshapes.

Specifically, the Opti-DCN algorithm cre-ates almost the same, if not better, cartogramsin terms of map aesthetics and readability, de-spite the fact that it only utilizes circles andsquares to approximate the original polygons(Figure 6). Visually inspected, cartograms pro-duced by Opti-DCN better preserve shapes.The square shape of Wyoming, for example,is transformed into a star shape with DCN;with Opti-DCN, Wyoming is largely kept in asquare shape with much less distortion at the

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Figure 6 Algorithmic convergence comparison. Opti-DCN iteration 1 (A), 2 (B), 3 (C); original DCNiteration 1 (D), 4 (E), 7 (F). Note: Opti-DCN only needs one third of the iterations as the original DCNalgorithm.

four corners. Another big difference betweenshapes generated by DCN and Opti-DCN isevident in viewing Long Island, New York. TheOpti-DCN algorithm well preserves its origi-nal shape, whereas DCN compresses this areato an unrecognizable narrow belt (Figure 6).

With optimization, the Opti-DCN algo-rithm works much faster than the original al-gorithm toward equal-density shapes. To mea-sure the global area errors of cartograms, threemeasures are calculated using similar metricsas in Keim, North, and Panse (2004) and

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Henriques, BaCao, and Lobo (2009). First, de-

fine ei = |Sdi −St

i |Sd

i +Sti

as the size error of the poly-gon i, in which St

i is the area of the ith trans-formed polygon and Sd

i is the ideal area calcu-lated from population. Then, mean quadratic

error is mqe = 1N

√∑Ni=1 e2

i , weighted mean

error is wme = 1S

∑Ni=1 (ei · St

i ), and simple av-erage error is se = 1

N

∑Ni=1 ei , where N is the

total number of polygons and S is the total areaof all polygons. In general, the Opti-DCN al-gorithm can achieve the same transformationrate in one step that would have required theoriginal DCN algorithm about three steps. Forexample, Opti-DCN only needs two steps toget the mean quadratic error to be lower than 2percent, whereas DCN takes six steps to achievethat (Table 1).

Most important, the Opti-DCN algo-rithm can theoretically guarantee topologicalequivalence for continuous space. With pointdensification using a maximum length ofone two-hundredth of the minimum mapdimension, the Opti-DCN algorithm does notcreate any topological errors, even for areasthat have complex spatial patterns, such asnarrow bays, rivers, islands, and peninsulas.At San Juan Islands of Washington, PotomacRiver, and Delaware Bay, Opti-DCN wellpreserves topologies (Figure 7). As somehave acknowledged, the DCN algorithm cansimilarly benefit from point densification, withwhich DCN can reduce many and even com-pletely avoid all topological errors (Du and Liu1999).

Discussion and Conclusion

This article presents an optimized algorithm—Opti-DCN—for continuous area cartogramconstruction, based on the rubber-sheet algo-rithm developed by Dougenik, Chrisman, andNiemeyer (1985). Opti-DCN gives the orig-inal DCN algorithm a more formal math-ematical treatment. It specifies a sufficientcondition for topological equivalence; that is, acontinuous, differentiable, and invertible trans-formation function. The essence of Opti-DCNis to specify and calibrate such a transforma-tion function through manipulating the partialderivatives of the basic negative exponentialforce equation. The transformation functionspecified in Opti-DCN also endogenously op-timizes the global elasticity coefficient, which isclosely related to topological integrity and con-vergence rate of the rubber-sheet algorithm.Additionally, Opti-DCN simplifies the trans-forming force generation mechanism. Usingcircles and squares to create force-generatingpoint pairs simplifies the algorithm and greatlyimproves its efficiency without sacrificing itsability to transform space and to preserveshapes.

Comparison with the original DCN algo-rithm shows that Opti-DCN can completelypreserve topology for the case of the continen-tal United States and is algorithmically faster toconverge to equal-density maps. More impor-tant, Opti-DCN provides a solid mathemati-cal foundation and a flexible framework for theapplication of rubber-sheet algorithms to geo-graphic transformation. Most elements in this

Table 1 Transformation efficiency and global area error

Opti-DCN algorithm DCN algorithm

StepsMean quadratic

errorWeighted

mean errorSimple

mean errorMean quadratic

errorWeighted

mean errorSimple

mean error

0 0.071 0.375 0.416 0.071 0.375 0.4161 0.040 0.191 0.226 0.062 0.305 0.3572 0.019 0.060 0.095 0.052 0.234 0.2933 0.014 0.032 0.052 0.042 0.169 0.2294 0.015 0.036 0.058 0.033 0.115 0.1695 0.009 0.024 0.037 0.024 0.074 0.1186 0.009 0.021 0.033 0.017 0.047 0.079

Note: DCN = Dougenik, Chrisman, and Niemeyer.

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Figure 7 Topology preservation with Opti-DCN. (A) San Juan Islands of Washington, (B) Potomac River,and (C) Delaware Bay. (Color figure available online.)

particular algorithm such as the approximateshapes, point densification, force exaggerationand cooling, and even the force formula itself,could be altered to achieve alternative algorith-mic features. Opti-DCN well illustrates that aslong as the sufficient condition of topologicalequivalence is mathematically met within theframework laid out in this article, the methodof generating transforming forces and specify-ing a rubber-sheet algorithm could be ratherflexible.

With such a mathematical framework, fur-ther improvement to Opti-DCN is possible.Transformation functions with better mathe-matical properties can be designed to make thecalculation of forces and their partial deriva-tives even more efficient. The simple circlesand squares used in Opti-DCN can be repo-sitioned to make force-generating points asclose to polygon vertices as possible. Theycould also be replaced by geometries thatbetter describe the original polygons using

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automatic cartographic generalization, whichcould further improve its ability to preserveshapes. �

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SHIPENG SUN is a Postdoctoral Fellow in theSchool of Planning, Faculty of Environment atthe University of Waterloo, 200 University AvenueWest, Waterloo, ON N2L3G1, Canada. E-mail:[email protected]. His research interests includenetwork analysis, geovisualization, agent-based mod-eling, intraurban migration, and land use and landcover change.

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an optimized rubber-sheet algorithm for continuous area...

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