A Riemannian Framework for Linear and Quadratic Discriminant Analysis on the Tangent Space of Shapes

Abstract

We present a Riemannian framework for linear and quadratic discriminant classification on the tangent plane of the shape space of curves. The shape space is infinite dimensional and is constructed out of square root velocity functions of curves. We introduce the notion of mean and covariance of shape-valued random variables and samples from a tangent space to the pre-shapes (invariant to translation and scaling) and then extend it to the full shape space (rotational invariance). The shape observations from the population are approximated by coefficients of a Fourier basis of the tangent space. The algorithms for linear and quadratic discriminant analysis are then defined using reduced dimensional features obtained by projecting the original shape observations on to the truncated Fourier basis. We show classification results on synthetic data and shapes of cortical sulci, corpus callosum curves, as well as facial midline curve profiles from patients with fetal alcohol syndrome (FAS).

Figure 1

Top row: 100 samples each from shapes of class A and B. Middle row: Mean shapes for the two classes. Bottom row: mean shape of the population.

Figure 2

Left to right: examples of the precentral and the central sulci on the cortex, population of all 175 precentral sulci, mean shape of the precentral sulci, population of all 175 central sulci, mean shape of the central sulci, and the mean shape of the overall population.

Figure 3

Top row: 20 corpus callosum shapes each for both healthy controls and patients with schizophrenia. Middle row: Mean shapes for the healthy controls and schizophrenia. Bottom row: mean shape of the population.

Figure 4

Left to right: midline facial profile curves for 24 healthy controls, mean shape of healthy controls, 21 facial profile curves for patients with FAS, mean shape of patients, and the mean shape of the overall population.