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.

女性在绝经后更容易患阿尔茨海默氏病和其他神经系统疾病，然而，将女性大脑健康与性激素波动联系起来的研究有限。本文旨在通过开发工具来量化性激素波动期间大脑发生的三维形状变化来研究这种联系。我们提出了适用于三维离散曲面形状空间的加速测地线回归的近似方法，并给出了各种近似方法适用的规则。通过在合成数据上测试我们的方法，量化了这些近似方法的速度和准确性的权衡，证明实践者可以在极大提速的同时只牺牲少量准确性。最后，我们将该方法应用于实际的脑形状数据，并首次描述了女性海马在月经周期中的形状变化特征与孕酮的功能联系，这些成果是我们的近似方法带来的实践可行性的结果。我们的工作为生物医学和计算机视觉领域的全面实用形状分析铺平了道路。我们的实现代码已在GitHub上公开。

地形回归描述女性脑部在月经期间的三维形状变化