A feature space for edgels in images with landmarks

In many current medical applications of image analysis, objects are detected and delimited by boundary curves or surfaces. Yet the most effective multivariate statistics available pertain to labeled points (landmarks) only. In the finite-dimensional feature space that landmarks support, each case of a data set is equivalent to a deformation map deriving it from the average form. This paper introduces a new extension of the finite-dimensional spline-based approach for incorporating edge information. In this implementation edgels are restricted to landmark loci: they are interpreted as pairs of landmarks at infinitesimal separation in a specific direction. The effect of changing edge direction is a singular perturbation of the thin-plate spline for the landmarks alone. An appropriate normalization yields a basis for image deformations corresponding to changes of edge direction without landmark movement; this basis complements the basis of landmark deformations ignoring edge information. We derive explicit formulas for these edge warps, evaluate the quadratic form expressing bending energies of their formal combinations, and show the resulting spectrum of edge features in typical scenes. These expressions will aid all investigations into medical images that entail comparisons of anatomical scene analyses to a normative or typical form.