We analytically determine Jacobi fields and parallel transports and compute
geodesic regression in Kendall's shape space. Using the derived expressions, we
can fully leverage the geometry via Riemannian optimization and thereby reduce
the computational expense by several orders of magnitude over common, nonlinear
constrained approaches. The methodology is demonstrated by performing a
longitudinal statistical analysis of epidemiological shape data. As an example
application we have chosen 3D shapes of knee bones, reconstructed from image
data of the Osteoarthritis Initiative (OAI). 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 alone.

本文在 Kendall 形状空间中确定 Jacobi 场和平行传输，以及计算测地回归。通过 Riemannian 优化，我们可以充分利用几何形式并将计算成本降低几个数量级，从而在流行病形状数据的纵向统计分析中实现了方法学的应用，具体例子为应用于 Osteoarthritis Initiative（OAI）图像数据重建的膝骨 3D 形状，发现患有骨关节炎的受试组和发展骨关节炎的受试组与正常对照组之间在股骨形态的时间发展方面存在明显差异，这为仅使用几何数据对发生膝骨关节炎进行早期预测铺平了道路。