On Partial Shape Correspondence and Functional Maps
Amit Bracha Thomas Dagès Ron Kimmel
While dealing with matching shapes to their parts, we often utilize an
instrument known as functional maps. The idea is to translate the shape
matching problem into ``convenient'' spaces by which matching is performed
algebraically by solving a least squares problem. Here, we argue that such
formulations, though popular in this field, introduce errors in the estimated
match when partiality is invoked. Such errors are unavoidable even when
considering advanced feature extraction networks, and they can be shown to
escalate with increasing degrees of shape partiality, adversely affecting the
learning capability of such systems. To circumvent these limitations, we
propose a novel approach for partial shape matching.
  Our study of functional maps led us to a novel method that establishes direct
correspondence between partial and full shapes through feature matching
bypassing the need for functional map intermediate spaces. The Gromov distance
between metric spaces leads to the construction of the first part of our loss
functions. For regularization we use two options: a term based on the area
preserving property of the mapping, and a relaxed version of it without the
need to compute a functional map.
  The proposed approach shows superior performance on the SHREC'16 dataset,
outperforming existing unsupervised methods for partial shape matching. In
particular, it achieves state-of-the-art result on the SHREC'16 HOLES
benchmark, superior also compared to supervised methods.