Geo-FARM: Geodesic Factor Regression Model for Misaligned Pre-Shape Responses in Statistical Shape Analysis

The problem of using covariates to predict shapes of objects in a regression setting is important in many fields. A formal statistical approach, termed Geodesic regression model, is commonly used for modeling and analyzing relationships between Euclidean predictors and shape responses. Despite its popularity, this model faces several key challenges, including (i) misalignment of shapes due to pre-processing steps, (ii) difficulties in shape alignment due to imaging heterogeneity, and (iii) lack of spatial correlation in shape structures. This paper proposes a comprehensive geodesic factor regression model that addresses all these challenges. Instead of using shapes as extracted from pre-registered data, it takes a more fundamental approach, incorporating alignment step within the proposed regression model and learns them using both pre-shape and covariate data. Additionally, it specifies spatial correlation structures using low-dimensional representations, including latent factors on the tangent space and isotropic error terms. The proposed framework results in substantial improvements in regression performance, as demonstrated through simulation studies and a real data analysis on Corpus Callosum contour data obtained from the ADNI study. However, the use of quotient space geometry in specifiy-