  In this article we study the problem of thoracic image registration, in
particular the estimation of complex anatomical deformations associated with
the breathing cycle. Using the intimate link between the Riemannian geometry of
the space of diffeomorphisms and the space of densities, we develop an image
registration framework that incorporates both the fundamental law of
conservation of mass as well as spatially varying tissue compressibility
properties. By exploiting the geometrical structure, the resulting algorithm is
computationally efficient, yet widely general.
