Curvature and Torsion estimation of 3D functional data: A geometric approach to build the mean shape under the Frenet Serret framework

Abstract: The analysis of curves has been routinely dealt with using tools from functional data analysis. However its extension to multi-dimensional curves poses a new challenge due to its inherent geometric features that are difficult to capture with the classical approaches that rely on linear approximations. We develop an alternative characterization of a mean that reflects shape variation of the curves. Based on a geometric representation of the curves through the Frenet-Serret ordinary differential equations, we introduce a new definition of mean curvature and mean torsion, as well as mean shape through the notion of mean vector field. This new formulation of the mean for multi-dimensional curves allows us to integrate the parameters for the shape features into the unified functional data modelling framework. We formulate the estimation problem of the functional parameters in a penalized regression and develop an efficient algorithm. We demonstrate our approach with both simulated data and real data examples.