Shape Correspondence by Aligning Scale-invariant LBO Eigenfunctions

When matching non-rigid shapes, the regular or scale-invariant Laplace-Beltrami Operator (LBO) eigenfunctions could potentially serve as intrinsic descriptors which are invariant to isometric transformations. However, the computed eigenfunctions of two quasi-isometric surfaces could be substantially different. Such discrepancies include sign ambiguities and possible rotations and reflections within subspaces spanned by eigenfunctions that correspond to similar eigenvalues. Thus, without aligning the corresponding eigenspaces it is difficult to use the eigenfunctions as descriptors. Here, we propose to model the relative transformation between the eigenspaces of two quasi-isometric shapes using a band orthogonal matrix, as well as present a framework that aims to estimate this matrix. Estimating this transformation allows us to align the eigenfunctions of one shape with those of the other, that could then be used as intrinsic, consistent, and robust descriptors. To estimate the transformation we use an unsupervised spectral-net framework that uses descriptors given by the eigenfunctions of the scale-invariant version of the LBO. Then, using a spectral training mechanism, we find a band limited orthogonal matrix that aligns the two sets of eigenfunctions.