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A fast, free-form rubber-sheet algorithm for contiguous area cartograms

INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 2013年第3期27卷 567-593页

作者： Sun, Shipeng Univ Michigan Sch Nat Resources & Environm Ann Arbor MI 48109 USA

This article presents a fast, free-form rubber-sheet (Carto3F) algorithm for the construction of contiguous area cartograms. rubber-sheet algorithms are influential and popular because of their conceptual simplicity. Existing rubber-sheet algorithms, however, tend to be algorithmically inefficient and computationally slow. More critically, they cannot completely preserve topology. Carto3F specifically improves rubber-sheet algorithms in these two aspects. First, with a spatial structure of quadtree and a mathematical condition for topological equivalence, Carto3F can prevent topological errors and guarantees topological integrity. Second, Carto3F is designed with efficiency as a priority. Its efficiency is primarily gained through using the auxiliary quadtree to reduce the number of points to be transformed. Furthermore, Carto3F allows parallel computation and can fully take advantage of the increasingly common multi-core, multi-thread processors. Based on the mathematical analysis, Carto3F also mitigates the effect of force cancellation that is inherent in rubber-sheet algorithms. On a computer equipped with a 2.4-GHz quad-core CPU, Carto3F can produce quality population cartograms of the United States, China, and the world within 1 second, 18 seconds, and 8 minutes, respectively. Quantitative measures show that Carto3F outperforms the optimized rubber-sheet algorithm and the diffusion algorithm in both transformation effectiveness and computational efficiency.

关键词： Carto3F fast free-form transformation contiguous area cartogram rubber-sheet algorithm

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An Optimized rubber-sheet algorithm for Continuous Area Cartograms

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PROFESSIONAL GEOGRAPHER 2013年第1期65卷 16-30页

作者： Sun, Shipeng Univ Waterloo Sch Planning Fac Environm Waterloo ON N2L 3G1 Canada

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

关键词： algorithm optimization continuous area cartogram rubber-sheet algorithm

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