最优传输制图：作为Brenier映射的世界人口分布
The optimal-transport cartogram: world population as a Brenier map
- Imperial College London（帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出基于最优传输的连续制图方法，利用Brenier映射计算2025年世界人口分布，相比扩散制图显著减少局部旋转和变形，并支持局部细化与半离散扩展。
AI中文摘要：
连续制图（contiguous cartogram）是一种地图，其面积与人口等数量成正比。其定义条件——形变的雅可比行列式等于密度——是包含两个未知函数的一个方程，因此每种制图方法都（通常是隐式地）添加一个决胜规则。最优传输使决胜规则显式化：在所有将框架映射到自身的密度均衡映射中，选取在均方意义下移动人口最少的那个映射。根据Brenier定理，该映射是某个凸势函数的梯度，因此它在任何地方都没有局部旋转，也不会折叠。我们使用Benamou、Froese和Oberman的定点Monge-Ampère迭代，结合谱泊松求解、人口份额连续化以及仅含海洋的缓冲带，在笔记本电脑GPU上，基于GHS-POP栅格数据，为2025年世界人口在4096×4096墨卡托网格（10公里单元）上计算该映射。与相同密度的Gastner-Newman扩散制图相比，我们发现两者的传输成本差异在0.4%以内，密度误差均为百分之几，但扩散制图的中位局部旋转为8.8度，而传输制图仅为0.01度；中位各向异性分别为6.30和3.93。该构造可扩展到局部细化，通过将城市100米人口传输到全局地图分配给该区域的面积度量，也可扩展到半离散对应物：8192个Laguerre单元，每个单元对应100万人。代码、数据来源和每张图的输入均公开。
英文摘要：
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.
