Metrics of the Laplace-Beltrami eigenfunctions for 2D shape matching

Assuming that a 1D curve can be represented as a graph embedded in a 2D-space, the metrics of the eigenfunctions of the weighted graph-Laplacian and diffusion operator of that graph are then a representation of the shape of that curve with invariance to rotation, scale, and translation. The diffusion operator is said to preserve the local proximity between data points by constructing a representation for the underlying manifold by an approximation of the Laplace-Beltrami operator acting on the graph of this curve. This work examines 2D shape clustering problems using a spectral metric of the Laplace-Betrami eigenfunctions for shape analysis of closed curves. Results demonstrate that the spectral metrics allow for good class separation over multiple targets with noise.