从特征学习到谱基学习：一个统一且灵活的高效鲁棒形状匹配框架
From Feature Learning to Spectral Basis Learning: A Unifying and Flexible Framework for Efficient and Robust Shape Matching
把函数映射流程中固定的拉普拉斯谱基换成可学习的抑制函数，使特征提取与基函数端到端联合优化，并用热扩散模块替代昂贵的求解器；做非刚性三维形状匹配、等距/拓扑噪声鲁棒性研究的人可以直接复用这一「谱基即谱卷积滤波器」的视角。
原文依据 we propose the first unsupervised spectral basis learning method for robust non-rigid 3D shape matching
摘要译文
由模型翻译 · 术语统一 · 请对照原文核对关键结论形状匹配是计算机图形学与视觉中的基础任务，深度函数映射已成为其中的主流范式。然而，现有方法主要通过约束逐点映射与函数映射来学习有信息量的特征表示，却忽视了谱基的优化——而谱基是函数映射流程中的关键组成部分。这一疏忽往往导致匹配结果次优。此外，许多现有方法依赖传统且耗时的函数映射求解器，带来显著的计算开销。为弥补这些不足，我们提出 Advanced Functional Maps，该框架用可学习的基函数替代固定基函数，从而推广了标准函数映射，并具有严格的理论保证。具体而言，谱基通过一组学习得到的抑制函数进行优化。在此基础上，我们提出了首个面向鲁棒非刚性三维形状匹配的无监督谱基学习方法，使特征提取与基函数能够联合、端到端地优化。我们的方法包含一种新颖的热扩散模块和一个无监督损失函数，并采用精简的架构，绕过了昂贵的求解器与辅助损失。大量实验表明，我们的方法显著优于最优的特征学习方法，尤其是在具有挑战性的非等距与拓扑噪声场景中，同时保持高效率。最后，我们揭示出优化基函数等价于谱卷积，其中抑制函数充当滤波器。这一洞见使得受谱图网络启发的表征增强成为可能，为未来研究开辟了新途径。我们的代码见 https://github.com/LuoFeifan77/Unsupervised-Spectral-Basis-Learning。
英文原文摘要 1861 字符
Shape matching is a fundamental task in computer graphics and vision, with deep functional maps becoming a prominent paradigm. However, existing methods primarily focus on learning informative feature representations by constraining pointwise and functional maps, while neglecting the optimization of the spectral basis--a critical component of the functional map pipeline. This oversight often leads to suboptimal matching results. Furthermore, many current approaches rely on conventional, time-consuming functional map solvers, incurring significant computational overhead. To bridge these gaps, we introduce Advanced Functional Maps, a framework that generalizes standard functional maps by replacing fixed basis functions with learnable ones, supported by rigorous theoretical guarantees. Specifically, the spectral basis is optimized through a set of learned inhibition functions. Building on this, we propose the first unsupervised spectral basis learning method for robust non-rigid 3D shape matching, enabling the joint, end-to-end optimization of feature extraction and basis functions. Our approach incorporates a novel heat diffusion module and an unsupervised loss function, alongside a streamlined architecture that bypasses expensive solvers and auxiliary losses. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art feature-learning approaches, particularly in challenging non-isometric and topological noise scenarios, while maintaining high efficiency. Finally, we reveal that optimizing basis functions is equivalent to spectral convolution, where inhibition functions act as filters. This insight enables enhanced representations inspired by spectral graph networks, opening new avenues for future research. Our code is available at https://github.com/LuoFeifan77/Unsupervised-Spectral-Basis-Learning.