  Although shape correspondence is a central problem in geometry processing,
most methods for this task apply only to two-dimensional surfaces. The
neglected task of volumetric correspondence--a natural extension relevant to
shapes extracted from simulation, medical imaging, and volume
rendering--presents unique challenges that do not appear in the two-dimensional
case. In this work, we propose a method for mapping between volumes represented
as tetrahedral meshes. Our formulation minimizes a distortion energy designed
to extract maps symmetrically, i.e., without dependence on the ordering of the
source and target domains. We accompany our method with theoretical discussion
describing the consequences of this symmetry assumption, leading us to select a
symmetrized ARAP energy that favors isometric correspondences. Our final
formulation optimizes for near-isometry while matching the boundary. We
demonstrate our method on a diverse geometric dataset, producing low-distortion
matchings that align closely to the boundary.
