FV1 is Gabor's feature vector

where

108 Machine Learning and Biometrics

and matrix A has a dimension of MN � q.

• Test/recognition phase

the G~ image are defined as

where v ¼ ð Þ v1; v2; ⋯; vk .

Approximated image is calculated as

We choose the k value according to the dependence:

infð Þk ≥ 0:99 means that we retain 99% of information.

We defined the vectors of features as follows:

where

tors account for over 90% of the variance of the data set (Figure 18).

By increasing the number of eigenvectors, we increase the recognition efficiency.

The covariance matrix has a dimension of MN � MN.

principal components corresponding to k largest eigenvalues.

A ¼ Φ1; Φ2; ⋯; Φ<sup>q</sup>

Then, we organize our eigenvectors according to their decreasing eigenvalues. We choose k

The new image is processed to obtain eigenvectors and eigenvalues. k the main components of

infð Þ¼ k

where k is the predefined number of eigenvectors and q the total number of eigenvectors.

A high value of k means that a large amount of input information will be stored, e.g.,

The variance of the first eigenvector is about 60% of the variance of the data set, the variance of the first 30 eigenvectors is about 85% of the variance of the data set, and 45 or more eigenvec-

P k 1 λi

P q 1 λi

Then, we calculate the eigenvalues and eigenvectors of the covariance matrix:

� � (23)

C vi ¼ λivi i ¼ 1, ⋯, q (24)

<sup>w</sup> <sup>¼</sup> vt <sup>G</sup><sup>~</sup> � <sup>Ψ</sup><sup>Þ</sup> � (25)

G ¼ vw þ Ψ (26)

FeatVect ¼ ð Þ FV1; FV2 (28)

(27)

FV2 is the co-occurrence feature vector

The quality of biometric systems is measured by two parameters: false acceptance rate (FAR) and false reject rate (FRR). FAR indicates the situation when the biometric input image is incorrectly accepted, and the FRR indicates the rejection of the user who should be correctly verified.

The size of the FV1 vector has been set to 60 eigenvectors. The featVect size is 100. For these parameters FRR is 1.16% and FAR is 0.26%.
