Product Manifold Filter: Non-rigid Shape Correspondence via Kernel Density Estimation in the Product Space

Abstract

Many algorithms for the computation of correspondences between deformable shapes rely on some variant of nearest neighbor matching in a descriptor space. Such are, for example, various point-wise correspondence recovery algorithms used as a post-processing stage in the functional correspondence framework. Such frequently used techniques implicitly make restrictive assumptions (e.g., nearisometry) on the considered shapes and in practice suffer from lack of accuracy and result in poor surjectivity. We propose an alternative recovery technique capable of guaranteeing a bijective correspondence and producing significantly higher accuracy and smoothness. Unlike other methods our approach does not depend on the assumption that the analyzed shapes are isometric. We derive the proposed method from the statistical framework of kernel density estimation and demonstrate its performance on several challenging deformable 3D shape matching datasets.

Topological Function Optimization for Continuous Shape Matching

“…Note the improvement in the continuity in the maps after topological optimization. We stress that unlike previous works, such as [MCSK*17, VLR*17], we never enforce continuity of point‐to‐point maps. Instead, the optimization is done purely in the “functional domain” by minimizing Eq.…”

“…These techniques have benefited from the computational advances in solving large‐scale transport problems, especially using the Sinkhorn method under entropic regularization [Cut13,SDGP*15]. For example, several recent methods in this category [MCSK*17,VLR*17,VLB*17] have been proposed to efficiently find bijective maps, while promoting continuity by iteratively solving optimal transport problems. While related to our work, in these techniques, continuity is measured using point‐to‐point correspondence or via metric distortion, most commonly by minimizing variance with respect to a previous solution in an iterative scheme.…”

“…Moreover, our functional map improvement procedure is not well‐adapted to partial shapes and it would be interesting to see how it can be used jointly with objectives such as promoting bijectivity or partiality, or with existing approaches that promote continuity of pointwise maps, e.g. [MCSK*17, VLR*17]. In addition, it would be interesting to study questions of convergence of our optimization problems, depending on the choice and size of the basis, as well as stability with respect to changes in the shape, taking advantage of the theoretical stability guarantees available for persistence diagrams.…”

Deblurring and Denoising of Maps between Shapes

“…Vestner et al [VLB*16, VLR*17] recover a bijective vertex‐to‐vertex map by solving a linear assignment problem. While vertex‐to‐vertex bijections are beneficial for shapes with a similar triangulation, they highly depend on the tessellation of the input shapes.…”

Localized Manifold Harmonics for Spectral Shape Analysis

“…Similar to the previous experiments, the standard Laplacian eigenbasis may not be the best choice in the presence of fine details: the low‐pass nature of the spectral representation of the map, embodied in matrix , negatively affects the quality of the representation at a point‐wise level. Indeed, recovering a point‐to‐point map from a functional map is considered a difficult problem in itself [RMC15, VLR*17], and is at the heart of several applications dealing with maps.…”