In-Air Signature Verification System Based on Beta-Elliptical Approach and Fuzzy Perceptual Detector

Believing that biometrics trends are moving to distant and contactless mode, in-air signature verification is nowadays considered as one of the principal users biometric identification in contactless mode allowing users identification by drawing their handwritten signature in the air. In-air signature verification is used in many applications like access control and forensic analysis. In this regard, we propose a novel system for in-air signature verification using Beta stroke segmentation. The Beta-elliptical approach and the fuzzy perceptual detector are used for features extraction. The proposed system defines a specific data acquisition protocol and uses preprocessing steps to prepare data. Finally, the verification phase is done based on Dynamic Time Warping. To evaluate our proposed system, we have created two in-air signature datasets with and without the use of a transparent glass plate, which we make publicly available at https://ieee-dataport.org/documents/air-signature-databases, termed respectively In-Air Signature dataset (IAS dataset) and In-Air Signature dataset using Glass Plate (IASGP dataset). Our verification system demonstrates promising outcomes, yielding an Equal Error Rate (EER) of 1.25% when applied to the IAS dataset and an EER of 2.00% when applied to the IASGP dataset in the skilled-forgery scenario. Extensive evaluations were conducted on both the 3DAirSig and the DeepAirSig datasets. The results confirm that the proposed system has a good performance compared to existing in-air signature verification systems for both skilled-forgery and random-forgery scenarios.


Biometric-based personal verification methods leverage individuals' inherent characteristics, avoiding the drawbacks
The associate editor coordinating the review of this manuscript and approving it for publication was Junggab Son .associated with token or knowledge-based approaches such as ID cards or passwords, which can be lost, stolen, or forgotten [1].The signature is one of the most biometric personality traits that can be acceptable legally [2].A person's signature exhibits distinct behavioral characteristics, making it a de facto identifier.One of the primary benefits of signature biometrics lies in its user-friendliness.Signature identification identifies the signature's owner, whereas signature verification that we will focus on in this paper finds whether a signature is genuine or forged [3].The research in the field of signature verification increased in the last years [4].Many applications use signature verification systems like mail voting, access controls, official communications, forensic analysis and banking services [5], [6].
Traditionally, researchers tend to treat individual verification based on their signatures as two distinct types namely offline and online.In the offline type, only the image of the signature path is available [7].Whereas, in the online type, data comprise information about the pen movement of the signature over time, besides, other physical measurements, are available [8].Due to the covid-19 situation, people prefer to not touch any screen devices because the latter can cause many infectious disease viruses.Consequently, contactless systems are emerging based on hygiene and maintenance reasons [9].For this reason, the in-air signature verification is spreading these days.In-air signatures are a new modality that permits a person to sign in the air by allowing free hand movements, that way dispensing with the use of a signing surface.Compared to offline and online signature verification, in-air signature verification systems have not yet reached maturity [10].Therefore, analyzing in-air signatures has been taken as the base modality of the present research work.
Indeed, in-air signature verification has posed real difficulties and challenges.One of the greatest challenges is the task of fingertip tracking.Therefore, it is difficult to get the correct path of the signature in the air.In addition, the occlusions of fingers and the viewpoint changes during signing freely in the air can affect the in-air signature verification systems' accuracy.Another difficulty with in-air signature verification is the factor that forgeries can be highly skillful as the other types of signatures.The limited number of reference signatures per signer represents another difficulty [11].Moreover, the neurophysiological characteristics (e.g.muscle strength and nervous system) and the psychological characteristics of an individual (e.g.stress and apprehension) influence the signature of the person.For example, because of the variability of the in-air signature speed, the same individual can have two signatures with dissimilar path distances.Thus, the person in a certain case is not able to provide a particular biometric.Consequently, an important challenge associated with in-air signature verification systems is to determine a set of characteristics able to represent the in-air signatures well.

B. CONTRIBUTIONS
Our proposed system aims to overcome the mentioned difficulties and challenges.Firstly, to get the correct path of the signature in the air even if the viewpoint changes during the signing, we will incorporate the MediaPipe Hands framework [12] into our work, offering accurate finger and hand tracking capabilities.Secondly, in order to well characterize the in-air signature, we will use the Beta-elliptical approach [13], [14] and the fuzzy perceptual detector [15], [16].Finally, to tackle the issue of having a limited number of reference signatures for each signer, we will utilize the Dynamic Time Warping algorithm (DTW).In fact, DTW plays a crucial role in this context as it offers a robust solution for aligning and comparing time-series data, allowing us to effectively analyze and verify signatures even with minimal reference samples.
Our contributions can be summarized as follows: • Two in-air signature datasets are created and publicly published with and without using a transparent glass plate.Each one contains 400 in-air signatures entered by 40 subjects.
• The Beta strokes segmentation is applied to segment the in-air signatures trajectory.
• The Beta-elliptical approach and the fuzzy perceptual detector are adopted for the first time in the features extraction to the in-air signature verification system.
• The proposed system which employs publicly accessible datasets, is illustrated and proves its superiority over existing methods.

C. PAPER ORGANISATION
The rest of the paper is organized as follows.Section II describes the related works of in-air signature verification systems.In Section III, our proposed datasets will be thoroughly presented.Section IV describes the methodology of the proposed system.The experimental setup and the performance evaluation are presented in Section V. Finally, in Section VI, conclusions and future works are discussed.

II. RELATED WORKS
Recently, with the widespread occurrence of an infectious disease in a community like the COVID-19 pandemic, various stone measures have been implemented to decrease the propagation of the virus.As a part of efforts, the predilection for touchless technology has been emerging [17], [18], [19].Consequently, many researchers have proposed the use of in-air signature verification systems in many real-life applications.Among these, we can cite the in-air signature verification system presented in [20].In this system, Ferrer et al. introduced a framework for the generation of synthetic 3D in-air signatures.This framework leverages the lognormality principle, which accurately replicates the intricate neuromotor control processes involved as the fingertip is in motion.To evaluate their system, the authors used real data and synthetic 3D signatures from Sigma-Lognormal model with DTW verification.
In [21], Diaz et al. presented a solution involving the synthesis of 3D signatures.In the initial stage, they focused on synthesizing the kinematics of 3D signatures drawing from the Kinematic Theory of Rapid Movements and the associated Sigma-Lognormal model in 3D.In the second stage, they extracted the velocity, the acceleration and the three position coordinates.Finally, they performed the evaluation using the DTW algorithm.
Li et al. [10] proposed the use of Siamese's recurrent neural networks to characterize signatures and to determine whether an in-air signature is from the claimed user or an imposter.To evaluate their system, the authors created a new in-air dataset using smartwatch motion sensors collected from 22 individuals.The obtained results indicate the usefulness of the proposed system.
Khoh et al. [22] used hand detection and localization algorithms to extract the region of interest from each of the depth in-air signature images.Then, the authors extracted several vector-based features.Finally, the support vector machine was used in the verification phase.
Malik et al. [23] propose three different features of the signature trajectory namely spatial(X,Y), depth(Z) and 3D(X,Y,Z).For classification, they used DTW, Image-based verification, and PointCloud-based verification.The latter method outperforms all the compared algorithms using their created dataset called DeepAirSig dataset.
Okawa et al. [24] suggested a new system improving signature verification.In features extraction, they used X and Y coordinates, pressure, path-tangent angle, path velocity magnitude, log curvature radius and total acceleration magnitude.In addition, they proposed a novel single-template strategy using mean templates and DTW by the local stability sequence (LS-DTW).Their system was evaluated, in both the random and the skilled forgery scenarios using two public online signature datasets, SVC2004 Task 2, MCYT 100, and using the 3DAirSig dataset.The obtained results demonstrate the effectiveness of the proposed system.
Khoh et al. [25] proposed a novel in-air hand gesture signature verification in order to study the feasibility of transfer learning in classifying a hand gesture-based signature.To release their system, the authors detected and segmented the hand region from each depth image.After that, they extracted the salient spatial and temporal features from various images.For classification, they used AlexNet model.
Guerra-Segura et al. [26] proposed a novel in-air signature verification system using a Leap Motion controller to characterize in-air strokes.To evaluate their system, they created a new database collected from 100 individuals, which consists of 10 genuine signatures from each user and 10 forgeries for every original user.The extracted features are the mean which represents the absolute value generated by the distribution, its standard deviation, correlation, Shannon entropy, Kurtosis and Skewness.For classification, they used Least Square Support Vector Machine.
Khoh et al. [27] proposed a palm detection and a predictive palm segmentation algorithm to segment the palm region from the frame sequence.Then, they applied a twodimensional representation of hand gesture signature based on a Motion History Image.The features have been extracted from this motion image subsequently.Lastly, the features matching was done using Euclidean distance, cosine distance, chi-square distance, and Manhattan distance in order to calculate the dissimilarity score of a test sample.
To verify a test of in-air signature as being genuine or forged, Malik et al. [28] propose four features extraction methods which are: -1) Depth-based Signature Verification method (DSV) based on the 1D depth Z of the signature trajectory.
For the verification process, the authors used the multidimensional dynamic time warping algorithm.The evaluation was made on their own public dataset which consists of 600 signatures performed by 15 volunteers.
Fang et al. [29] presented a new video-based system for in-air signature verification.To do their system, the authors firstly used fingertip tracking in order to generate the trajectory of signature from the in-air signing videos.Secondly, they proposed an improved dynamic time warping algorithm, and they proved the validity of the Fast Fourier Transform technique in the field of track signature verification.Finally, they explored the feasibility of the fusion of the dynamic time warping algorithm with the Fast Fourier Transform technique based on Gaussian probability distribution.Table 1 summarizes recent works on in-air signature verification.
In reviewing the presented related works, several technical gaps and shortcomings emerge, which have motivated the design of our proposed methodology.In fact, in-air signature verification involves analyzing dynamic and continuous hand movements, and existing approaches often struggle to account for the natural variability in individual handwriting styles and speeds.The techniques rely on simplistic feature representations, which may not capture the intricate nuances of in-air signatures, potentially leading to accuracy issues.Moreover, achieving interoperability with various hardware devices and platforms is a challenge in in-air signature verification.Compatibility issues can hinder the adoption of such systems.The proposed methodology aims to bridge these technical gaps by offering a more comprehensive solution that addresses the complexities of in-air signature verification and improves user acceptance, making it a valuable contribution to this evolving field.
In the realm of in-air signature verification systems, a significant challenge lies in defining a set of features capable of effectively discriminating between genuine and forged inair signatures of individuals.In this study, we investigate the effectiveness of employing the Beta-elliptical approach and the fuzzy perceptual detector for feature extraction in the context of in-air signature verification.To the best of our knowledge, the Beta-elliptical approach and the fuzzy perceptual detector have not been previously used for in-  air signature verification system.Our choice of using the Beta-elliptical approach and the fuzzy perceptual detector justified by the fact that these models are characterized by several types of information that is to say several features that can be used to well describe the signer.Moreover, we are interested in using Dynamic Time Warping in the verification phase.More details about our in-air signature datasets and our proposed verification system are presented in the next sections.

III. PROPOSED IN-AIR SIGNATURE DATASETS
Indeed, there is a lack of in-air signature datasets available for researchers [28].The existing datasets employ various devices like the Leap Motion and the Microsoft Kinect sensor camera.However, these devices are not without their drawbacks and have certain limitations.Challenges include the expenses associated with their implementation, the need for technical expertise to operate them, and potential difficulties for users in adjusting their finger movements to fit within the device's restricted field of view, especially if they lack familiarity with such equipment.
To mitigate these issues, we have created two in-air signature datasets exclusively using a laptop's camera, eliminating the necessity for any additional specialized equipment.we make our new created datasets publicly available at https://ieee-dataport.org/documents/air-signature-databases.
Forty participants voluntarily took part in each of the two datasets' construction.Written informed consent was obtained from each participant.Their age ranged from 21 to 40 years.Table 2 shows the characteristics of the participants in data collection.
Each volunteer signs five signatures in the air and imitates five signatures of five other volunteers in one session.We choose to take only five signatures to make in-air verification more challenging.Moreover, in real life generally, we take only a few signatures of users in the applications.Table 3 presents a short description of the constructed datasets.
The proposed protocol for data acquisition consists of two ways for the two in-air signature datasets respectively.In IAS dataset, the volunteer signs in the air directly in front of the camera.In IASGP dataset and in order to make the task of in-air signature verification more challengeable, the volunteers sign in the air using a transparent glass plate between them and the camera.

A. IN -AIR SIGNATURE DATASET (IAS DATASET)
During data acquisition, volunteers were seated in a comfortable chair directly in front of the camera leaving a distance of 60 cm between their dominant hand and the camera.The volunteers were asked to raise their dominant hand in front of the camera.After that, they were asked to close all the fingers of the hand leaving only the index finger.Then they were asked to sign in the air.When the in-air signature is over, the volunteers were asked to open all the fingers of the used hand.The purpose of this protocol is to control the beginning and the end of the signature in the air for each volunteer.
To assure that the volunteers had accurately comprehended the demanded task, the experimenter performed the task.The volunteers were permitted to realize the requested task before the recording of the data for five minutes.The volunteers were asked to sign in the air at their preferred speed.Custommade software was developed in Python using the MediaPipe Hands framework, offering accurate finger and hand tracking capabilities, to record the data on two files: -The first file is a CSV file that contains the 2D coordinates of the signature in the air (x(t) and y(t)).-The second is an image file of the signature in the air.Figure 1 shows an example of data acquisition for IAS dataset collected from the first author of this paper.

B. IN-AIR SIGNATURE DATASET USING GLASS PLATE (IASGP DATASET)
The experimental protocol for IASGP dataset acquisition is the same as in IAS dataset.The only difference is that we put a transparent glass plate in front of the camera, away 30 cm.This glass plate, with dimensions of 60 cm in both length and width, is placed inside a wooden frame and fixed on the table in front of the volunteers as shown in Figure 2.
During the in-air signature task, the volunteers touch the transparent glass plate in front of the camera.Figure 3 shows an example of data acquisition for IASGP dataset.
Figure 4 presents some examples of in-air signatures extracted from IAS and IASGP datasets.We can note that the forged in-air signatures in (B) and in (D) have more tremors than the genuine in-air signatures in (A) and (C) respectively.Moreover, the forged in-air signature in (B) extracted from IAS dataset without the use of a transparent glass plate has few tremors compared to the forged in-air signature in (D)   extracted from IASGP dataset with the use of a transparent glass plate.

IV. PROPOSED IN-AIR SIGNATURE VERIFICATION
The proposed methodology for in-air signature verification builds upon several foundational methods and techniques from the field of biometrics and pattern recognition.Here are the key methods that serve as the foundation for the design and development of the proposed methodology: 1) Data preprocessing: Robust data preprocessing steps are applied to address issues related to noise.Techniques such as normalization and Chebyshev type 2 filter are utilized to enhance data quality.2) Segmentation: Data segmentation step is applied to divide the in-air signature into Beta strokes.
3) Features extraction techniques: The methodology incorporates advanced features extraction techniques based on Beta-elliptical approach and fuzzy perceptual detector.These methods aim to capture the salient information from the dynamic in-air signature trajectories.
4) Dynamic Time Warping (DTW): DTW is used as a similarity measure to compare the in-air signature trajectories.DTW helps account for variations in signature execution speed and timing, making it a crucial component for accurate verification.
5) Decision threshold: To establish an appropriate threshold for verification, threshold selection algorithms like Equal Error Rate (EER) optimization are integrated into the methodology.This algorithm helps determine the decision boundary between genuine and forgery samples.
The main components of the proposed system are shown in Figure 5.
A detailed explanation of the proposed system components will be presented in the following sub-sections.

A. PREPROCESSING
The input of our in-air signature verification system is the coordinates in time of the index fingertip positions x(t) and y(t).To perform the preprocessing step, we have adjusted the vertical dimension h of the in-air signatures lines to obtain a normalized size to keep the original height-to-width ratio.The h value is set to 128 after many experimental tests.For each point of the hand movement trajectory, the normalization is done by computing its normalized coordinates x_norm and y_norm attained by the following formulas: where m = max(max x − min x , max y − min y ) where (x,y) are the original point coordinates, x_norm and y_norm are the corresponding ones after the normalization procedure, and h is the value chosen to achieve the normalization.
Thereafter, a Chebyshev type 2 filter with a cut-off frequency equal to 12Hz is performed to eliminate the noise.The choice of using the Chebyshev type 2 filter among many filters is based on the following factors: 1) Chebyshev Type 2 filters are known for their variable ripple in the stopband, which means they can provide strong attenuation for certain frequencies while allowing others to pass with minimal distortion.Depending on the frequency characteristics of in-air signatures, this filter type is chosen to emphasize specific frequency components relevant to the task.
2) Since in-air signature signals deal with low Signal-to-Noise Ratio (SNR), Chebyshev Type 2 filters are preferred due to their ability to provide deep stopband attenuation, which can help reduce noise and interference from unwanted frequencies.
3) Chebyshev Type 2 filters offer flexibility in designing filter responses by adjusting parameters like the amount of ripple and the filter order.This allows designers to tailor the filter to the specific requirements of the in-air signature verification task, balancing the trade-off between passband ripple and stopband attenuation.Figure 6 shows an application of preprocessing to an in-air signature extracted from IAS dataset.

B. SEGMENTATION
To segment the in-air signatures, firstly, we represent the curvilinear velocity by the following equation: Afterward, we detect two types of points from V σ (t) as depicted in Figure 7 below, which are: -The double inflection point represents the variation of speed that indicates the transition of the movement drive from a neuromuscular subsystem to another or from a neurophysiologic impulse to the following which corresponds to the change in convexity in the path.
-The local minimum represents the local minima of the curvilinear velocity.
Thereafter, we divided the curvilinear velocity into small elements called Beta strokes delimited between two successive velocity local minimums or double inflection points.After that, we divided the in-air signature path into strokes delimited between the correspondents' points used in the curvilinear velocity segmentation.An example of segmentation of in-air signature extracted from IAS dataset is depicted in Figure 8.

C. FEATURES EXTRACTION
we are interested to use Beta-elliptical approach in the in-air signature verification system, which sufficiently represents the hand movements in real-time, by describing together its profile parts: the Beta impulses and the elliptic arcs [13], [14], [30].Besides, we are interested to utilize the fuzzy perceptual detector in order to adequately represent the hand movements path [15], [16].
Hand movement is considered as the consequence of the activation of many neuromuscular subsystems performed by the neuronal system and transferred by the nerves motor to the muscles to activate the arm of the hand.The research done on the impact of these neuromuscular subsystems on the dynamic profile called also velocity profile indicates that each subsystem produces an impulsive signal converged to a Beta function [31] expressed as follows.
pulse β (K , t, q, p, t 0 , t 1 ) where t 0 and t 1 denote respectively the start and the end times of the constructed impulse, t c is the time when the impulse achieves its highest amplitude K , p and q are intermediate parameters.The velocity profile can be calculated as given in formula 6.
Boubaker et al. [14] suggest a new method for modeling the velocity profile based on the Beta function with the addition of a continuous training component called Extended Beta-Elliptic model which we used in this work.In this model, the curvilinear velocity inside the interval time [t 0 , t 1 ] is divided into two elements: where t 0 and t 1 denote respectively the start and the end times of the constructed impulse, t c is the time when the impulse achieves its highest amplitude K , p and q are intermediate parameters.
2) A CONTINUOUS TRAINING ELEMENT, CALCULATED USING ( 8) where where: -t 0 and t 1 denote respectively the start and the end times of the constructed impulse.
-V i and V f are respectively the velocities at the start and the end times of the constructed impulse.
Finally, the curvilinear velocity is the sum of the impulsive element and the continuous training element as defined in (10) In the geometric profile, each stroke can be fitted utilizing two adjacent elliptic arcs E 1 (a 1 , b 1 , θ 1 , θ p1 ) and E 2 (a 2 , b 2 , θ 2 , θ p2 ) as presented in Figure 9.
These two arcs have the same inclination angles (θ 1 = θ 2 = θ).To guarantee the continuity of curvature when departing from the first to the second elliptic arc, the link between the two small and large axis lengths should confirm the condition shown in (11). where: a 1 and a 2 are respectively the major axis half-length of the ellipse including the first arc and the second arc.-b 1 and b 2 are respectively the small axis half-length of the ellipse supporting the first arc and the second arc.
In addition, we integrate into the features extraction phase the coefficient of logarithmic proportionality between the curvilinear velocity and the curvature radius λ known also as the two-thirds power law [32].We compute this parameter as the average absolute gradient of the parametric curve describing the variation of the logarithm of the curvilinear velocity with respect to the logarithm of the curvature radius as expressed in ( 12): -where: -N is the points number in the current stroke.
-n is the current index of points in the trajectory and t n represents its execution time.
On the other hand and inspired by the PerTOHS theory (Perceptual Theory for On-line Handwriting Segmentation) [15], [16], we propose to use the fuzzy perceptual detector for our in-air signature verification system.Thus, we use the inclination angle of the ellipse major axis to attach each Beta stroke to one of the four types of Elementary Perceptual Codes (EPC) as shown in Table 4.We divide the trigonometric circle into eight regions corresponding to the EPC as displayed in Figure 10 Indeed, EPCs have a problem with ambiguity and indecision caused by several conditions such as hand disorder.To overcome this drawback, we used the fuzzy logic theory to assign a membership degree for each EPC.Therefore, we achieved four features which are FEPC 1 , FEPC 2 , FEPC 3 , and FEPC 4 denoting respectively the membership degree of EPC 1 , EPC 2 , EPC 3 , and EPC4.
To recapitulate, each one of the Beta strokes is represented by 22 features as given in Table 5.

TABLE 5.
Extracted features based on beta-elliptical approach and fuzzy perceptual detector.
The first seven features depict the neuromuscular excitations affected during the in-air signature actions while the fourteen following features express the geometric characteristics in the path of the in-air signatures.The last parameter allows a precise description of the shape of the path curvature radius variation in the in-air signatures.

D. VERIFICATION BASED ON DYNAMIC TIME WARPING 1) DYNAMIC TIME WARPING MATCHING
Dynamic Time Warping algorithm (DTW) is a well-known algorithm in numerous fields of research like gestures recognition [33], eye movement analysis [34], pattern recreation [35] and signature verification [4].
134066 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
DTW permits a non-linear mapping of one time series to another by minimizing the distance between the two.Therefore, to calculate the distance between two in-air signatures, a simple design of the DTW algorithm is used, which can align two non-linear temporal sequences via dynamic programming.The distance matrix between a reference signature R = {r (i)} described by N Beta strokes and a questioned signature Q = {q (j)} j=1..M described by M Beta strokes, can be computed by filling a DTW N +1×M +1 matrix following equation ( 13) after initialization of DTW [0, 0] = 0 and DTW [i, 0] = DTW [0, j] = ∞∀i, j ∈ [1, N ]: The distance between reference and questioned signatures will be saved at the upper right corner of the DTW matrix: Moreover, the local distance dist in ( 13) was the usual Euclidean vector distance as shown in (15).
where k denotes the k th feature dimension.Let us remember that 22 is the total number of extracted features.

2) SELECTION OF A TEMPLATE SIGNATURE
The DTW distances reflect in a way the similarity of a questioned signature with a reference genuine signature of a signer.For the training, we use G reference genuine signatures {S 1 , S 2 , . . ., S G } for each user.Thus, we compute the DTW distance between each one of the G reference genuine signatures with the remaining reference genuine signatures.Therefore, we choose one template genuine signature for DTW S indexDTW for that user as detailed in Algorithm 1.Thus, only the signatures S indexDTW will be used in the distance measurement for the testing.Therefore, for the final verification decision, we calculate the distance measurement between the signatures S indexDTW and the questioned signature.If the distance measurement is inferior to a specific threshold, the questioned signature will be considered as genuine.Otherwise, the questioned signature will be considered forgery.

A. EVALUATION PROTOCOL
We used three metrics to evaluate the performance of our inair signature verification system which are: • False Acceptance Rate (FAR) is the probability that forged in-air signatures are incorrectly accepted as genuine • False Rejection Rate (FRR) is the probability that genuine in-air signatures are incorrectly rejected as forged inair signatures by the system.

FRR =
Misclassified forgery in − air signatures Total number of forgery in − air signatures * 100 • Equal Error Rate (EER) To determine EER, we follow these steps: 1) We collect genuine and forged scores from the DTW using the template signature.
2) We sort the scores by arranging the genuine and forged scores in ascending order.
3) We define a range of decision thresholds that we'll evaluate.These thresholds represent the point at which similarity scores are considered a match or non-match.
4) We calculate the FRR and the FAR based on the number of genuine scores incorrectly rejected and forged scores incorrectly accepted, respectively.5) We find the threshold at which the FRR equals the FAR.EER is the point at which our system's false rejection rate matches the false acceptance rate.
6) We report the EER which is typically expressed as a percentage and represents our system's overall performance.Lower EER values indicate better performance, as they imply a smaller gap between the FRR and FAR.
The relationship between FAR, FRR and EER and the curve are shown in Figure 11.

B. RESULTS AND DISCUSSION
To evaluate the effectiveness of our in-air signature verification system, we performed the experiments using our datasets, the 3DAirSig dataset [28] and the DeepAirSig dataset [23].

1) EXPERIMENTS ON OUR DATASETS
During the training of the forty users for both IAS and IASGP datasets, we randomly selected 3 genuine signatures for each signer as the reference set in each experiment.The remaining 2 genuine signatures and 5 skillfully forged signatures were used for the test samples as presented in Table 6.To prevent a selection bias, we repeated all the experiments five times in both IAS and IASGP datasets.Figure 12 (a, b) and Table 7 show the average in-air signature verification results of the proposed system on both IAS and IASGP datasets for the skilled-forgery scenario.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The analysis of the results reveals that our proposed system, using the Beta-elliptical approach and the fuzzy perceptual detector in features extraction and the DTW in verification, is effective to distinguish between genuine and forged in-air signatures.It can be noted that the FAR in IAS and IASGP datasets belong to different users, which indicates that our proposed system has relatively low fault tolerance.As can be noticed also, the results of FRR in IAS dataset IASGP dataset are inferior to 1.50%.In other words, we can say that our in-air signature verification system can detect a misclassified forgery with a good performance in the two cases with and without the use of a transparent glass plate.
We also observe that the results on IASGP dataset using the transparent glass plate give us an EER higher than the results on IAS dataset.This can be explained by the fact that IASGP dataset is more challenging.In fact, the use of a transparent glass plate in IASGP dataset adds a constraint to the in-air signature, which can affects the quality of information of the in-air signature verification system.Some of the anterior works studied not only the skilledforgery scenario, but also the random-forgery scenario.For this reason, we conducted an experiment on randomforgery scenario.The random-forgery scores were achieved by comparing the selected template genuine signature to the two remaining genuine signatures from the target signer as genuine signatures and all the genuine signatures from each of the remaining signers as forged signatures in the verification phase.Figure 12 (c, d) and Table 8 show the in-air signature verification results of the proposed system on both IAS and IASGP datasets for the random-forgery scenario.From the obtained results, we can conclude that our proposed system is effective for in-air signature verification for both skilled-forgery and random-forgery scenarios.

2) EXPERIMENTS ON THE 3DAIRSIG DATASET
We made use of the publicly available dataset called 3DAirSig dataset created by Malik et al. [28] and which was contributed by 15 volunteers.For each volunteer, the dataset contained 5 genuine signatures as the reference set for the training and 10 genuine signatures and 25 skilled forgeries obtained from five impostors for the testing process as presented in Table 9.   10 show the in-air signature verification results of the proposed system using 3DAirSig dataset for the skilled-forgery scenario.The analysis of the results reveals that our proposed system, using the Betaelliptical approach and the fuzzy perceptual detector in features extraction and the DTW in verification, gives a comparative result, with an EER equal to 0.08, compared to the work presented in [24].This proves the effectiveness of the proposed system to distinguish between genuine and forged in-air signatures.
We also conducted the experiment on random-forgery scenario.The random-forgery scores were achieved by comparing the selected template genuine signature to the remaining 10 genuine signatures from the target signer as genuine signatures and all the genuine signatures from each of the remaining signers as forged signatures in the verification phase as in the protocol evaluation presented in [24].Figure 13 (b) and Table 11 show the in-air signature verification results of the proposed system using 3DAirSig dataset for random-forgery.The results reveal that our proposed system outperforms the in-air signature verification systems presented in [24], suggesting that the Beta-elliptical approach and the fuzzy perceptual detector in features extraction serve as powerful tools for the in-air signature verification process.

3) EXPERIMENTS ON THE DEEPAIRSIG DATASET
We utilized also the publicly accessible DeepAirSig dataset, established by Malik et al. [23].This dataset consisted of contributions from 40 volunteers.Within this dataset, each volunteer's data included 10 genuine signatures for training as the reference set and 10 genuine signatures along with 25 skilled forgeries obtained from five impostors for the testing phase, as outlined in Table 12.   13 show the in-air signature verification results of the proposed system using DeepAirSig dataset for the skilled-forgery scenario.
The analysis of the obtained results reveals that our proposed system, using the Beta-elliptical approach and the 134070 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.fuzzy perceptual detector in features extraction and the DTW in verification, gives a comparative result, with an EER equal to 0.34%.
We additionally carried out an experiment under a randomforgery scenario.In scenario, the random-forgery scores were obtained by comparing the selected template genuine signature to the remaining 10 genuine signatures from the target signer considered as genuine, and all the genuine signatures from each of the other signers were treated as forged during the verification phase.The results of the in-air signature verification using the DeepAirSig dataset for the random-forgery scenario are presented in Figure 14 (b) and Table 14.The results we acquired indicate that our proposed system surpasses the performance of the in-air signature verification systems introduced in [20] and [21].This suggests that the utilization of the Beta-elliptical approach and the fuzzy perceptual detector in extraction demonstrates their effectiveness in the in-air signature verification process.
By utilizing a diverse dataset and implementing state-ofthe-art techniques, the system is capable of recognizing subtle nuances in signature dynamics, ensuring reliable verification even in challenging real-world scenarios.

4) ANALYSIS OF VERTICAL DIMENSION VALUES OF THE IN-AIR SIGNATURES LINES
During the preprocessing phase, specifically within the normalization process, we conducted numerous experiments.These experiments involved varying the vertical dimension h of the in-air signature lines, as illustrated in Figure 15.As depicted in Figure 15, the lowest EER is achieved when the value of h is set to 128.

VI. CONCLUSION AND FUTURE WORKS
A novel in-air signature verification system is proposed based on the Beta-elliptical approach and the fuzzy perceptual detector.DTW is used in verification.To evaluate the proposed system, two in-air signature datasets are created.Forty voluntary participants took part in both of the two datasets' construction.Experimental results indicate that our proposed system using the Beta-elliptical approach and the fuzzy perceptual detector in features extraction and DTW in verification is effective to distinguish between genuine and forged in-air signatures.The effectiveness of our proposed system was also examined using both the 3DAirSig and the DeepAirSig datasets, which achieves interesting results compared to those of the existing systems for both skilled-forgery and random-forgery scenarios.Our proposed system can be used in many applications like mail voting, access controls, official communications, forensic analysis, banking services and secure communication with IoT Devices.
The exploration of other alternative techniques for improving features extraction like the use of Jerk, which represents the time rate of change of acceleration, is planned in the next experimentations.Moreover, there are potential applications of the proposed method outside the e-security domain like the diagnosis of Parkinson's disease using in-air signatures.

DECLARATION
Financial Interests: The authors declare they have no financial interests.
Non-Financial Interests: None.

Conflict of Interest:
The authors declare that they have no conflict of interest.

FIGURE 1 .
FIGURE 1. Example of data acquisition for IAS dataset.

FIGURE 2 .
FIGURE 2. Experimental setup using the transparent glass plate.

FIGURE 3 .
FIGURE 3. Example of data acquisition for IASGP dataset.

FIGURE 4 .
FIGURE 4. Examples of in-air signatures extracted from IAS and IASGP datasets for the same user.(A): genuine in-air signatures extracted from IAS dataset; (B): forged in-air signatures extracted from IAS dataset; (C): genuine in-air signatures extracted from IASGP dataset; (D): forged in-air signatures extracted from IASGP dataset.

FIGURE 5 .
FIGURE 5. General architecture of the proposed in-air signature verification system.

FIGURE 6 .
FIGURE 6. Application of preprocessing on an example of an in-air signature extracted from IAS dataset.

FIGURE 7 .
FIGURE 7. Detection of segmentation points from the curvilinear velocity profile.

FIGURE 8 .
FIGURE 8. Segmentation of an example of an in-air signature extracted from IAS dataset.

FIGURE 9 .
FIGURE 9. Parameters of the geometric profile.

FIGURE 10 .
FIGURE 10.Presentation of EPCs on the trigonometric circle.
in-air signatures by the system.FAR =Misclassified genuine in − air signatures Total number of genuine in − air signatures * 100

FIGURE 12 .
FIGURE 12. (a): Results using IAS dataset for the skilled-forgery scenario; (b): Results using IASGP dataset for the skilled-forgery scenario; (c): Results using IAS dataset for the random-forgery scenario; (d): Results using IASGP dataset for the random -forgery scenario.

Figure 13 (
Figure13(a) and Table10show the in-air signature verification results of the proposed system using 3DAirSig dataset for the skilled-forgery scenario.The analysis of the results reveals that our proposed system, using the Betaelliptical approach and the fuzzy perceptual detector in features extraction and the DTW in verification, gives a comparative result, with an EER equal to 0.08, compared

Figure 14 (
Figure14(a) and Table13show the in-air signature verification results of the proposed system using DeepAirSig dataset for the skilled-forgery scenario.The analysis of the obtained results reveals that our proposed system, using the Beta-elliptical approach and the

FIGURE 15 .
FIGURE 15.Equal Error Rate results using different vertical dimension h of the in-air signatures lines.

TABLE 1 .
Summaries of recent works on in-air signature verification.
VOLUME 11, 2023134061 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE 2 .
Characteristics of the participants in data collection.

TABLE 3 .
Details of our in-air signature datasets.

TABLE 6 .
Enrollment protocol using IAS and IASGP datasets.

TABLE 7 .
Results using IAS and IASGP datasets for the skilled-forgery scenario.

TABLE 8 .
Results using IAS and IASGP datasets for the random -forgery scenario.

TABLE 10 .
Results using 3DAirSig dataset for the skilled-forgery scenario.

TABLE 11 .
Results using 3DAirSig dataset for the random -forgery scenario.

TABLE 13 .
Results using DeepAirSig dataset for the skilled-forgery scenario.

TABLE 14 .
Results using DeepAirSig dataset for the random -forgery scenario.