transport, Animation and Morphisms


Computational models are used everywhere, from games to movies—e.g., special effects, animations—and deformations are the basic tool for animating objects, so we focus on a particular aspect of the vast morphing research field. The key point is: given two different meshes X and Y undergoing two deformations (called diffeomorphisms in this research area), we would like to answer this question: “when it is possible to say that X and Y are undergoing the same deformation”?

If we can answer this question, at least approximately, we could then transport deformations from a mesh to a different one, a typical task of the shape analysis research. In particular, it is necessary to perform such a kind of transport when performing group-wise statistical analyses in shape or size and shape spaces. A naive possible answer could be that the displacement field is the same, but this gives different results from the one an animator expects. Take the following example:

Let’s assume we deform an object, constituted by three triangles, as in the first two steps; next we want to apply the same deformation to object in the third step. As we’re in the plane, and we’re dealing with triangles, everything should work just fine, except it does not work at all even in a plane, as we can see in the fourth step. Copy and pasting animations is not easy as it might seem.

We assume a different point of view: two templates undergo the same deformation if, given a bijective map between the two underlying affine spaces (i.e., we can map points on one mesh to points in another), the local metric (non linear strain) induced by the two diffeomorphisms is the same for corresponding points.