  In this paper, we describe in detail a model of geometric-functional
variability between fshapes. These objects were introduced for the first time
by the authors in [Charlier et al. 2015] and are basically the combination of
classical deformable manifolds with additional scalar signal map. Building on
the aforementioned work, this paper's contributions are several. We first
extend the original $L^2$ model in order to represent signals of higher
regularity on their geometrical support with more regular Hilbert norms
(typically Sobolev). We describe the bundle structure of such fshape spaces
with their adequate geodesic distances, encompassing in one common framework
usual shape comparison and image metamorphoses. We then propose a formulation
of matching between any two fshapes from the optimal control perspective, study
existence of optimal controls and derive Hamiltonian equations and conservation
laws describing the dynamics of geodesics. Secondly, we tackle the discrete
counterpart of these problems and equations through appropriate finite elements
interpolation schemes on triangular meshes. At last, we show a few results of
metamorphosis matchings on synthetic and several real data examples in order to
highlight the key specificities of the approach.
