Online signature verification based on null component analysis and principal component analysis

THEORETICAL ADVANCES

Bin Li Æ David Zhang Æ Kuanquan Wang

Online signature verification based on null component analysisand principal component analysis

Received: 31 January 2005 / Accepted: 4 September 2005 / Published online: 24 November 2005� Springer-Verlag London Limited 2005

Abstract This paper describes a method for stroke-basedonline signature verification using null componentanalysis (NCA) and principal component analysis(PCA). After the segmentation and flexible matching ofthe signature, we extract stable segments from eachreference signature in order that the segment sequenceshave the same length. The reference set of feature vectorsare transformed and separated into null components(NCs) and principal components (PCs) by K-L trans-form. Online signature verification is a special two-cat-egory classification problem and there is not a singleavailable forgery set in an actual system. Therefore, it isdifferent from the typical application of PCA in patternrecognition that both NCA and PCA are used torespectively analyze stable and unstable components ofgenuine reference set. Experiments on a data set con-taining a total 1,410 signatures of 94 signers show thatthe NCA/PCA-based online signature verificationmethod can achieve better results. The best result yieldsan equal error rate of 1.9%.

Keywords NCA Æ PCA Æ K-L transform Æ Signatureverification Æ Stable segment extraction

1 Introduction

E-transactions involving legal or financial documentsrequire a highly secure, reliable, and legally acceptableway to approve or authenticate contents or authorship.Some personal devices such as pocket PC and tablet PC

also require a high secure access authentication whichcan replace the conventional password to resist theinvasion of privacy well. Along with the development ofcomputer science and technology, biometrics is an activetopic in the research of secure authentication. Of themany possible biometrics schemes, online signatureverification is a strong candidate for technology, sincehandwriting is a skill that is personal to individuals anda handwritten signature is commonly used to authenti-cate the contents of a document or a financial transac-tion. Especially, online signature verification can providea new secure access authentication for pocket PC andtablet PC without any additional devices. With manypeople engaged in research on online signature verifi-cation, the results have been a wide range of reportedmethods.

The methods of online signature verification can begenerally classified into two categories: function-basedand parameter-based [1]. In the function-based method,a signature is usually taken as some time-dependentfunction and each original point (or resample point) ofthe signature is used for verification [1–4]. Many meth-ods of online signature verification in literature areparameter-based [5–12]. In parameter-based method, theenrollment data size and computation is very small; theprivacy problems of the users are also considered, sinceonly parameters are enrolled and original signaturecannot be constructed [12]. Rhee et al. extract segmentsby Brault’s method [5], present 11 features to define asegment and use dynamic time warping (DTW) algo-rithm and Euclidean distance for verification [9]. Quet al. distinguish strokes from a signature by the pointwhich is decreased in pressure or velocity etc. Theyintroduce a definition of the significant stoke into veri-fication to denote the stable component in references [10,11]. Lee et al. [12] propose an approach for segmenting asignature by geometric extrema. They achieve segment-to-segment correspondence by DTW algorithm and takea BP neural network as the classifier.

Online signature verification is a special two-categoryclassification problem. Only genuine signature references

B. Li Æ K. WangDepartment of Computer Science and Technology,Harbin Institute of Technology, Harbin, China

D. Zhang (&)Biometrics Research Centre, Department of Computing,The Hong Kong Polytechnic University,Hung Hum Kowloon, Hong KongE-mail: [email protected].: +852-2766-7271Fax: +852-2774-0842

Pattern Anal Applic (2006) 8: 345–356DOI 10.1007/s10044-005-0016-4

of a signer can be employed in an actual online signatureverification system. There is not a single available forg-ery set for each signer. All the research on online sig-nature verification is based on a set of genuinereferences. It will improve the performance of verifica-tion if we take the stability analysis of the reference setinto account, since there are stable and unstable com-ponents in the reference set.

In order to analyze the stable and unstable compo-nents of signatures, we introduce a novel method basedon null component analysis (NCA) and principal com-ponent analysis (PCA) for online signature verification.PCA or PCA-based approaches have been very successfulin image representation and recognition. In 1987,Sirovich andKirby [14, 15] used PCA to represent humanfaces. Subsequently, Turk and Pentland [16] proposed aPCA-based face recognition method, Eigenfaces. PCAhas now been widely investigated and has been success-fully applied to other image recognition tasks [17–19].

It is different from the typical application of PCAthat both PCA and NCA are used for the stabilityanalysis of signatures in this paper. We first divide asignature into a sequence of segments and find an opti-mal segment-to-segment correspondence by DTWalgorithm. Stable segments are extracted by searchingthe corresponding path among each pair of same signer’ssignatures. Feature matrices of same signer’s referencescan be composed by these stable segments. The PCs andthe NCs of each reference feature matrix are producedby K-L transform. For the use of PCs, we introduce anonlinear transform to decrease the effect of unstablecomponents in the reference set. For the use of NCs, wedefine a concept, energy in NCs (ENC), to measure therest energy of a testing signature projecting on the NCspace. At last, we verify a testing signature by combiningNCA and PCA into a final distance and comparing itwith a writer-dependent threshold.

The organization of this paper is as follows: Sect. 2describes the feature extraction and flexible matching inwhich the algorithm of stable segment extraction is alsointroduced. Section 3 proposes NCA/PCA-based meth-od for online signature verification. Experiments andcomparisons are presented in Sect. 4. Section 5 offersour conclusion.

2 Feature extraction and flexible matching

2.1 Feature extraction

Since the device of different clients in E-transaction andthe most personal devices can not be consistent in cap-turing a special writing feature such as pressure, we onlyuse the spatial and temporal information of a signaturein our method. With a fixed sample frequency, a signa-ture can be described by a series of points {pi|pi = (xi,yi, ti), i = 1, ..., N)}. After the preprocessing includingsmoothness and normalizing, an original signaturecaptured by a general device is shown in Fig. 1.

There are many segmentation algorithms [5, 6, 10, 12,13]. In this paper, we adopt the algorithm of Brault [5]for segmentation, since this algorithm is more effectiveand stable in our experiments.

Figure 2 illustrates a segment i, pi and pi+1 are thestart and end points of segment i, ci is the center ofgravity of segment i, and C is the center of gravity of thewhole signature. We define a set of features to describe asegment as follows:

1. The curve length of a segment i: li.2. The direct length of two neighboring critical points pi

and pi+1: lid.

3. The horizontal angle of the line connecting pi andpi+1: ai.

4. The horizontal angle of the angular bisector betweentangents of critical point pi’s two neighboring points:bi.

5. The distance between ci and C: lic.

6. The horizontal angle of the line connecting ci and C:ci.

7. The average writing velocity in segment i: �vi:8. The writing acceleration in segment i: ai.

Feature bi denotes the directional relation of twoneighboring segments. Features li

d and ci denote therelative location of segment i to the center of signature.Features �vi and ai describe the dynamic information of asegment. Now, we obtain a sequence of feature vectorsto describe a whole signature as follows:

W ¼ ðw1;w2 . . . ;wN Þ;wi ¼ ðli; ldi ; ai; bi; l

ci ; ci;�vi; aiÞT ;

ð1Þ

where N is the segment number of a signature.

2.2 Flexible matching

As we know, segment numbers of two signaturesproduced by a same signer are often different. Wemust try to find an optimal matching path betweensegment sequences of two signatures before calculatingtheir similarity. DTW is a technique that is well suit-able for this matching. Using DTW, we can obtain anoptimal path in which the summation of distancesbetween two signature segment sequences achieves theminimum. During the production of a genuine signa-ture or a forgery, the writer usually pays more atten-tion to its shape and then to its dynamic information.This means it is stabler and more suitable that we onlyuse the static feature for this flexible matching.Therefore, we select six static features to reconstruct anew sequence:

S ¼ ðs1; . . . ; sN Þ; si ¼ ðli; ldi ; ai; bi; l

ci ; ciÞT ; ð2Þ

where N is the segment number of a signature.Set Sp = (s1

p, ..., sIp) and Sq = (s1

q, ..., sJq) are two

segment sequences which belong to different signatures.

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In order to find an optimal path between these two se-quences, a DTW algorithm is introduced as following:

Di;j ¼ minDi;j�1 þ di;j

Di�1;j þ di;j

Di�1;j�1 þ di;j þ Cp

0@

1A;

di;j ¼ w � ðspi � sq

j ���

���; 1 � i � I ; 1 � j � J ;

ð3Þ

where di,j is the distance between element sip and element

sjq which belong to different sequences, w is a 1·6 coef-ficient vector of which distributes different weights to thedistances of six different features, Di,j is the accumula-tion of distances, and Cp is a punishment parameter.

To improve the speed of calculation, a restriction ofthe searching window is appended [6] as follows:

ji� jj � rW; ð4Þ

where r W is the maximum width of the searching win-dow. In our method, rW is set to (I+J)/4.

Figure 3 shows an optimal matching path betweentwo genuine signatures.

2.3 Stable segment extraction

All corresponding relations in optimal matching pathsof different signature pairs can not be entirely consistent,although these signatures are produced by the samesigner. It is necessary to extract stable segments fromeach signature in terms of these matching paths. Thereare two reasons: (1) weeding out unstable segments can

improve the performance of verification; (2) K-L trans-form needs a reference feature matrix which is composedby feature vectors with the same length. Now, we discussthe stable segment extraction in training and verifying,respectively.

2.3.1 Stable segment extraction in training

Let R denote a set which includes M genuine referencesignatures produced by the same signer. We select sig-nature ri (ri 2R, i = 1, ...,M) which has Ni segments as atemplate. We can obtain M�1 optimal matching pathsbetween the template ri and each of the rest M�1 sig-natures rj (rj 2R, j „ i) which has Nj segments by theDTW algorithm (Sect. 2.2), We define the followingstructure to record a matching path.

Pathri;j ¼ fpair1; . . . ; pairKg; pairk ¼ ðsegt; pstartm ; pendn Þ;

k ¼ 1; . . . ;K m; n ¼ 1; . . . ;Nj

ð5Þ

where pairk is a matching segment pair and includesthree parameters. segt denotes the tth segment of the

Fig. 1 An original signature isshown in 2D with spatialinformation and in 3D withspatial and temporalinformation

Fig. 2 Feature description of a segmentFig. 3 An optimal matching path between two genuine signatures.Asterisk is the critical point

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template ri. pmstart and pn

end belong to signature rj anddenote the start point of a segment and the end point ofa segment, respectively. Here we need point out thatpmstart and pn

end may belong to a same segment m (i.e.,m=n) or two different segments m and n (i.e., m „ n).Nj is the total segment number of rj.

We search all M�1 matching paths for the segmentsegt which belongs to the template ri and satisfies thecondition: there must be one of matching segment pairsin each path Pathri;j which contains segt. This kind ofsegments is recorded as the stable segment. We assumethe number of these segments is N¢i.

We want to search for the segments that belong toother M�1 reference signatures and correspond to thestable segments of ri. i.e., we search each of M�1matching path for the point pair (pm

start, pnend) which

corresponds to stable segments of ri. There are two in-stances of this search which need to be explained:

1. pmstart and pn

end belong to a same segment m (i.e.,m=n): Directly record the segment i as a stable one.

2. pmstart and pn

end belong to different segments m and n(i.e., m „ n): Combine segments form m to n into anew segment and record the new segment as a stableone. Here, we need point out that features of the newsegment need to be recalculated.

Taking ri as a template, we can obtain M stablesegment sequences which belong to M referencesrespectively. Their length is N¢i. In order to ensure thesearch robust, we need to take each reference as thetemplate respectively. In terms of the above searching

algorithm, we can achieve the best result that the num-ber of stable segments is the maximum. Here, we lettemplate r* to denote the template with the best result. r*

and its corresponding set of stable segments are recordedfor the next steps. For the convenience of the followingdescription, we set the number of stable segments to beN.

For each feature of a segment which is defined inSect. 2.1, we can construct a feature matrix [real]M · N ofthe reference set. Figure 4 shows an example of a seriesof stable segments extracted from five reference signa-tures. We can see that most stable segments can becorrectly matched.

2.3.2 Stable segment extraction in verifying

We have obtained a stable segment sequence of eachreference signature in the training. We need to searchtesting signature for a segment sequence which cancorrespond to the stable segment sequence of a refer-ence. Here, we also set r* as the template. By usingDTW, we achieve a matching path patht between testingsignature t and the template r*. The structure of patht isredefined as following:

Patht ¼ fpair1; . . . ; pairKg;pairk ¼ ðrpstartm ; rpendn ; tpstart

i ; tpendj Þ; k ¼ 1; . . . ;K

ð6Þ

where pairk is a matching segment pair. Start point rpmstart

and end point rpnend belong to the template r*. They may

belong to the same segment m (i.e., m=n) or differentsegments m and n (i.e., m „ n). Start point tpi

start andend point tpj

end belong to the testing signature t. Theymay belong to the same segment i (i.e., i=j) or differentsegments i and j (i.e., i „ j).

We implement a searching algorithm in terms of thefollowing seven instances:

1. rpmstart and rpn

end belong to a stable segment m (i.e.,m=n), tpi

start and tpjend belong to a same segment i

(i.e., i=j): Directly record the segment i as the onewhich corresponds to the stable segment m of thetemplate r*.

2. rpmstart and rpn

end belong to a stable segment m (i.e.,m=n), but tpi

start and tpjend belong to different seg-

ments i and j (i.e., i „ j): Combine segments from i tosegment j into a new segment and record it as thesegment which corresponds to the stable segment mof the template r*. Features of the new segment needto be recalculated.

3. rpmstart or rpn

end belongs to a stable segment x, tpistart

and tpjend belong to a same segment i (i.e., i=j):

Directly record the segment i as the one whichcorresponds to the stable segment x of the templater*.

4. rpmstart or rpn

end belongs to a stable segment x, tpistart

and tpjend belong to different segments i and j (i.e., i

„ j): Combine segments from i to segment j into anew segment and record it as the segment which

Fig. 4 A series of stable segments are extracted from five referencesignatures

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corresponds to the stable segment of the template r*.Features of the new segment need to be recalculated.

5. rpmstart and rpn

end belong to different stable segments mand n (i.e., m „ n), tpi

start and tpjend belong to the

same segment i (i.e., i=j): Directly record the segmenti as two segments which correspond to stable seg-ments m and n of the template r*.

6. rpmstart and rpn

end belong to different stable segments mand n (i.e., m „ n), tpi

start and tpjend belong to dif-

ferent segments i and j (i.e., i „ j): Combine seg-ments from i to segment j into a new segment andrecord it as the segment which corresponds to stablesegment m and n of the template r*. R Features of thenew segment need to be recalculated.

7. There is not a stable segment of the template in patht,we record a NULL segment as the one which corre-sponds to the stable segment of the template r*.

After the above corresponding segment searching, asegment sequence of test signature can be obtained.

3 Null component analysis and principal componentanalysis for online signature verification

3.1 Null component analysis and principal componentanalysis

Principal component analysis is an essential techniquefor feature extraction and recognition. It has been widelyused in the field of pattern recognition. In most appli-cations of PCA, people usually throw away minorcomponents (MCs) and only care about principal com-ponents (PCs), since PCs contain most information ofthe reference data. There is a little difference in thedefinition of PCs in this paper. Here, PCs are the com-ponents that contain all information of the referencedata, and the rest of the components are named as nullcomponents (NCs). Not only PCA but NCA are used inthe proposed method for online signature verification.

Principal component analysis is typically used inpattern recognition (for example face recognition andpalmprint recognition) to solve the multi-category clas-sification problem. Samples of each class can be reallyacquired. A reference set is constructed with samples ofall classes. PCA can make the inter-class classificationmore legible, when the reference samples are projected tothe PC’s space. The PCA is used to analyze the distri-bution of the inter-class difference. Whereas, onlinesignature verification is a two-category classificationproblem: one is the genuine signature and the other isthe forgery. There is not a single available forgery set ofeach signer in an actual system and only the set ofgenuine signatures can be used. We cannot really ana-lyze the distribution of the difference between genuinesignatures and forgeries. We can only analyze the uni-form characteristic of each genuine signature in a ref-erence set and find the difference between each genuinesignature in the reference set. Eigenvectors with larger

eigenvalues indicate that there is higher intra-class var-iability in the reference set on the space of these eigen-vectors. Contrarily, the reference samples are stable onthe space of eigenvectors with smaller eigenvalues. Spe-cially, eigenvectors with zero eigenvalues (NCs) can beused to indicate the uniform characteristic of referencesamples. NCA is just used to analyze the stable com-ponents of genuine signatures in the reference set, sincethe space of NCs do not contain any information on thereference set. We define the ENC to measure the dif-ference between a testing sample and the referencesample. The difference (intra-class distance) amonggenuine signatures is analyzed by PCA. For the appli-cation of PCs, a nonlinear transform is introduced todecrease the effect of unstable components and increasethe effect of stable components of references. We discussour method with following two subsections.

3.2 Training

After the flexible matching and stable segment extrac-tion, M reference signatures have been replaced by Mstable segment sequences whose lengths are equal to N.There is a restriction that M should be less than N. If Mis more than N, we need change the algorithm of seg-mentation to achieve more segments, or decrease thenumber of reference samples, otherwise, the algorithmwill be invalid. In this paper, we assume that M is lessthan N.

We separate a segment sequence into eight N ·1feature vectors in accordance with the feature definitionin Sect. 2.1. Each feature vector is normalized andreconstructed to M· N feature reference matrices. Foreach M· N matrix [real]i

M· N which corresponds to theith feature of Eq. 1, we calculate the matrix Ri as fol-lowing:

Ri ¼ ðri1 � �ri; ri

2 � �ri; . . . ; riM � �riÞ; �ri ¼ 1

M

XMm¼1

rim;

rim � <M�N

i ; ð7Þ

where rmi is a N·1 column vector which describes the mth

reference by the ith feature, and �ri is the N· 1 averagecolumn vector.

By K-L transform, we obtain eigenvectors Ui={uni ,

n = 1, ..., N} and eigenvalues {kni , n = 1, ..., N} of the

matrix Ri. The space constructed by fore M�1 eigen-vectors contains all information on reference signatures.People usually name the eigenvector with large eigen-value as principal component (PC) and name theeigenvector with very small or zero eigenvalue as MC. Inthis paper, a little shift of this concept is made in that wename the eigenvector with a non-zero eigenvalue as PCand name the eigenvector with a zero eigenvalue as NC.

We separate eigenvectors Ui={uin, n = 1, ..., N} intotwo parts, one is UPC

i constituted of PCs and the other isUNC

i constituted of NCs.

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UiPC ¼ fui

n; n ¼ 1; . . . M � 1g;Ui

NC ¼ fuin; i ¼ M ; . . . ;Ng; ð8Þ

Now, we analyze the effect of different eigenvectors inUi

PC. The eigenvector with a large eigenvalue implicatesthat the variety of references projecting on this eigen-vector is very acute, i.e., the unstable component of asignature can be separated by projecting the signature onthis eigenvector. The larger the eigenvalue is, the greaterthe variety of the reference on this eigenvector is, andvice versa. Apparently, the effect of unstable componentsshould be decreased, and the effect of stable componentsshould be increased. Therefore, we give a large coeffi-cient to the eigenvector with a small eigenvalue, anda small coefficient to the eigenvector with a largeeigenvalue. In accordance with the eigenvalue of eacheigenvector, this transform can be defined as follows:

U iPC ¼

ui1ffiffiffiffiffiki1

q ;ui2ffiffiffiffiffiki2

q ; . . . ;ui

M�1ffiffiffiffiffiffiffiffiffiffiffiki

M�1

q

8><>:

9>=>;; ui

m � UiPC; ð9Þ

where umi is the eigenvector corresponding to the eigen-

value kmi .

Reference signature sets produced by different writersmay have different stabilities. During the training, wecalculate a writer-dependent threshold Ti for feature i. Itis given as follows:

T i¼Direfþr1;

Diref¼

Xp;q¼1p 6¼q

Mri

p� riq

������

MðM�1Þ2

� �,MðM�1Þ

2

� �;

ripðqÞ ¼ U i

PCT � ri

pðqÞ;

ð10Þ

where Drefi is the average distance of intra-reference of

ith feature, and r 1 is a shift coefficient. By adjusting r 1,we can achieve different FAR and FRR.

There must be less information of references on theeigenvector with a smaller eigenvalue. Further, theremust be nothing in the projection of references on ei-genvectors with zero eigenvalues. In accordance withthis characteristic, we introduce NCA into online sig-nature verification and give a new concept, ENC space.Since all information is contained in the space of UPC

i ,the energy of references must be zero in the space ofUNC

i . We can verify a test signature as genuine or as aforgery by its ENC. A smaller ENC of a testing signa-ture indicates that this signature is more similar with thereference signatures, and vice versa. The definition ofENC will be given in the next subsection.

3.3 Verifying

Let s be a testing signature. We separate the testing sig-nature into eight N·1 feature vectors {si, i = 1, ..., 8},after the flexible matching and stable segment extraction.

Firstly, we calculate the distances between the testingsignature and references based on PCA. The distance ofthe ith feature is defined as follows:

DiPCA ¼

1

M

XMm¼1

U iPC

T � si � rim

�� ��; ð11Þ

where rim is given by Eq. 10.

Secondly, we calculate the ENC of the ith feature of atesting signature as follows:

EiNC ¼ ðsi � �riÞT � Ui

NC

�� ��; �ri ¼ 1

M

XMm¼1

rim; ri

m � <M�Ni ;

ð12Þ

We combine DPCAi and ENC

i into a distance Di toevaluate the total similarity of the ith feature betweenthe testing signature and the reference.

Di ¼ e1DiPCA þ e2Ei

NC; ð13Þ

where e1 and e2 are two weight coefficients. Experi-mentally, we achieve better results when setting e1=1and e2=4.

At last, we assign eight different weight coefficients toeach distance Di respectively and then obtain a finaldistance between the testing signature and references.

DIS ¼ cfea � D; D ¼ ðDi; i ¼ 1; . . . ; 8Þ; ð14Þ

where cfea is a weight coefficient vector for differentfeatures of the segment. By adjusting cfea to achieve abetter result in our experiments, we finally set cfea=(4, 4,4, 1, 3, 3, 2, 2).

Since the ENC of each reference is zero, there is onlya uniform threshold for NCA. Therefore, we verify asignature as genuine under the following discriminant:

D ¼ cfea � ðD� T Þ � r > 0; D ¼ ðDi; i ¼ 1; . . . ; 8Þ;T ¼ ðT i; i ¼ 1; . . . ; 8Þ;

ð15Þ

otherwise, as a forgery. Here, r is a shift coefficient. Byadjusting r, we can achieve different FAR and FRR.

4 Experiments

4.1 Experimental database

A digital tablet with 100 Hz sampling frequency wasused for capturing the signatures. In order to make thesigner feel more comfortable, we put a kind of carbonpaper on the tablet. When writing on the tablet, thesigner feels just like writing on the paper and he can seethe feedback of signing from the paper instead of theCRT. Over a period of 4 months, 94 Chinese volunteerstook part in the data acquisition. Each writer was askedto write his/her signature ten times. Altogether 940genuine signatures were recorded in our database.

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About the forgery, six forgers were asked to imitate fivesignatures for each genuine signer, after they observedand were familiar with both the shape and the wholeproducing process of genuine signatures. No signaturesfrom professional imitators were available, so these 470forgeries were called skilled forgeries. A total of 1,410signatures in our experimental database was used toevaluate our method. Figure 5 shows several examplesof genuine signatures and skilled forgeries in our data-base.

4.2 Implementation and evaluation of the proposedmethod

We select 5�7 genuine signatures of a signer as refer-ences respectively, the rest 5�3 genuine signatures and 5

forgeries as testing samples. We implement the proposedmethod and evaluate the performance of the systembased on PCA, NCA and the combination of NCA andPCA, respectively. Figures 6, 7 and 8 show the perfor-mance of our method which used 5�7 genuine signa-tures as references. When increasing the number ofreferences from 5 to 7, the equal error rate (EER) ofPCA-based method decreased from 23 to 17.2%, andthe EER of NCA-based method decreased from 3.9 to3.3%. We can combine NCA- and PCA-based methodby adjusting coefficients e1 and e2 (in Eq. 13). When weset e1=1 and e2=4, better results can be achieved. TheEER of NCA/PCA-based method is decreased from 3.6to 1.9%, when increasing the number of references from5 to 7.

From the experimental result we can see that NCAhas better classification performance than PCA. It is

Fig. 5 Examples of genuinesignatures and skilled forgeriesin our database

Fig. 6 Error tradeoff curves formethods based on PCA, NCAand the combination of PCAand NCA. The number ofreferences is 5

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because that the intra-class stable component canbe easily separated by NCA. The intra-class stablecomponent plays a very important role in online signa-ture verification. PCA has classification performance toa certain extent, since PCA mainly describes the intra-class variability. We can easily combine NCA and PCAby Eq. 13 and specially give a larger weight to NCA, asthe PC and the NC are linearly uncorrelated.

We also implement two stroke-based methods tocompare with our method: (1) Without using NCA and

PCA, we directly calculate Euclidean distance of featuresto verify the signature. (2) The method of Lee et al. [12].

1. In this compared method, the segmentation, featureextraction and flexible matching algorithms aresame with our proposed method (Subsects. 2.1 and2.2). Without using stable segment extraction andNCA/PCA, we directly calculate Euclidean distancebetween two patterns. A writer-dependent thresholdis determined by the distance of intra-reference.

Fig. 7 Error tradeoff curves formethods based on PCA, NCAand the combination of PCAand NCA. The number ofreferences is 6

Fig. 8 Error tradeoff curves formethods based on PCA, NCAand the combination of PCAand NCA. The number ofreferences is 7

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This method is just similar with Jain’s method [3],expect for the definition of segment (he use theresample points in his paper) and the feature set.Here, since our purpose is to compare the methodsbetween NCA/PCA-based and non-NCA/PCA-based. The segmentation and feature extractionalgorithms are consistent with the proposed meth-od. In this experiment, we select 5�7 genuine sig-natures of a signer as references respectively, therest 5�3 genuine signatures and 5 forgeries astesting samples. Figure 9 shows the performancemeasure of non-NCA/PCA method. When wechoose seven genuine signatures as references, thismethod achieves the better result that the EER isabout 6.6%. Actually, with the number of refer-ences increasing from 5 to 7, the performance ofthis method cannot be improved distinctly.

2. According to Lee’s method [12], we implement hissystem for comparing with our method. In thisexperiment, the selection of reference samples andtesting samples is same with above experiments.Figure 10 shows the performance measure of Lee’smethod. When we choose seven genuine signaturesas references and rest three genuine signatures andfive forgeries as testing, this method achieves thebetter result that the EER is about 7%. Since theclassifier of their method is the neural network ofback error propagation model, the authors calcu-late each feature average of all segments as theinput in order to ensure the input layer has a fixedlength. This kind of average would induce to de-crease the distinguishing performance between thegenuine signature and the skilled-forgery. However,

it is suitable for distinguishing the genuine signa-ture and random-forgery, i.e., it is more suitable foronline signature recognition, since there is anavailable random-forgery set for each signer.

The comparison of these three methods is shown inTable 1. From the experiments, we can see that theNCA/PCA-based method is an effective method. Onlybased on NCA-method, we can obtain better resultsthan based on non-NCA/PCA-based method. Com-pared with NCA-based method, the result of PCA-basedmethod seems not good. Although the precision of PCA-based method is not very high, we can combine NCAand NCA to improve the performance of whole system,since PC and NC are linearly uncorrelated. Comparedwith non-NCA/PCA method and Lee’s method, NCA/PCA-based method achieves a better result, since thismethod can easily analyze the intra-class stable andunstable component in genuine signature set. It is moresuitable for the discrimination between genuine signa-tures and skilled forgeries.

5 Conclusions

We proposed a novel method based on NCA/PCA foronline signature verification. Different from theapplication of PCA in other fields of pattern recog-nition, both NCA and PCA are used in the proposedmethod. An algorithm of stable segment extraction isdescribed to construct reference feature matrices. Forthe use of NCA, we present a new concept, ENCspace to measure the rest energy of a testing signature

Fig. 9 Error tradeoff curves forthe method based non-NCA/PCA

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in the space of NCs. For the use of PCA, we intro-duce a nonlinear transform to decrease the effect ofunstable components and increase the effect of stablecomponents of references. All experiments are basedon a data set containing 1,410 signatures of 94 writers.Compared with non-PCA/NCA-based method andLee’s method, the method presented in this paper canobtain better results in distinguishing between thegenuine signature and the skilled forgery. The bestresult of the proposed method yields an EER of 1.9%.The experimental results reasonably demonstrate thatthe method described in this paper is an effectivemethod and can be successfully applied to online sig-nature verification.

Since the proposed method can well analyze the intra-class stable and unstable component of one-class sampleset, it is also suitable for those two- or multi-categoryclassification problems in which there is only a one-classsample set can be used for training. It needs to be furtherinvestigated that how to apply the NCA/PCA-basedmethod for this kind of classification problems.

6 Originality and contribution

This paper introduces NCA and PCA into stroke-based online signature verification. Different fromthe typical application of PCA, online signature veri-fication is a two-category classification problem andthere is not an available forgery set in an actualsystem. Therefore, we introduce NCA and PCA toanalyze uniform characteristic and difference in thereference set of genuine signatures. We make theuniform characteristic play an important roleand decrease the effect of intra-class difference. Wedefine a new concept, ENC space (ENC), to analyzethe uniform characteristic in the reference. The intra-class difference is analyzed by PCA. Experimentsshow that NCA/PCA-based methods can achievebetter results. This method is also suitable for thosetwo- or multi-category classification problems in whichthere is only a one-class sample set can be used fortraining.

Fig. 10 Error tradeoff curvesfor Lee’s method [12]

Table 1 The comparisonamong NCA/PCA-basedmethod, non-NCA/PCA-basedand Lee’s methods

Methods EER (%)

Five references Six references Seven references

NCA/PCA 3.6 3.3 1.9Non-NCA/PCA 7.1 7 6.6Lee’s method 11.2 8.1 7

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7 About the authors

Acknowledgements This work is supported by Sci-Tech. Project ofHarbin, China (2003AA1CG055-10).

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Bin Li received his BE and ME degrees in computer science fromHarbin Institute of Technology (HIT), Harbin, China in 1996 and2000, respectively. He is currently working toward the Ph.D. degreeat the Biocomputing Research Center of Harbin Institute ofTechnology. In 2001 and 2004, he worked as a research assistant inthe Hong Kong Polytechnic University. His research interestsinclude pattern recognition, biometrics, etc

David Zhang graduated in computer science from Peking Univer-sity in 1974 and received his MSc and Ph.D. degrees in computerscience and engineering from the Harbin Institute of Technology(HIT), Harbin, China, in 1983 and 1985, respectively. He receivedthe second Ph.D. degree in electrical and computer engineering atthe University of Waterloo, Waterloo, ON, Canada, in 1994. From1986 to 1988, he was a Postdoctoral Fellow at Tsinghua University,Beijing, China, and became an Associate Professor at AcademiaSinica, Beijing, China. Currently, he is a Professor with the HongKong Polytechnic University, Hong Kong. He is the Founder andDirector of Biometrics Research Centers supported by theGovernment of the Hong Kong SAR (UGC/CRC) and theNational Nature Scientific Foundation (NSFC) of China, respec-tively. He is also the Founder and Editor-in-Chief of theInternational Journal of Image and Graphics (IJIG), Book Editor,The Kluwer International Series on Biometrics and an AssociateEditor of several international journals, such as IEEE TRANS-ACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PARTA: SYSTEMS AND HUMANS AND IEEE TRANSACTIONSON SYSTEMS, MAN, AND CYBERNETICS—PART C:APPLICATIONS AND REVIEWS, and Pattern Recognition. Hisresearch interests include automated biometrics-based authentica-tion, pattern recognition, biometric technology and systems. As aprincipal investigator, he has finished many biometrics projectssince 1980. So far, he has published over 200 papers and ten books

Kaunquan Wang received his BE and ME degrees in computerscience from Harbin Institute of Technology (HIT), Harbin, China,and his Ph.D. degree in computer science from ChongqingUniversity, Chongqing, China, in 1985, 1988 and 2001, respec-tively. From 1988 to 1998, he worked in the Department ofComputer Science, Southwest Normal University, Chongqing,China as a tutor, lecturer and associate professor, respectively.From 1998, he has been working in the Biocomputing ResearchCentre, HIT as a professor, a supervisor of Ph.D. candidates andan associate director. Meanwhile, from 2000 to 2001 he was avisiting scholar in Hong Kong Polytechnic University supported byHong Kong Croucher Funding and from 2003 to 2004 he was aresearch fellow in the same university. So far, he has published over70 papers. His research interests include biometrics, imageprocessing and pattern recognition. He is a member of the IEEE,an editorial board member of International Journal of Image andGraphics. In addition, he is a reviewer of IEEE Trans. SMC andPattern Recognition

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