Optimization-Based Construction of Quadrilateral Table Cartograms

Abstract

A quadrilateral table cartogram is a rectangle-shaped figure that visualizes table-form data; quadrilateral cells in a table cartogram are transformed to express the magnitude of positive weights by their areas, while maintaining the adjacency of cells in the original table. However, the previous construction method is difficult to implement because it consists of multiple operations that do not have a unique solution and require complex settings to obtain the desired outputs. In this article, we propose a new construction for quadrilateral table cartograms by recasting the construction as an optimization problem. The proposed method is formulated as a simple minimization problem to achieve mathematical clarity. It can generate quadrilateral table cartograms with smaller deformation of rows and columns, thereby aiding readers to recognize the correspondence between table cartograms and original tables. In addition, we also propose a means of sorting rows and/or columns prior to the construction of table cartograms to reduce excess shape deformation. Applications of the proposed method confirm its capability to output table cartograms that clearly visualize the characteristics of datasets.

What are Table Cartograms Good for Anyway? An Algebraic Analysis

“…Changes to these values might be invisible (yielding a Confuser ) if the change is small or the cells are not labeled with their corresponding value. This suggests that, like both geographic cartograms [NK16] and tree maps [Fri94], TACOs are more effective when used in conjunction with a secondary visual encoding, such as color or text, as it facilitates easier value retrieval than simply using area alone [IL20]. We argue that, given the low accuracy of the perceptual system for understanding numeric values through area [Mac86], that value retrieval hinges on the presence of secondary annotations, and possesses a Confuser otherwise.…”

“…One way to address this failure mode is to impose additional constraints-such as by minimizing bearing angle differenceswhich causes there to be a single "correct" layout [IL20]. While these criteria can yield more rectangular displays, their selection is arbitrary and an artifact only of designers' preferences, an ambi- guity which may in turn hold another Hallucinator.…”

A Minimally Constrained Optimization Algorithm for Table Cartograms

“…The visual effect is that of a shaded matrix that has been "area-ed" rather than merely colored. A given dataset can produce multiple equally accurate table cartograms because the problem is under-constrained [3].…”

“…They demonstrate that every dataset admits a table cartogram by providing a constructive computational geometry algorithm. Subsequent work by Inoue and Li [3,4] found that it was more effective to generate them through mathematical optimization procedures. Unfortunately, both approaches are capable of generating only a single solution for a given dataset, which precludes the aesthetic control of the output or design-focused exploration of the solution space.…”

“…Unfortunately, both approaches are capable of generating only a single solution for a given dataset, which precludes the aesthetic control of the output or design-focused exploration of the solution space. Further, Inoue and Li's [3] approach relies on a MATLAB numerical solver, which is impractical for web-based applications, a common setting for contemporary visualizations.…”