An approach for on-line signature authentication using Zernike moments 2011.pdf

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An approach for on-line signature authentication using Zernike moments

K.R. Radhika a,, M.K. Venkatesha b, G.N. Sekhar a

a B.M.S. College of Engineering, Basavangudi, Bangalore 560 019, Karnataka, Indiab R.N.S. Institute of Technology, Bangalore 560 062, Karnataka, India

a r t i c l e i n f o

Article history:

Received 17 December 2009

Available online 21 December 2010

Communicated by G. Borgefors

Keywords:

On-line signature authentication

Zernike moments

Polynomial order selection

a b s t r a c t

In this work, shape analysis of the acceleration plot, using lower order Zernike moments is performed for

authentication of on-line signature. The on-line signature uses time functions of the signing process. The

lower order Zernike moments represent the global shape of a pattern. The derived feature, acceleration

vector is computed for the sample signature which comprises on-line pixels. The Zernike moment repre-

sent the shape of the acceleration plot. The summation value of a Zernike moment for a signature sample

is obtained on normalized acceleration values. This type of substantiation decreases the influence of pri-

mary features with respect to translation, scaling and rotation at preprocessing stage. Zernike moments

provide rotation invariance. In this investigation it was evident that the summation of magnitude of a

Zernike moment for a genuine sample was less as compared to the summation of magnitude of a impos-

ter sample. The number of derivatives of acceleration feature depends on the structural complexity of the

signature sample. The computation of best order by polynomial fitting and reference template of a sub-

ject is discussed. The higher order derivatives of acceleration feature are considered. Signatures with

higher order polynomial fitting and complex structure require higher order derivatives of acceleration.

Each derivative better represents a portion of signature. The best result obtained is 4% of False Rejection

Rate [FRR] and 2% of False Acceptance Rate [FAR].

2010 Elsevier B.V. All rights reserved.

1. Introduction

Anincreasing numberof banks andcompaniesaim to move secu-

rity from simple static passwords to more dynamic security mea-

sures to suit the comfort level of the user in mobile-commerce and

web-commerce. The most personal way for authentication is sign-

ing. By providing the signature, an identity-conclusion function is

observed. The conclusion function signals the finality of providing

authentication via self-certification (Coats et al., 2007). Signatures

are non-intrusive, culturally accepted and understood proof of

authentication all over the world. New concepts that can mine

behavioral knowledgein large datasets arevery much essential. Like

fingerprint, face, irisand other biometricfeatures, signature alsohas

some common characteristics. The ubiquitous signature pattern is

storedin numberof applications andis notdependenton age(Guest,

2006). Being a behavioral pattern, it is non-invasive.

The major challenge in signature authentication is that, it is

strongly affected by user-dependencies, as it varies from one sign-

ing instance to another in a known way depending on an available

signing space. Discrimination power is achieved to deal low per-

manence and vulnerability due to forgery. In simple forgery, a

forged signature is produced with the knowledge about genuine

writers name only. In simulated forgery, a forged signature imitat-

ing reasonably a genuine signature is captured. In unskilled forger-

ies, signatures are produced by inexperienced forgers without the

knowledge of the spelling of name but are done after having ob-

served the genuine specimens closely for some time. Skilled forger-

ies are those where forgers can see the genuine signature and have

time to practice the imitations. It is well known that no two genu-

ine signatures of a person are precisely the same and some signa-

ture experts note that if two signatures written on paper were

same, then they could be considered as forgery by tracing ( Gupta

and Joyce, 2007). Any authentication system should consider sta-

tistical matching because different features are evaluated with dif-

ferent user specific thresholds. The robust acquisition procedure,

feature extraction process and classifier procedure are required

for applications such as crypto-biometrics and bio-hashing. Even

if the forger takes great pain in remembering the styles and con-

tours of the strokes, it is extremely unlikely that he/she would be

able to match the velocity profile or any other dynamic character-

istics of the original signature (Kiran et al., 2001). The work done

by Julian Fierrez provided explicit listing of 100 on-line features

(Fierrez-Aguilar et al., 2005). Using statistical parametric method,

Luan proposed a maximin distance for a subject, by ordering fea-

tures (Lee et al., 1996). k best features for which distance defined

by a criterion is largest from the rest of the entire population are

selected. Symbolic representation is a feature based approach

where, signature is described compactly to reduce enrollment data

0167-8655/$ - see front matter 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.patrec.2010.12.008

Corresponding author. Tel.: +91 9845387862; fax: +91 80 26614357.

E-mail address:[email protected](K.R. Radhika).

Pattern Recognition Letters 32 (2011) 749760

Contents lists available at ScienceDirect

Pattern Recognition Letters

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p a t r e c
http://dx.doi.org/10.1016/j.patrec.2010.12.008mailto:[email protected]://dx.doi.org/10.1016/j.patrec.2010.12.008http://www.sciencedirect.com/science/journal/01678655http://www.elsevier.com/locate/patrechttp://www.elsevier.com/locate/patrechttp://www.sciencedirect.com/science/journal/01678655http://dx.doi.org/10.1016/j.patrec.2010.12.008mailto:[email protected]://dx.doi.org/10.1016/j.patrec.2010.12.008


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size (Guru and Prakash, 2007). These capture feature variations in

the form of interval type data, to form a statistical parametric mod-

el. A holistic vector representation consisting of global features is

derived from the signature trajectories. Each feature value is repre-

sented in the form of an interval with the aid of their respective

mean and standard deviation values. The symbolic representation

proposed by Guru using MCYT signature corpus achieved equal er-

ror rate [ERR] of 5.35%. Hundred features provided by Kashi et al.

(1997) were used. The work has tabulated FRR and FAR for the

range of feature dependent threshold values. In the work done

by Oscar Miguel-Hurtado, time sequences of signature acquired

by the sensor are pre-processed by a dynamic time warping algo-

rithm and Gaussian mixture models (Migual-hurtado et al., 2007).

Analysis for discriminative power is done by Fishers ratio, with

Gaussian components ranging from 4, 8 and 16. The experiment

was on MCYT signature database of 2500 genuine and 2500 skilled

forgeries from 100 users. The performance of 8% EER for skilled

forgeries was achieved for a subset of 26 features. Shape matching

through particle dynamics warping (Agam and Suresh, 2007) mea-

sures similarity by an energy term which depends only on the

shape context of the particles and not on the distance between

them. Hidden Markov Model has ability to absorb both variability

and the similarity between patterns. In empirical risk minimiza-

tion principle, the decision rule is based on a finite number of

known examples in training set. It represents a doubly stochastic

process governed by an underlying markov chain with finite num-

ber of states and a set of random functions, each of which is asso-

ciated with the output-observation of one state. The best results in

terms of learning probability can be implemented with a proper

selection of a global feature. The classification, decoding and train-

ing problems are solved with forwardbackward algorithm, the

Viterbi algorithm and BaumWelch algorithm respectively. The

vector quantization for input vector is achieved by k-means algo-

rithm. Forward algorithm is used to determine the verification

probability. The work proposed by Fierrez et al. (2007) achieved

0.74% and 0.05% EER for skilled and random forgeries respectively

with a posteriori user-dependent decision threshold. The databaseconsisted 145 subjects comprising 3625 client signatures, 3625

skilled forgeries and 41,760 random imposter attempts. In an an-

other regional based statistical method, one dimensional fractal

coding was performed for persian on-line signature recognition.

The n spatially uniform points with range segment centriod, do-

main segment start point number and range to domain transfor-

mation parameters were sampled (Mozaffari et al., 2006). The

mapping vector accumulator recorded the angle and magnitude

of domain-range mapping vector. Multiple mapping vector accu-

mulator and multiple domain-range co-location matrix [MDRCLM]

was used for a signature locus. The work done by Saeed Mozaffari

achieved 83% recognition rate using MDRCLM on entropy and mo-

ment. This was tested on a database of 15 signature classes, each

containing 10 unconstraint samples gathered by a hyper pen1200u digitizer tablet. Each of the sample was normalized into

300 spatially uniform points.

The synergistic method definition of a ballistic stroke using

kinematic theory of rapid human movements was given by

Plamondon et al. (2007). The model lead to trajectory reconstruc-

tion, both in spatial domain and in the kinematic domain. The

study of outlier patterns for interactive comparative analysis by

Djioua et al. (2006) represented rapid human movements in vecto-

rial version. The delta-log normal model can produce smooth con-

nections by ensuring the deformed strokes which are consistent.

The kinematics theory describes the basic properties of a single

stroke and how strokes can be added vectorially to generate a sig-

nature, from a given series of input commands. The commands are

fed alternatively in two competing systems, agonist system andantagonist system to the direction of movement. The overall syn-

ergy controls the pen tip velocity to produce a signature. Panel

interface was used to display simulated signals of the test pattern

and to compare with standard pattern visually.

1.1. Zernike moments

Orthogonal Zernike momentsaredefinedby the projection of the

imagefunctionf

(x

,y

) within theunitcircle onto thecomplex Zernike

polynomials. The Zernike invariants are the magnitudes of the real

and imaginary componentsof the resultingmoments. When images

are normalized in terms of regular low-level central geometric mo-

ments, the derived Zernike moments will be invariant to rotation,

translation and scale. The image can be decomposed into a sum of

basic structures. The polynomial set covers the interior of the unit

circlex2 +y2 = 1. Zernike moments are orthogonally stable and pro-

vide image reconstruction. The choice of the Zernike method will

avoid skeletonization as compared to Fourier descriptors (Milgram

et al., 1990). Rotation in spatial domain implies a phase shift to the

Zernike moments. The orthogonal property enables to separate out

the individual contribution of each order moment to reconstruction

process (Amin and Subbalakshmi, 2004).

Some of the applications of Zernike Moments are robust image

water marking, iris, face, palm print, gait and Devanagiri handwrit-

ten numeral identifications (Xiao and Yang, 2008; Hse and Richard

Newton, 2004; Wiliem et al., 2007; Hanmandlu and Murthy, 2007;

El-ghazal et al., 2007). The optimum automatic thresholding is

achieved using the phase of Zernike moments for edge detection

(Belkasim et al., 2004). Wave aberrations in an optical system with

a circular pupil are accurately described by a weighted sum of Zer-

nike polynomials. The better classification accuracy and noise

robustness to white Gaussian noise are achieved compared to blur

invariant analysis methods based on complex moments (Zhu et al.,

2009). The Zernike moment provide expression efficiency, fast

computation and multi-level representation for describing the var-

ious shapes of pattern (Bin and Xiong, 2002). Haddadnia et al.

(2002) and Pang et al. (2004) explored the use of pseudo Zernike

moment invariants as facial features for face recognition. In thework done byChen and Srihari (2005)for off-line signature verifi-

cation with an on-line flavor, Zernike is applied for upper and low-

er contours of signature. The signature is segmented into 20 small

curves linearly so that shape feature can be separately computed

for each contours. Sixteen moments, up to order six are extracted.

16 2 20 feature values were extracted from a signature. Apply-ing harmonic distance as similarity measure, 83.7% acceptance rate

and 83.4% rejection rate was achieved. The shape characteristic of

derived feature acceleration under continuous dynamic program-

ming provided genuine acceptance rate of 97% and imposter rejec-

tion rate of 92% using MCYT-100 on-line signature database

(Radhika et al., 2009). The acceleration vector is fed as input to Zer-

nike moment summation value computations. The imposter rejec-

tion rate of 90% and genuine acceptance rate of 80% is achieved(Radhika et al., 2009). This paper proposes higher order derivatives

of the acceleration vector with a novel reference template selection

method.

Some challenges exist while implementing Zernike moments in

an application. The scale and translation invariance cannot be

explicitly achieved. One of the indirect approaches is through

expressing moments using centralized and normalized regular mo-

ments (Binand Xiong, 2002; Belkasim et al., 2007). The square to

circular mapping during the computation of moments lead to geo-

metrical error. It is minimized by mapping all the pixels inside the

unit disk. The numerical error is caused by double integral and the

use of the truncated approximation series of Zernike moments

(Wee and Paramesran, 2007). It is eliminated by integrating poly-

nomials over corresponding intervals of pixels. Less shape informa-tion is used for skewed and stretched shapes (Zhang and Lu, 2002).

750 K.R. Radhika et al. / Pattern Recognition Letters 32 (2011) 749760


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1.2. Feature extraction using zernike moments

The lower order moments represent the global shape of a pat-

tern and the higher order, the detail (Padilla-Vivanco et al.,

2003). High order Zernike moments are more sensitive to noise(Milgram et al., 1990). The zeroth order moment represents the

mean intensity value in an image neighbourhood and first-order

moments are related to the centre of gravity of the intensity sur-

face. The second-order moment captures the variance of the inten-

sity levels present in the local neighbourhood, so on (Chen et al.,

2005). In this paper, we aim to work on magnitude of summation

of lower order Zernike moments with a derived feature. This leads

in reducing response time of the system as higher order moments

take significantly longer time to compute. The Euclidean norm is

applied to each of the vector representing the acceleration value

for each on-line pixel. The maximum euclidean norm value of

acceleration values is used for normalization. The magnitudes of

a rotated image are same as compared to the magnitudes of the

image before rotation as shown inFig. 1.

2. Formulation

The Zernike functions are a product of the radial polynomials

with sine and cosine functions. As stated earlier, Zernike moments

have simple rotational symmetry properties, which lead to a poly-

nomial product of formz[q]g[h], whereq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 y2

p andg[h] is a

continuous function that repeats every 2p radians. This satisfiesthe requirement that, rotating the co-ordinate system by an angle

adoes notchange the formof the polynomialg[h+ a] (Wyant,1992).

The other property of Zernike polynomials is that the radial

function must be a polynomial in q of degree 2n and contain noexponent ofq less than m. n is order number and m is a variablewith a value less than are equal to n. Bothn andm are non-nega-

tive integers. Further,z[q] must be even ifmis even and odd ifmisodd.z[n, m,q] is a polynomial of order 2n and it can be written as

Xnms0

1s 2nms!

s!ns!nms!q2nsm: 1

The radial polynomials are combined with sines and cosines as

z[n, m,q]cos[mh], z[n, m,q]sin[mh] where cosh xffiffiffiffiffiffiffiffiffix2y2

p andsinh yffiffiffiffiffiffiffiffiffi

x2y2p .

3. Proposed system

3.1. Feature vector calculation

From signature samples, on-line features such as x,y, pressure,azimuth and inclination in three dimensional space are extracted.

Azimuth is the angle between thez-axis and the radius vector con-

necting the origin and any point of interest. Inclination is the angle

between the projection of the radius vector onto thexyplane and

thex-axis. z-axis is the pressure axis. The range ofx, y,z, azimuth

and inclination are [012,699], [09649], [01024], [0360] and

[090], respectively. The different combinations of the five time

functions in presence or absence of time variability is studied

(Houmani et al., 2009).

P is the point of interest. O is the origin. The radial distance of

a point from origin is calculated asr= (x2 +y2 +z2)1/2.u and h are

Fig. 1. Rotation invariant property.

(a) (b) (c)

Fig. 2. (a) Representation of spherical coordinates. (b) 2D cross sectional view ofXYplane. (c) Representation of unit vectors.

K.R. Radhika et al./ Pattern Recognition Letters 32 (2011) 749760 751


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measured.Sis the distance of the projection from point of interest

to thez-axis as shown in Fig. 2(a). In Fig. 2(b), S is the displacement

vector from the origin to projection of point of interest to xy

plane. We can equate the two values of S as (x2 +y2)1/2 = r sinh.

UsingFig. 2(b), x= S cosu andy= S sinu. Substituting for S, x= rsinhcosu andy = rsinhsinu. FromFig. 2(a),z= rcosh.

The direction of rcan be analysed by keeping h andu valuesfixed.

qis the position vector of

P. ^

rpoints vertically upwards as

shown inFig. 2(c). Therefore unit vector ris @q@r

@q@r

. The directionof hcan be analysed by keeping rand u values fixed. The value ofhis increased. h points towards southwards. The unit vector h is@q@h

@q@h

. The direction of the u can be analysed by keeping randhvalues fixed. The value ofu is increased. upoints towards east-

wards. The unit vector u is @q@u

h i. @q@u

. r, h, u forms the orthogonal

basis. The three components ofrare [x,y,z] = [r sinhcosu, r sinhsinu,rcosh]. q rsinhcosu^x rsinhsinu^y rcoshz. The position vectorq rr. So, the velocity is dq

dt dr

dtr rr0 ~v rr0 hrh0 urusinh

(http://www.scar.utoronto.ca/pat/fun/NEWT3D/PDF/VECTRS3D.

PDF). Differentiating,

~a rr00 rh02 ru02sin2h h 1

rddt

r2h0 ru02sinhcosh

u 1

rsinh

d

dt r2u0sin2h

: 2

The derivation of~a is given inAppendix A.

3.2. Lower order Zernike moment selection

The moment should provide, a descriptive power with a simi-

larity ordering for genuine samples to achieve classification and

compact in size to minimize the storage requirement. The transfor-

mation gives an infinite sequence of moments, while separability

of feature space is guaranteed (Nassery and Faez, 1996). Signature

is a ballistic motion without feedback.

Fourty-eight Zernike polynomials are generated in polar co-

ordinates for n< m values substituted in (2). In these set of polyno-

mials, the six Zernike polynomials with n = m values are selected.

These are of the order (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6). Each

of the training sample is subjected to Zernike polynomial order and

moment summation values are calculated as explained in Section

1.2. It was evident that, the magnitude of summation values of

genuine samples were less than forged samples summation values

(vice versa).Fig. 3(a) and (b) depict the moment summation values

of genuine and imposter samples for a person. The summation val-

ues are in the range of [10281029] for genuine samples and for

imposter samples they are in the range of [10321033].

In the proposed work,Pttraining samples are selected for each

subject. Pt ranges from 5 to 10. The two measures of threshold,

average and maximum average difference are considered. The

average value [ag] is found using the moment summation values,

Ztotal for Pt training samples. The maximum average difference

[agmax] measure is the highest difference value between Ztotal

andag, considering all the Ptraining samples. The testing samples

are compared with threshold to find FRR and FAR. The summation

value less than or equal to threshold leads to acceptance. The sum-

mation value greater than threshold lead to rejection. The experi-

ment was conducted for 30 people on all six polynomials with

n= musing (1). Considering only the sine components for n= mor-

der, an optimistic order with lesser FRR and FAR for same row was

not indicated as shown inTable 1.

Continuing the experiment for only cosine components for

n= m orders, sixth row inTable 2led to lesser values in both FRR

and FAR columns. The experiment generated (6,6) as an optimistic

order under agmax threshold.

The results are shown in Fig. 4(a) and (b). After selecting the or-

der 6 as best fit Zernike polynomial for the acceleration values cal-

culated using (2), the other variation for n= 6 and m ranging from 0

to 6 were considered (Wyant, 1992).Table 3shows FRR and FAR

Fig. 3. (a) Zernike summation values for 25 genuine samples of order (6,1). (b) Zernike summation values for 25 forged samples of order (6,1).

Table 1

Zernike moments ofn = m order with sine components.

n m Zernike

coefficient

FRR in %

using ag

measure

FAR in %

using ag

measure

FRR in %

using agmax

measure

FAR in %

using agmax

measure

1 1 qsin[h] 12.53 2.83 6.23 6.562 2 q2sin[2h] 11.7 2.83 7 5.53 3 q3sin[3h] 8.96 4.73 6.8 5.134 4 q4sin[4h] 7.93 5.26 7.3 4.835 5 q5sin[5h] 4.9 6.5 6.83 5.566 6 q6sin[6h] 5.56 7.2 6.73 5.16

Table 2

Zernike moments ofn = m order with cosine components.

n m Zernike

coefficient

FRR in %

using ag

measure

FAR in %

using ag

measure

FRR in %

using agmax

measure

FAR in %

using agmax

measure

1 1 qcos[h] 13.43 1.7 7.4 5.72 2 q2cos[2h] 7.9 5.2 8.3 63 3 q3cos[3h] 7.16 5.9 6.9 5.64 4 q4cos[4h] 6.7 6.5 6.8 5.65 5 q5cos[5h] 5.96 6.83 6.36 5.766 6 q6cos[6h] 5.76 6.06 6.23 5.3

http://www.scar.utoronto.ca/pat/fun/NEWT3D/PDF/VECTRS3D.PDFhttp://www.scar.utoronto.ca/pat/fun/NEWT3D/PDF/VECTRS3D.PDFhttp://www.scar.utoronto.ca/pat/fun/NEWT3D/PDF/VECTRS3D.PDFhttp://www.scar.utoronto.ca/pat/fun/NEWT3D/PDF/VECTRS3D.PDF


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values for the two measures of threshold ofn= 6 order. The exper-

iment generated (6,1) as an best fitting order.

3.3. Polynomial order and reference template selection procedure for a

subject

In the proposed system the reference template of a person is se-

lected according to leave-one-out method. The genuine signaturesamples of a subject vary in length. ~xt and~yt represent on-linepixel co-ordinates. In the set of normalization techniques such as

minmax, decimal scaling,z-score, median, median absolute devi-

ation, etc., minmax normalization is best suited for the case

where the bounds are known (Jain et al., 2005). The minmax nor-

malization is achieved using(3) and (4).

~ynti ~yti minyQ

maxy miny; i2 1;2;. . .; length~ytf g andQ 512;

3

~xnti ~xti minxQ

maxxminx; i2 1;2;. . .; length~xtf g andQ 512;

4

~ar;ot is the reference sample acceleration vector, ~as;jt is trainingsample acceleration vector as shown in Fig. 5(a) and (b). r, sindicate

reference and training sample. o,j indicate {1, 2,3,4} order ofrand s,

respectively. The value ofQrepresents the maximumx andy co-

ordinate value. The sample is normalized to Q Qsize. Dynamic

time warping is utilized to match two signature sequences so that

correspondent characteristic point pair can be extracted from the

matching result (Chen et al., 2009). Dynamic time warping proce-

dure is applied on the acceleration vectors as shown in Fig. 5(c).

The mean of acceleration values according to position are calculated

leading to 101 parts to form ~cr;ot and~cs;jt vectors as shown inFig. 5(d). The normalization technique applied is z-score. z-score

is calculated using the arithmetic mean and standard deviation of

the respective vector values. The resulting vectors are further nor-

malized to improve numerical properties. The translation transfor-

mation is achieved by the difference value with mean. The scaling

transformation is achieved using standard deviation.

pcoeffr;s;o pf ~cs;jt;~cr;ot

; 5

wherej ranges from 1 to the best polynomial fitting order of refer-

ence sample. The experiment checks the variance of residual values

obtained after performing polynomial fitting process using(5). Var-

iance of residual values is computed up to fourth-order polynomial

fitting for each subject. The sample leading to minimum variance is

selected as reference template. The order of acceleration derivative

for a subject is one less than the best polynomial fitting order of ref-

erence sample.Acceleration vector of a sample represents the transformed ver-

sion of the sample. The proposed work supports the renewability

capacity by achieving only the transformed sequence derivatives.

These incorporates the protection and replacement of features

which will be useful in cancelable biometrics (Maiorana et al.,

2010).

1 2 3 4 5 6 7 8 9 102

1.5

1

0.5

0

0.5

1

1.5x 10

7

SlopeValues

Training Samples

0 5 10 152

1.5

1

0.5

0

0.5

1

1.5x 10

7

Testing Samples

SlopeValues

1 2 3 4 5 6 7 8 9 102

1.5

1

0.5

0

0.5

1

1.5

2x 10

4

Training Samples

Y

InterceptValues

0 5 10 151.5

1

0.5

0

0.5

1

1.5x 10

4

Testing Samples

Y

InterceptValues

a b

c d

Fig. 6. Variation inp1(1) andp1(2) values (green-genuine and red-imposter). (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)



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3.4. Higher order acceleration values

~at ~atix;~atiy;~atiz

i2 1;2;. . . ;nf g; 6~a0t ~a0tix;~a

0tiy;~a0tiz

i2 1;2;. . . ;nf g 7(6) and (7) represent ~at and~a0t. The direction cosines of~at,~a0t,~a00t and~a000t are given in Appendix A. The higher order deriv-atives are implemented using symbolic mathematical tool (http://

www.mathworks.com/help/toolbox/symbolic/brvfu8o-1.html).

3.5. Algorithm: computation of summation value for a sample

1. ForNpeople

2. For each on-line signature sample ofm on-line pixels

2.1. For 2 to m

a. The derivative vectors ofr0,h0, u0,r2h0,r2u0sin2h, r00arecalculated.

2.2. For 1 to m

a.[r[1], r[2], r[3]], [h[1], h[2], h[3]], [u[1], u[2], u[3]],

[d^r[1], d^r[2],dr[3]], [dh[1], dh[2], dh[3]], [du[1], du[2], du[3]] vectors are

calculated.

b. The acceleration vector is calculated using (2).

c. [acc[1], acc[2], acc[3]] The order of derivative

equal to the polynomial order-1.

d. accnorm = Euclidean norm (acc[1],acc[2],acc[3]).

2.3. The maximum-accnorm is found.

2.4. For 1 to m

a. The order of Zernike moment is selected.

b. norm-acc-value = accnorm/maximum-accnorm.

c. new-radial-distance = applying Euclidean distance

measure between positional value, norm-acc-value.

d. Zmoment-of-each-pixel = applying new-radial-

distance to selected Zernike moment order, momentvalue is calculated.

2.5. Ztotal = Summation of Zmoment-of-each-pixel is

calculated.

4. Evaluation of order accuracy

~p1 pf~x;~y;order; ~p1 p11;p12;. . . ;p1order 1f g; 8

~y p11~xp12: 9

From (5)the best fit order of the subject is found using acceleration

vector of the signature as the feature vector. To further evaluate the

classification, cartesian domain behavior of on-line signature signal

is considered. The selected order from (5)is applied to~x, ~yfor poly-

nomial fitting in (8). The first two coefficients are used for hypothet-

ical linear plot.

Fig. 6(a) and (b) shows the variation in p1(1) values in training

and testing samples. The imposter training samples have lesser

slope values than genuine training samples represented in red col-

or.Fig. 6(c) and (d) shows the variation in p1(2) values in training

and testing samples. The imposter training samples achieved high-

ery intercept values than genuine training samples represented in

red color. The value ofxis used to evaluate the value ofyusing (9).

xis independent variable and

yis dependent variable. The plot ob-

tained by applying(9) for training samples is shown by Fig. 7(a).

This indicates that the direction corresponding to genuine samples

are parallel to each other. The direction of imposter samples differ

from genuine samples. From testing samples as shown inFig. 7(b)

and (c) it is evident that, the selected order provided same slope

directional variations in genuine samples.

5. Experimental results

The experiment was conducted for 100 25 2 samples usingMCYT on-line signature database. It consists of 2500 genuine sam-

ples and 2500 forged samples of 100 subjects.

5.1. Using acceleration vector

For the training samples Pt= 10, the rejection rate of 90% and

acceptance rate of 80% is achieved for (6,1) order as shown in

Fig. 8(a) (Radhika et al., 2009).

In the work done by Fierrez-Aguilar et al. (2005), the system

based on global analysis outperformed the local approach when

training sample size was 5. Continuing the experiment for Pt= 5

9 the FRR value decreased and FAR value increased with increase

in training samples. FAR and FRR are trade off against one another

(Jain et al., 2002).

Keeping track of receiving operating curve characteristics, pro-

posed system suggests Pt= 7 with EER = 4.5, as reliable value for

authentication applications using (6,1) order as shown in Table 4

andFig. 8(b). Fig. 8(c) depicts, a promising EER can be obtained

by increasing the number of training samples with ag as measure

of threshold for EER lesser than 4.5.

5.2. Using derivatives of acceleration vector

Considering training samples Pt= 7, Table 5 depicts the best

order for each subject (Radhika et al., 2011). The order represented

in table depicts the best polynomial fitting order as explained in

Section 3.3. The best polynomial order depicts the maximum deriv-

ative of acceleration vector to be considered for the subject.

~aorder1person t ~aorder1person tix; ~a

order1person tiy; ~a

order1person tiz

i2 1;2;. . . ;nf g: 10

The three components of~aorder1person are computed using(10). It is pro-vided as input to (6,1) Zernike order as explained in Section 3.2. The

subject wise best fit polynomial order depicting maximum acceler-

ation derivative is saved. The best result obtained is 4% of False

Table 4

Performance for various values ofP.

Pt FRR in % using agmax

measure

FAR in % using agmax

measure

FRR in % using ag

measure

FAR in % using ag

measure

Total time for 100 people in

seconds

Pt= 5 8.12 3.63 7.63 4.27 2.0175E+2

Pt= 6 5.2 4.33 7.25 4.25 2.020E+2

Pt= 7 4.61 4.71 7.2 4.21 2.011E+2

Pt= 8 4.56 4.99 6.72 4.16 2.013E+2

Pt= 9 4.49 5.47 6.11 4.25 2.016E+2

Pt= 10 4.42 5.58 5.46 4.32 2.010E+2

http://www.mathworks.com/help/toolbox/symbolic/brvfu8o-1.htmlhttp://www.mathworks.com/help/toolbox/symbolic/brvfu8o-1.htmlhttp://www.mathworks.com/help/toolbox/symbolic/brvfu8o-1.htmlhttp://www.mathworks.com/help/toolbox/symbolic/brvfu8o-1.html


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Rejection Rate [FRR] and 2% of False Acceptance Rate [FAR] using

agmaxthreshold.

6. Conclusion and future work

6.1. Conclusions

In this paper a simple linear classification approach using lower

order Zernike moments involving only a derived feature such as

acceleration is proposed for the authentication of on-line hand

written signature with less computational time. Although previous

work done have lead to better results with lot of preprocessing

steps like segmentation, warping and probability structures, an at-

tempt is made for arriving at a simple robust system. The less

memory method will aid in pre-embedding authentication soft-

ware to the acquisition device. The feature vector represent bio-

metric characteristics which is unique to every individual and

cannot be reproduced by an imposter.

6.2. Future work

Kinematic properties involved to perform the joint angle trajec-

tory obeys an elliptic form. Extraction principle states that, when

humans draw planar curves, the instantaneous tangential accelerationof hand decrease as the curvature increase. The general structure of

contour curvature is elliptical. The different derivatives of acceler-

ation can be considered to reconstruct the different parts of the ref-

erence template sample. Signatures with higher order polynomial

fitting and complex pole structure require higher order derivatives

of acceleration (Papaj and Hermanowicz, 2010). Each derivative

better represents a portion of reconstruction.

Acknowledgements

The authors thank J. Ortega-Garcia for the provision of MCYT

Signature database from Biometric Recognition Group, B-203, Uni-

versidad Autonoma de Madrid SPAIN (Ortega-Garcia et al., 2003).

Appendix A

r, h, u forms the orthogonal basis. The three components ofrare[rsinhcosu, rsinhsinu, rcosh]. The position vector is written as:

q rsinhcosu^xrsinhsinu^yrcosh^z: 11

Differentiating the position vector with respect to r:

@q

@r sinhcosu^xsinhsinu^ycosh^z;

@q

@r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin

2hcos2u sin2hsin2ucos2h

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2hcos2usin2u cos2hq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin

2hcos2h

q 1;

r sinhcosu^xsinhsinu^ycosh^z:

Differentiating the position vector with respect to h:

@q

@h rcoshcosu^xrcoshsinu^yrsinh^z;

@q

@h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffir2cos2hcos2ur2cos2hsin2ur2sin2h

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2cos2hcos2u sin2u r2sin2h

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2sin

2h r2cos2h

q r;

h coshcosu^xcoshsinu^ysinh^z:

Differentiating the position vector with respect tou:

@q

@u rsinhsinu^xrsinhcosu^y;

@q

@u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2sin

2hsin

2u r2sin2hcos2uq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffir2sin

2hsin

2ucos2uq

rsinh;

u sinu^xcosu^y:

The position vectorq rr. The velocity vector (~v) is represented as:dq

dt

dr

dtr rr0 rr0 hrh0 uru0sinh:

Table 5

ordersubjectfor 100 subjects of MCYT database.

Person Reference

sample

Best

order

Person Reference

sample

Best

order

Person1 1 2 Person51 5 3




Person5 6 2 Person55 4 3Person6 1 2 Person56 1 3
















Person22 4 1 Person72 9 4Person23 3 2 Person73 1 1




























K.R. Radhika et al./ Pattern Recognition Letters 32 (2011) 749760 757


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