Cortical Morphometry Analysis based on Worst Transportation Theory

TL;DR

This paper introduces a novel biomarker based on worst transportation theory to analyze cortical morphometry, effectively distinguishing Alzheimer's disease from normal aging with improved statistical performance.

Contribution

It generalizes the Brenier theorem to worst transportation, develops an efficient computation algorithm, and demonstrates its effectiveness in AD analysis.

Findings

WT cost effectively differentiates AD from normal aging

Proposed method outperforms existing optimal transportation-based methods

WT provides a new perspective for cortical morphometry analysis

Abstract

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.