Effective finger vein-based authentication - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 376

(4)

which means the first M − 1 eigenvectors λ and eigenvalues v can be obtained by calculat-
ing WW T .
When we have M eigenvectors and eigenvalues, the images could be projected onto L M
dimensions by computing

(5)

where Ω is the projected value. Finally, to determine which face provides the best description
of an input image, the Euclidean distance is calculated using Equation (6) .

(6)

And finally, the minimum ∈ k will decide the unknown data into k class.

4 Kernel principal component analysis
4.1 KPCA Algorithm
Unlike PCA, KPCA extracts the features of the data nonlinearly. It obtains the principal com-
ponents in F which is a high-dimensional feature space that is related to the feature spaces
nonlinearly. The main idea of KPCA is to map the input data to the feature space F first us-
ing a nonlinear mapping Φ when input data have nonlinearly been mapped, the PCA will be

performed on the mapped data [ 3 ] . Assuming that F is centered, where M

is the number of input data. The covariance matrix of F can be defined as

(7)