Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds

We address the problem of estimating full curves/paths on certain nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. These curves are analogous to smoothing splines in Euclidean spaces as they are optimal under a similar objective function, which is a weighted sum of a fitting-related (data term) and a regularity-related (smoothing term) cost functions. The search for smoothing splines on manifolds is based on a Palais metric-based steepest-decent algorithm developed in Samir et al. [38]. Using three representative manifolds: the rotation group for pose tracking, the space of symmetric positive-definite matrices for DTI image analysis, and Kendall's shape space for video-based activity recognition, we demonstrate the effectiveness of the proposed algorithm for optimal curve fitting. This paper derives certain geometrical elements, namely the exponential map and its inverse, parallel transport of tangents, and the curvature tensor, on these manifolds, that are needed in the gradient-based search for smoothing splines. These ideas are illustrated using experimental results involving both simulated and real data, and comparing the results to some current algorithms such as piecewise geodesic curves and splines on tangent spaces, including the method by Kume et al. [24].