Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

Spectral Generalized Multi-Dimensional Scaling

Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances between each pair of points in the set. Distances in the target space can be computed analytically in this setting. Generalized MDS is an extension that allows mapping one metric space into another, that is, multidimensional scaling into target spaces in which distances are evaluated numerically rather than analytically. Here, we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition, where the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS procedure enables efficient embedding by implicitly incorporating smoothness of the mapping into the problem, thereby substantially reducing the complexity involved in its solution while practically overcoming its non-convex nature. The method is compared to existing techniques that compute dense correspondence between shapes. Numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state-of-the-art approaches

A Low-Dimensional Representation for Robust Partial Isometric Correspondences Computation

Intrinsic isometric shape matching has become the standard approach for pose invariant correspondence estimation among deformable shapes. Most existing approaches assume global consistency, i.e., the metric structure of the whole manifold must not change significantly. While global isometric matching is well understood, only a few heuristic solutions are known for partial matching. Partial matching is particularly important for robustness to topological noise (incomplete data and contacts), which is a common problem in real-world 3D scanner data. In this paper, we introduce a new approach to partial, intrinsic isometric matching. Our method is based on the observation that isometries are fully determined by purely local information: a map of a single point and its tangent space fixes an isometry for both global and the partial maps. From this idea, we develop a new representation for partial isometric maps based on equivalence classes of correspondences between pairs of points and their tangent spaces. From this, we derive a local propagation algorithm that find such mappings efficiently. In contrast to previous heuristics based on RANSAC or expectation maximization, our method is based on a simple and sound theoretical model and fully deterministic. We apply our approach to register partial point clouds and compare it to the state-of-the-art methods, where we obtain significant improvements over global methods for real-world data and stronger guarantees than previous heuristic partial matching algorithms.Comment: 17 pages, 12 figure

Geometric deep learning for shape analysis: extending deep learning techniques to non-Euclidean manifolds

The past decade in computer vision research has witnessed the re-emergence of artificial neural networks (ANN), and in particular convolutional neural network (CNN) techniques, allowing to learn powerful feature representations from large collections of data. Nowadays these techniques are better known under the umbrella term deep learning and have achieved a breakthrough in performance in a wide range of image analysis applications such as image classification, segmentation, and annotation. Nevertheless, when attempting to apply deep learning paradigms to 3D shapes one has to face fundamental differences between images and geometric objects. The main difference between images and 3D shapes is the non-Euclidean nature of the latter. This implies that basic operations, such as linear combination or convolution, that are taken for granted in the Euclidean case, are not even well defined on non-Euclidean domains. This happens to be the major obstacle that so far has precluded the successful application of deep learning methods on non-Euclidean geometric data. The goal of this thesis is to overcome this obstacle by extending deep learning tecniques (including, but not limiting to CNNs) to non-Euclidean domains. We present different approaches providing such extension and test their effectiveness in the context of shape similarity and correspondence applications. The proposed approaches are evaluated on several challenging experiments, achieving state-of-the- art results significantly outperforming other methods. To the best of our knowledge, this thesis presents different original contributions. First, this work pioneers the generalization of CNNs to discrete manifolds. Second, it provides an alternative formulation of the spectral convolution operation in terms of the windowed Fourier transform to overcome the drawbacks of the Fourier one. Third, it introduces a spatial domain formulation of convolution operation using patch operators and several ways of their construction (geodesic, anisotropic diffusion, mixture of Gaussians). Fourth, at the moment of publication the proposed approaches achieved state-of-the-art results in different computer graphics and vision applications such as shape descriptors and correspondence