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SWARTHMORE CS 97 - A Constraint-Based Approach to Constructing Continuous Cartograms

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A Constraint-Based Approach to Constructing Continuous CartogramsChristopher J. Kocmoud†Donald H. House‡Texas Center for Applied Technology Visualization LaboratoryTexas Engineering Experiment Station Texas A&M UniversityKeywords: cartogram, value-by-area map, map transformation, anamorphosis, thematic cartography.AbstractWe present a new constraint-based continuous area cartogram construction method that isunique in its ability to preserve essential cues for recognition of region shapes. It automaticallyachieves desired region areas while maintaining correct map topology. The algorithm iscompared with a number of existing methods, and results are shown to be superior in bothaccuracy and preservation of shape recognition cues. Through hierarchical resolution, we firstperform gross adjustments upon a coarsely resampled map and later refine the map atprogressively higher levels of detail.1 IntroductionThe area cartogram is a useful tool for visualizing the geographic distribution of “routine”data in a variety of disciplines, including politics, social demographics, epidemiology, andbusiness. Through the spatial transformation of map regions relative to the data, the cartogramprominently emphasizes data distribution instead of territorial size.For example, the results of the popular vote in the 1996 U.S. presidential race are visualized inFigure 1a using traditional thematic mapping. There is a significant problem with thisvisualization. Without prior knowledge of population density, the viewer has no clear indicator asto who actually won the election. This map produces an intrinsic distortion of the data. Theresults would be better visualized on a map more representative of population. Figure 1b is anequal population cartogram of the same data generated using the Constraint-Based Methoddescribed in this paper. It clearly shows the winner.A continuous area cartogram is one in which the topology, or connectivity, of the map regionsis retained while areas are resized. Accurately resizing regions relative to a data variable whilemaintaining continuity and region recognition is a challenging task [8], that has not been solvedeffectively to date. This paper presents a new approach to this problem, that has promise to makethe continuous area cartogram a common tool for cartographers.2 BackgroundSeveral computer algorithms have already been developed to construct continuous areacartograms [1, 2, 4, 7, 9, 10, 11]. We present here five of the most effective of these methodsand make comparisons of their results with ours at the end of the paper. We chose the names ofthe methods discussed here to be descriptive in nature, they may not necessarily be the namesused by the authors.† 245 Wisenbaker − M.S. 3407, College Station, Texas 77843-3407, Tel: (409) 862-3272, Fax: (409) 862-3336,[email protected].‡ 216 Langford − M.S. 3137, College Station, Texas 77843-3137, Tel: (409) 845-3465, Fax: (409) 845-4491,[email protected]) Thematic map.b) Cartogram.Figure 1: 1996 U.S. presidential return visualizations (Data Source: Federal Election Commission).2.1 Continuous cartogram methodsTobler’s Pseudo-Cartogram Method creates an equal density approximation by compressingor expanding the lines of latitude and longitude until a least root mean square error solution isobtained [10]. This method provides an effective way to “preprocess” a map prior to cartogramconstruction but the cartograms produced can contain extensive area error. Dorling’s CellularAutomaton Method is adapted from the “Game of Life,” where a map has a grid superimposed onit and individual grid cells are traded until every geographic region obtains its desired number ofcells [1]. While this method is very effective at achieving area, regions tend to lose their uniquecontours and acquire a shape reflecting the grid.The three other methods are radial in nature. The Radial Expansion Method of Selvin et al.applies radial transformations from each region upon all map vertices such that the selected regionexpands or shrinks while leaving the area of all other regions unchanged [7]. The Rubber SheetMethod of Dougenik et al. exerts radial forces from each region upon all map vertices at amagnitude proportional to region area error and inversely proportional to distance [2]. Gusein-Zade and Tikunov’s Line Integral Method applies radial transformations such that the density of aselected cell is made uniform while leaving all other cells unchanged, with the vector sum oftransformations applied as a line integral around each of the region boundaries [4]. While theradial methods produce reasonable results in terms of area error, they produce both a “ballooning”effect that can render regions unrecognizable and a “pinching” of originally rectangular regioncorners.CharacteristicsRadialExpansionRubberSheetPseudo-CartogramCellularAutomatonLineIntegral1. Independent of region traversal orderü ü ü ü2. Independent of coordinate axesü ü ü ü3. Conformal mappingü ü ü ü4. Global displacements per iterationü ü ü5. Intersection preventionü ü6. Ability to fix (pin down) pointsü7. User controls on area vs. shapeTable 1: Desired cartogram method characteristics.2.2 Desired cartogram method characteristicsIn Table 1 we have expanded upon a previous review of methods by Gusein-Zade andTikunov [5] and quantified seven characteristics that we feel, if attained, would lead to acomprehensive and versatile cartogram algorithm. Independence of results from vertex traversalorder and the coordinate axes enables reproducible results, generating the same cartogramregardless of the organization of the map’s source data base and orientation. In order to be aneffective communication tool, the method should be conformal in its preservation of angles locallyso that detailed areas on the cartogram are similar to the original map. Globality assures that allregions influence every map vertex, generally resulting in faster convergence to a solution. Inorder to guarantee quality results, the algorithm should also prevent the intersection and self-overlapping of regions. The last two characteristics provide user control of aesthetics by allowingvertex locations to be pinned down or modified and by enabling the user to make shape versusaccuracy trade-offs. None of the methods described above possesses all of the first five technicalrequirements, and none, with the exception of the Cellular


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SWARTHMORE CS 97 - A Constraint-Based Approach to Constructing Continuous Cartograms

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