SyncSpecCNN: Synchronized Spectral CNN for 3D Shape Segmentation Supplementary Material

A set of precomputed functional maps are used for SpecTN pretraining. We mentioned in the main paper that the functional map C from S to the average shape S̄ could be induced from the spatial correspondences between S and S̄, by the primal-dual relationship. Once we have the bases of S and S̄, as well as the rough spatial correspondences between them from the volumetric occupancy, the functional map can then be discovered by the approach proposed in [3]. To be specific, we use Bv to denote the volumetric reparametrization of graph Laplacian eigenbases B for each shape S, and use B̄v to denote the graph Laplacian eigenbases of S̄. Bv and B̄v both lie in the volumetric space and their spatial correspondence is natural to acquire. The functional map Cpre aligning Bv with B̄v could be computed through simple matrix multiplication Cpre = B̄ v Bv . The computed functional map will serve as supervision and SpecTN is pretrained to minimize the loss function ||C − Cpre||F . It is worth mentioning that, if the shapes under consideration are diverse in topology and geometry, i.e. shapes from different categories, aligning every shape to a single “average” shape might cause unwanted distortion. Therefore we leverage multiple “average” shapes {S̄i}i=1 and use a combination of their spectral domains as the canonical domain. Specifically, we assign each shape S to its closest “average” shape under some global similarity measurement (i.e. lightfield descriptor) and use {ai}i=1 to represent such assignment, namely ai = 1 if S is assigned to S̄i and ai = 0 otherwise. Also we use B̄vi to denote the spectral bases of S̄i. Then the functional mapCpre for each shape S could be computed through Cpre = [a1B̄v1 a2B̄v2 ... anB̄vn]Bv . The SpecTN is pretrained to predict a functional map which only synchronizes spectral domain of each shape to its most similar “average” shape.