Deformotion: Deforming motion, shape average and the joint registration and approximation of structures in images

What does it mean for a deforming object to be "moving"? How can we separate the overall motion ( a finite-dimensional group action) from the more general deformation ( a diffeomorphism)? In this paper we propose a definition of motion for a deforming object and introduce a notion of "shape average" as the entity that separates the motion from the deformation. Our definition allows us to derive novel and efficient algorithms to register nonidentical shapes using region-based methods, and to simultaneously approximate and align structures in greyscale images. We also extend the notion of shape average to that of a "moving average" in order to track moving and deforming objects through time. The algorithms we propose extend prior work on landmark-based matching to smooth curves, and involve the numerical integration of partial differential equations, which we address within the framework of level set methods.