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

Abstract

We analyse the shapes of star-shaped objects which are prealigned. 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 whereas in the former case the angles were free. Subsequently, this leads to different shape spaces, related to different concepts of size, for the analysis. Whereas 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 that the shapes evolve mainly along the first principal component during growth; this is the 'geodesic hypothesis' that was formulated by Le and Kume. Moreover, we could link change-points of this evolution to significant changes in environmental conditions.

Quantifying the Weight of Evidence from a Forensic Fingerprint Comparison: A New Paradigm

“…In fact, entire polygons comprising ordered vertices, or being even star shaped as considered here, may be represented in a Euclidean space, rendering their analysis particularly simple (Hobolth et al. , 2002; Hotz et al. , 2010), as all methods of multivariate analysis (Mardia et al.…”

“…However, there is a well-developed theory within shape analysis (Dryden and Mardia, 1998) allowing us to compare triangles and model distributions of these; see Gottschlich et al (2011) for an application of shape analysis to fingerprints. In fact, entire polygons comprising ordered vertices, or being even star shaped as considered here, may be represented in a Euclidean space, rendering their analysis particularly simple (Hobolth et al, 2002;Hotz et al, 2010), as all methods of multivariate analysis (Mardia et al, 1979;Anderson, 2003) become directly applicable. Such models will intrinsically respect correlations of the individual variables, e.g.…”

Reconstructing leaf growth based on non-destructive digitizing and low-parametric shape evolution for plant modelling over a growth cycle

“…A unit-speed geodesic is uniquely determined by initial offset x 0 ∈ S 5 and initial velocity v 0 ∈ S 5 orthogonal to x 0 . Our geodesic model thus is specified as As has been shown previously in Hotz et al (2010), shapes of poplar leaves during growth closely follow geodesics in shape space. As explained below, the proportional model and the geodesic model can be used in conjunction to model rather diversified growth and development.…”

“…Every such 3-landmark configuration x = (x (1) , x (2) , x (3) ) is then viewed as a matrix x / ǁxǁ in the pre shape space S 5 ⊆ R 2×3 which carries the canonical structure of a non-flat 5-dimensional unit-sphere (Hotz et al 2010). Within this setup, in order to model growth over one-dimensional time, most parsimonious one-dimensional data descriptors are sought for.…”

Estimating Multiple Pathways of Object Growth Using Nonlongitudinal Image Data

“…Many studies which have proposed mathematical models for estimating object growth trajectories using longitudinal landmark data (Kent et al , 2001Morris et al 2000;Kume, Dryden, and Le 2007;Barry and Bowman 2008;Hotz et al …”