Evolutionary Morphing

Abstract

We introduce a technique to visualize the gradual evolutionary change of the shapes of living things as a morph between known three-dimensional shapes. Given geometric computer models of anatomical shapes for some collection of specimens - here the skulls of the some of the extant members of a family of monkeys - an evolutionary tree for the group implies a hypothesis about the way in which the shape changed through time. We use a statistical model which expresses the value of some continuous variable at an internal point in the tree as a weighted average of the values at the leaves. The framework of geometric morphometrics can then be used to define a shape-space, based on the correspondences of landmark points on the surfaces, within which these weighted averages can be realized as actual surfaces. Our software provides tools for performing and visualizing such an analysis in three dimensions. Beginning with laser range scans of crania, we use our landmark editor to interactively place landmark points on the surface. We use these to compute a "tree-morph" that smoothly interpolates the shapes across the tree. Each intermediate shape in the morph is a linear combination of all of the input surfaces. We create a surface model for an intermediate shape by warping all the input meshes towards the correct shape and then blending them together. To do the blending, we compute a weighted average of their associated trivariate distance functions and then extract a surface from the resulting function. We implement this idea using the squared distance function, rather than the usual signed distance function, in a novel way.