Correlation filters: Face recognition systems based on the correlation filter (CF) have given good results in terms of robustness, location accuracy, efficiency, and discrimination. In the field of facial recognition, the correlation techniques have attracted great interest since the first use of an optical correlator [

47]. These techniques provide the following advantages: high ability for discrimination, desired noise robustness, shift-invariance, and inherent parallelism. On the basis of these advantages, many optoelectronic hybrid solutions of correlation filters (CFs) have been introduced such as the joint transform correlator (JTC) [

48] and VanderLugt correlator (VLC) [

47] techniques. The purpose of these techniques is to calculate the degree of similarity between target and reference images. The decision is taken by the detection of a correlation peak. Both techniques (VLC and JTC) are based on the “

$4f$ ” optical configuration [

37]. This configuration is created by two convergent lenses (

Figure 4). The face image

$F$ is processed by the fast Fourier transform (FFT) based on the first lens in the Fourier plane

${S}_{F}$. In this Fourier plane, a specific filter

$\mathrm{P}$ is applied (for example, the phase-only filter (POF) filter [

2]) using optoelectronic interfaces. Finally, to obtain the filtered face image

${F}^{\prime}$ (or the correlation plane), the inverse FFT (IFFT) is made with the second lens in the output plane.

For example, the VLC technique is done by two cascade Fourier transform structures realized by two lenses [

4], as presented in

Figure 5. The VLC technique is presented as follows: firstly, a 2D-FFT is applied to the target image to get a target spectrum

$S$. After that, a multiplication between the target spectrum and the filter obtain with the 2D-FFT of a reference image is affected, and this result is placed in the Fourier plane. Next, it provides the correlation result recorded on the correlation plane, where this multiplication is affected by inverse FF.

The correlation result, described by the peak intensity, is used to determine the similarity degree between the target and reference images.

where

$FF{T}^{-1}$ stands for the inverse fast FT (FFT) operation, * represents the conjugate operation, and ∘ denotes the element-wise array multiplication. To enhance the matching process, Horner and Gianino [

49] proposed a phase-only filter (POF). The POF filter can produce correlation peaks marked with enhanced discrimination capability. The POF is an optimized filter defined as follows:

where

${S}^{\ast}\left(u,v\right)$ is the complex conjugate of the 2D-FFT of the reference image. To evaluate the decision, the peak to correlation energy (PCE) is defined as the energy in the correlation peaks’ intensity normalized to the overall energy of the correlation plane.

where

$i$,

$j$ are the coefficient coordinates;

$M$ and

$N$ are the size of the correlation plane and the size of the peak correlation spot, respectively;

${E}_{peak}$ is the energy in the correlation peaks; and

${E}_{correlation-plane}$ is the overall energy of the correlation plane. Correlation techniques are widely applied in recognition and identification applications [

4,

37,

50,

51,

52,

53]. For example, in the work of [

4], the authors presented the efficiency performances of the VLC technique based on the “4f” configuration for identification using GPU Nvidia Geforce 8400 GS. The POF filter is used for the decision. Another important work in this area of research is presented by Leonard et al. [

50], which presented good performance and the simplicity of the correlation filters for the field of face recognition. In addition, many specific filters such as POF, BPOF, Ad, IF, and so on are used to select the best filter based on its sensitivity to the rotation, scale, and noise. Napoléon et al. [

3] introduced a novel system for identification and verification fields based on an optimized 3D modeling under different illumination conditions, which allows reconstructing faces in different poses. In particular, to deform the synthetic model, an active shape model for detecting a set of key points on the face is proposed in

Figure 6. The VanderLugt correlator is proposed to perform the identification and the LBP descriptor is used to optimize the performances of a correlation technique under different illumination conditions. The experiments are performed on the Pointing Head Pose Image Database (PHPID) database with an elevation ranging from −30° to +30°.