Two-Dimensional Quaternion PCA and Sparse PCA

• Published in: IEEE transaction on neural networks and learning systems 2019-07, Vol.30 (7), p.2028-2042
• Main Authors: Xiao, Xiaolin, Zhou, Yicong
• Format: Article
• Language: English
• Subjects: 2D-principle component analysis (2D-PCA); 2D-quaternion PCA (2D-QPCA); 2D-quaternion sparse PCA (2D-QSPCA); Algorithms; Color; color face recognition; Color imagery; Covariance matrix; Eigenvectors; Equivalence; Face recognition; Feature extraction; Image color analysis; Image processing; Mathematical analysis; Mathematical model; Matrix methods; Outliers (statistics); partial occlusions; Pattern recognition; Principal component analysis; Principal components analysis; QPCA; Quartz crystal microbalance; Quaternions; Regression analysis; Regression models; Robustness (mathematics); Two dimensional displays; Two dimensional models
• ISSN: 2162-237X; 2162-2388
• DOI: 10.1109/TNNLS.2018.2872541

Benefited from quaternion representation that is able to encode the cross-channel correlation of color images, quaternion principle component analysis (QPCA) was proposed to extract features from color images while reducing the feature dimension. A quaternion covariance matrix (QCM) of input samples was constructed, and its eigenvectors were derived to find the solution of QPCA. However, eigen-decomposition leads to the fixed solution for the same input. This solution is susceptible to outliers and cannot be further optimized. To solve this problem, this paper proposes a novel quaternion ridge regression (QRR) model for two-dimensional QPCA (2D-QPCA). We mathematically prove that this QRR model is equivalent to the QCM model of 2D-QPCA. The QRR model is a general framework and is flexible to combine 2D-QPCA with other technologies or constraints to adapt different requirements of real-world applications. Including sparsity constraints, we then propose a quaternion sparse regression model for 2D-QSPCA to improve its robustness for classification. An alternating minimization algorithm is developed to iteratively learn the solution of 2D-QSPCA in the equivalent complex domain. In addition, 2D-QPCA and 2D-QSPCA can preserve the spatial structure of color images and have a low computation cost. Experiments on several challenging databases demonstrate that 2D-QPCA and 2D-QSPCA are effective in color face recognition, and 2D-QSPCA outperforms the state of the arts.