Expert Systems with Applications 41 (2014) 4170–4180

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Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

On-line signature verification using vertical signature partitioning

0957-4174/$ - see front matter � 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.eswa.2013.12.047

⇑ Corresponding author. Tel./fax: +48 343250546.E-mail addresses: krzysztof.cpalka@iisi.pcz.pl (K. Cpałka), marcin.zalasinski@

iisi.pcz.pl (M. Zalasinski).

Krzysztof Cpałka, Marcin Zalasinski ⇑Institute of Computational Intelligence, Czestochowa University of Technology, Al. Armii Krajowej 36, 42-200 Czestochowa, Poland

a r t i c l e i n f o a b s t r a c t

Keywords:BiometricsOn-line signatureDynamic signatureFuzzy sets theoryNeuro-fuzzy systemsInterpretability

In this paper we propose a new approach to identity verification based on the analysis of the dynamicsignature. Considered problem seems to be particularly important in terms of biometrics. Effectivenessof signature verification significantly increases when dynamic characteristics of the signature are consid-ered (e.g. velocity, pen pressure, etc.). These characteristics are individual for each user and difficult toforge. The effectiveness of the verification on the basis of an analysis of the dynamics of the signaturecan be further improved. A well-known way is to consider the characteristics of the signature in the sec-tions called partitions. In this paper we propose a new method for identity verification which uses par-titioning. Partitions represent time moments of signing of the user. In the classification process thepartitions, in which the user created more stable reference signatures during acquisition phase, are moreimportant. Other important features of our method are: using capabilities of fuzzy set theory and devel-opment on the basis of them the flexible neuro-fuzzy systems and interpretable classification system forfinal signature classification. In this paper we have included the simulation results for the two currentlyavailable databases of dynamic signatures: free SVC2004 and commercial BioSecure database.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction process, e.g. pressure. In this case the shape of the signature is

Signature is a biometric attribute used to verify identity of theindividual. It belongs to the behavioural biometric attributes whichare related to the pattern of behaviour of a person. Verificationbased on these attributes is more difficult than verification basedon the physiological ones, but this process is less invasive than ver-ification with use of the physiological attributes (e.g. fingerprint,iris). Moreover, signature is very interesting behavioural attributewhich is commonly accepted in the society. Thus if the effective-ness of verification systems based on the signature is high enough,they may be used for commercial purposes, for example as the ac-cess verification systems to the workplace or as the systems sup-porting identity authentication in banks. Moreover, they mayreplace many commonly used methods of authorization, e.g. pass-word authorization, PIN code authorization. Disadvantage of thesetraditional methods is possibility of forgetting or losing passwordor PIN code. The use of dynamic signature in authorization processeliminates these dangers.

Signature verification systems may be classified into two cate-gories – static (off-line) and dynamic (on-line). Systems usingoff-line work on the basis of information about the shape of thesignature and can use, for example, signatures from documentsin verification process. Systems using on-line signature duringverification can use also information about dynamics of the signing

represented by the horizontal and vertical trajectories. Identityverification based on the on-line signature is more reliable thanverification using the off-line signature, because the dynamic fea-tures make the signature more unique and more characteristicfor the individual.

Approaches to identity verification based on dynamic signaturemay be categorized into few groups:

– Global feature based methods. Some methods base on the glo-bal features which are extracted from signature and used duringtraining and classification phase. Examples of these features aresignature total duration and number of pen-ups. Approachbased on global features may be found in many research papers.In Nanni and Lumini (2005) authors have proposed method ofsignature verification based on the ensemble of Parzen windowclassifiers. In Nanni and Lumini (2006) system based on two-class problem is presented. The system uses Support VectorMachine to classify vector of signature global features as genu-ine or forgery. In Nanni (2006a) method using combination oftwo-class classifier and one-class classifier is proposed. Fusionof the classifiers is realized using the radial basis function-sup-port vector machine. In Lumini and Nanni (2009) algorithm forbuilding an ensemble of on-line signature verification systemsbased on one-class classifiers and using ‘‘artificial features’’ isshown. These features are extracted with use of so calledOverComplete global feature combination, and a small subsetof features used during classification is selected by sequential

K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180 4171

forward floating selection. In Fierrez-Aguilar, Nanni, Lopez-Penalba, Ortega-Garcia, and Maltoni (2005) authors have pro-posed a set of features sorted by their individual discriminativepower for the group of signers. The best results achieves fusionof classifier based on Principal Component Analysis method andParzen window classifier.

– Function based methods. Another approach commonly used inidentity verification based on dynamic signature is function-based approach. This approach bases on comparison of timefunctions, which contains information about changes of signa-ture features over time. Time functions extracted from the signa-ture are compared to the time functions of the other signatureand classification is made on the basis of this process result.Comparison is performed using elastic distance measures, e.g.Dynamic Time Warping (see Jeong, Jeong, & Omitaomu, 2011).In this approach one can use time functions acquired duringsigning process on the digital tablet (e.g. x-trajectory, y-trajec-tory, pressure), their derivatives (velocity, altitude) or combina-tions (difference between the values of two consecutive points ofx-trajectory or y-trajectory). In Jain, Griess, and Connell (2002)authors have proposed verification based on the set of featurescomputed for each discretization point. This set contains fea-tures which describe shape of the signature and the ones basedon dynamics of the signing process. In Kholmatov and Yanikoglu(2005) classification based on the distances of the test signatureto the nearest, farthest and template reference signatures, storedin a three-dimensional feature vector, was presented. The dis-tances are computed by Dynamic Time Warping algorithm andthe feature vector is classified into one of the two classes (genu-ine or forgery) by classifier based on Principal Component Anal-ysis. In Faundez-Zanuy (2007) classification by combination ofvector quantization and Dynamic Time Warping was presented.The combination is performed by means of score fusion. In Nanniand Lumini (2008) dynamic signature verification system basedon the Linear Programming Descriptor classifier was presented.In this method the time functions extracted from the signaturesare transformed by discrete 1-D wavelet transform and next theDiscrete Cosine Transform is used to reduce the approximationcoefficients vector to a feature vector of a given dimension. InMaiorana (2010) author has presented function based systemwith use of cryptographic techniques to provide protection andcancelability to signature templates.

– Regional based methods. The literature contains alsoapproaches relying on segmentation of signature into someregions, which are used during training and verification phase.Methods based on regional information of signature use oftenclassifier based on Hidden Markov Models (see e.g. Fierrez,Ortega-Garcia, Ramos, & Gonzalez-Rodriguez, 2007). Manyauthors propose also different methods of classification. InHuang and Hong (2003) signatures are segmented into strokesand for each of them reliability measure is computed on thebasis of the features values which belong to the current stroke.In Khan, Khan, and Khan (2006) a stroke-based algorithm thatsplits velocity signal into three bands was proposed. Thisapproach assumes that low and high-velocity bands of the sig-nal are unstable, whereas the medium-velocity band is useablefor discrimination purposes. In our previous works we have alsoconsidered a method from regional based methods group (seeZalasinski & Cpałka, 2011, 2012, 2013). The idea of this methodwas consideration of signature areas characteristic for eachuser. These areas (known as horizontal partitions) were associ-ated with high and low velocity of signature, and a low and asmall value of pen pressure.

– Hybrid methods. In the literature one can also find the hybridmethods which are based on combination of the describedapproaches. For example in Nanni, Maiorana, Lumini, and

Campisi (2010) system for dynamic signature verification basedon an ensemble of local, regional, and global matchers was pre-sented. The system uses fusion of two methods employingDynamic Time Warping, a Hidden Markov Model approachand a Linear Programming Descriptor classifier trained by glo-bal features. In Moon, Lee, Cho, and Kim (2010) authors havepresented system based on combination of global featuresbased methods and function based methods. Feature basedmodule calculates the distance between the multi-dimensionalvectors of reference signature and test signature. Functionbased module calculates the accumulated distance betweeninput time of reference signature and test signature. Next, cal-culated distances are combined in accordance with an appropri-ate form.

In this paper we propose new method for the on-line signatureverification based on partitioning. The method belongs to the re-gional based methods group and it bases on partitioning of wave-forms which describe the signature. The idea of partitioning hasbeen considered in the literature:

– In Ibrahim et al. (2010) authors have proposed a very interest-ing and effective algorithm. It assumes using in classificationprocess vertical and horizontal trajectories of signature whichlies in the regions of it, extracted on the basis of the values ofpressure and velocity signals. The regions of signature are calledpartitions. After division of signature trajectories into partitions,the template which represents information about the signer iscreated for each partition on the basis of the training signatures.Next, selection of the most discriminative partition (called sta-ble partition) is performed. Stable partition is selected on thebasis of similarities between each training signature of the userand the template. The template from selected partition is usedduring verification process to determine whether the test signa-ture of the user is true or not.

– In our previous work Zalasinski and Cpałka (2012) we have pro-posed a new algorithm for dynamic signature verification usinghorizontal partitioning. Partitions corresponded to high and lowvelocity of the signature and high and low pressure. For eachuser partitions and importance of them in the classification pro-cess were selected individually. This allowed for more efficientverification of identity. On the basis of this method the methodproposed in Zalasinski and Cpałka (2013) was created, which ischaracterized by interpretability of the signature classificationprocess. Using that methods (differ in the way of final classifica-tion) we obtained a very good accuracy and come to some inter-esting conclusions. It turned out that in the classification ofdynamic signature more important role than pressure signalplays velocity signal, and more characteristic for the users werethese areas of the signature, which were created at a higherspeed and larger pen pressure. The results of simulationsencouraged us to continue to use the idea of partitioning inthe process of signature verification. The method proposed inthis paper also uses partitioning, but it is decisively differentfrom our previously proposed methods. This is due to the factthat in this paper partitions have a completely different inter-pretation and represent time moments of signing by the user.

It should be noted, that the method proposed in this paper ischaracterized by important advantages over other methods avail-able in the literature which use partitioning of dynamic signature.These advantages include, among others, use of the partitionimportance in the classification process of dynamic signature,interpretability of knowledge acquired in the classifier of the signa-ture and using the abilities of fuzzy sets and fuzzy systems in thesignature classification. The combination of these characteristics

4172 K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180

makes that our method is characterized by both unconventionalapproach to the classification of the signature and the appropriateaccuracy of work. Primary characteristics of this method can besummarized as follows:

– Proposed method bases on four signature signals: x-trajectoryand y-trajectory are used in classification phase to compare sig-natures, pressure and velocity signals are used during pre-pro-cessing phase to develop the one-to-one correspondencebetween the signatures.

– Signature is divided into partitions on the basis of time indexesvalues. We assume that some regions of the signature acquiredin certain timeframe can be more characteristic for the userthan other regions. It is a distinctive feature of the proposedmethod.

– Each partition contains fragment of signature trajectories,which will be used during verification process. It is a distinctivefeature of the proposed method due to the fact that in otherworks many other features are used during classification phase.

– Weights of importance which values correspond to discrimina-tive power of user’s partition are computed for each partition. Itis a distinctive feature of the proposed method due to the factthat in other works use of weighted signature feature in classi-fication process is not applied.

– All partitions are used during verification process. It is a distinc-tive feature of the proposed method due to the fact that in otherworks, which use partitions of the signature, only one partitionis used during classification phase.

– Our method takes advantage of Zadeh’s fuzzy sets theory (seee.g. Zadeh, 1965). The basic element of this theory is fuzzy setwhich in a flexible manner describes the values assumed by lin-guistic variables (e.g. the ‘‘low’’, ‘‘high’’, ‘‘medium’’). Signaturescreated by each user in the acquisition phase are not the sameand also may differ from the signatures being under verifica-tion. Thus, the use of the fuzzy set theory to signature verifica-tion seems to be very appropriate. Of course, the theory of fuzzysets may be used to signature verification if the signaturesstored in database will be appropriate processed. It should benoted that in the group of methods that use partitioning, thetheory of fuzzy sets has not been used so far.

– Neuro-fuzzy classifier of the Mamdani-type is used in classifica-tion process. Neuro-fuzzy systems are characterized by workbased on a set of rules in the form of if-then and possibility ofcalculating the parameters of the system. These systems unitethe advantages of both neural networks and fuzzy systems(Garcia, 2012; Horzyk & Tadeusiewicz, 2004; Hyunsoo,Cheong-Sool, Jun Seok, & Jun-Geol, 2013; Korytkowski, Gabryel,Rutkowski, & Drozda, 2008, 2006; Pérez-Sánchez, Fontenla-Romero, Guijarro-Berdinas, & Martínez-Rego, 2013; Peteiro-Barral, Guijarro-Berdinas, & Perez-Sanchez, 2012; Pławiak &Tadeusiewicz, 2014; Rutkowski, 2008). In addition, a flexiblesystem allows to take into account the importance hierarchyof the rules antecedents and the importance of the whole rules(Cpałka, 2009a, 2009b; Rutkowski & Cpałka, 2003; Rutkowski &Cpałka, 2005a; Cpałka & Rutkowski, 2005a). It is worth notingthat used rules may be interpretable. This is due to the depen-dence of the rules parameters on the appropriate parameters ofthe dynamic signature verification process. This led to the use ofits in the signature classification process which is one of thesteps of proposed method.

– The proposed method allows for immediate consideration of thesignatures of the new user, because it does not require a processof learning. Addition of new user is associated only with thedetermination of descriptors of the user’s training signatures.

– Limitations of the proposed method result from the specificityof considered problem, namely the characteristics of a signaturebiometric feature. The most important of them is the lack ofresistance of the method to change the signature by the user.This change may be intentional or physiological, resulting froma change in the style of signing over time. Please note that thisrestriction applies to all of the methods of behavioural biomet-rics which analyses the dynamics of behaviour.

We would like to emphasize that approach to partitioning pro-posed in this paper (partitioning in the time domain) was not pre-viously considered in the literature. This prompted us to preparethis paper. An additional motivation for the development of ourmethod was elaboration of the concept to adapt our previous ideasfor partitioning in the time domain. It is about ideas related to: theassignment of weights to partitions, taking into account theweights in the classification process, and an interesting approachto classification. Of course, those components have to be imple-mented differently than in our previous works (due to the differenttype of problem).

To test the proposed method we used the database of dynamicsignatures which are admitted sources of data using in this field.The performance of the proposed method was tested using twotest data sets:

– Public SVC 2004 database (see Homepage of SVC, 2003) which iscommonly used by researchers to evaluate the effectiveness ofthe signature verification systems.

– Commercial BioSecure database (BMDB) distributed by the Bio-Secure Association (see Homepage of Association BioSecure,2013).

This paper is organized into 4 sections. Section 2 contains de-tailed description of the proposed algorithm. Simulation resultsare presented in Section 3. Conclusions are drawn in Section 4.

2. Description of the new method of dynamic signatureverification

Assumptions of our method are based on selection of signatureregions which are the most reliable for the signer. Each region rep-resents a period of time in which signature is created. Value ofimportance level is created for each region individually for eachuser. Higher stability of signature created by the user in the parti-tion means that this partition will be more important in the finalclassification of the signature. It should be noted that our methodis also distinguished by the process of final classification. As men-tioned earlier, capabilities of neuro-fuzzy system are used in theclassification. Another advantage of the system is the ability tointerpret the mode of action.

Algorithm proposed in this paper may be described by the fol-lowing steps:

– Step 1. Partitioning of signatures. Signatures are partitionedwith use of the method which creates vertical partitions, select-ing best discretization points groups (see Fig. 2).

– Step 2. Templates generation. In this step templates for eachpartition are generated (see Fig. 3).

– Step 3. Calculation of distances between signatures and tem-plate in each partition. The distances are treated as descriptorsof the signature.

– Step 4. Computation of the weights of importance. Duringthis step weights of importance for each partition are created.The weights are used in classification process.

K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180 4173

– Step 5. Creation decision boundary for each partition. Dur-ing this step linear decision boundary in each partition iscreated.

– Step 6. Determination of the fuzzy rules used in classificationphase. Fuzzy rules describe a way of test signature classifica-tion. The fuzzy sets in the rules are based on decision bound-aries determined in the step 5.

– Step 7. Classification. In this step signature is classified as gen-uine or forgery. In the verification process flexible neuro-fuzzysystem of the Mamdani type is used.

Steps 1–6 are performed during training phase, while steps1,3,7 are performed during test phase.

Please note that in the algorithm working in practice, templates,weights of partitions, decision boundaries and fuzzy rules can beupdated in test phase. Algorithm designed in such a way would al-low the implementation of changes in the way of signing by eachuser.

matched to each other training signaturesused in training and testing phases

mismatched trainingsignatures of the user

obtained in theacquisition phase matching the training

signatures usingDTW algorithm

final matchingof training signatures

selection of base signature

signals of trajectories

signals of penvelocity and signals

of pen pressure

Fig. 1. Matching of signatures using Dynamic Time Warping.

2.1. Pre-processing of signatures

In our method we use four signals of signature over time: x-tra-jectory, y-trajectory, pressure and velocity. Signatures and all fea-tures of them are acquired using a digital graphic tablet, which isa computer peripheral. Data transmitted from the tablet to thecomputer contain, among others, information about the currentposition of the pen and the value of the pen pressure on the tabletsurface. Velocity of the signature is first derivative of its trajectorysignal.

Measurement of time begins when pen first time contactswith the tablet and ends with the cessation of the user ac-tions on the graphic tablet surface. Having the values of thetime step for the subsequent samples and the sampling fre-quency of the tablet (which is the technical parameter ofthe device), you can determine the time characteristics ofthe signature.

Before beginning of main phase of the method, all training sig-natures of the ith signer should be pre-processed by commonlyused methods (see e.g. Jain et al., 2002; Ibrahim et al., 2010; Lei& Govindaraju, 2005; O’Reilly & Plamondon, 2009). Pre-processingis used to remove some intra-class variations. The signer creates afew signatures (5 in our simulations). All his signatures must bepre-processed with reference to one signature, called base signa-ture. The base signature is the most similar to all training signa-tures, the average Euclidean distance between it and other fourtraining signatures is the lowest. Next, signatures are subjectedto pre-processing, in order to match length, rotation, scale andoffset. Training signatures are matched to the base signatureusing the Dynamic Time Warping algorithm (see e.g. Bankó &János, 2012; Fernandez de Canetea, Garcia-Cerezoa, Garcia-Morala, Del Saza, & Ochoa, 2013; Svalina, Galzina, Lujic, & Šimunovic,2013), which operates on the basis of matching velocity and pres-sure signals. The result of matching of two signatures is a map oftheir corresponding points. On the basis of the map, trajectoriesof the signatures are matched. Matching using DTW could notbe done directly with the use of trajectories, because this wouldremove the differences between the shapes of the signatures. Itwould have a very negative impact on training. Elimination of dif-ferences in rotation of signatures is performed by the PCA algo-rithm which in the literature is commonly used to make theimages rotation invariant (see e.g. Gonzalez & Woods, 2002).The scale and offset are compensated by the standard geometrictransformations.

After a pre-processing, main phase of training process isperformed.

2.2. Partitioning of signatures

The new approach presented in this paper assumes partitioningbased on selected time intervals of signing. This approach is possi-ble to implement because lengths of the all signature signals arethe same through the pre-processing (see Fig. 1).

Each signal of the signature is divided into parts of the samewidth. Membership of the kth sample of the jth signature of theith user to the pth partition is described as follows:

partfsgi;j;k ¼

1 for 0 < k 6 Li

Pfsg

2 for Li

Pfsg< k 6 2Li

Pfsg

..

.

Pfsg for ðPfsg�1ÞLi

Pfsg< k 6 Li

;

8>>>>>><>>>>>>:

ð1Þ

where s is a signal type (velocity or pressure) used during alignmentphase, i is the user number (i ¼ 1;2; . . . ; I), j is the signature number(j ¼ 1;2; . . . ; J), Li is a number of samples of the ith user, k is thesample number (k ¼ 1;2; . . . ; Li) and Pfsg is a number of partitions(Pfsg � Li). In this method we assumed that Pfvg ¼ Pfzg. Partitioningprocedure is shown in Fig. 2.

Next, templates of the signatures for each partition aregenerated.

2.3. Templates generation

Generation of the templates is based on the training signatures.Templates are concerned with the user and assigned to the parti-

tion. Generation of an element of template tafsgp;i;k; p ¼ 1;2; . . . ; Pfsg,is calculated by the formula:

tafsgp;i;k ¼1J

XJ

j¼1

afsgp;i;j;k; ð2Þ

where afsgp;i;j;k is trajectory (x or y) value. Template tafsgp;i is described bythe following equation:

tafsgp;i ¼ tafsgp;i;1; tafsgp;i;2; . . . ; tafsgp;i;Kp;i

h i: ð3Þ

Idea of the template generation for the ith user is shown inFig. 3.

Next, distances between each template and each signature tra-jectory are calculated.

2.4. Calculation of distances between signatures and template in eachpartition

Distance dafsgp;i;j between template and a trajectory is described asfollows:

Fig. 2. Illustration of the process of partitions creation for the ith user for the ssignal.

Fig. 4. Illustration of selection of decision boundary location. Genuine trainingsignatures of the user are described as grey circles, genuine training signatures ofother users are described as diamonds, test signature is described as white circle.

4174 K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180

dafsgp;i;j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXK

k¼1

tafsgp;i;k � afsgp;i;j;k

� �2

vuut : ð4Þ

Next, distances between templates and signatures in twodimensional space are calculated. Distance dfsgp;i;j between the trajec-tory of signature and template is calculated by the formula:

dfsgp;i;j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dxfsgp;i;j

� �2þ dyfsgp;i;j

� �2r

: ð5Þ

In the next step weights of importance for partitions arecalculated.

2.5. Computation of the weights of importance

First step to compute weights of importance is calculation ofmean distances between signatures and template in partitions.Mean distance between signatures �dfsgp;i is calculated by theformula:

�dfsgp;i ¼1J

XJ

j¼1

dfsgp;i;j: ð6Þ

Then, standard deviation of distances in each partition shouldbe calculated. Standard deviation of signatures rfsgp;i is calculatedusing the following equation:

rfsgp;i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1J

XJ

j¼1

�dfsgp;i � dfsgp;i;j

� �2

vuut : ð7Þ

Next, weights of importance are calculated. Weight w0fsgp;i is cal-culated by the following formula:

w0fsgp;i ¼ �dfsgp;i � rfsgp;i : ð8Þ

Fig. 3. Illustration of the templates crea

After that, weights should be normalized. Normalization ofweight is used to simplify the classification phase. Weight wfsgp;i isnormalized by the following equation:

wfsgp;i ¼ 1�0:9 �w0fsgp;i

max w0fsg1;i ; . . . ;w0fsgPfsg ;i

n o : ð9Þ

Use of coefficient 0.9 in formula (9) causes that partition withthe lowest value of weight of importance is also used in classifica-tion process.

Next, selection of location of decision boundary is performed.

2.6. Creation decision boundary for each partition

Selection of location of decision boundary and determination of

the value dlrnmaxfsgp;i is performed for each partition (see Zalasinski& Cpałka, 2012). Idea of this process, which is performed once foreach user in the learning phase, is presented in Fig. 4. The decisionboundary is located between genuine signatures and forgery signa-tures. During this process genuine signatures of the other users areconsidered as forged signatures. The determined values have animpact on distribution of fuzzy sets, which represent valuesflow;highg assumed by the ðPfvg þ PfzgÞ linguistic variables ‘‘thetruth of the ith user signature from pth partition of s signal’’. It isshown in Fig. 5.

Next, determination of the fuzzy rules used in classificationphase is performed.

2.7. Determination of the fuzzy rules used in classification phase

In classification of the dynamic signature we used a flexibleneuro-fuzzy system of Mamdani type. This system is based onthe clear rules in the form if-then. The fuzzy rules contains fuzzy

tion for partition p of the ith user.

K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180 4175

sets which represent the values low, high of the input and outputlinguistic variables. In our method the input linguistic variables aredepended on the similarity between the test signature and thetemplate of the signature, and output linguistic variables describethe reliability of the signature. In our method input parameters offuzzy sets are individually selected for each user, so we can see forexample what is the stability of the training signatures of the user.Please note that if training signatures are more similar to eachother, the tolerance of our classifier is lower. Of course, similarityrefers to the shape of the signature in some regions selected onthe basis of velocity and pressure signals values. It significantlyhinders falsification of signatures. The flexibility of the classifier re-sults from the possibility of using in the classification the impor-tance of signature regions, which are selected individually foreach user. Taking into account the weights of importance of thepartitions is possible thanks to the use of proposed by us earlier(see e.g. Rutkowski & Cpałka, 2003) aggregation operators namedweighted triangular norms. In theory of fuzzy systems it is as-sumed that the fuzzy rules can be defined by an expert or can beadapted to the training data in an iterative way as a result of learn-ing (see e.g. Cpałka, 2009b; Cpałka & Rutkowski, 2005b; Rutkowski,2008; Gabryel, Cpałka, & Rutkowski, 2005; Przybyłand Cpałka,2002). In our method, the rules are very precisely adjusted individ-ually for each user without the need to learn the system (for eachuser input parameters of fuzzy sets are stored in the database). Onthe one hand, it allows the interpretation of the classification of theuser’s signature, on the other hand improves the accuracy ofclassification.

Our system works on the basis of two fuzzy rules presented asfollows:

R1 :IF dtstfsg1;i is A1fsg

1;i

� �wfsg1;i

��� AND . . .

dtstfsgPfsg ;i

is A1fsg

Pfsg ;i

� �wfsg

Pfsg ;i

��� THEN yi is B1

264

375

R2 :IF dtstfsg1;i is A2fsg

1;i

� �wfsg1;i

��� AND . . .

dtstfsgPfsg ;i

is A2fsg

Pfsg ;i

� �wfsg

Pfsg ;i

��� THEN yiisB2

264

375

8>>>>>>>><>>>>>>>>:

; ð10Þ

where

– dtstfsgp;i are input linguistic variables, whose numeric value is adistance between the test signature trajectory of the ith signerand decision boundary in the pth partition for signaturesaligned with use of signal s.

– A1fsg

p;i ;A2fsg

p;i are input fuzzy sets related to the signal s 2 fv ; zg of

the ith signer shown in Fig. 5. Fuzzy sets A1fsg

p;i

A1fvg

1;i ; . . . ;A1fvg

Pfvg ;i;A1fzg

1;i ; . . . ;A1fzg

Pfzg ;i

� �represent values ‘‘low’’ assumed

by input linguistic variables dtstfsgp;i . Analogously, fuzzy sets A2fsg

p;i

Fig. 5. Input and output fuzzy sets of the flexible neuro-fuzzy system of theMamdani type for signature verification.

A2fvg

1;i ; . . . ;A2fvg

Pfvg ;i;A2fzg

1;i ; . . . ;A2fzg

Pfzg ;i

� �represent values ‘‘high’’

assumed by input linguistic variables dtstfsgp;i . Thus, each rule

contains ðPfvg þ PfzgÞ antecedents.– yi is output linguistic variable interpreted as reliability of signa-

ture considered to be created by the ith signer.– B1;B2 are output fuzzy sets shown in Fig. 5. Fuzzy set B1 repre-

sents value ‘‘low’’ of output linguistic variable determining thereliability of signature. Analogously, fuzzy set B2 representsvalue ‘‘high’’ of output linguistic variable determining the reli-ability of signature.

– wfsgp;i are weights of the pth partition of the ith signer related tosignal s.

In the next subsection, signature verification is described.

2.8. Signature verification

In this step flexible Mamdani-type neuro-fuzzy system is used(see e.g. Cpałka & Rutkowski, 2005b; Rutkowski, 2008). A signatureis true if the following assumption is satisfied:

�yi ¼

T�l

A21;ifsg dtstfsg1;i

� �; . . . ;l

A2fsgPfsg ;i

dtstfsgPfsg ;i

� �;

wfsg1;i ; . . . ;wfsgPfsg ;i

8><>:

9>=>;

T�l

A2fsg1;i

dtstfsg1;i

� �; . . . ;l

A2fsgPfsg ;i

dtstfsgPfsg ;i

� �;

wfsg1;i ; . . . ;wfsgPfsg ;i

8><>:

9>=>;þ

T�l

A1fsg1;i

dtstfsg1;i

� �; . . . ;l

A1fsgPfsg ;i

dtstfsgPfsg ;i

� �;

wfsg1;i ; . . . ;wfsgPfsg ;i

8><>:

9>=>;

0BBBBBBBBB@

1CCCCCCCCCA

> cthi;

ð11Þ

where

– T� �f g is a weighted t-norm (see Rutkowski & Cpałka, 2003) inthe form:

T�fa1;a2;w1;w2g¼ Tf1�w1 � 1�a1ð Þ;1�w2 � ð1�a2Þg

¼e:g:ð1�w1 � ð1�a1ÞÞ � ð1�w2 � ð1�a2ÞÞ; ð12Þ

where t-norm Tf�g is a generalization of the usual two-valuedlogical conjunction (studied in classical logic), w1 andw2 2 ½0;1� mean weights of importance of the argumentsa1; a2 2 ½0;1�. Please note that T�fa1; a2; 1;1g ¼ Tfa1; a2g andT�fa1; a2; 1;0g ¼ a1. More details about weighted t-norm canbe found in Cpałka (2009b), Cpałka and Rutkowski (2005b)and Rutkowski (2008).

– lAð�Þ is a triangular membership function expressed by the fol-lowing formula (see e.g. Rutkowski, 2008):

lAðxÞ ¼

0 for x 6 ax�ab�a for a 6 x 6 bc�xc�b for b 6 x 6 c

0 for x P c

8>>><>>>:

: ð13Þ

Please note that for rule 1 from the base of rules (10) parame-ters a ¼ b ¼ 0 and parameter c corresponds to parameter

dlrnmaxfsgp;i . Analogously, for rule 2 from the base of rules (10)

parameter a ¼ 0 and parameters b ¼ c ¼ dlrnmaxfsgp;i .– �yi; i ¼ 1;2; . . . ; I, is the value of the output signal of applied

neuro-fuzzy system described by rules (10). Detailed descrip-tion of the system can be found in Cpałka (2009b), Cpałka andRutkowski (2005b), Rutkowski (2008) and Rutkowski andCpałka (2005a). Formula (11) was created by taking intoaccount in the description of system simplification resulting

4176 K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180

from the spacing of fuzzy sets, shown in Fig. 5. The simplifica-tions are presented below:

lA1

p;ifsg dlrnmaxfsgp;i

� �¼ 0

lA1

p;ifsg ð0Þ ¼ 1

lA2

p;ifsg dlrnmaxfsgp;i

� �¼ 1

lA2

p;ifsg ð0Þ ¼ 0

8>>>>>>>><>>>>>>>>:

: ð14Þ

– cthi 2 ½0;1� - coefficient determined experimentally duringtraining phase for each user to eliminate disproportion betweenFAR and FRR error (see e.g. Yeung et al., 2004). The parameterscthi 2 ½0;1�, computed individually for the ith user are used dur-ing verification process in the test phase.

Remarks relating to the approach used to classification can besummarized as follows:

– Our method uses authorial flexible neuro-fuzzy classifier inappraisal of the reliability of the tested signatures. The methodsused to verify identity on the basis of analysis of the dynamicsof signature must stand out by their ability to process large dat-abases of dynamic signatures belonging to multiple users. Inaddition, these methods must be characterized also by possibil-ity of immediate consideration of new signatures (belonging tonew users).

– The specific character of the problem, particularly the need forimmediate adaptation of method for verification of signaturesof new users, determined our approach to the classification.We decided to use a method of classification which does notrequire a training process, taking some inspiration from thecommon solution in this area (see e.g. Jain et al., 2002; Kholma-tov & Yanikoglu, 2005; Nanni, 2006b; Khan et al., 2006; Nanni &Lumini, 2008; Ibrahim et al., 2010).

– The approach used for the classification does not use the capa-bilities of a common classification methods known in the liter-ature. With this in mind we used a solution based on the theoryof fuzzy sets. Using fuzzy sets we built fuzzy rules in the form‘‘if . . .then . . .’’, which define the appraisal of the signature’s reli-ability. Therefore, used fuzzy sets determine the similarity ofthe signature to the template associated with the user withineach of the separated partition. In addition, with every anteced-ent of the rules (expressed using the appropriate fuzzy set) weassociated weight of importance individually for each user. Itsvalue is proportional to the stability of the signing by the userin the corresponding partition. In this way, a flexible neuro-fuzzy classifier was created, which is the original solution pro-posed by the authors of the paper. During development of thisclassifier we took into account our experience with neuro-fuzzysystems (mentioned earlier) and experience in the field ofdynamic signature verification.

– The use of flexible neuro-fuzzy classifier in appraisal of the reli-ability of the tested signatures also changes the look on the pro-cess of classification. Fuzzy rules of the classifier define the wayof classification of the signature which depends on the locationof the test signature descriptors in relation to the boundary ofthe inclusion of the training user’s signatures. The sample(white circle in Fig. 4, representing the distance of the test sig-nature from the template in a given partition) does not have tobe classified as false even if it is situated over the border ofinclusion of user’s training signatures in the partition (withinthe inclusion of false signatures). This happens when: (a) sam-ple in the other partitions is sufficiently similar to the template,(b) the reliability of the partition is small (denoted by the lowvalue of the weight of the partition). It is a distinctive feature

of our method against the methods presented in the abovementioned works.

2.9. Aspects of interpretability

Aspects of interpretability used in the method are based on ourprevious experience with interpretability of rule-based systems(see e.g. Cpałka, Łapa, Przybył, & Zalasinski, 2014; Cpałka, 2009a).We use the fuzzy rules described by the formula (10) to evaluatereliability of the signature. Values of linguistic variables in the fuzzyrules are represented by fuzzy sets, as mentioned earlier. These setsare shown in Fig. 5. Substituting the labels of fuzzy sets (shown inthe figure) and considering the name of linguistic variables in Eq.(10), mechanism of inference may be presented in a more readableway (weights of importance of the partitions have been omitted forreadability of the notation):

Rð1Þ :

IF ð‘similarity to template of partition 1’ IS ‘low’ÞAND ð‘similarity to template of partition 2’ IS ‘low’Þ . . .

AND ð‘similarity to template of partition last’ IS ‘low’ÞTHEN ð‘reliability of signature’ IS ‘low’Þ

Rð2Þ :

IF ð‘similarity to template of partition 1’ IS ‘high’ÞAND ð‘similarity to template of partition 2’ IS ‘high’Þ . . .

AND ð‘similarity to template of partition last’ IS ‘high’ÞTHEN ð‘reliability of signature’ IS ‘high’Þ

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

: ð15Þ

It is worth noting that so-defined fuzzy rules perfectly meet theassumptions of rule-based systems interpretability (also semanticinterpretability), summarized in detail e.g. in the paper Gacto,Alcala, and Herrera (2011). In the cited paper interpretability crite-ria were grouped as follows: Q1: complexity at the rule base level(number of rules, number of conditions), Q2: complexity at the le-vel of fuzzy partitions (number of membership functions, numberof features or variables), Q3: semantics at the rule base level (rulesfired at the same time), Q4: semantics at the fuzzy partition level(completeness or coverage, normalization, distinguish ability, com-plementarity). The rules defined by us meet all of the levels in-cluded in the Q1-Q4 and described in detail in the cited paper.

The authors would like point out that the parameters of therules are individually selected for each user on the basis of analysisof training signatures created in the acquisition phase, but theinference mechanism is consistent. In addition, in the selection ofrules the supervised learning algorithm is not used, as is usuallye.g. in case of the selection of the parameters of neuro-fuzzyclassifiers.

3. Simulation results

The simulation was performed with use of two currently avail-able databases:

– Public SVC2004 database (see Homepage of SVC, 2003). Thedatabase contains signatures of 40 users, acquired in two ses-sions using the digitizing tablet. Set of each user contains 20genuine signatures from one signature contributor and 20skilled forgeries from five other contributors. During trainingphase 5 genuine signatures of each signer were used. Duringtest phase 10 genuine signatures and 20 forged signatures ofeach signer were used.

– Commercial BioSecure database, distributed by the BioSecureAssociation (see Homepage of Association BioSecure, 2013).Simulations were performed using DS2 Signature database

Table 1Results of simulation performed by our system using SVC2004 database for differentnumber of partitions (row describing partition number with the best value of averageerror is bold).

Pfsg Average FAR (%) Average FRR (%) Average error (%)

2 11.52 10.45 10.993 10.95 10.45 10.704 6.32 24.5 15.41

Table 2Results of simulation performed by our system using BioSecure database for differentnumber of partitions (row describing partition number with the best value of averageerror is bold).

Pfsg Average FAR (%) Average FRR (%) Average error (%)

2 3.13 4.15 3.643 2.57 5.50 4.044 1.94 8.85 5.40

Table 3Comparison of simulation results for the SVC2004 database (method describingmethod with the best value of average error is bold).

Method Average FAR(%)

Average FRR(%)

Average error(%)

Ibrahim et al. (2010) 11.05 13.75 12.40Zalasinski and Cpałka

(2013)12.15 11.00 11.58

Our method 10.95 10.45 10.70

K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180 4177

which contains signatures of 210 users. This database containstwo sessions, acquired two weeks apart. Each session contains15 genuine signatures and 10 skilled forgeries per person. Dur-ing training phase 5 genuine signatures of each signer from ses-sion number one were used. During test phase 10 genuinesignatures and 10 forged signatures of each signer from sessionnumber two were used.

The test was carried out five times for all signers, with use ofrandomly chosen training and test signatures. Results of the simu-lations are described as:

– Values of FAR (False Acceptance Rate) and FRR (False RejectionRate), which are commonly used in biometrics (see e.g. Jain &Ross, 2008), determined for different numbers of partitionsPfsg (see Tables 1–4).

– Values of weights of importance set for each partition, averagedin the context of the users (see Fig. 6), which describe reliabilityof the signature in the partitions.

– Values dlrnmax averaged for all the users, describing whichareas of the signature are the most characteristic for the user(see Fig. 7).

All tests were performed using the authorial testing environ-ment implemented in C# language.

Table 4Comparison of simulation results for the BioSecure database (row describing methodwith the best value of average error is bold).

Method Average FAR(%)

Average FRR(%)

Average error(%)

Ibrahim et al. (2010) 2.96 6.20 4.58Zalasinski and Cpałka

(2013)2.92 4.46 3.69

Our method 3.13 4.15 3.64

3.1. Comparison of methods to signature verification

The accuracy of the methods described in the literature in thefield of dynamic signature verification expressed by the arithmeticmean of the FAR and FRR errors for both considered in this paperdatabases varies within very wide limits. For SVC2004 base rangesfrom 2.89% to 16.34% (see Yeung et al., 2004), and the base BioSe-cure is from 1.71% to 27.76% (see Houmani, Garcia-Salicetti,Mayoue, & Dorizzi, 2009). It should be pointed out a very impor-tant fact: a direct comparison of accuracy of the various methodsis very difficult and cannot be objective. This is due to the followingreasons:

– Selection of pre-processing methods (see e.g. Jain et al., 2002;Ibrahim et al., 2010; Lei & Govindaraju, 2005; O’Reilly &Plamondon, 2009) for signatures included in the test databasehas a huge impact on the results of the signature verificationmethods. Pre-processing methods are used among others tomake the signatures rotation invariant and scale invariant.

– Methods for signature verification proposed in the literature byother authors were tested using a databases which are not pub-lic available. Such databases are usually developed by authors ofthose methods in local communities and they are not beingmade available (e.g. Huang & Hong, 2003; Khan et al., 2006;Ibrahim et al., 2010).

– Methods for signature verification proposed in the literature byother authors were tested using a databases which were publicavailable, but they are no longer available for use (e.g.MCYT-100, see Fierrez et al., 2007; Ortega-Garcia et al., 2003;Homepage of ATVS, 2013).

We eliminated above problems as follows:

– Among regional based methods available in the literature wehave chosen the one which shows the best accuracy for dat-abases which are not currently available. Next, we implemented

Fig. 6. Average values of users’ weights determined for each partition of dynamicsignature for: (a) SVC2004 database, (b) BioSecure database and for: (1) Pfsg ¼ 2, (2)Pfsg ¼ 3, (3) Pfsg ¼ 4.

Fig. 7. Average values of users’ dlrnmax determined for each partition of dynamicsignature for: (a) SVC2004 database, (b) BioSecure database and for: (1) Pfsg ¼ 2, (2)Pfsg ¼ 3, (3) Pfsg ¼ 4.

4178 K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180

it in our authorial test environment to compare its accuracy.This ensures that our simulations were independent of the dat-abases, which are no longer available, and eliminates impact ofpre-processing on the resulting accuracy. The results of thiscomparison are presented in Table 3 for SVC2004 databaseand in Table 4 for BioSecure database.

– For comparison of the methods for signature verification wealso took into account the main characteristics of each method.The results of this comparison are presented in Table 5.

The conclusions from our comparison of methods to signatureverification based on the regional approach can be summarizedas follows:

– Our method works with a very good accuracy for the SVC2004database (see Table 3) and BioSecure database (see Table 4).

– Our method in the classification phase does not use the so-called skilled forgeries (false signatures created by skilled forg-ers on the basis of knowledge about genuine signatures of the

Table 5Main characteristics of the algorithms for the on-line signature verification based onregional approach.

Characteristic of the method f1 f2 f3 f4 f5

Fierrez et al. (2007) yes no no no noKhan et al. (2006) yes yes no no noIbrahim et al. (2010) yes yes no no noHuang and Hong (2003) yes yes yes no noZalasinski and Cpałka (2012) yes yes yes no noZalasinski and Cpałka (2013) yes yes yes yes noOur method yes yes yes yes yes

f1- Does the method divide the signature into the parts in order to increase theefficiency of signature verification accuracy? f2- Does the method evaluate thestability of the signature in selected parts of the signature? f3- Does the methodtake into account a hierarchy of selected parts of the signature in the classificationprocess? f4- Is a way of classification interpretable? f5- Does the method promotepartitions created in the time intervals which are the most characteristic for theuser?

users). This kind of signatures are used only in test phase toevaluate the ability to generalize the knowledge accumulatedin our system.

– Our method for each individual user extracts moments of time,during which the signature is created and assigns weights ofimportance to the moments (partitions).

– Our method during classification of signatures takes intoaccount all moments of time, not only the ones which are asso-ciated with the weights of the highest value. This is because wedo not want to lose any detailed information in the verificationprocess.

– Our method uses capabilities of flexible fuzzy classifier. In theconstruction of the classifier we used all our experience withdynamic signature verification and the experience with devel-oped by us flexible neuro-fuzzy classifiers (see e.g. Cpałka,2009a, 2009b) and proposed by us so called weighted triangularnorms (see e.g. Rutkowski & Cpałka, 2005a, 2003).

– Our method ensures the interpretability of knowledge accumu-lated in the fuzzy classifier. In this respect, we used our experi-ence in the interpretability of expert knowledge of rule-basedsystems (see e.g. Cpałka et al., 2014). As a result, the proposedsystem does not require training and works only on the basisof properly designed by us descriptors of the signatures (basedon templates of the signatures).

– Our method uses flexible approach to the classification of signa-tures. The methods which are used to signatures verificationmust take into account quick addition of new signatures ofnew users. This means that the vast majority of well-knownclassification methods cannot be used because they require tocarry out supervised learning. In our work we use the classifierdeveloped by us in order that signature verification, which doesnot require a prior learning.

– Our method can be used almost immediately in other areas ofbehavioural biometrics, in which the dynamics of user behav-iour is analysed in the context of identity verification.

– The information presented in Table 6 shows that the complexityof the proposed method is slightly higher than the complexityof the method presented in Ibrahim et al. (2010) and compara-ble with the complexity of the method presented in Zalasinskiand Cpałka (2013). This results from the fact that the proposedmethod has some important features in comparison with othermethods for dynamic signature verification based on the regio-nal approach (see Table 5). Consideration of these features (suchas the importance of the partition and a clear verification mech-anism using the capabilities of flexible fuzzy systems) requires aslightly higher quantity of calculations. Despite that, it shouldbe noted (see Table 6) that proposed algorithm does not requireusing computation of high complexity (e.g., recursive calls oractions of exponential complexity). In addition, most of thesteps of our method are executed only once (in the context of

Table 6The computational complexity of the proposed method for the on-line signatureverification using vertical signature partitioning.

Stepnumber

Ibrahim et al.(2010)

Zalasinski and Cpałka(2013)

Our method

1 2PJKi 2PJKi 2PJKi

2 PJKi PJKi PJKi

3 PJðKi þ 2Þ PJðKi þ 2Þ PJðKi þ 2Þ4 – 4PðJ þ 1Þ 4PðJ þ 1Þ5 – JðPKi þ 2Þ þ nb JðPKi þ 2Þ þ nb

6 – J J7 1 1 1

J is the number of training signatures of the user (in our simulations J ¼ 5), Ki is thenumber of samples of the ith signer, P is the number of partitions, nb is the numberof steps of procedure for determining the decision boundary for each partition.

K. Cpałka, M. Zalasinski / Expert Systems with Applications 41 (2014) 4170–4180 4179

each user) in the training phase. This increases its speed in thephase of new signatures verification (test phase). These consid-erations allow the use of the proposed method for processing alarge number of signatures.

3.2. Conclusions from the simulations

Conclusions from performed simulations may be summarizedas follows:

– For both tested databases (SVC2004 and BioSecure) proposedalgorithm worked with very good accuracy.

– In the process of verifying the signatures from both databases,signature velocity was more important than the value of penpressure (Fig. 6). This is due to the fact that values of weightsof importance averaged for all users, associated with the parti-tions created on the basis of velocity signal, have a greater val-ues than the weights of importance associated with thepartitions created on the basis of pressure signal.

– In the process of verification of signatures from the databaseSVC2004 pen velocity in the early and middle stages of signingprocess was slightly more important than in the final stage(Fig. 6(a.1)–(a.3)). In the process of verification of signaturesfrom the database BioSecure this effect was noticeable onlyfor four partitions (Fig. 6(b.3)).

– In the verification process of the signatures from both dat-abases, pen pressure in the middle and final phase of signingprocess was more important than in the initial phase (Fig. 6).

– In the process of verification of the signatures from the bothdatabases, pen pressure in the initial and final phase of signingprocess is more stable than the velocity of the pen (Fig. 7). Inturn, in the middle phase of signing process for two and threepen partitions, velocity of the pen is more stable than the penpressure.

4. Conclusions

Significant characteristic of proposed method is realization ofdivision of signing time into time moments, called partitions, indi-vidually for each user. Partitioning in the time domain was notconsidered in the literature. Created partitions are processed insuch a way as to increase the importance of the ones, in whichthe stability of signing is greater in the acquisition phase. This pro-cess takes into account the dynamics of training signatures, i.e.velocity of the pen and the pen pressure on the surface of the gra-phic tablet. This information is used to determine the weights ofimportance for the partitions, individually for each users The val-ues of these weights are proportional to the stability of signatureand have a big impact on the classification phase. An importantfeature of the proposed method is also that it allows to extractknowledge about the course of dynamic signature verification indi-vidually for each user. This knowledge is stored in the form ofinterpretable rules of flexible fuzzy classifier, taking into accountthe importance of the weights of each partition. Fuzzy sets fromthe rules are set individually for each user, depending on theparameters describing the stability of the signing in the partitions(descriptors of signatures).

The main conclusions from simulation research carried out forthe SVC2004 database and BioSecure database are: (a) approachbased on partitioning in the time domain proved to be an effectivemethod of the dynamic signature verification, (b) in the classifica-tion process of dynamic signature more important role than pres-sure signal played velocity signal, (c) in the initial and final phaseof signing pen pressure is more stable (reproducible in the contextof the individual user) than the velocity of the pen, and in middlephase of signing velocity of the pen is more stable than the pen

pressure, (d) proposed approach can be successfully used in anyfield, in which dynamics of behaviour is verified.

In further works in the field of dynamic signature verificationwe plan to give attention to, among others, limitation of the impactof changes in the way of signing to work of signature verificationmethods, approach that integrates partitioning in the time domainand vertical partitioning (approach known in the literature, wedealt with this approach) and using the potential of the methodsfrom the field of image processing in the context of dynamic signa-ture verification based on the partitioning.

Acknowledgment

The project was financed by the National Science Centre (Po-land) on the basis of the decision number DEC-2011/01/N/ST6/06964.

The authors would like to thank the reviewers for very helpfulsuggestions and comments in the revision process.

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