*6.3. Simulations for a First Series of Faces from the PHPID Database*

With a face image of person 0 chosen beforehand, our goal is to determine if person 0 belongs to a database or not by making use of our classification procedure. The first training set we considered was made of 126 images, where 9 were coming from person 0. We found that when using this training set, a mean square error (MSE) of 4.8% on the whole testing set (541 planes of correlation), corresponding to the first series of the PHPID database, and only one false positive, i.e., a false positive that is an error of prediction when the person must not be recognized, but the person is recognized by the algorithm as person 0, it is a type I error. The false negative is of a type II error, occurring when the result of the algorithm is negative while the true response must be positive. We found that 13 out of 39 images from the person of reference were recognized. If the entire database was used, the MSE was 0.92%, and only 5 images from person 0 were not recognized.

In Figure 6, we plot the MSE calculated with different numbers of images in the training set. The training set was made of 14 × (2*m* + 1) images, where *m* is the number of images used from either side of the image centered on the face (horizontal shift) for all 14 persons. 2*m* + 1 is the total number of images used for one person: One for the centered face, m images with horizontal shift on the right, and m images with horizontal shift on the left. We observe that when *m* is in the range 1–11, the MSE decreases linearly, but for larger values of *m*, the MSE is almost constant at a value of 1%, indicating the good level of performance of our method. Thus, it is not necessary to build a learning process of the algorithm on the whole database. This avoids the so-called problem of overfitting.

**Figure 6.** The mean square error (MSE) versus the number of images in the training set.

Figure 7 shows the Receiver Operating Characteristic (ROC) curve to check the ability of this algorithm to classify person 0 and the others correctly. The area under the curve is 0.9996. In order to compare with existing methods, we plotted the ROC curve with the peak-to-correlation energy (PCE) criterion for the same data. We observe that our method leads to much better results. We then have a near perfect classifier which clearly outperforms the standard algorithm using the PCE criterion.

**Figure 7.** ROC curves on the testing set of our method (KSR) and peak-to-correlation energy (PCE) criterion. Plot of true positive rate (TPR) vs false positive rate (FPR).
