Extracting a functional representation from a dictionary for non-rigid shape matching

Abstract

Shape matching is a fundamental problem in computer graphics with many applications. Functional maps translate the point-wise shape-matching problem into its functional counterpart and have inspired numerous solutions over the last decade. Nearly all the solutions based on functional maps rely on the eigenfunctions of the Laplace–Beltrami Operator (LB) to describe the functional spaces defined on the surfaces and then convert the functional correspondences into point-wise correspondences. However, this final step is often error-prone and inaccurate in tiny regions and protrusions, where the energy of LB does not uniformly cover the surface. We propose a new functional basis Principal Components of a Dictionary (PCD) to address such intrinsic limitation. PCD constructs an orthonormal basis from the Principal Component Analysis (PCA) of a dictionary of functions defined over the shape. These dictionaries can target specific properties of the final basis, such as achieving an even spreading of energy. Our experimental evaluation compares seven different dictionaries on established benchmarks, showing that PCD is suited to target different shape-matching scenarios, resulting in more accurate point-wise maps than the LB basis when used in the same pipeline. This evidence provides a promising alternative for improving correspondence estimation, confirming the power and flexibility of functional maps.