Extracting a functional representation from a dictionary for non-rigid shape matching
This addresses a specific limitation in non-rigid shape matching for computer graphics applications, offering an incremental improvement over existing functional map methods.
The paper tackles the problem of inaccurate point-wise shape matching in tiny regions and protrusions by proposing a new functional basis called Principal Components of a Dictionary (PCD), which results in more accurate point-wise maps than the traditional Laplace-Beltrami basis on established benchmarks.
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.