Towards Efficient Time Stepping for Numerical Shape Correspondence
This work addresses efficiency in shape analysis for computational geometry applications, but it is incremental as it focuses on optimizing existing PDE-based methods.
The paper tackled the problem of improving time stepping schemes for numerical shape correspondence by assessing their dependence on time step size, and found that specific l0 stable schemes are favorable, as demonstrated through experiments on TOSCA datasets.
The computation of correspondences between shapes is a principal task in shape analysis. To this end, methods based on partial differential equations (PDEs) have been established, encompassing e.g. the classic heat kernel signature as well as numerical solution schemes for geometric PDEs. In this work we focus on the latter approach. We consider here several time stepping schemes. The goal of this investigation is to assess, if one may identify a useful property of methods for time integration for the shape analysis context. Thereby we investigate the dependence on time step size, since the class of implicit schemes that are useful candidates in this context should ideally yield an invariant behaviour with respect to this parameter. To this end we study integration of heat and wave equation on a manifold. In order to facilitate this study, we propose an efficient, unified model order reduction framework for these models. We show that specific $l_0$ stable schemes are favourable for numerical shape analysis. We give an experimental evaluation of the methods at hand of classical TOSCA data sets.