On the optimality of shape and data representation in the spectral domain

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA

Shape Generation using Spatially Partitioned Point Clouds

We propose a method to generate 3D shapes using point clouds. Given a point-cloud representation of a 3D shape, our method builds a kd-tree to spatially partition the points. This orders them consistently across all shapes, resulting in reasonably good correspondences across all shapes. We then use PCA analysis to derive a linear shape basis across the spatially partitioned points, and optimize the point ordering by iteratively minimizing the PCA reconstruction error. Even with the spatial sorting, the point clouds are inherently noisy and the resulting distribution over the shape coefficients can be highly multi-modal. We propose to use the expressive power of neural networks to learn a distribution over the shape coefficients in a generative-adversarial framework. Compared to 3D shape generative models trained on voxel-representations, our point-based method is considerably more light-weight and scalable, with little loss of quality. It also outperforms simpler linear factor models such as Probabilistic PCA, both qualitatively and quantitatively, on a number of categories from the ShapeNet dataset. Furthermore, our method can easily incorporate other point attributes such as normal and color information, an additional advantage over voxel-based representations.Comment: To appear at BMVC 201

Efficient Deformable Shape Correspondence via Kernel Matching

We 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. Surprisingly, in spite of the highly non-convex nature of the resulting quadratic assignment problem, our method converges to a semantically meaningful and continuous mapping in most of our experiments, and scales well. We provide preliminary theoretical analysis and several interpretations of the method.Comment: Accepted for oral presentation at 3DV 2017, including supplementary materia

Functional correspondence by matrix completion

In this paper, we consider the problem of finding dense intrinsic correspondence between manifolds using the recently introduced functional framework. We pose the functional correspondence problem as matrix completion with manifold geometric structure and inducing functional localization with the $L_1$ norm. We discuss efficient numerical procedures for the solution of our problem. Our method compares favorably to the accuracy of state-of-the-art correspondence algorithms on non-rigid shape matching benchmarks, and is especially advantageous in settings when only scarce data is available

On Nonrigid Shape Similarity and Correspondence

An important operation in geometry processing is finding the correspondences between pairs of shapes. The Gromov-Hausdorff distance, a measure of dissimilarity between metric spaces, has been found to be highly useful for nonrigid shape comparison. Here, we explore the applicability of related shape similarity measures to the problem of shape correspondence, adopting spectral type distances. We propose to evaluate the spectral kernel distance, the spectral embedding distance and the novel spectral quasi-conformal distance, comparing the manifolds from different viewpoints. By matching the shapes in the spectral domain, important attributes of surface structure are being aligned. For the purpose of testing our ideas, we introduce a fully automatic framework for finding intrinsic correspondence between two shapes. The proposed method achieves state-of-the-art results on the Princeton isometric shape matching protocol applied, as usual, to the TOSCA and SCAPE benchmarks