  This paper introduces a new mathematical and numerical framework for surface
analysis derived from the general setting of elastic Riemannian metrics on
shape spaces. Traditionally, those metrics are defined over the infinite
dimensional manifold of immersed surfaces and satisfy specific invariance
properties enabling the comparison of surfaces modulo shape preserving
transformations such as reparametrizations. The specificity of the approach we
develop is to restrict the space of allowable transformations to predefined
finite dimensional bases of deformation fields. These are estimated in a
data-driven way so as to emulate specific types of surface transformations
observed in a training set. The use of such bases allows to simplify the
representation of the corresponding shape space to a finite dimensional latent
space. However, in sharp contrast with methods involving e.g. mesh
autoencoders, the latent space is here equipped with a non-Euclidean Riemannian
metric precisely inherited from the family of aforementioned elastic metrics.
We demonstrate how this basis restricted model can be then effectively
implemented to perform a variety of tasks on surface meshes which, importantly,
does not assume these to be pre-registered (i.e. with given point
correspondences) or to even have a consistent mesh structure. We specifically
validate our approach on human body shape and pose data as well as human face
scans, and show how it generally outperforms state-of-the-art methods on
problems such as shape registration, interpolation, motion transfer or random
pose generation.
