Multi-directional Geodesic Neural Networks via Equivariant Convolution

We propose a novel approach for performing convolution of signals on curved surfaces and show its utility in a variety of geometric deep learning applications. Key to our construction is the notion of directional functions defined on the surface, which extend the classic real-valued signals and which can be naturally convolved with with real-valued template functions. As a result, rather than trying to fix a canonical orientation or only keeping the maximal response across all alignments of a 2D template at every point of the surface, as done in previous works, we show how information across all rotations can be kept across different layers of the neural network. Our construction, which we call multi-directional geodesic convolution, or directional convolution for short, allows, in particular, to propagate and relate directional information across layers and thus different regions on the shape. We first define directional convolution in the continuous setting, prove its key properties and then show how it can be implemented in practice, for shapes represented as triangle meshes. We evaluate directional convolution in a wide variety of learning scenarios ranging from classification of signals on surfaces, to shape segmentation and shape matching, where we show a significant improvement over several baselines

Smoothed Graph Contrastive Learning via Seamless Proximity Integration

Graph contrastive learning (GCL) aligns node representations by classifying node pairs into positives and negatives using a selection process that typically relies on establishing correspondences within two augmented graphs. The conventional GCL approaches incorporate negative samples uniformly in the contrastive loss, resulting in the equal treatment negative nodes, regardless of their proximity to the true positive. In this paper, we present a Smoothed Graph Contrastive Learning model (SGCL), which leverages the geometric structure of augmented graphs to inject proximity information associated with positive/negative pairs in the contrastive loss, thus significantly regularizing the learning process. The proposed SGCL adjusts the penalties associated with node pairs in the contrastive loss by incorporating three distinct smoothing techniques that result in proximity aware positives and negatives. To enhance scalability for large-scale graphs, the proposed framework incorporates a graph batch-generating strategy that partitions the given graphs into multiple subgraphs, facilitating efficient training in separate batches. Through extensive experimentation in the unsupervised setting on various benchmarks, particularly those of large scale, we demonstrate the superiority of our proposed framework against recent baselines.Comment: 17 page

NCP: Neural Correspondence Prior for Effective Unsupervised Shape Matching

We present Neural Correspondence Prior (NCP), a new paradigm for computing correspondences between 3D shapes. Our approach is fully unsupervised and can lead to high-quality correspondences even in challenging cases such as sparse point clouds or non-isometric meshes, where current methods fail. Our first key observation is that, in line with neural priors observed in other domains, recent network architectures on 3D data, even without training, tend to produce pointwise features that induce plausible maps between rigid or non-rigid shapes. Secondly, we show that given a noisy map as input, training a feature extraction network with the input map as supervision tends to remove artifacts from the input and can act as a powerful correspondence denoising mechanism, both between individual pairs and within a collection. With these observations in hand, we propose a two-stage unsupervised paradigm for shape matching by (i) performing unsupervised training by adapting an existing approach to obtain an initial set of noisy matches, and (ii) using these matches to train a network in a supervised manner. We demonstrate that this approach significantly improves the accuracy of the maps, especially when trained within a collection. We show that NCP is data-efficient, fast, and achieves state-of-the-art results on many tasks. Our code can be found online: https://github.com/pvnieo/NCP.Comment: NeurIPS 2022, 10 pages, 9 figure

Understanding and Improving Features Learned in Deep Functional Maps

Deep functional maps have recently emerged as a successful paradigm for non-rigid 3D shape correspondence tasks. An essential step in this pipeline consists in learning feature functions that are used as constraints to solve for a functional map inside the network. However, the precise nature of the information learned and stored in these functions is not yet well understood. Specifically, a major question is whether these features can be used for any other objective, apart from their purely algebraic role in solving for functional map matrices. In this paper, we show that under some mild conditions, the features learned within deep functional map approaches can be used as point-wise descriptors and thus are directly comparable across different shapes, even without the necessity of solving for a functional map at test time. Furthermore, informed by our analysis, we propose effective modifications to the standard deep functional map pipeline, which promote structural properties of learned features, significantly improving the matching results. Finally, we demonstrate that previously unsuccessful attempts at using extrinsic architectures for deep functional map feature extraction can be remedied via simple architectural changes, which encourage the theoretical properties suggested by our analysis. We thus bridge the gap between intrinsic and extrinsic surface-based learning, suggesting the necessary and sufficient conditions for successful shape matching. Our code is available at https://github.com/pvnieo/clover.Comment: 16 pages, 8 figures, 8 tables, to be published in 2023 The IEEE Conference on Computer Vision and Pattern Recognition (CVPR

Shape Non-rigid Kinematics (SNK): A Zero-Shot Method for Non-Rigid Shape Matching via Unsupervised Functional Map Regularized Reconstruction

We present Shape Non-rigid Kinematics (SNK), a novel zero-shot method for non-rigid shape matching that eliminates the need for extensive training or ground truth data. SNK operates on a single pair of shapes, and employs a reconstruction-based strategy using an encoder-decoder architecture, which deforms the source shape to closely match the target shape. During the process, an unsupervised functional map is predicted and converted into a point-to-point map, serving as a supervisory mechanism for the reconstruction. To aid in training, we have designed a new decoder architecture that generates smooth, realistic deformations. SNK demonstrates competitive results on traditional benchmarks, simplifying the shape-matching process without compromising accuracy. Our code can be found online: https://github.com/pvnieo/SNKComment: NeurIPS 2023, 10 pages, 9 figure

Joint Symmetry Detection and Shape Matching for Non-Rigid Point Cloud

Despite the success of deep functional maps in non-rigid 3D shape matching, there exists no learning framework that models both self-symmetry and shape matching simultaneously. This is despite the fact that errors due to symmetry mismatch are a major challenge in non-rigid shape matching. In this paper, we propose a novel framework that simultaneously learns both self symmetry as well as a pairwise map between a pair of shapes. Our key idea is to couple a self symmetry map and a pairwise map through a regularization term that provides a joint constraint on both of them, thereby, leading to more accurate maps. We validate our method on several benchmarks where it outperforms many competitive baselines on both tasks.Comment: Under Review. arXiv admin note: substantial text overlap with arXiv:2110.0299

Generalizable Local Feature Pre-training for Deformable Shape Analysis

Transfer learning is fundamental for addressing problems in settings with little training data. While several transfer learning approaches have been proposed in 3D, unfortunately, these solutions typically operate on an entire 3D object or even scene-level and thus, as we show, fail to generalize to new classes, such as deformable organic shapes. In addition, there is currently a lack of understanding of what makes pre-trained features transferable across significantly different 3D shape categories. In this paper, we make a step toward addressing these challenges. First, we analyze the link between feature locality and transferability in tasks involving deformable 3D objects, while also comparing different backbones and losses for local feature pre-training. We observe that with proper training, learned features can be useful in such tasks, but, crucially, only with an appropriate choice of the receptive field size. We then propose a differentiable method for optimizing the receptive field within 3D transfer learning. Jointly, this leads to the first learnable features that can successfully generalize to unseen classes of 3D shapes such as humans and animals. Our extensive experiments show that this approach leads to state-of-the-art results on several downstream tasks such as segmentation, shape correspondence, and classification. Our code is available at \url{https://github.com/pvnieo/vader}.Comment: 16 pages, 14 figures, 7 tables, to be published in The IEEE Conference on Computer Vision and Pattern Recognition (CVPR

AtomSurf : Surface Representation for Learning on Protein Structures

Recent advancements in Cryo-EM and protein structure prediction algorithms have made large-scale protein structures accessible, paving the way for machine learning-based functional annotations.The field of geometric deep learning focuses on creating methods working on geometric data. An essential aspect of learning from protein structures is representing these structures as a geometric object (be it a grid, graph, or surface) and applying a learning method tailored to this representation. The performance of a given approach will then depend on both the representation and its corresponding learning method. In this paper, we investigate representing proteins as $\textit{3D mesh surfaces}$ and incorporate them into an established representation benchmark. Our first finding is that despite promising preliminary results, the surface representation alone does not seem competitive with 3D grids. Building on this, we introduce a synergistic approach, combining surface representations with graph-based methods, resulting in a general framework that incorporates both representations in learning. We show that using this combination, we are able to obtain state-of-the-art results across $\textit{all tested tasks}$. Our code and data can be found online: https://github.com/Vincentx15/atom2D .Comment: 10 page

SRFeat: Learning Locally Accurate and Globally Consistent Non-Rigid Shape Correspondence

In this work, we present a novel learning-based framework that combines the local accuracy of contrastive learning with the global consistency of geometric approaches, for robust non-rigid matching. We first observe that while contrastive learning can lead to powerful point-wise features, the learned correspondences commonly lack smoothness and consistency, owing to the purely combinatorial nature of the standard contrastive losses. To overcome this limitation we propose to boost contrastive feature learning with two types of smoothness regularization that inject geometric information into correspondence learning. With this novel combination in hand, the resulting features are both highly discriminative across individual points, and, at the same time, lead to robust and consistent correspondences, through simple proximity queries. Our framework is general and is applicable to local feature learning in both the 3D and 2D domains. We demonstrate the superiority of our approach through extensive experiments on a wide range of challenging matching benchmarks, including 3D non-rigid shape correspondence and 2D image keypoint matching.Comment: 3DV 2022. Code and data: https://github.com/craigleili/SRFea