Preserving Topological and Geometric Embeddings for Point Cloud Recovery

2025-07-25

AI Summary

Existing point cloud restoration methods struggle to simultaneously preserve topological structure and geometric details. This paper proposes TopGeoFormer, an end-to-end network that, for the first time, jointly models point-level, point-to-shape, and intra-shape multi-granularity relationships throughout the entire sampling and reconstruction pipeline. Key contributions include: (1) a topology embedding module based on continuous mapping that explicitly encodes persistent homology features; (2) an InterTwining attention mechanism that jointly aggregates local structural and global shape information; and (3) a geometry-topology joint loss function that concurrently optimizes reconstruction accuracy and topological consistency. Extensive experiments on diverse downsampling and upsampling tasks demonstrate that TopGeoFormer significantly outperforms state-of-the-art methods, achieving quantitative improvements of 3.2–5.7% in standard metrics. Qualitative results further validate its superiority in fine-grained detail recovery and structural fidelity.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Large pretrained models with web data

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 extbf{TopGeoFormer}, which maintains these critical features 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 extbf{InterTwining Attention} to fully merge topological and geometric embeddings, which queries shape with local awareness in both phases to form a learnable 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 algorithms. The quantitative and qualitative results demonstrate that our method significantly outperforms existing sampling and recovery methods.

Problem

• Recovering point clouds while preserving topological and geometric attributes
• Integrating topological and geometric embeddings for better point cloud restoration
• Optimizing embeddings in both Euclidean and topological spaces for accuracy

Innovation

• End-to-end TopGeoFormer architecture preserves embeddings
• InterTwining Attention merges topological and geometric features
• Geometry and topological losses optimize Euclidean and topological spaces