Geodesic Regression Characterizes 3D Shape Changes in the Female Brain
  During Menstruation
Adele Myers Caitlin Taylor Emily Jacobs Nina Miolane
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.