**4. Overview of the Method**

The method was organized in two parts, see Figures 3–5. The first part (Figure 3) was the construction of a new database containing computed correlation planes. This part was realized by calculating the correlation plane corresponding to a reference person (person 0) and all or a part of the series 1 of the PHPID dataset. The different ways of dividing the series 1 constitute the different training sets, which will be described in Section 6. The testing set is either the whole series 1 or the whole series 2.

**Figure 4.** Flowchart illustrating the decision part.

**Figure 5.** Flowchart illustrating the method.

The second part of our method was the decision part (Figure 4). Considering the correlation planes and the corresponding label indicating if they are the same person or not (in the training set), we made our decision on the testing set by estimating the probability of recognition via a kernel smoothing procedure.

Let us comment on the two-step algorithm represented in Figure 5.

Step 1: Step 1 began with the target image, which is the image to be recognized. The target image was introduced in a VLC correlator to be compared with all reference images of a database. This reference image database contains n different persons (X1,..., Xn) (Xi is the ith person); Xij represents the different variations that the person Xi can have (m is the number of variations considered).

A set of reference images were used with the target image to generate different correlation planes (Px11,...,Pxnm). These correlation planes were then compared in step 2 with pre-computed correlation planes, known as the reference database.

Step 2: The correlation planes (Px11,...,Pxnm) were compared with a correlation plane database realized in step 1. This database was divided in two parts: The first part contained the good correlation plane of references and another part listed the bad correlations, i.e., the false correlation plane of references. We will compare the good and bad correlation planes in Section 5. The construction of these correlation planes of references was made as follows: The correlation planes of various images of person A, the correlation planes of various images of person B, ..., and the correlations planes of various images of person Z, which constitute the good correlation planes of reference. The reference database of bad correlation planes was constructed as follows: We calculated the correlation planes between various images of person A and various images of person B, etc.

The comparison shown in Figure 5 was then realized with the Hausdorff distance and by making use of the kernel smoothing method, which realizes an estimation of the probability of belonging to the class of a known person (Section 5).

#### **5. Nonparametric Model**

The Hausdorff distance is widely used in the context of image recognition, see e.g., [19–21] for reviews. A modified version of the Hausdorff distance has been also applied to matching objects [22]. The Hausdorff distance can be defined as follows: Let *E* and *F* be two non-empty subsets of a metric space (*M*, *d*). The Hausdorff distance is given as:

$$d\_H(E, F) = \max\left\{ \sup\_{\substack{\mathbf{x} \in E \ \mathbf{y} \in F}} \inf\_{y \in F} d(\mathbf{x}, y), \sup\_{\substack{\mathbf{x} \in F \ \mathbf{y} \in E}} \inf\_{y \in E} d(\mathbf{x}, y) \right\}.$$

Here, for the purpose of comparing the target and database correlation planes, the Hausdorff distance between two planes (one with the unknown image and the other one calculated beforehand for the image present in the database) was evaluated. Once the distance was known, a nonparametric classification for decision making was performed.

Next, the decision part illustrated in Figure 4 is described. A kernel smoothing estimate of the regression function was employed. This estimate was used to perform a classification with a given threshold set to 0.5. Here, we made use of the Nadaraya–Watson estimator of the regression function [15] for classification. The principle is described as follows: Assume we have (*Y*1, *X*<sup>1</sup> ), ... , (*Yn*, *Xn*) independent and identically distributed (i.i.d.) random variables coming from (*Y*, *X*) where *Y* is the variable labeled by 1, if the person is detected, and 0 otherwise, *X* is the corresponding correlation plane, and *n* denotes the sample size. Let us comment briefly on this i.i.d. sample: This collection of correlation planes was obtained from the learning database with person 0. Considering a new face image from the testing dataset, we computed the correlation plane with person 0. We then performed a nonparametric classification with a kernel estimate of the regression function *E*(*Y*|*X*). Assuming that *Y* is a Bernoulli random variable, then *P* (*Y* = 1 *<sup>X</sup>* <sup>=</sup> *<sup>x</sup>*) = <sup>E</sup>(*<sup>Y</sup> <sup>X</sup>* <sup>=</sup> *<sup>x</sup>*), where *<sup>P</sup>* represents the probability measure. Thus, *P* (*Y* = 1 *<sup>X</sup>* <sup>=</sup> *<sup>x</sup>*) is the probability of detection, knowing the autocorrelation plane *<sup>x</sup>*. <sup>E</sup>(*<sup>Y</sup> <sup>X</sup>* <sup>=</sup> *<sup>x</sup>*) is the expected value of *Y*, knowing the autocorrelation plane *x*. Now let us define an estimator of this probability as:

$$\hat{Y} \;= \begin{cases} \begin{array}{c} \frac{\sum\_{i=1}^{n} Y\_i K\_h(d(\mathbf{x}, \mathbf{X}\_i))}{\sum\_{i=1}^{n} K\_h(d(\mathbf{x}, \mathbf{X}\_i))}\\ 0 \; if \sum\_{i=1}^{n} K\_h(d(\mathbf{x}, \mathbf{X}\_i)) = 0 \end{array} \tag{1} \\\ \end{cases} \tag{1}$$

where *Y*ˆ is a prediction of *Y*, knowing the correlation plane *x*, keeping in mind that in our case it is also an estimate of the probability that *Y* is equal to 1 at the correlation plane *x*. Knowing *Y*ˆ, we can decide if the face image is identical or not. If *Y*ˆ is close to 1, there is a high probability that there is a good match between the two persons, and if *Y*ˆ is close to zero, the probability that the two persons are not the same is large. In Equation (1), *K* is a real asymmetrical kernel, *h* is the bandwidth (calibration parameter), *Kh*(.) = <sup>1</sup> *<sup>h</sup>K*( . *h* , and *d* is the Hausdorff distance between images.

For the asymmetrical kernel, we used the asymmetrical version of the Epanechnikov kernel, namely *K*(*x*) = <sup>3</sup> 2 1 − *x*<sup>2</sup> 1[0,1)(*x*), where 1[0,1)(.) stands for the indicator function on the set [0, 1). The use of an asymmetrical kernel is standard in functional data analysis (see Ferraty and Vieu [17]) because the distance is positive for all planes of correlation. Other type of kernels can be used, but our numerical results show that the Epanechnikov kernel performs better than others. From Equation (1), we also see that the value of *Y*ˆ ranges from 0 to 1 and represents an estimation of the probability of *Y* to

be of class 1, knowing that the *X* is at the target point *x*: *P* (*Y* = 1 *<sup>X</sup>* <sup>=</sup> *<sup>x</sup>*). Assume that *Yi* is 1 and *x* is close to *Xi* -, then the procedure in Equation (1) will affect a value close to 1 to *Y*ˆ.

For convenience, a threshold is set to 0.5, i.e., all values of *Y*ˆ larger to this threshold are recognized, and vice versa. This threshold value corresponds to an estimation of the Bayes classifiers [23]. The Bayes classifiers maximizes the probability *P* (*Y* = 1 *<sup>X</sup>* <sup>=</sup> *<sup>x</sup>*). In our case, it corresponds to set *<sup>Y</sup>*<sup>ˆ</sup> <sup>=</sup> 1 to the plane of correlation with *Y*ˆ ≥ <sup>1</sup> <sup>2</sup> and *<sup>Y</sup>*<sup>ˆ</sup> <sup>=</sup> 0 to *<sup>Y</sup>*<sup>ˆ</sup> <sup>&</sup>lt; <sup>1</sup> <sup>2</sup> . In the next section, we illustrate this approach by considering two series of faces from the PHPID database.

#### **6. Numerical Results and Discussion**
