Women are at higher risk of Alzheimer's and other neurological diseases after menopause, and yet research connecting female brain health to sex hormone fluctuations is limited. We seek to investigate this connection by developing tools that quantify 3D shape changes that occur in the brain during sex hormone fluctuations. Geodesic regression on the space of 3D discrete surfaces offers a principled way to characterize the evolution of a brain's shape. However, in its current form, this approach is too computationally expensive for practical use. In this paper, we propose approximation schemes that accelerate geodesic regression on shape spaces of 3D discrete surfaces. We also provide rules of thumb for when each approximation can be used. We test our approach on synthetic data to quantify the speed-accuracy trade-off of these approximations and show that practitioners can expect very significant speed-up while only sacrificing little accuracy. Finally, we apply the method to real brain shape data and produce the first characterization of how the female hippocampus changes shape during the menstrual cycle as a function of progesterone: a characterization made (practically) possible by our approximation schemes. Our work paves the way for comprehensive, practical shape analyses in the fields of bio-medicine and computer vision. Our implementation is publicly available on GitHub: https://github.com/bioshape-lab/my28brains.

翻译：女性在更年期后患阿尔茨海默病及其他神经系统疾病的风险更高，然而关于女性大脑健康与性激素波动之间联系的研究仍十分有限。我们旨在通过开发量化性激素波动期间大脑三维形态变化的工具来探索这一关联。基于三维离散表面空间的测地线回归为刻画大脑形态演变提供了严谨的方法，但其现有计算成本过高，难以实际应用。本文提出加速三维离散表面形状空间测地线回归的近似方案，并给出了不同近似方法适用场景的经验法则。我们在合成数据上测试了这些方法，量化了近似计算中速度与精度的权衡关系，结果表明实际应用可在仅牺牲少量精度的前提下获得显著加速。最终，我们将该方法应用于真实大脑形态数据，首次实现了对月经周期中女性海马体形态随孕酮变化的表征——这一表征正是通过我们的近似方案才得以（实际）可行。本研究为生物医学与计算机视觉领域开展全面实用的形态分析铺平了道路。我们的代码已开源发布于GitHub：https://github.com/bioshape-lab/my28brains