Shape spaces for pre-aligned star-shaped objects – studying the growth of plants by principal components analysis

We analyse the shapes of star-shaped objects which are pre-aligned. This is motivated from two examples studying the growth of leaves, and the temporal evolution of tree rings. In the latter case measurements were taken at fixed angles while in the former case the angles were free. Subsequently, this leads to different shape spaces, related to different concepts of size, for the analysis. While several shape spaces already existed in the literature when the angles are fixed, a new shape space for free angles, called spherical shape space, needed to be introduced. We compare these different shape spaces both regarding their mathematical properties, and in their adequacy to the data at hand; we then apply suitably defined principal component analysis on these. In both examples we find the shapes to evolve mainly along the first principal component during growth; this is the “geodesic hypothesis” formulated by Le, H. and Kume, A. (Detection of Shape Changes in Biological Features, Journal of Microscopy, 2000 (200), 140–147). Moreover, we were able to link change points of this evolution to significant changes in environmental conditions.