We consider the problem of non-rigid shape matching, and specifically the functional maps
framework that was recently proposed to find correspondences between shapes. A key step in this
framework is to formulate descriptor preservation constraints that help to encode the information
(e.g., geometric or appearance) that must be preserved by the unknown map. We show
that considering descriptors as linear operators acting on functions through multiplication,
rather than as simple scalar-valued signals, allows to extract significantly more information from
a given descriptor and ultimately results in a more accurate functional map estimation. Namely, we
show that descriptor preservation constraints can be formulated via commutativity with
respect to the unknown map, which can be conveniently encoded by considering relations between
matrices in the discrete setting. As a result, when the vector space spanned by the descriptors
has a dimension smaller than that of the reduced basis, our optimization may still provide a
fully-constrained system leading to accurate point-to-point correspondences, while previous
methods might not. We demonstrate on a wide variety of experiments that our approach leads to
significant improvement for functional map estimation by helping to reduce the number of necessary
descriptor constraints by an order of magnitude, even given an increase in the size of the reduced
basis.

Dorian Nogneng studied at Ecole Polytechnique and did his master in Paris Diderot (master of research in computer science: https://wikimpri.dptinfo.ens-cachan.fr/doku.php).
He is now a PhD student of Pr. Maks Ovsjanikov in Ecole Polytechnique (Palaiseau, FRANCE) and works on shape matching, in particular using functional maps.