Statistics on Manifolds applied to Shape Theory

In this report, we use a variety of tools from differential geometry to propose a nonlinear extension of the principal components analysis (PCA) into manifolds setting. This extension, that we shall call principal geodesics analysis (PGA), attempts to find analogs of the principal components by introducing the principal geodesic components. We then construct the shape space of triangles Σ^3_2 and find a convenient parametrization of it. Finally, we apply the PGA procedure previously designed to analyze the variability of a sample of shapes, randomly chosen onto the shape space of triangles.