Esfandiar Nava-Yazdani, Christoph von Tycowicz, Hans-Christian Hege, and I have just published a preprint of our work “A Shape Trajectories Approach to Longitudinal Statistical Analysis”.
For Kendall's shape space we determine analytically Jacobi fields and parallel transport, and compute geodesic regression. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data.
As application example we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data only.