  Biomarkers play an important role in early detection and intervention in
Alzheimer's disease (AD). However, obtaining effective biomarkers for AD is
still a big challenge. In this work, we propose to use the worst transportation
cost as a univariate biomarker to index cortical morphometry for tracking AD
progression. The worst transportation (WT) aims to find the least economical
way to transport one measure to the other, which contrasts to the optimal
transportation (OT) that finds the most economical way between measures. To
compute the WT cost, we generalize the Brenier theorem for the OT map to the WT
map, and show that the WT map is the gradient of a concave function satisfying
the Monge-Ampere equation. We also develop an efficient algorithm to compute
the WT map based on computational geometry. We apply the algorithm to analyze
cortical shape difference between dementia due to AD and normal aging
individuals. The experimental results reveal the effectiveness of our proposed
method which yields better statistical performance than other competiting
methods including the OT.
