Shape representation by metric interpolation

Abstract

Coordinates of vertices in a triangulated surface can be efficiently represented as a set of coefficients that multiply a given basis of functions. One such natural orthonormal basis is provided by the eigenfunctions of the Laplace-Beltrami operator of a given shape. The coefficients in this case are nothing but the result of the scalar inner product of the coordinates treated as a smooth function on the surface of the shape and the eigenfunctions that form the orthonormal basis. Keeping only the significant coefficients allows for efficient representation of a given shape under practical transformations. Selecting the regular metric for the construction of the Laplace-Beltrami operator we notice that while the general shape is preserved, important fine details are often washed out. At the other end, using a scale invariant metric for defining the operator and the corresponding basis, preserves the fine details at the potential expense of loosing the general structure of the shape. Here, we adopt the best of both worlds. By finding the right mix between scale invariant and a regular one we select the metric that serves as the best representation-basis generator for a given shape. We use the mean square error (MSE) to select the optimal space for shape representation, and compare the results to classical spectral shape representation techniques.