Deforming Motion , Shape Average and the Joint Registration and Segmentation of 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 non-identical 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 involve the numerical integration of partial differential equations, which we address within the framework of level set methods. Figure 1: A jellyfish is “moving while deforming.” What exactly does it mean? How can we separate its “global” motion from its “local” deformation?