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.

翻译：本文提出了一种基于单纯复形的多维持续层拉普拉斯算子（MPSL）框架，用于图像分析。该方法的提出动机在于常用降维技术（如主成分分析（PCA））对降维维度选择的高度敏感性。不同于选择单一降维维度或跨维度平均结果，我们充分利用多个降维维度的互补优势。在给定维度下，图像样本被视为单纯复形，并利用持续层拉普拉斯算子为单个图像样本提取多尺度局部拓扑谱表示。所得谱的统计摘要随后跨尺度和维度进行聚合，形成多尺度多维图像表示。我们使用标准分类协议在COIL20和ETH80图像数据集上评估所提出的框架。实验结果表明，所提出的方法在广泛的降维维度范围内表现出更稳定的性能，并在中等维度范围内相对基于PCA的基线方法实现了持续的性能提升。