**2. Modeling Correlation**

In essence, a Vander Lugt Correlator (VLC) compared a target image (input plane) with a reference image. The result of this comparison is presented in the form of a correlation plane. More precisely, the spectrum of a target image was obtained with a FT and was multiplied by a correlation filter made from the reference image [1–5]. An inverse FT (FT<sup>−</sup>1) was then applied to get the output plane containing a noised correlation peak. The measure of the highest peak (i.e., the peak-to-correlation energy (PCE)) characterized the similarities between the reference and the target images. To validate our approach, we used a classical phase-only filter (POF), see Figure 1.

**Figure 1.** Illustrating the Vander Lugt Correlator (VLC) principle.

#### **3. Dataset**

Simulation results were obtained using the Pointing Head Pose Image Database (PHPID) [18]. This dataset includes 1302 face pictures: 14 different persons (93 images per person) with different

orientations (from −90◦ to +90◦ with respect to the horizontal direction and from −10◦ and +10◦ with respect to the vertical direction). The resolution of each image is 314 × 238 pixels. It is also worth noting that this database includes a variety of persons (Figure 2), e.g., various skin colors, person with glasses or not, etc.

**Figure 2.** Three selected images from the Pointing Head Pose Image Database (PHPID) dataset.

In this numerical study, different training/testing databases from the PHPID dataset were chosen in order to demonstrate the efficiency of our method. Once the specific dataset was chosen, tests were performed as follows: Firstly, a person from the testing database was chosen as a reference person (person 0). A classical POF filter and an autocorrelation plane were computed for each person in this dataset. The correlation technique, whose principle is shown in Figure 1, was applied to get the output correlation plane using the corresponding POF filter for each person in the database. A classification algorithm based on the Hausdorff distance was then used (Section 4). If the person was recognized, the analysis ended. If the person was not recognized, the procedure was repeated with another person from the training database until the training database was empty. Thus, only two possibilities exist, i.e., either the person is recognized or not.

Several remarks are in order, concerning Figure 3, which is organized in two parts. Part 1 defines the VLC, whose output is a plane of correlation. Part 2 is the decision part. For each face of the database, the VLC is used with the target image, thus resulting in a large collection of correlation planes. Next, the decision-making procedure relies on the Hausdorff distance between the target correlation plane and the correlation planes coming from the database by selecting a specific bandwidth (hereinafter described in Section 4).

**Figure 3.** Flowchart illustrating the transition between the database and the correlation planes database.

By adopting a kernel smoothing classification algorithm, we took care of the shape, location, and denoising of the peak of correlation. From this algorithm, we learned which correlation plane was good, and we filtered bad correlation planes. The result of the algorithm was the variable *Y*ˆ, the value of which is 0 for non-recognition decision, i.e., the person is not in the database and is equal to 1 for recognition, i.e., the person is in the database.
