  Intrinsic and parametric regression models are of high interest for the
statistical analysis of manifold-valued data such as images and shapes. The
standard linear ansatz has been generalized to geodesic regression on manifolds
making it possible to analyze dependencies of random variables that spread
along generalized straight lines. Nevertheless, in some scenarios, the
evolution of the data cannot be modeled adequately by a geodesic. We present a
framework for nonlinear regression on manifolds by considering Riemannian
splines, whose segments are B\'ezier curves, as trajectories. Unlike
variational formulations that require time-discretization, we take a
constructive approach that provides efficient and exact evaluation by virtue of
the generalized de Casteljau algorithm. We validate our method in experiments
on the reconstruction of periodic motion of the mitral valve as well as the
analysis of femoral shape changes during the course of osteoarthritis,
endorsing B\'ezier spline regression as an effective and flexible tool for
manifold-valued regression.
