Two-Dimensional Principal Component Analysis and Its Extensions

Normally in Principal Component Analysis (PCA) (Sirovich & Kirby, 1987; Turk & Pentland, 1991), the 2D image matrices are firstly transformed to 1D image vectors by vectorization. The vectorization of a matrix is the column vector obtain by stacking the columns of the matrix on top of one another. The covariance or scatter matrix are formulated from the these image vectors. The covariance matrix will be well estimated if and only if the number of available training samples is not far smaller than the dimension of this matrix. In fact, it is too hard to collect this the number of samples. Then, normally in 1D subspace analysis, the estimated covariance matrix is not well estimated and not full rank.