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. 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.

中文翻译：

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肯德尔形状空间中的测地分析及流行病学应用

我们分析性地确定Jacobi场和平行传输，并在Kendall形状空间中计算测地线回归。使用导出的表达式，我们可以通过黎曼优化充分利用几何形状，从而与常见的非线性约束方法相比，可以将计算费用减少几个数量级。通过对流行病学形状数据进行纵向统计分析证明了该方法。作为示例应用程序，我们选择了从骨关节炎倡议组织的图像数据重建的3D膝盖骨形状。将受试者与突发和发展中的骨关节炎与正常对照组进行比较，我们发现股骨形状在时间上的发展存在明显差异。这为单独使用几何数据为早期预测膝关节骨关节炎的发生铺平了道路。