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Women are at higher risk of Alzheimer's and other neurological diseases aftermenopause, and yet research connecting female brain health to sex hormonefluctuations is limited. We seek to investigate this connection by developingtools that quantify 3D shape changes that occur in the brain during sex hormonefluctuations. Geodesic regression on the space of 3D discrete surfaces offers aprincipled way to characterize the evolution of a brain's shape. However, inits current form, this approach is too computationally expensive for practicaluse. In this paper, we propose approximation schemes that accelerate geodesicregression on shape spaces of 3D discrete surfaces. We also provide rules ofthumb for when each approximation can be used. We test our approach onsynthetic data to quantify the speed-accuracy trade-off of these approximationsand show that practitioners can expect very significant speed-up while onlysacrificing little accuracy. Finally, we apply the method to real brain shapedata and produce the first characterization of how the female hippocampuschanges shape during the menstrual cycle as a function of progesterone: acharacterization made (practically) possible by our approximation schemes. Ourwork paves the way for comprehensive, practical shape analyses in the fields ofbio-medicine and computer vision. Our implementation is publicly available onGitHub: https://github.com/bioshape-lab/my28brains.

Geodesic Regression Characterizes 3D Shape Changes in the Female Brain During Menstruation