Search

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.

3D acquisition pipeline delivers 3D digital models accurately representing real-world objects, improving the geometric accuracy and realism of virtual reconstructions. However, even after intensive clean-up, the captured models fall short of many of the requirements imposed by the downstream application, such as video-games, virtual reality, digital movies, etc. Often, the captured 3D model can only serve as a starting point for a cascade of subsequent phases, by either digital artists or geometry processing algorithms, such as a complete remeshing (or retopology), surface parameterization, and skinning for animation. In contrast, we propose a novel remeshing-by-matching approach, where we automatically combine the accurate 3D geometry of the captured model with the tessellation of a target pre-existing template which already satisfies all the professional requirements. At the core of this process, there is a matching strategy based on the functional mapping framework. To this end, we introduce a new set of basis functions designed for this context: termed Coordinates Manifold Harmonics (CMH). We evaluate this strategy (quantitatively and qualitatively) over models of different classes, obtaining a favourable comparison with existing methods.