**1. Introduction**

The use of correlation methods [1–4] remains very competitive despite the abundance of purely numerical methods, such as Support Vector Machines and neural networks. Correlation is easy to use in practice because it is based on two Fourier transforms (FTs) and one multiplication in the frequency domain [5].

For comparison, a deep learning-based method has generally good performance but also significant drawbacks due to algorithm complexity, implementation difficulty, time-consuming learning processes, and a high number of computational resources [6]. Most of the developments are devoted to increasing the performance of correlation methods concentrated in the Fourier plane [7–11] by designing innovative correlation filters. On the other hand, there exists a growing scientific community dealing with biometric issues, such as face recognition, fingerprint detection, and early automatic disease detection [12,13]. The primary focus of this paper is to deal with an authentication problem using a database. There are two kinds of issue, i.e., identification and verification. Here, our primary goal is to optimize the solution to verification.

In order to improve the decision performance, our model uses a statistical learning method, i.e., a supervised classification method. The regression function between the binary output (class of person) and the input (correlation plane) is nonparametrically estimated for the learning database by making use of the modified kernel smoothing Nadaraya–Watson algorithm [13,14].

Functional data analysis has recently been developed to statistically analyze curves or objects, see e.g., Reference [15] for a good introduction to this subject. The interested reader may also refer to [16] for an overview of nonparametric estimation with functional data. In [17], the authors defined an extension of the Nadaraya–Watson estimator for objects such as curves by introducing a distance in the kernel between two functional objects. Here, we propose the use of kernel smoothing estimation to cope with the correlation plane, and we choose an appropriate distance, i.e., the Hausdorff distance, to plug in the kernel for estimating the regression function. As a result, it is possible to propose a decision-making protocol which has dual effects for increasing good decision rates and reducing false alarm rates.

The rest of this paper is organized as follows. Section 2 provides the correlation principle. After a short description of the database in Section 3, our overall method is explained in Section 4. Our model is implemented in Section 5. The method's accuracy is checked in Section 6, which provides two series of simulation studies. Section 7 briefly concludes.
