Optimal transport map reshapes world population maps accurately
The optimal-transport cartogram: world population as a Brenier map
Computational GeometryGraphics
Summary
Maps that resize countries based on population can look distorted or twisted. The authors used math from optimal transport theory to create a smooth map where areas are proportional to population without local twisting or folding. They tested this method on a high-resolution world population map and compared it to a popular diffusion method, finding similar accuracy but much less distortion. Their approach can also zoom in on cities and group populations into cells, and they provide all data and code publicly.
What this means in practice
- •For geographic information system teams: Produce population cartograms with minimized distortion and accurate local scaling for better visual analytics and planning.
- •For urban planning departments: Generate refined city-scale population maps that allocate space proportional to population density without overlap or folding.
Authors
Philipp Bogdan
Abstract
A contiguous cartogram is a map whose area is proportional to a quantity such as population. The defining condition, that the Jacobian determinant of the deformation equals the density, is one equation for two unknown functions, so every cartogram method adds a tie-breaker, usually implicitly. Optimal transport makes the tie-breaker explicit: among all density-equalising maps of the frame onto itself, take the one that moves the population least in the mean-square sense. By Brenier's theorem that map is the gradient of a convex potential, so it has no local rotation anywhere and cannot fold. We compute this map for the world population of 2025 on a 4096 by 4096 Mercator grid (10 km cells) from the GHS-POP raster, using the fixed-point Monge-Ampère iteration of Benamou, Froese and Oberman with spectral Poisson solves, continuation in the population share and an ocean-only buffer, on a laptop GPU. Against a Gastner-Newman diffusion cartogram of the same density we find the same transport cost to within 0.4 per cent and density errors of a few per cent for both, but a median local rotation of 8.8 degrees for diffusion against 0.01 for transport, and a median anisotropy of 6.30 against 3.93. The construction extends to local refinement, by transporting a city's 100 m population onto the area measure the global map assigns to it, and to a semi-discrete counterpart: 8,192 Laguerre cells of 1.00 million people each. Code, data provenance and every figure's inputs are public.