On the Stability of Functional Maps and Shape Difference Operators

In this paper, we provide stability guarantees for two frameworks that are based on the notion of functional maps – the shape difference operators introduced in [?] and the framework of [?] which is used to analyze and visualize the deformations between shapes induced by a functional map. We consider two types of perturbations in our analysis: one is on the input shapes and the other is on the change in scale. In theory, we formulate and justify the robustness that has been observed in practical implementations of those frameworks. Inspired by our theoretical results, we propose a pipeline for constructing shape difference operators on point clouds and show numerically that the results are robust and informative. In particular, we show that both the shape difference operators and the derived areas of highest distortion are stable with respect to changes in shape representation and change of scale. Remarkably, this is in contrast with the well-known instability of the eigenfunctions of the Laplace-Beltrami operator computed on point clouds compared to those obtained on triangle meshes.