arXiv 2407.048442024-07-05cs.CVcs.AI

神经变分流：一种用于量化点云几何的聚合表示

Neural varifolds: an aggregate representation for quantifying the geometry of point clouds

School of Electronics and Computer Science, University of Southampton（南安普顿大学电子与计算机科学学院）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Institut Camille Jordan, Université Claude Bernard Lyon 1（里昂第一大学卡米尔·约旦研究所）

Juheon Lee, Xiaohao Cai, Carola-Bibian Schönlieb, Simon Masnou

备注：: The first author, Juheon Lee, is an unaffiliated, independent researcher. This work is a personal endeavor, unrelated to his current job

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AI总结：

本文提出神经变分流表示，通过点位置与切空间上的测度刻画点云表面几何，并利用神经网络与神经正切核计算变分流范数，在形状匹配、少样本分类和重建任务中取得优越或竞争性表现。

摘要：

点云因其对基于表面的几何结构具有精细且紧凑的表示能力，成为现实物体（如LiDAR和Kinect中）常用的三维表示形式。近年来，一些方法通过将基于深度学习的技术与几何保真度度量（如Chamfer距离和Wasserstein度量等最优传输代价）相结合来刻画点云的几何特征。本文在这一研究领域中提出了一种新的表面几何刻画方法，即点云的神经变分流（neural varifold）表示。在该表示中，表面被表示为点云中点位置和切空间上的测度/分布。变分流表示不仅通过基于流形的判别来量化点云的表面几何，还借助组合积空间捕捉表面上细微的几何一致性。本研究提出了神经变分流算法，利用点云上的神经网络及其神经正切核（neural tangent kernel）表示来计算两个点云之间的变分流范数。所提出的神经变分流在三个备受关注的任务上进行了评估——形状匹配、少样本形状分类和形状重建。详细的评估以及与最先进方法的比较表明，所提出的通用神经变分流在形状匹配和少样本形状分类上表现优越，在形状重建上具有竞争力。

英文摘要：

Point clouds are popular 3D representations for real-life objects (such as in LiDAR and Kinect) due to their detailed and compact representation of surface-based geometry. Recent approaches characterise the geometry of point clouds by bringing deep learning based techniques together with geometric fidelity metrics such as optimal transportation costs (e.g., Chamfer and Wasserstein metrics). In this paper, we propose a new surface geometry characterisation within this realm, namely a neural varifold representation of point clouds. Here the surface is represented as a measure/distribution over both point positions and tangent spaces of point clouds. The varifold representation quantifies not only the surface geometry of point clouds through the manifold-based discrimination, but also subtle geometric consistencies on the surface due to the combined product space. This study proposes neural varifold algorithms to compute the varifold norm between two point clouds using neural networks on point clouds and their neural tangent kernel representations. The proposed neural varifold is evaluated on three different sought-after tasks -- shape matching, few-shot shape classification and shape reconstruction. Detailed evaluation and comparison to the state-of-the-art methods demonstrate that the proposed versatile neural varifold is superior in shape matching and few-shot shape classification, and is competitive for shape reconstruction.

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