  This paper introduces and studies a metamorphosis framework for geometric
measures known as varifolds, which extends the diffeomorphic registration model
for objects such as curves, surfaces and measures by complementing
diffeomorphic deformations with a transformation process on the varifold
weights. We consider two classes of cost functionals to penalize those combined
transformations, in particular the LDDMM-Fisher-Rao energy which, as we show,
leads to a well-defined Riemannian metric on the space of varifolds with
existence of corresponding geodesics. We further introduce relaxed formulations
of the respective optimal control problems, study their well-posedness and
derive optimality conditions for the solutions. From these, we propose a
numerical approach to compute optimal metamorphoses between discrete varifolds
and illustrate the interest of this model in the situation of partially missing
data.
