  In this paper we present a novel representation for deformation fields of 3D
shapes, by considering the induced changes in the underlying metric. In
particular, our approach allows to represent a deformation field in a
coordinate-free way as a linear operator acting on real-valued functions
defined on the shape. Such a representation both provides a way to relate
deformation fields to other classical functional operators and enables analysis
and processing of deformation fields using standard linear-algebraic tools.
This opens the door to a wide variety of applications such as explicitly adding
extrinsic information into the computation of functional maps, intrinsic shape
symmetrization, joint deformation design through precise control of metric
distortion, and coordinate-free deformation transfer without requiring
pointwise correspondences. Our method is applicable to both surface and
volumetric shape representations and we guarantee the equivalence between the
operator-based and standard deformation field representation under mild
genericity conditions in the discrete setting. We demonstrate the utility of
our approach by comparing it with existing techniques and show how our
representation provides a powerful toolbox for a wide variety of challenging
problems.
