Essence of Two-Dimensional Principal Component Analysis and Its Generalization: Multi-dimensional PCA

Abstract

This paper examines the connection between two-dimensional principal component analysis (2DPCA) and traditional one-dimensional principal component analysis (PCA) and theoretically reveals the reason why 2DPCA outperforms PCA. Our finding provides new insights into the computation of 2DPCA and give a new equivalent algorithm for performing 2DPCA based on row vectors of original matrices. Based on the new algorithm, we extend the existing 2DPCA algorithm to its multi-dimensional case by developing a new feature extraction technique for multi-dimensional data called multi-dimensional principal component analysis (MDPCA). Different from 2DPCA and PCA, MDPCA is based on multi-dimensional data rather than 2D image matrices or 1D vectors so the range of PCA-based applications is significantly enlarged. The experimental results also demonstrate that MDPCA can extract more effective and robust multi-dimensional image features than 2DPCA.