Informative Descriptor Preservation via Commutativity for Shape Matching

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. In this paper, 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.