用多维拓扑谱分析图像，比传统降维方法更稳定可靠。
Multi-dimensional Persistent Sheaf Laplacians for Image Analysis
- 在单纯复形上构建多维持久层拉普拉斯算子，提取图像局部拓扑特征
- 在中等降维范围内相比PCA基线性能提升，且跨维度表现更稳定
- 适合需要高鲁棒性图像表示的场景，如小样本或噪声干扰环境
我们提出一种在单纯复形上用于图像分析的多维持久层拉普拉斯（MPSL）框架。该方法旨在克服主成分分析（PCA）等降维技术对选定降维维度的高度敏感性。不同于选择单一维度或平均多个维度的结果，我们利用多个降维维度的互补优势。在特定维度下，将图像样本视为单纯复形，并使用持久层拉普拉斯算子为每个图像样本提取多尺度局部拓扑谱表示。随后，将这些谱的统计信息在不同尺度和维度间聚合，形成多尺度多维图像表示。我们在COIL20和ETH80图像数据集上采用标准分类协议进行评估。实验结果表明，所提方法在宽范围降维条件下表现出更稳定的性能，并在中等降维区间持续优于基于PCA的基线方法。
原文摘要 · Abstract (English)
We propose a multi-dimensional persistent sheaf Laplacian (MPSL) framework on simplicial complexes for image analysis. The proposed method is motivated by the strong sensitivity of commonly used dimensionality reduction techniques, such as principal component analysis (PCA), to the choice of reduced dimension. Rather than selecting a single reduced dimension or averaging results across dimensions, we exploit complementary advantages of multiple reduced dimensions. At a given dimension, image samples are regarded as simplicial complexes, and persistent sheaf Laplacians are utilized to extract a multiscale localized topological spectral representation for individual image samples. Statistical summaries of the resulting spectra are then aggregated across scales and dimensions to form multiscale multi-dimensional image representations. We evaluate the proposed framework on the COIL20 and ETH80 image datasets using standard classification protocols. Experimental results show that the proposed method provides more stable performance across a wide range of reduced dimensions and achieves consistent improvements to PCA-based baselines in moderate dimensional regimes.
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