Use of Polynomial Classifiers for On-line Signature ... - IEEE Xplore

Use of Polynomial Classifiers for On-line Signature Recognition Emanuele Maiorana1 , Patrizio Campisi1 , Daria La Rocca1 , Gaetano Scarano2 1

2 Department of Applied Electronics, University Roma Tre, DIET, University Sapienza Via della Vasca Navale 84, 00146, Rome, Italy. Via Eudossiana 18, 00184, Rome, Italy.

Abstract

sent an issue for centralized databases, they may represent a problem when distributed databases have to be employed. Specifically, few hundreds of bytes can be used to represent a given biometrics when the templates have to be stored in the magnetic strips of credit cards, in barcodes for machine readable travel documents (MRTD), or even in smart cards [7]. Template size may represent an even more severe issue when a multi-biometric system exploiting different modalities has to be designed, being the space assigned to each trait further reduced. In case the size of the template should be kept as small as possible, global features should be therefore exploited to perform signature recognition. Several matching processes have been proposed to compare signatures represented through sets of parameters, with the most commonly employed strategy relying on the computation of the Mahalanobis distance between the vectors acquired during enrollment and verification [2]. Also Support Vector Machine (SVM) [8] and Fisher Linear Discriminant Analysis (FLD) [9] have been recently proposed to perform signature recognition based on global features. Our approach defines a novel strategy to perform user verification based on global signature features. Specifically, the typical characteristics of the forgeries trying to imitate each user’s signatures are taken into account when performing the matching phase, in order to make to process more robust against possible false acceptances, while polynomial classifiers are employed to model the classes of both the genuine signatures and the possible forgeries when performing user recognition. The proposed system should be thus able to guarantee high verification performance while requiring low complexity and low storage capacity for the employed users’ templates. The paper is organized as follows: an introduction to polynomial classifiers is provided in Section 2, while the details of the proposed on-line signature recognition system are given in Section 3. The tests performed to evaluate the proposed system are then discussed in Section 4, while conclusions are eventually drawn in Section 5.

In this paper we propose a system exploiting polynomial classifiers, typically employed in identification scenarios, for the case of user verification based on on-line signature. In order to accomplish this task, a novel strategy for generating synthetic classes of signature features is proposed. The proposed system guarantees high verification performance while requiring low complexity and low storage capacity for the employed users’ templates. Experimental tests conducted over the public MCYT database show the effectiveness of the proposed approach.

1. Introduction People recognition based on signatures, being part of everyday life and relying on a behavioral biometrics, is perceived as a non-invasive and non-threatening process by the majority of its users, thus representing one of the most wellaccepted biometric authentication method [1]. It can be performed according to two different modalities by extracting either local or global features from the acquired data. Local features provide rich information in terms of temporal sequences describing the geometrical and topological behavior of a signature. Elastic matching procedures like Dynamic Time Warping (DTW) [3] or statistical recognition approaches such as those relying on Hidden Markov Models (HMMs) [4] are employed to perform the matching between signatures acquired in different sessions. On the other hand, global features are parameters representing static information like the height and the width of a signature, or dynamic information like signature velocity, acceleration, or pressure. While approaches relying on local features are typically able to provide better recognition performance than those achievable through global parameters, it is also worth remarking that the former require a much higher computational complexity and a much larger size of the needed templates. The size of the templates employed for representing the local behavior of a signature is one of the largest among the employed biometric modalities [5]. Specific compression algorithms have been proposed to reduce the size of this kind of templates [6]. Although the storage requirements for biometric templates do not repre-

978-1-4673-1228-8/12/$31.00 ©2012 IEEE

2. Polynomial Classifiers A brief outline regarding polynomial-based classification [10] is provided in this Section. This kind of classifiers is typically employed when dealing with scenarios

265

However, the assumptions made in (1) in defining a linear classifier require decision regions represented through plane intersections in the space of the observations x, while the actual boundaries of the decision regions are typically non-linear curves whose form depend on the pdfs at hand. As indicated in [10], a linear classification in the form (1) can be still retained by employing a suitably non-linearly expansion to an observation space with an increased cardinality. Specifically, given the feature-based representation z = [𝑧1 , . . . , 𝑧𝑀 ]T of the observed data, its polynomial counterpart expanded to the 𝑂-th order is expressed as:

involving non-linear clustering, where the decision regions in the space of the observed features cannot be determined through plane intersections. Their main characteristic resides in the expansion of the available feature-based data representations into an observation space where the decision regions can be defined through linear borders, thus simplifying and optimizing the classification process. Specifically, let us consider a generic classification problem where an observation can be represented through an 𝑀 -dimensional vector x = [𝑥1 , . . . , 𝑥𝑀 ]T , which is supposed (to be drawn from a probability distribu) tion 𝑝X∣ℋ(𝑛) x∣ℋ(𝑛) belonging to a finite set of cardinality (𝑁 . The ) probability density functions (pdfs) 𝑝X∣ℋ(𝑛) x∣ℋ(𝑛) , 𝑛 = 1, . . . , 𝑁 , represent the stochastic description of the 𝑁 admissible classes or hypotheses ℋ(𝑛) among which an observer has to choose the one to which the considered instance x belongs. The a priori probability that x belongs to the 𝑛-th class can be indicated as 𝒫 (𝑛) . Having defined as y(𝑛) = [0, . . . , 0, 1, 0, . . . , 0]T an 𝑁 -dimensional vector where the unique 1 is in the 𝑛-th position and employing such vector to indicate that x belongs to the 𝑛-th class, i.e. it has ( been)drawn from the probability distribution 𝑝X∣ℋ(𝑛) x∣ℋ(𝑛) , it is possible to define as linear a classifier which estimates the class which x belongs to after a linear transformation: ˆ T (Q) = xT ⋅ Q, (1) y T ˆ (Q) identifies the estimation obtained when apwhere y plying the transformation matrix Q to the observed vector x. Specifically, Q should be designed according to a suitable optimality criterion. When considering the Minimum Mean Square Error (MMSE) criterion, Q can be generated as: ∑𝑁 (𝑛) Q = arg min {[Γ 𝑛=1 𝒫 ]⋅T [ ]} (2) ˆ (Γ) ⋅ y(𝑛) − y ˆ (Γ) . ⋅Ex∣ℋ(𝑛) y(𝑛) − y It is worth noting that the expectation in (2) is carried out jointly on x and over all the 𝑁 considered classes. A common solution [10] to this optimization process can be obtained by solving the set of normal equations given by (3) Rx ⋅ Q = Rxy where Rx = E {x ⋅ xT } collects the auto-correlations of the elements in the observation vector x, while 𝑁 ∑ 𝒫 (𝑛) ⋅ Ex∣ℋ(𝑛){x} ⋅ (y(𝑛) )T (4) Rxy =

x(𝑂) (z) =

2 [1, 𝑧1 , . . . , 𝑧𝑀 , 𝑧12 , . . . , 𝑧𝑀 , 𝑧1 𝑧2 , . . . , 3 3 𝑂 𝑧1 , . . . , 𝑧𝑀 , 𝑧1 𝑧2 𝑧3 , . . . , 𝑧12 𝑧2 , . . . , 𝑧1𝑂 , . . . , 𝑧𝑀 ]. (5) This polynomial consists of a constant unit term, followed by 𝑀 linear terms 𝑧𝑚 , 𝑚 = 1, . . . , 𝑀 , followed by all the quadratic terms, all the possible cubic terms, and so forth till the 𝑂-th degree. In general, the 𝑂-th order polynomial expansion of an 𝑀(-dimensional vector z yields to a vector ) with size 𝑀 (𝑂) = 𝑀𝑂+𝑂 . When operating on a polynomially-expanded vector, the linear classifier in (1) can be written as follows:

ˆ T (Q) = (x(𝑂) (z))T ⋅ Q, (6) y where the matrix Q is obtained as in (3) by solving the linear system of equations: (7) Rx(𝑂) ⋅ Q = Rx(𝑂) y . By evaluating the optimal matrix Q through the MMSE criˆ (Q) sum up to terion, the elements of the estimated vector y one, and therefore each value can be interpreted as the computed probabilities of z belonging to the 𝑛-th class [10]. ˆ (Q), several decision criteria can be Hence, stemming on y used to obtain an estimation of which class z belongs to. ˆ (Q) selects the position of A commonly employed use of y its maximum value to estimate the class from which the observed instance x should have been extracted. It is worth remarking that a cardinality explosion may arise when 𝑀 is large and the employed expansion order 𝑂 is large too. However, successfully classification examples are typically reported for small values of both 𝑀 and 𝑂. Specifically, polynomial classifiers with 𝑀 = 12 and 𝑂 ∈ {3, 4} have been employed to perform speaker recognition in [11] and in [12]. Polynomial classifiers have been also used for applications different from people recognition as in [13], where the best performance achievable for isolated word recognition is obtained with 𝑀 = 19 characteristics expanded to the order 𝑂 = 2. It is worth observing that polynomial classifiers have been employed so far to deal with identification problems, where it has to be determined which class, within a previously defined closed set, the observed data belongs to. Such approach necessarily requires the knowledge of information extracted from all the 𝑁 admissible classes in order to analyze the samples coming from one of them. In the follow-

𝑛=1

provides the probabilistically averaged products between the conditional mean values of the observations x and the assignment vector y(𝑛) . The matrix Q can be therefore evaluated once the matrices Rx and Rxy are available. ( )In absence of the knowledge of the pdfs 𝑝X∣ℋ(𝑛) x∣ℋ(𝑛) , these matrices can be obtained after a suitable Monte Carlo-based stage, which has to properly account also for the different a priori probabilities 𝒫 (𝑛) .

266

ing, Section 3 proposes a novel approach exploiting polynomial classifiers for realizing user verification, which is the main modality adopted when considering on-line signature recognition, never performed till now by means of polynomial classifiers.

performed verification modality is proposed in Section 3.3. In order to keep as small as possible the size of the template representing user’s specific information, the most of the processing is performed during the verification phase, while the enrollment only computes the main statistics of the presented user.

3. Proposed Approach

3.1. Training Phase

As outlined in Section 1, the proposed on-line signature recognition system relying on polynomial classifiers is here designed in order to guarantee high recognition performances while keeping the size of the requested user’s specific template as small as possible. Specifically, in the following it will be detailed how polynomial classifiers are here employed to perform on-line signature recognition by projecting a feature-based on-line signature representation on a space where linear classification can be performed, and by providing the means to take into account information regarding possible signature forgeries in the recognition process. As observed in the previous Section, the use of polynomial classifiers allows to determine the class an observed sample belongs to by selecting the most probable one in a set of 𝑁 possible choices. This is a closed-set identification process, in which data from 𝑁 different classes are initially acquired and a decision on the class which a test data has been extracted from is taken during the recognition phase. In fact, the transformation matrix Q in (7), responsible of modeling the considered classification problem, can be estimated only when the matrices Rx(𝑂) and Rx(𝑂) y have been computed, and this can be done only when samples belonging to all the considered 𝑁 classes are available for processing. Unfortunately, being on-line signature recognition typically performed in verification modality, observations from classes different from the one of the considered user are not available during the enrollment stage. Moreover, an on-line signature recognition system should be robust against the possibility that malicious subjects may perform forgeries of the signatures of a legitimate user in order to claim access to the system. Obviously, it is not possible to produce skilled forgeries of a given user at the same time of his enrollment. In order to employ polynomial classifiers for on-line signature recognition it is therefore necessary to define a method with which samples belonging to classes different from the one of the user being enrolled can be generated. Moreover, in order to guarantee high recognition performances, the synthetically generated classes should be able to properly represent plausible imitations of the original signatures taken from a legitimate user. The following Sections describe how on-line signature recognition can be performed by means of polynomial classifiers. Specifically, Section 3.1 describes how a training dataset should be employed to evaluate the statistics needed to generate the classes employed during a user’s enrollment. This latter phase is then described in Section 3.2, while the

It is assumed that a dataset comprising 𝑆 genuine signatures and 𝐷 forged signatures for each of 𝑇 subjects is available during a training phase. These data are employed to estimate the general statistics needed to generate the classes with which the polynomial classifiers have to be modeled during the enrollment of a user. Specifically, having defined for each available subject 𝑡 ∈ {1, . . . , 𝑇 } two subsets 𝒮 and 𝒟 of genuine and forged signatures respectively, with 𝒮 ⊆ {1, . . . , 𝑆} and 𝒟 ⊆ {1, . . . , 𝐷}, and having indi(𝑡) cated with z𝐺,𝑠 the 𝑀 -dimensional vectorial representation of the 𝑠-th genuine (G) signature taken from the subset 𝒮 (𝑡) of the 𝑡-th user, and with z𝐹,𝑑 the representation of the 𝑑-th forged (F) signature of the subset 𝒟 of 𝑡-th user, the following values are computed:

)2  ∑∣𝒮∣ ( (𝑡) (𝑡)  ∣𝒮∣ z − 𝝁 ∑ ⎷ 𝐺,𝑠 𝐺 𝑠=1 1 (𝑡) (𝑡) (𝑡) z , 𝝈𝐺 = 𝝁𝐺 = ∣𝒮∣ 𝑠=1 𝐺,𝑠 ∣𝒮∣ − 1 (8) and

)2  ∑∣𝒟∣ ( (𝑡) (𝑡)  ∣𝒟∣ z𝐹,𝑑 − 𝝁𝐹 ∑ ⎷ 𝑑=1 1 (𝑡) (𝑡) (𝑡) , z𝐹,𝑑 , 𝝈𝐹 = 𝝁𝐹 = ∣𝒟∣ ∣𝒟∣ − 1 𝑑=1 (9) from which it is then possible to evaluate the following measures: (𝑡) (𝑡) (𝑡) 𝝁 −𝝁 𝝈 (𝑡) (𝑡) 𝝀𝒮,𝒟 = 𝐹 (𝑡) 𝐺 , 𝝎𝒮,𝒟 = 𝐹(𝑡) . (10) 𝝈𝐺 𝝈𝐺 The values in (10) are computed for each subject 𝑡 ∈ {1, . . . , 𝑇 } varying the subsets 𝒮 and 𝒟. From the set of computed values it is then possible to estimate the pdfs 𝑝Λ (𝝀) and 𝑝Ω (𝝎) by collecting all the measures computed according to (10). The two evaluated statistics will be used during the verification phase, as described in Section 3.3, to generate different synthetic classes of simulated forgeries for each user whose identity has been claimed.

3.2. Enrollment Phase As already outlined, in order to keep the size of the requested user’s specific template as small as possible, the most of the requested processing is performed during the verification phase. As for the enrollment of a user 𝑢, it is assumed that 𝐸 signatures are collected from him and that (𝑢) each of them is represented through a feature vector z𝑒 having size 𝑀 , with 𝑒 = 1, . . . , 𝐸. The collected data are then processed in order to generate two vectors 𝝁(𝑢) and

267

𝝈 (𝑢) providing the estimated mean and standard deviation for each considered features, that is,

( )2 ∑ (𝑢)  𝐸 (𝑢) 𝐸 z − 𝝁 ∑ 𝑒 ⎷ 𝑒=1 1 . z(𝑢) , 𝝈 (𝑢) = 𝝁(𝑢) = 𝐸 𝑒=1 𝑒 𝐸−1 (11) The template which has to be stored for a specific user 𝑢 comprises just the two vectors 𝝁(𝑢) and 𝝈 (𝑢) , being thus characterized by a very limited size of just 2 ⋅ 𝑀 parameters. The considered template is then employed during the verification phase to carry out the whole process of generating the classes required to perform recognition by means of polynomial classifiers, as described in Section 3.3.

then formed after solving the system of normal equations given by ˆ (𝑢) ˆ (𝑢) )−1 ⋅ R . (15) Q(𝑢) = (R x(𝑂) x(𝑂) y The test signature provided by the subject claiming the 𝑢-th identity can be then processed in order to derive its feature-based representation z, which is employed to comˆ = (x(𝑂) (z))T ⋅ Q(𝑢) . The value in pute the score vector y ˆ should be related to the probability the first position of y that the provided sample x belongs to the class representing the claimed identity. The score resulting from the proposed verification process is then evaluated as ˆ (1) y , (16) ˆ (𝑛) − min𝑛 y ˆ (𝑛)) (max𝑛 y ˆ (𝑛) and min𝑛 y ˆ (𝑛) respectively the maxibeing max𝑛 y ˆ. mum and the minimum value of the vector y It is worth observing that the samples generated during verification to represent the class of the legitimate user, as well as those generated to take into account plausible forgeries, are extracted from distributions characterized by independent features. Although this assumption is made in order to keep small the size of the user’s template generated during the enrollment, the resulting performance are not significantly affected by this simplification. Moreover, it can also be noticed that all the possible classes, both the genuine and the forgeries ones, are considered equiprobable ˆ (𝑢) in (14). It is also worth when computing the matrix R x(𝑂) y ˆ (𝑢) specifying that a controlled pseudo-inversion of R is x(𝑂) strongly advised when computing (15), in order to avoid numerical instabilities in the matrix inversion operation.

3.3. Verification Phase The user’s specific template, composed by 𝝁(𝑢) and 𝝈 , is employed together with the general statistics estimated during the training phase 𝑝Λ (𝝀) and 𝑝Ω (𝝎), which are common for all the users enrolled in the system, to perform the verification phase, once the identity of user (1) 𝑢 is claimed by a subject. Specifically, 𝑊 vectors z𝑤 , with 𝑤 = 1, . . . , 𝑊 , are first generated by sampling an 𝑀 -dimensional Gaussian distribution having mean equal to 𝝁(𝑢) , and a diagonal covariance matrix with standard deviation 𝝈 (𝑢) . These 𝑊 vectors are assumed to be representative of the class comprising the original signatures of user 𝑢. In addition, other 𝑁 − 1 classes, each comprising 𝑊 fea(𝑛) ture vectors z𝑤 , with 𝑤 = 1, . . . , 𝑊 and 𝑛 = 2, . . . , 𝑁 , are generated in order to characterize different possibilities of forgeries. In more details, the samples belonging to the 𝑛-th class are generated by randomly sampling an 𝑀 -dimensional Gaussian distribution having mean equal to (𝑢)

4. Experimental Results In order to verify the effectiveness of the proposed approach, an extensive set of experimental tests has been performed using the public MCYT on-line signature corpus [14]. This database comprises 25 genuine signatures and 25 skilled forgeries, captured during five different sessions, for each of 100 users. The employed database has been divided in two disjoint data sets: a training set, containing the genuine and forged signatures of the first 30 subjects, and a test set, which includes the genuine and forged signatures of the remaining 70 users. The training set has been employed to estimate the general pdfs 𝑝Λ (𝝀) and 𝑝Ω (𝝎) introduced in Section 3.1. Specifically, the above statistics have been evaluated for all the 100 global features presented in [2], while the considered sets 𝒮 and 𝒟 have been generated by comprising signatures taken during one or two sessions, being thus the cardinality of 𝒮 and 𝒟 equal to 5 or 10. The training set has been also employed to determine the 𝑀 signature features which have to be used to represent the considered on-line signatures. More in detail, the feature selection process described in [15] is performed to select the most discriminative features to be employed in the proposed systems, out of

(12) 𝝁(𝑛) = 𝝀(𝑛) ⋅ 𝝈 (𝑢) + 𝝁(𝑢) and a diagonal covariance matrix with standard deviations 𝝈 (𝑛) = 𝝎 (𝑛) ⋅ 𝝈 (𝑢) (𝑛)

(13)

(𝑛)

being 𝝀 and 𝝎 two 𝑀 -dimensional random values extracted respectively from the probability density functions 𝑝Λ (𝝀) and 𝑝Ω (𝝎) estimated during the training. The 𝑊 vectors available for each of the 𝑁 generated classes are then polynomially expanded to the 𝑂-th order by applying the operator x(𝑂) (⋅) introduced in (5). In order to compute the transformation matrix Q(𝑢) specific of the 𝑢-th user, the following matrices are first evaluated: ˆ (𝑢) = R x(𝑂) ˆ (𝑢) R x(𝑂) y

𝑁 𝑊 1 ∑ ∑ (𝑂) (𝑛) T x (z𝑤 ) ⋅ (x(𝑂) (z(𝑛) 𝑤 )) 𝑁 𝑊 𝑛=1 𝑤=1

𝑁 𝑊 1 ∑ ∑ (𝑂) (𝑛) = x (z𝑤 ) ⋅ (y(𝑛) )T 𝑁 𝑊 𝑛=1 𝑤=1

(14)

where y(𝑛) = [0, . . . , 0, 1, 0, . . . , 0]T possesses a unique 1 in the 𝑛-th position. The discriminant matrix Q(𝑢) can be

268

14

16 Mahalanobis distance Polynomial classifiers, O = 1 Polynomial classifiers, O = 2 Polynomial classifiers, O = 3

14

FAR with Mahalanobis distance and real skilled forgeries FAR with Mahalanobis distance and synth. skilled forgeries FAR with polynomial classifiers and real skilled forgeries FAR with polynomial classifiers and random forgeries

12

12 10

FAR (in %)

EER (in %)

10 8

8

6

6 4

4

2

2 0 1

2

3

4

5

6

7

8 M

9

10

11

12

13

14

0 0

15

2

4

6 8 FRR (in %)

10

12

14

Figure 1. Performance achievable with polynomial classifiers and Mahalanobis distance, for different numbers of features 𝑀 and orders of expansion.

Figure 2. Performance achievable with polynomial classifiers and Mahalanobis distance, considering real skilled forgeries, synthetic skilled forgeries and random forgeries.

the available ones. The employed feature selection procedure is iterative and requires 𝑀 steps: having defined with 𝒦 = {1, . . . , 100} the set of all available features, and with ℒ𝑚 the set of features selected at the 𝑚-th step, being ℒ0 the initialization empty set, the following actions are performed at step 𝑚, 𝑚 = 1, . . . , 𝑀 : 1. the performances achievable over the training set with the proposed approach, by exploiting the feature set ∪ given by ℒ𝑚−1 𝑘 are computed for any feature 𝑘 ∈ {𝒦 ∖ ℒ𝑚−1 }, being “∖” the set difference operator; 2. the set ∪ of selected features is updated as ℒ𝑚 = ˜ being 𝑘˜ the feature providing the best {ℒ𝑚−1 𝑘}, performances in terms of Equal Error Rate (EER). The feature selection process has been performed by considering feature expansions at the third order, while considering 𝐸 = 𝑊 = 10 during the enrollment and verification phases. Once the training phase has been accomplished, we have evaluated the performance of the proposed on-line signature recognition system based on polynomial classifiers over the test data set. The number of classes employed for each user in the verification phase is 𝑁 = 100, therefore comprising one class representing signatures taken from a legitimate user, and 99 classes associated to possible forgery attempts. Figure 1 shows the obtained results in terms of EER, evaluated for different values of 𝑀 and different orders of expansions. As can be seen, an EER = 4.22% can be obtained by using only 13 features expanded to the third order. Also the expansion to the fourth order, not reported in Figure 1 for the sake of image clarity, allows achieving performance similar to those of the third order for 12 ≤ 𝑀 ≤ 15. In Figure 1 it is also reported a comparison between the proposed approach relying on polynomial classifiers and a system us-

ing the Mahalanobis distance during the matching phase as in [2]. The feature selection process described in [15] and reported above is also applied to estimate the best feature set for this kind of approach. As can be seen, the proposed system has better performance than the one relying on Mahalanobis distance, resulting in an improvement in recognition rates close to the 20%, while maintaining the same size for the employed user’s specific template. We have also verified the effectiveness of the method proposed for the synthesis of features representing skilled forgeries for a given user. Specifically, Figure 2 shows the Receiver Operating Characteristics (ROC) which can be obtained when performing verification by means of the scalar Mahalanobis distance as in [2], while considering both the forgeries originally available in the considered test data set from MCYT, and synthetic skilled forgeries generated according to the approach described in Section 3. The 𝑀 = 13 features providing the best results when using Mahalanobis distance for classification have been used in both cases. Moreover, also the ROC curves related to the use of the proposed approach with the same feature set are reported in Figure 2. Specifically, both the False Acceptance Rate (FAR) computed when considering skilled forgeries, and the FAR related to the use of random forgeries, are displayed with respect to the False Recognition Rate (FRR). As can be seen, the proposed approach for the generation of 𝑁 synthetic classes to be used by a polynomial classifier guarantees good recognition rates in both cases. Eventually, we have also reported in Figure 3 the performances achievable by using the DTW approach [3] over the same test data set. As can be seen, better performance can be obtained with a local method such DTW, yet the requested template size can be also thousands of times larger

269

14

equiprobable a priori probabilities for the belonging of a sample to a specific class in (14), as well as the use of notdiagonal matrixes for the generation of the synthetic forged signature features.

Polynomial classifiers DTW Fusion of Polynomial classifiers and DTW Fusion of Mahalanobis distance and DTW

12

FAR (in %)

10

References [1] P. Campisi, E. Maiorana, A. Neri “Signature Biometrics,” in Encyclopedia of Cryptography and Security, H.C.A van Tilborg and S. Jajodia, editors, Springer, 2011.

8

6

[2] J. Fierrez-Aguilar, L. Nanni, J. Lopez-Penalba, J. OrtegaGarcia, D. Maltoni, “An On-Line Signature Verification System Based on Fusion of Local and Global Information,” AVBPA, 2005.

4

2

0 0

2

4

6 8 FRR (in %)

10

12

[3] A. Kholmatov, B. Yanikoglu, “Identity authentication using improved online signature verification method,” Pattern Recognition Letters, Vol. 26, No. 15, 2005.

14

[4] L. Yang, B.W. Widjaja, R. Prasad, “Application of hidden Markov models for signature verification”, Pattern Recognition, Vol. 28, No. 2, 1995.

Figure 3. Performance achievable with polynomial classifiers, DTW, and the fusion of both.

[5] P.S. Sandhu, I. Kaur, A. Verma, S. Jindal, S. Singh, “Biometric Methods and Implementation of Algorithms”, Int. Journal of Electrical and Electronics Engineering, Vol. 3, No. 8, 2009.

than the one employed in the proposed methods, which requires the storage of just 2 ⋅ 𝑀 = 26 parameters, while the time required to perform a comparison is in average 8 times greater, when both approaches are implemented in MatlabⓇ . However, Figure 3 shows that the proposed polynomial classifiers can be employed to enhance the performance of a system relying on DTW too, by fusing the scores generated by the two approaches through the tanh normalization and the sum fusion rule [16]. The obtained performance are better than those achieved by fusing a DTW- and a Mahalanobis distance-based classifier, where the best results are provided through median normalization and sum fusion rule.

[6] H.S.M. Beigi, “Aggressive compression of the dynamics of handwriting and signature signals,” IEEE Int. Conf. on Multimedia and Expo (ICME), 2004. [7] S. Schimkea, S. Kiltza, C. Vielhauer, T. Kalkerb, “Security Analysis for Biometric Data in ID Documents”, Security, Steganography, and Watermarking of Multimedia Contents VII, 2005. [8] X. Ling, Y. Wang, Z. Zhang, Y. Wang, “On-line Signature Verification Based on Gabor Features,” Wireless and Optical Communications Conf. (WOCC), 2010. [9] M.T. Ibrahim, M. Kyan, L. Guan, “On-line Signature Verification Using Global Features,” Canadian Conf. on Electrical and Computer Engineering (CCECE), 2009.

5. Conclusions In this paper we have proposed an on-line signature recognition system relying on polynomial classifiers. The proposed approach is able to guarantee accurate recognition rates while requiring the storage of very compact user’s specific templates. We have proposed a strategy for providing the employed classifiers with data able to effectively represent features extracted from synthetic forgeries, thus allowing to use polynomial classification in a verification modality and realizing a system able to properly partition the observation space of each user in decision regions separating the genuine representations from the forged ones. The achieved experimental results show the effectiveness of the proposed recognition system based on polynomial classifiers, as well of the proposed means for generating synthetic skilled forgeries. Further improvements of the proposed system may regard the use of user-dependent thresholds, as well of different and more discriminant features to be extracted from on-line signatures. As for the employed modeling, it would be possible to investigate the effects of assigning not-

[10] J. Schurmann, Pattern classification: a unified view of statistical and neural approaches, Wiley, 1996. [11] W.M. Campbell, K.T. Assaleh, C.C. Broun, “Speaker recognition with polynomial classifiers,” IEEE Transactions on Speech and Audio Processing, Vol. 10, No. 4, 2002. [12] X.Y. Zhang, J. Wu, Q. Zhang, “Speaker identification based on modified polynomial classifier,” Int. Conf. on Machine Learning and Cybernetics, 2003. [13] N.S. Nehe, R.S. Holambe, “Isolated word recognition using polynomial classifier,” Int. Conf.on Control, Automation, Communication and Energy Conservation (INCACEC), 2009. [14] J. Ortega-Garcia et al. “MCYT baseline corpus: A bimodal biometric database,” IEE Proceedings Vision, Image and Signal Processing, Vol. 150, No. 6, 2003. [15] E. Maiorana, P. Campisi, A. Neri, “Feature Selection and Binarization for On-line Signature Recognition”, IAPR/IEEE Int. Conf. on Biometrics (ICB), 2009. [16] A. Ross, K. Nandakumar, A.K. Jain, Handbook of Multibiometrics, Springer, 2006.

270