Point Cloud Upsampling with Geometric Algebra Driven Inverse Heat Dissipation

摘要

Point cloud upsampling is crucial for 3D reconstruction, with recent research significantly benefitting from the advances in deep learning technologies. The majority of existing methods, which focus on a sequence of processes including feature extraction, augmentation, and the reconstruction of coordinates, encounter significant challenges in interpreting the geometric attributes they uncover, particularly with respect to the intricacies of transitioning feature dimensionality. In this paper, we delve deeper into modeling Partial Differential Equations (PDEs) specifically tailored for the inverse heat dissipation process in dense point clouds. Our goal is to detect gradients within the dense point cloud data distribution and refine the accuracy of interpolated points' positions along with their complex geometric nuances through a systematic iterative approximation method. Simultaneously, we adopt multivectors from geometric algebra as the primary tool for representing the geometric characteristics of point clouds, moving beyond the conventional vector space representations. The use of geometric products of multivectors enables us to capture the complex relationships between scalars, vectors, and their components more effectively. This methodology not only offers a robust framework for depicting the geometric features of point clouds but also enhances our modeling capabilities for inverse heat dissipation PDEs. Through both qualitative and quantitative assessments, we demonstrate that our results significantly outperform existing state-of-the-art techniques in terms of widely recognized point cloud evaluation metrics and 3D visual reconstruction fidelity.