On the optimality of shape and data representation in the spectral domain

摘要: A proof of the optimality eigenfunctions Laplace-Beltrami operator (LBO) in repre- senting smooth functions on surfaces is provided and adapted to field applied shape data analysis. It based Courant-Fischer min-max principle our case. The theorem we present supports new trend geometry processing treating geometric structures by using their projection onto leading decomposition LBO. Utilization this result can be used for constructing numerically efficient algorithms process shapes spec- trum. We review a couple applications as possible practical usage cases proposed criteria. refer scale invariant metric, which also bending manifold. This novel pseudometric allows an LBO eigenspace surface defined. demonstrate efficiency intermediate defined interpolation be- tween regular one, representing while capturing both coarse fine details. Next, numerical acceleration technique classical scaling, member family flattening methods known multidimensional scaling (MDS). There, exploited efficiently approximate all geodesic distances between pairs points given thereby match compare almost isometric surfaces. Finally, revisit principal component analysis (PCA) definition coupling its variational form with Dirichlet energy By pairing PCA handle that go beyond scope observation set handled PCA.