17 Region Matching and Tracking under Deformations or Occlusions Stefano Soatto, Anthony Yezzi and Alessandro Duci Abstract We present a variational approach to the problem of registering non- equivalent shapes with missing parts. Registration is achieved through the evolution of partial differential equations that simultaneously estimates the shape of the missing region, the underlying “average complete shape” and the collection of group elements (Euclidean or affine) performing the reg- istration. Our techniques apply both to shapes, for instance represented as characteristic functions (binary images), and to grayscale images, where all intensity levels evolve simultaneously in a system of partial differential equations. 17.1 Introduction Consider a sheet of paper falling. If it were a rigid object, one could describe its motion by providing the coordinates of one particle and the orientation of an orthogonal reference frame attached to that particle. That is, 6 numbers would be sufficient to describe the object at any instant of time. However, being a non-rigid object, in order to describe it at any instant of time one should really specify the trajectory of each individual particle on the sheet [21]. That is, if represents the initial collection of particles, one could provide a function that describes how the entire set of particles evolves in time: . Indeed, if each particle can move independently, there may be no notion of “overall motion,” and a more appropriate description of is that of a “deformation” of the sheet. That includes as a special case a rigid motion, described collectively by a rotation matrix and a translation vector , so that with and independent of the particle in . In practice, however, that is not how one usually describes a sheet of paper falling. Instead, one may say that the sheet is “moving” downwards along the vertical direction