Efficient Deformable Shape Correspondence via Kernel Matching

Abstract

Figure 1: Qualitative examples on FAUST models (left), SHREC'16 (middle) and SCAPE (right). In the SHREC experiment, the green parts mark where no correspondence was found. Notice how those areas are close to the parts that are hidden in the other model. The missing matches (marked in black) in the SCAPE experiment are an artifact due to the multiscale approach. AbstractWe present a method to match three dimensional shapes under non-isometric deformations, topology changes and partiality. We formulate the problem as matching between a set of pair-wise and point-wise descriptors, imposing a continuity prior on the mapping, and propose a projected descent optimization procedure inspired by difference of convex functions (DC) programming.

Functional Maps Representation On Product Manifolds

“…The refinement process described above simultaneously improves the correspondence and reduces the support of the density around the most likely bijective map. This is similar in spirit to the kernel matching approaches of [VLR*17, VLB*17], however, with the additional step of ‘carving out’ the relevant portion throughout the iterations.…”

“…A similar approach was applied in [LRS*16] for 2D‐to‐3D matching. In [VLR*17], correspondence is formulated as kernel density estimation on the product manifold, interpreted as an alternating diffusion‐sharpening process in [VLB*17]. A product between more than two shapes is considered in [CRA*17], but the resulting optimization problem is restricted to yield only sparse correspondences.…”

Spectral coarsening of geometric operators

“…For example in , the authors used the standard QSlim [Garland and Heckbert 1997] to simplify the meshes before matching them using PMF. Unfortunately, since standard appearance-based simplification methods can severely distort the spectral properties this can cause problems for spectral methods such as [Vestner et al 2017] both during matching between coarse domains and while propagating back to the dense ones. Instead our spectral-based coarsening, while not resulting in a mesh provides all the necessary information to apply a spectral technique via the eigen-pairs of the coarse operator, and moreover provides an accurate way to propagate the information back to the original shapes.…”

“…Our main observation is that if the original function space is preserved during the coarsening, less error will be introduced when moving across domains. We tested this approach by evaluating a combination of our coarsening with [Vestner et al 2017] and compared it to several baselines on a challenging non-rigid non-isometric dataset containing shapes from the SHREC 2007 contest [Giorgi et al 2007], and evaluated the results using the landmarks and evaluation protocol from [Kim et al 2011] (please see the details on both the exact parameters and the evaluation in the Appendix). Figure 26 shows the accuracy of several methods, both that directly operate on the dense meshes [Kim et al 2011;Nogneng and Ovsjanikov 2017] as well as using kernel matching [Vestner et al 2017] with QSlim and with our coarsening.…”

“…We tested this approach by evaluating a combination of our coarsening with [Vestner et al 2017] and compared it to several baselines on a challenging non-rigid non-isometric dataset containing shapes from the SHREC 2007 contest [Giorgi et al 2007], and evaluated the results using the landmarks and evaluation protocol from [Kim et al 2011] (please see the details on both the exact parameters and the evaluation in the Appendix). Figure 26 shows the accuracy of several methods, both that directly operate on the dense meshes [Kim et al 2011;Nogneng and Ovsjanikov 2017] as well as using kernel matching [Vestner et al 2017] with QSlim and with our coarsening. The results in Figure 26 show that our approach produces maps with comparable quality or superior quality to existing methods on these non-isometric shape pairs, and results in significant improvement compared to coarsening the shapes with QSlim.…”