Multi-dimensional Persistent Sheaf Laplacians for Image Analysis
This work addresses a specific problem in image analysis for researchers by providing a more robust alternative to PCA, though it is incremental as it builds on existing topological methods.
The authors tackled the sensitivity of dimensionality reduction methods like PCA to the choice of reduced dimension by proposing a multi-dimensional persistent sheaf Laplacian framework that aggregates topological features across scales and dimensions, achieving more stable performance and consistent improvements over PCA-based baselines on COIL20 and ETH80 datasets.
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.