Preserving Topological and Geometric Embeddings for Point Cloud Recovery

TL;DR

The paper tackles point cloud recovery with a focus on preserving both topological and geometric structure. It introduces TopGeoFormer, an end-to-end architecture featuring Down-Preservation, InterTwining Attention, and two-stage Up-Preservation, augmented by geometry and topological constraint losses to supervise multi-resolution reconstructions. The method achieves state-of-the-art performance on recovery and sampling benchmarks, with strong ablations confirming the importance of topology-aware embeddings and joint optimization. The work also emphasizes efficiency, showing favorable complexity and runtime characteristics suitable for remote or embedded deployment.

Abstract

Recovering point clouds involves the sequential process of sampling and restoration, yet existing methods struggle to effectively leverage both topological and geometric attributes. To address this, we propose an end-to-end architecture named \textbf{TopGeoFormer}, which maintains these critical properties throughout the sampling and restoration phases. First, we revisit traditional feature extraction techniques to yield topological embedding using a continuous mapping of relative relationships between neighboring points, and integrate it in both phases for preserving the structure of the original space. Second, we propose the \textbf{InterTwining Attention} to fully merge topological and geometric embeddings, which queries shape with local awareness in both phases to form a learnable 3D shape context facilitated with point-wise, point-shape-wise, and intra-shape features. Third, we introduce a full geometry loss and a topological constraint loss to optimize the embeddings in both Euclidean and topological spaces. The geometry loss uses inconsistent matching between coarse-to-fine generations and targets for reconstructing better geometric details, and the constraint loss limits embedding variances for better approximation of the topological space. In experiments, we comprehensively analyze the circumstances using the conventional and learning-based sampling/upsampling/recovery algorithms. The quantitative and qualitative results demonstrate that our method significantly outperforms existing sampling and recovery methods.

Figures (6)

• Figure 1: Overview of an aircraft and front views of recovered details. More details such as continuity and curvature (propeller, fuel tank, and wing) are preserved by our method.
• Figure 2: Architecture of our TopGeoFormer (a) consists of a Down-Preservation (DP) phase and two cascaded Up-Preservation (UP) phases, to preserve both topological and geometric attributes. (b) The learnable 2D shape code $S$ is (c) intertwined with latent embeddings to reflect tangent-space structures. (d) Assisted with ITA, UPA preserves similar features for restoration.
• Figure 3: Qualitative visualizations. (a-c) guitar, piano, and plant in the random set. (d-e) chair and table in the partial set. Both our sampling (DP) and recovery methods preserve plausible details and overall shape.
• Figure 4: Recovery comparison on large scenes.
• Figure 5: Sampling precisions. Track fairings are preserved.