Extracting a functional representation from a dictionary for non-rigid
  shape matching
Michele Colombo Giacomo Boracchi Simone Melzi
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.