Memory-Scalable and Simplified Functional Map Learning

Deep functional maps have emerged in recent years as a prominent learning-based framework for non-rigid shape matching problems. While early methods in this domain only focused on learning in the functional domain, the latest techniques have demonstrated that by promoting consistency between functional and pointwise maps leads to significant improvements in accuracy. Unfortunately, existing approaches rely heavily on the computation of large dense matrices arising from soft pointwise maps, which compromises their efficiency and scalability. To address this limitation, we introduce a novel memory-scalable and efficient functional map learning pipeline. By leveraging the specific structure of functional maps, we offer the possibility to achieve identical results without ever storing the pointwise map in memory. Furthermore, based on the same approach, we present a differentiable map refinement layer adapted from an existing axiomatic refinement algorithm. Unlike many functional map learning methods, which use this algorithm at a post-processing step, ours can be easily used at train time, enabling to enforce consistency between the refined and initial versions of the map. Our resulting approach is both simpler, more efficient and more numerically stable, by avoiding differentiation through a linear system, while achieving close to state-of-the-art results in challenging scenarios.

To Supervise or Not to Supervise: Understanding and Addressing the Key Challenges of 3D Transfer Learning

Transfer learning has long been a key factor in the advancement of many fields including 2D image analysis. Unfortunately, its applicability in 3D data processing has been relatively limited. While several approaches for 3D transfer learning have been proposed in recent literature, with contrastive learning gaining particular prominence, most existing methods in this domain have only been studied and evaluated in limited scenarios. Most importantly, there is currently a lack of principled understanding of both when and why 3D transfer learning methods are applicable. Remarkably, even the applicability of standard supervised pre-training is poorly understood. In this work, we conduct the first in-depth quantitative and qualitative investigation of supervised and contrastive pre-training strategies and their utility in downstream 3D tasks. We demonstrate that layer-wise analysis of learned features provides significant insight into the downstream utility of trained networks. Informed by this analysis, we propose a simple geometric regularization strategy, which improves the transferability of supervised pre-training. Our work thus sheds light onto both the specific challenges of 3D transfer learning, as well as strategies to overcome them.

PoNQ: a Neural QEM-based Mesh Representation

Although polygon meshes have been a standard representation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning-based applications. In this work, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neural shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the boundary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target surface alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.

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/SNK

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.

Back to 3D: Few-Shot 3D Keypoint Detection with Back-Projected 2D Features

With the immense growth of dataset sizes and computing resources in recent years, so-called foundation models have become popular in NLP and vision tasks. In this work, we propose to explore foundation models for the task of keypoint detection on 3D shapes. A unique characteristic of keypoint detection is that it requires semantic and geometric awareness while demanding high localization accuracy. To address this problem, we propose, first, to back-project features from large pre-trained 2D vision models onto 3D shapes and employ them for this task. We show that we obtain robust 3D features that contain rich semantic information and analyze multiple candidate features stemming from different 2D foundation models. Second, we employ a keypoint candidate optimization module which aims to match the average observed distribution of keypoints on the shape and is guided by the back-projected features. The resulting approach achieves a new state of the art for few-shot keypoint detection on the KeyPointNet dataset, almost doubling the performance of the previous best methods.

Unsupervised Representation Learning for Diverse Deformable Shape Collections

We introduce a novel learning-based method for encoding and manipulating 3D surface meshes. Our method is specifically designed to create an interpretable embedding space for deformable shape collections. Unlike previous 3D mesh autoencoders that require meshes to be in a 1-to-1 correspondence, our approach is trained on diverse meshes in an unsupervised manner. Central to our method is a spectral pooling technique that establishes a universal latent space, breaking free from traditional constraints of mesh connectivity and shape categories. The entire process consists of two stages. In the first stage, we employ the functional map paradigm to extract point-to-point (p2p) maps between a collection of shapes in an unsupervised manner. These p2p maps are then utilized to construct a common latent space, which ensures straightforward interpretation and independence from mesh connectivity and shape category. Through extensive experiments, we demonstrate that our method achieves excellent reconstructions and produces more realistic and smoother interpolations than baseline approaches.

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 .

Spatially and Spectrally Consistent Deep Functional Maps

Cycle consistency has long been exploited as a powerful prior for jointly optimizing maps within a collection of shapes. In this paper, we investigate its utility in the approaches of Deep Functional Maps, which are considered state-of-the-art in non-rigid shape matching. We first justify that under certain conditions, the learned maps, when represented in the spectral domain, are already cycle consistent. Furthermore, we identify the discrepancy that spectrally consistent maps are not necessarily spatially, or point-wise, consistent. In light of this, we present a novel design of unsupervised Deep Functional Maps, which effectively enforces the harmony of learned maps under the spectral and the point-wise representation. By taking advantage of cycle consistency, our framework produces state-of-the-art results in mapping shapes even under significant distortions. Beyond that, by independently estimating maps in both spectral and spatial domains, our method naturally alleviates over-fitting in network training, yielding superior generalization performance and accuracy within an array of challenging tests for both near-isometric and non-isometric datasets. Codes are available at https://github.com/rqhuang88/Spatiallyand-Spectrally-Consistent-Deep-Functional-Maps.

VoroMesh: Learning Watertight Surface Meshes with Voronoi Diagrams

In stark contrast to the case of images, finding a concise, learnable discrete representation of 3D surfaces remains a challenge. In particular, while polygon meshes are arguably the most common surface representation used in geometry processing, their irregular and combinatorial structure often make them unsuitable for learning-based applications. In this work, we present VoroMesh, a novel and differentiable Voronoi-based representation of watertight 3D shape surfaces. From a set of 3D points (called generators) and their associated occupancy, we define our boundary representation through the Voronoi diagram of the generators as the subset of Voronoi faces whose two associated (equidistant) generators are of opposite occupancy: the resulting polygon mesh forms a watertight approximation of the target shape's boundary. To learn the position of the generators, we propose a novel loss function, dubbed VoroLoss, that minimizes the distance from ground truth surface samples to the closest faces of the Voronoi diagram which does not require an explicit construction of the entire Voronoi diagram. A direct optimization of the Voroloss to obtain generators on the Thingi32 dataset demonstrates the geometric efficiency of our representation compared to axiomatic meshing algorithms and recent learning-based mesh representations. We further use VoroMesh in a learning-based mesh prediction task from input SDF grids on the ABC dataset, and show comparable performance to state-of-the-art methods while guaranteeing closed output surfaces free of self-intersections.