Biometric analysis for finger vein data - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 331

The total scater matrix in kernel 2DPCA can be calculated as
(8)
Thus,
(9)
In kernel 2DPCA, after achieving B i = [ Y 1 i , Y 1 i , …, Y d i ] and G t Φ , obtaining projecting axes and
the projection and classification procedures are same as in 2DPCA.
4.2 Kernel Mapping in Row and Column Directions and 2DPCA
Equations (7) - (9) demonstrate how the kernel mapping is performed on the input data. Our
argument here is that by applying this mapping in different directions (along row and column
directions), we will end up having two different data having different dimensions. To further
elaborate this, let us assume we have m × n input matrixes ( n > m ). By applying the kernel map-
ping function along the row direction (we have m rows meaning that there are m elements for
the kernel matrix to be made of), the kernel matrix will be ( m × m ). In this case, by applying
kernel mapping, we reduce the number of input data and its dimension in kernel space which
is ( n × n ). However, if the kernel function is applied along column direction, the kernel matrix
will be ( n × n ). In this case, we have expanded the input data to a higher dimensional space