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Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2009, Article ID 260516, 13 pages doi:10.1155/2009/260516 Research Article Online Signature Veriﬁcation Using Fourier Descriptors Berrin Yanikoglu1 and Alisher Kholmatov2 1Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul 34956, Turkey 2National Research Institute of Electronics and Cryptology (UEKAE), Scientiﬁc and Technological Research Council of Turkey (TUBITAK), Gebze, Kocaeli 41470, Turkey Correspondence should be addressed to Berrin Yanikoglu, email@example.com Received 27 October 2008; Revised 25 March 2009; Accepted 25 July 2009 Recommended by Natalia A. Schmid We present a novel online signature veriﬁcation system based on the Fast Fourier Transform. The advantage of using the Fourier domain is the ability to compactly represent an online signature using a ﬁxed number of coeﬃcients. The ﬁxed-length representation leads to fast matching algorithms and is essential in certain applications. The challenge on the other hand is to ﬁnd the right preprocessing steps and matching algorithm for this representation. We report on the eﬀectiveness of the proposed method, along with the eﬀects of individual preprocessing and normalization steps, based on comprehensive tests over two public signature databases. We also propose to use the pen-up duration information in identifying forgeries. The best results obtained on the SUSIG-Visual subcorpus and the MCYT-100 database are 6.2% and 12.1% error rate on skilled forgeries, respectively. The fusion of the proposed system with our state-of-the-art Dynamic Time Warping (DTW) system lowers the error rate of the DTW system by up to about 25%. While the current error rates are higher than state-of-the-art results for these databases, as an approach using global features, the system possesses many advantages. Considering also the suggested improvements, the FFT system shows promise both as a stand-alone system and especially in combination with approaches that are based on local features. Copyright © 2009 B. Yanikoglu and A. Kholmatov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Signature veriﬁcation is the task of authenticating a person based on his/her signature. Online (dynamic) signatures are signed on pressure sensitive tablets that capture dynamic properties ofa signaturein addition to itsshape,whileoﬄine (static) signatures consist of only the shape information. Dynamic features, such as the coordinates and the pen pressure at each point along the signature’s trajectory, make online signatures more unique and more diﬃcult to forge compared to oﬄine signatures. In online signature veriﬁcation systems, like in any other biometric veriﬁcation system, users are ﬁrst enrolled to the system by providing reference samples. Later, when a user presents a signature claiming to be a particular individual, thequerysignatureiscomparedwiththereferencesignatures of the claimed individual. If the dissimilarity is above a certain ﬁxed threshold, the user is rejected. As a behavioral biometric, online signatures typically show more intrapersonal variations compared to physical biometrics (e.g., iris, ﬁngerprint). Furthermore, forging a signature may be relatively easy if the signature is simple and its timing can be guessed from its static image (e.g., short signature showing a strictly left to right progression). Despite these shortcomings, signature is a well-accepted biometric and has potential niche applications such as identity veriﬁcation during credit card purchases. Also, forging the shape and timing at the same time proves to be diﬃcult in reality, as evidenced by the success of automatic veriﬁcation algorithms . In this work, we present an online signature veriﬁcation system based on the spectral analysis of the signature using the Fast Fourier Transform (FFT). The advantage of using the Fourier domain is the ability to compactly represent an online signature using a ﬁxed number of coeﬃcients, which leads to fast matching algorithms. More importantly, the ﬁxed-length is better suited or even necessary in certain applications related to information theory and biometric cryptosystems. For instance, the template protection scheme 2 EURASIP Journal on Advances in Signal Processing by Tuyls et al.  requires a ﬁxed-length feature representa- tion of the biometric signal. Similarly, an earlier version of the proposed system was used for assessing the individuality of online signatures, where the ﬁxed-length representation was important for simplifying the analysis . Approaches using global and local features are called feature-based and function-based in literature, respectively. In this work, we also refer to them shortly as global and local approaches. The challenge of using the Fourier domain representa- tion, on the other hand, is to ﬁnd the right preprocessing steps and matching algorithm for this representation. We report on the eﬀectiveness of the proposed method, along with the eﬀects of individual preprocessing and normal- ization steps, on the overall system performance, based on comprehensive tests over two public signature databases. While the current error rates are higher than state-of-the- art results for the used databases, this is to be expected since approaches based on global features of the signature normally underperform those using local information. On the other hand, in addition to the aforementioned advan- tages, global approaches are good complements to local approaches such as Dynamic Time Warping (DTW) or Hidden Markov Models (HMMs). In fact, we show that the fusion of the proposed system improves the performance of our DTW system by up to about 25%. With regard to the preprocessing, we show that the proposed incorporation of the pen-up durations signiﬁcantly improves veriﬁcation performance, while subsampling which is commonly used to obtain equal-length signatures, has the opposite eﬀect. Finally,wediscusspotentialimprovementsandconcludethat the proposed system has potential both as a stand-alone system and especially in combination with approaches that are based on local features. This paper is organized as follows. Section 2 describes the previous work in the general area of online signature veriﬁcation problem, along with some speciﬁc work that are morecloselyrelatedtoours.Section 3describestheproposed method, including preprocessing, feature extraction, and matching steps. Sections 4 and 5 present and discuss the experimental results using the SUSIG and MCYT databases. Finally Section 6 mentions future work to improve the current performance. 2. Previous Work Signature veriﬁcation systems diﬀer both in their feature selection and in their decision methodologies. In fact, more than 70 diﬀerent feature types have been used for signature veriﬁcation [4–7]. These features can be classiﬁed in two types: global and local. Global features are those related to the signature as a whole, including the signature bounding boxdimensions,averagesigningspeed,andsigningduration. Fourier Descriptors studied in this work are also examples of global features. Genuine signatures of a person often diﬀer in length due to the natural variations in signing speed. The advantage of global features is that there are a ﬁxed number of measurements (features) per signature, regardless of the signature length; this makes the comparison of two signatures a relatively straightforward task. The ﬁxed-length representation is also better suited or even necessary in certain applications. Xu et al. use the Fourier transform to obtain a ﬁxed-length representation of ﬁngerprint minutiae . Similarly, Yi et al. use the phase information of the Gabor ﬁlter to align online signatures and use the temporal shift and the shape dissimilarity measures to represent online signatures using a ﬁxed-length feature vector . In contrast to global features, local features are measured or extracted at each point along the trajectory of the signature and thus vary in number even among genuine signatures. Examples of local features include position, speed, curvature, and pressure at each point on the signature trajectory. In [5, 10], some of these features are compared in order to ﬁnd the more robust ones for signature veriﬁcation purposes. When local features are used, one needs to use methods which are suitable to compare feature vectors of diﬀerent lengths: for instance, the Dynamic Time Warping algorithm [4, 5, 11–13] or Hidden Markov Models [14– 19]. These methods are more complicated compared to the relatively simple metrics used with global features but they are generally more successful as well. Methods using global and local features are called feature-based and function- based approaches in literature . Comprehensive surveys of the research on signature veriﬁcation, including a recent one, can be found in [20–22]. The performance of biometric veriﬁcation systems are evaluated in terms of false reject rate (FRR) of genuine samples, false accept rate (FAR) of impostors, and equal error rate (EER), where the two types of errors are equal. Due to the diﬀerences in databases and forgery quali- ties, comparing reported performance results is diﬃcult. The First International Signature Veriﬁcation Competition (SVC2004), organized in 2004, provided a common test set and tested more than 15 online signature veriﬁcation systems from industry and academia. The results of this competition indicate state-of-the-art results of 2.6% equal error rate in skilled forgery detection and 1.85% equal error rate in randomforgerydetectiontasks,usingonlypositionsequence (x, y) of a signature [ 1]. Our DTW-based system using only positional information, later described in , was declared as the winning system (Team 6) for its performance in the skilled forgery tests. We will refer to this system as our DTW system from now on. Many diﬀerent features and matching algorithms have been used to compare two signatures but the use of the Fourier Transform has not been widely explored [23–25]. In the work by Lam et al. , the signature is ﬁrst resampled to a ﬁxed-length vector of 1024 complex numbers consisting of the x- andy-coordinates of the points on the signature trajectory. This complex signal then undergoes various preprocessing steps, some of which are suggested by Sato and Kogure , including normalization for duration, drift, rotation, and translation, prior to the application of the Fast Fourier Transform (FFT). Feature extraction involves calculating the Fourier Descriptors of the normalized signa- ture and selecting the 15 Fourier Descriptors with the highest magnitudes, normalized by sample variances. Discriminant analysis is then used with the real and imaginary parts of EURASIP Journal on Advances in Signal Processing 3 the 15 selected harmonics, to ﬁnd the most useful features and their weights. The proposed system was tested using a verysmallsignaturedataset(8genuinesignaturesofthesame user and 152 forgeries provided by 19 forgers), achieving a 0% FRR and 2.5% FAR. In a similar work, Quan et al.  use windowed FFT to avoid the discontinuities in the signal, also using discriminant analysis to pick the important FFTcoeﬃcients.Theauthorsshowthatwindowingimproves performance, resulting in an EER of 7% EER on the MCYT- 100 database, using 15 reference signatures. Similar to the Fourier transform, the Discrete Wavelet Transform (DWT) is recently used for online signature veriﬁcation by Nanni and Lumini . The results of this system on the MCYT-100 database are 11.5% equal error rate on skilled forgeries, when using only the coordinate information (x- andy-coordinates as a function of time) of the signature. The DWT is also used by Nakanishi et al. , with about 4% EER on a small private database. Recent research on signature veriﬁcation has concen- trated on the fusion of multiple experts [7, 26, 28]. These systems typically combine new methods with proven ones such as DTW and HMMs (e.g., [13, 19] which received the ﬁrst and second place in the SVC2004 competition). Fusion systems have some of the best results obtained for their respective databases; this is not very surprising because online signature is a complex signal of several dimensions and one method may concentrate on one aspect of the signal (e.g., shape), while another method may focus on another (e.g., timing). In this paper, we present a novel online signature veriﬁcation system based on the Fast Fourier Transform. Our work diﬀers from previous work using Fourier analysis [23–25] in preprocessing and normalization steps as well as the matching algorithm. Furthermore, the results of the proposed algorithm and the individual preprocessing steps are comprehensively tested on two large, public databases. The results show the potential of the proposed system and also highlight the importance of the timing information for online signatures, in contrast to previous work where the timing information was discarded to a large extent [23–25]. 3. Proposed Method 3.1. Input Signal. An online signature S, collected using a pressure-sensitive tablet, can be represented as a time sequence: S(n)=x(n) y(n) p(n) t(n)T (1) for n=1,2,...,N, whereN is the number of points sampled along the signature’s trajectory; x(n) andy(n) denote the coordinates of the points on the signature trajectory, while p(n) andt(n) indicate the pen pressure and timestamp, at sample point n. A pressure-sensitive tablet typically samples 100 points in a second (100HZ) and captures samples only during the interaction of the pen tip with the tablet. Depending on the tablet capabilities, pen azimuth (az(n)) and pen altitude (al(n)), indicating the angle of the pen with respect to the writing surface, can also be collected. Other features such as local velocity and acceleration may be calculated using the above features, as done by many signature veriﬁcation systems [5–7, 12, 14]. The positional information consisting of x(n) andy(n) is important because it describes the shape of the signature and it is common to all tablets. The pressure information, on the other hand, had not seem very useful in some previous studies [10, 13, 26], while others have found it useful . In particular, our DTW system  using just the positional information achieved the lowest error rates in the skilled forgery tasks of SVC2004, including the task where pressure, azimuth, and altitude were available to participating systems . On the other hand, Muramatsu and Matsumoto  tested the discriminative power of the component signals of an online signature both alone and in groups and achieved 10.4% EER when they included the pressure and azimuth information, compared to 12.7% without them, using the SVC2004 database. In the current work, we have also observed that the pressure, azimuth, and altitude information improves the performance, although not drastically. In addition, we propose to use the timestamp information to identify and use the pen-up periods in identifying forgeries. In the remainder of the paper, we use the sequence index n as if it refers to time (see Section 3.2.1) and describe the methodology concentrating on the positional information, denoted as s(t), while the other input components are used as available. 3.2. Preprocessing. Preprocessing of online signatures is commonly done to remove variations that are thought to be irrelevant to the veriﬁcation performance. Resampling, size, and rotation normalization are among the common preprocessing steps. While useful in object recognition, our previous research  had suggested that preprocessing may decrease biometric authentication performance by removing individual characteristics of the user. Therefore, we keep the amount of preprocessing done to a minimum, preserving as much of the discriminatory biometric information as possible. In the previous work on online signature veriﬁcation using FFT [23–25], the signature undergoes various prepro- cessing steps, consisting of spike and minor element removal to remove noise and extraneous segments; adding ligatures to connect consecutive strokes to reduce discontinuities that would aﬀect FFT results; equi-time subsampling to obtain a ﬁxed-length signature; drift removal; and rotation, translation, and scale normalization. In , the eﬀects of drift removal and ligature processing are analyzed and authors report that drift removal signiﬁcantly improves veriﬁcation performance, while ligature processing only brings a marginal improvement. They guess that ligature processing that is done to reduce discontinuities is not very helpful because the high-frequency components aﬀected by the discontinuities are discarded in the matching process. We tested the individual eﬀects of the preprocessing steps foundtobeimportantin,usingtwolargedatabases.The results described in Section 4.4 show that subsampling which 4 EURASIP Journal on Advances in Signal Processing is commonly done to normalize the length of a signature signiﬁcantly reduces veriﬁcation performance by removing most of the timing information. This was also conﬁrmed in our previous research. On the other hand, mean and drift removal are found to be useful, while scale removal is not needed since our features (Fourier Descriptors) are normalized to be invariant to translation, rotation, and scale changes. In addition to the steps described above, we propose to use the timestamp information to identify and use the pen-up periods in identifying forgeries. The next sections describe the preprocessing steps used in this work. 3.2.1. Pen-up Durations. Pen-up periods indicate the times when the pen is not in contact with the tablet. These periods may be detected using discontinuities between the timestamps of consecutive points (t(n) andt(n + 1)) and actualpen-updurationscanbecalculatedusingthesampling rate of the tablet and the diﬀerence between timestamps. Forgery signatures often have longer pauses between strokes, compared to genuine signatures, which may help in identifying forgeries. Thus, while the pen-up durations can be useful for veriﬁcation, such as in detecting a forger’s hesitation or recomposition, it is often discarded, keeping just the order of the sampled points. In fact, the timing information is discarded to a large extent by many systems that use resampling to obtain a ﬁxed-length signature, including the previous work using FFT [23–25]. Note that resampling results in keeping only the relative order of the points on the trajectory, while other timing information is discarded. We propose to ﬁll the pen-up durations with imaginary points, which has a twofold beneﬁt: (i) it incorporates pen- up durations directly into the signature trajectory; (ii) it reduces trajectory discontinuities, which enhances the FFT analysis. For example, if there is a 50ms wait between two consecutive points of the trajectory using a 100Hz tablet (corresponding to 10ms between consecutive samples), we add 4 imaginary points. Imaginary points can be generated through (a) interpolation between the last and ﬁrst points of the two strokes corresponding to the pen-up event or (b) as if the pen was actually left on the tablet after the stroke prior to the pen-up event. In order for the pen-up events not to dominate the signal, we place imaginary points sparingly (every 30ms for the 100Hz tablet). Both methods of adding imaginary points improve the system performance, though the more sophisticated method of interpolation obtains better results, as expected. Note that after this process, the timestamp information (t(n)) itself is basically redundant and discarded. We use the sequence index n and time t interchangeably in the rest of the paper. 3.2.2. Drift and Mean Removal. In signatures that go from left to right, x(t) has a signiﬁcant drift as time increases and the same can be said for signatures being signed top to bottom and y(t). Drift removal step aims to remove the baseline drift component of a signal, so as to keep only the important information in the signal. We use a linear regression using least squares ﬁt to estimate the drift. Given a discrete time signal y of length n, the drift removed version y can be computed as y = y−β×t−t, (2) where β= Σyt−nyt Σt2 −nt2 . (3) Mean removal on the other hand is simply achieved by subtracting the mean of the signal from itself: y = y− y. 3.3. Feature Extraction. We use the Fourier Transform to analyze the spectral content of an online signature. The details of the Fourier transform are out of the scope of this paper but can be found in many references (e.g., ). Below we give the basic idea and necessary deﬁnitions. 3.3.1. Fourier Transform. Any periodic function can be expressed as a series of sinusoids of varying amplitudes, called the Fourier Series. If the signal is periodic with fundamental frequency ω, the frequencies of the sinusoids that compose the signal are integer multiples of ω and are called the harmonics. The Fourier Transform is used to ﬁnd the amplitude of each of the harmonic component, which is called the frequency spectrum of the signal. It thus converts a signal from the time domain into the frequency domain. The Discrete Fourier Transform discrete time signal f (t) is deﬁned as follows: Ck = 1 N N−1 t=0 f (t)e−i2πkt/N k =0,1,...,N −1, (4) where f (t) is the input signal; N is the number of points in the signature; k indicates the frequency of the particular harmonic; eix =cos(x)+isin(x). The amplitude of the kth harmonic found by the Fourier transform is referred to as the kth Fourier Coeﬃcient. Given a complex Fourier coeﬃcient Ck = ak + ibk, the magnitude and phase corresponding to the kth harmonic are given by |Ck|=a2 k + b2 k and tan−1(bk/ak), respectively. The Fourier coeﬃcients are normalized to obtain the Fourier Descriptors which are the features used in this study, as described in Section 3.3.3. The Inverse Fourier Transform is similarly deﬁned as f (t)= N−1 k=0 Ckei2πkt/N t =0,1,...,N −1. (5) The Fourier transform has many uses in signal processing. For instance, reconstructing a time signal using the inverse Fourier transform by discarding the high-frequency compo- nents of a signal can be done for noise removal. 3.3.2. Input Signal Components. An online signature consist- ing of x- andy-coordinates can be represented as a complex EURASIP Journal on Advances in Signal Processing 5 0 50 100 y 0 100 200 300 400 Index 0 500 1000 x 0 100 200 300 400 Index 0 50 100 y 0 100 200 300 Index 0 200 400 x 0 100 200 300 Index (a) 0 50 100 y 0 50 100 150 200 Index (b) 0 200 400 x 0 50 100 150 200 Index Figure 1:The y-coordinate(a)andx-coordinate(b)proﬁlesbelongingtogenuinesignaturesof3diﬀerentsubjectsfromtheSUSIGdatabase. signal s(t)=x(t)+iy(t) wherex(t) andy(t) are the x- andy- coordinates of the sampled points. The Fourier transform of the signature trajectory can then be directly computed using the complex signal s(t) as the input, as described in (4). In signatures which are signed from left to right or right to left, x(t) is a monotonic function for the most part and carries little information, as shown in Figure 1. Based on this observation, we ﬁrst evaluated the discriminative power of y(t) alone, discarding x(t) for simplicity. Later, we also did the reverse and used only x(t) for completeness. Similarly, we assessed the contribution of other input signal components to the veriﬁcation performance, by concatenating features extracted from individual component signals (e.g., x(t), y(t), p(t)), to obtain the ﬁnal feature vector. We denote these feature vectors by indicating the individual source signals used in feature extraction: for instance, x | y | p denotes a feature vector obtained from the x-, y-coordinates and pressure component, respectively. The input signal f (t) in (4) can be any one of these signals (s(t), y(t), x(t), p(t), etc.). 3.3.3. Fourier Descriptors. The extracted Fourier coeﬃcients are normalized to obtain the Fourier Descriptors, using normalization steps similar to the ones used in 2D shape recognition. In particular, the Fourier coeﬃcients obtained by applying the Fourier Transform to the object contour (x(t), y(t)) can be normalized to achieve invariance against translation, rotation, and scaling of the original shape . Speciﬁcally, translation of a shape corresponds to adding a constant term to each point of the original shape and aﬀects (only) the ﬁrst Fourier coeﬃcient. By discarding C0, deﬁned in (4), one obtains translation invariance in the remaining coeﬃcients. Rotation of a shape results in a phase change in each of the Fourier coeﬃcients; rotation invariance is automatically obtained when one uses only the magnitude information of the Fourier Transform. Alternatively, each coeﬃcient can be normalized such that the phase of one of the coeﬃcients (e.g., C1) is zero; this is equivalent to assuming a canonical rotation that gives a zero phase to C1. Finally, scaling of a shape corresponds to multiplying all coordinate values of the shape by a constant factor and results in each of the Fourier coeﬃcients being multiplied by the same factor. Therefore, scale normalization is achieved by dividing each coeﬃcient by the magnitude of one of the components, typically|C1|. An online signature must show adequate match to the reference signatures of the claimed identity in both shape and dynamic properties, in order to be accepted. As with the above normalization steps, it is easy to see that by discarding C0 and using the magnitudes of the remaining coeﬃcients as features, we obtain invariance to translation (position of the signature on the tablet) and rotation (orientation relative to the tablet). Scale invariance is more complicated, due to the additional dimension of time. If a signature is only 6 EURASIP Journal on Advances in Signal Processing 0 20 40 60 80 100 y 0 100 200 300 400 x (a) 0 20 40 60 80 100 y 0 50 100 150 Index 0.02 0.04 0.06 0.08 0.1 0.12 Amplitude 0 5 10 15 20 25 30 Fourier descriptor 0 20 40 60 80 100 y 0 100 200 300 400 x (b) 0 20 40 60 80 100 y 0 100 200 300 400 Index 0.01 0.02 0.03 0.04 0.05 Amplitude 0 5 10 15 20 25 30 Fourier descriptor Figure 2: A veriﬁcation case is shown for illustration, using only the y-proﬁle. From left to right: (a) Genuine signature, its y-proﬁle and its Fourier Descriptors. (b) Forgery signature, its y-proﬁle and its Fourier Descriptors. The Fourier Descriptors of genuine and forgery signatures (shown as dots) are overlaid on top of the envelope showing the min and max values of the reference signatures’ descriptors, while the line in the middle denotes the mean reference feature. scaled in space, while keeping the signing duration the same, dividing each coeﬃcient’s magnitude by|C1| achieves scale normalization. However for the more general case involving both scale and time variations, we have found that a more robust approach is to divide each coeﬃcient by the total magnitude of the Fourier spectrum: m= N−1 k=0 |Ck|= N−1 k=0 Ck ∗C∗ k , (6) where N is the length of the signature;|Ck|is the magnitude of the complex coeﬃcient Ck; C∗ k is the complex conjugate of Ck. The total energy of the Fourier spectrum is also com- monly used for normalization of the Fourier coeﬃcients: e= N−1 k=0 |Ck|2. (7) In our experiments, we have found that the normaliza- tion by the total amplitude has outperformed normalization done either by dividing each component by |C1| or by the total energy of the Fourier Transform (about 3% and 1% percent points less error, resp.). Using (7), our ﬁnal features or the Fourier Descriptors Fk are thus obtained as Fk = |Ck| m k =1,..., N 2 . (8) Notice here that k goes from 1 to N/2 since we discard half of thecoeﬃcientsduetothesymmetryoftheFouriertransform spectrum. 3.3.4. Zero-Padding. Due to the natural variation in the signing process, genuine signatures of the same user almost never have equal lengths. The length variation results in Fourier domain representation with varying number of components, hence feature vectors of varying lengths. While onecancutoutthehigh-frequencycomponents,leavingonly the ﬁrst k Fourier coeﬃcients, when the signatures are of diﬀerent lengths, these components do not correspond to the same frequencies. In order to obtain an equal number of Fourier Descrip- tors which correspond to the same frequencies, we pad each signature to be compared (reference set + query) with zeros, to match the length of the longest signature in the set, prior to the application of the Fourier Transform. This process is called zero-padding and does not aﬀect the amplitudes of the Fourier coeﬃcients but changes the frequency resolution. 3.3.5. Smoothing. We smooth the computed Fourier descrip- tors Fk by averaging two consecutive descriptors, to account for the normal timing variations between genuine signatures that would result in energy seeping into the neighboring harmonics. The smoothing is found to have a signiﬁcant eﬀect (roughly 2% point) in overall system performance in both tested databases. Sample signatures and their forgeries, along with the resultant Fourier descriptors, are shown in Figure 2, using only the y-dimension for simplicity. The ﬁgure shows the envelope of the reference set descriptors to indicate the diﬀerencebetweenqueryandreferencesignaturedescriptors, while in matching we only use the distance to the mean. The diﬀerence in the Fourier descriptors of the reference signatures for the genuine and forgery queries is due to zero- padding used in this example. As explained before, zero- padding does not change the frequency content of a signal but increases the frequency resolution (here note that the forgery signature that is used in determining the padding amount is much longer than the references). EURASIP Journal on Advances in Signal Processing 7 Table 1: The summarizing characteristics of the public databases used in this study. In both of them, the genuine signatures are collected in multiple sessions and there are 5 reference signatures per user. Dataset Subjects Genuine Skilled forgeries Input SUSIG-Visual 100 2000 1000 x, y, p, timestamp MCYT-100 100 2500 2500 x, y, p, az, al Table 2: Equal error rates obtained using diﬀerent components of the input signal. The timestamp is discarded after incorporating the pen-up durations into the trajectory, for the SUSIG database. Dataset x + iy y x x| yx | y | px | y | p |az x | y | p |az |al SUSIG-Visual 8.37% 9.90% 8.42% 6.20% — — — MCYT-100 17.62% 17.38% 17.42% 14.53% 12.99% 12.61% 12.11% 3.4. Matching. When a query signature is input to the system along with a claimed ID, the dissimilarity of its Fourier Descriptors from those of the reference signatures of the claimed person is calculated. Then, this distance is normalized using the reference set statistics of the user, and the query signature is accepted as genuine if this normalized distance is not too large. These steps are explained in detail in the following subsections. 3.4.1. Distance Between Query and Reference Set. During enrollment to the system, the user supplies a number of reference signatures that are used in accepting or rejecting a query signature. To ﬁnd the dissimilarity between a query signature q and the reference set Ri of the claimed user i, we compute the Euclidian distance between the query features Fq obtained from q and the vector FRi which is the mean of the feature vectors of the reference signatures in Ri: dq,Ri= Fq −FRi . (9) We have also evaluated diﬀerent matching algorithms, suchasthe number of matching Fourier Descriptors between the compared signatures but the presented matching algo- rithm gave the best results. Ideally, one can apply machine learning algorithms to ﬁnd the most important descriptors or to decide whether the query is genuine or forgery given the Fourier descriptors of the query and reference set. 3.4.2. User-Dependent Distance Normalization. In order to decide whether the query is genuine or forgery, the distance computed in (9) should be normalized, in order to take into account the variability within the user’s signatures. We use a normalization factor computed only from the reference signatures of the user. The normalization factor Di which is separately calculated for each user i, is the average dissimilarity of a reference signature r to the rest of the reference signatures: Di =meanr∈Rid(r,Ri/r), (10) where Ri/r indicate the set Ri without the element r. The normalization factor Di is calculated by putting a reference signature aside as query and calculating its dissimilarity d to the remaining reference signatures (Ri/r). The resulting normalized distance d(x,Ri)/Di is compared to a ﬁxed, user- independent threshold. We have previously found that this normalization is quite robust in the absence of training data . Results of similar methods of normalization using slightly diﬀerent statistics of the reference signatures are shown in Table 5. More conventional normalization techniques using client and impostor score distributions can be used when training data is available  and are expected to perform better. 3.4.3. Removing Outliers. Often, there are some important diﬀerences (in timing or shape) among the reference sig- natures of a user. In this work, we experimented with the removal of outliers from the reference set. While the template selection is a research area by itself, we found that eliminating up to one of the outlier from the reference set in a conservative fashion brings some improvement. For this, we sort the reference set distances of a user, as calculated using (9), and discard the last one (the one with the highest distance to the remaining references) if there is a big diﬀerence between the last two. 4. Experimental Results 4.1. Databases. The system performance is evaluated using the base protocols of the SUSIG  and MCYT [ 33] databases. The SUSIG database is a new, public database consisting of real-life signatures of the subjects and including “highly skilled” forgeries that were signed by the authors attempting to break the system. It consists of two parts: the Visual subcorpus obtained using a tablet with a built-in LCD display providing visual feedback and the Blind Subcorpus collected using a tablet without visual feedback. The Visual subcorpus used in this study contains a total of 2000 genuine signatures and 1000 skilled (half are highly skilled) forgeries collected in two sessions from 100 people. The data in SUSIG consists of x, y, andtimestamp, collected at 100Hz. The MCYT database is a 330-people database of which a 100-user subcorpus is made public and is widely used for evaluation purposes. The database contains 25 genuine signatures and 25 skilled forgeries signed by 5 diﬀerent 8 EURASIP Journal on Advances in Signal Processing forgers, for each user. The data in MCYT database consists of consists of x, y, pressure, azimuth, and altitude, collected at 100Hz. Table 1 summarizes these datasets, while the details can be found in their respective references. 4.2. Results of the Proposed System. We evaluated the use- fulness of various preprocessing steps and the diﬀerent components of the input signal, on the overall veriﬁcation performance. The results obtained using the best set of preprocessing steps, while varying the input signal, are sum- marized in Table 2. As can be seen in this table, using only the coordinate information of the signature, we obtained minimum equal error rates of 6.20% and 14.53% for SUSIG and MCYT databases,respectively. These resultsare obtained using the concatenation of the Fourier descriptors obtained from y(t) andx(t). The pressure, azimuth, and altitude information available in the MCYT-100 database further reduced the EER to 12.11% EER. In addition to the EER results, the DET curves showing how FAR and FRR values change according to changing acceptance thresholds are given for the databases used in the evaluation, in Figure 3. These results are obtained using 30 normalized Fourier descriptors per signal component (i.e., 30 for y(t), 60 for y(t) | x(t), etc.) and the preprocessing steps described in Section 4.4. However, very similar results were obtained with 20 and 25 descriptors. As described in Section 4.4, up to one reference signature was removed from the reference set, if deemed as an outlier. Timestamp information was not available for the MCYT database, and subsequently the pen- up durations were not used for this database. Considering the eﬀects of the diﬀerent input signal components, we see that each information source brings the error rate down, from 14.53% using x | y to 12.11% using x | y | p | az | al, for the MCYT database. Notice that the diminishing improvement is not necessarily and indication of the value of an input signal by itself. As for the positional information, we observe that the signature encoded as a complex signal (i.e., s(t) = x(t)+iy(t)) which was used in  gave signiﬁcantly worse results compared to the concatenation of the features obtained from the x- andy-components separately (i.e., x | y). Another interesting observation is that our initial assumption about the x-component being mostly useless was not reﬂected in the results. While the x-component indeed contains little information in signatures signed strictly from left to right, the results show that it contains enough discriminative information to separate genuine and forgery signatures to a large extent, for the particular databases used. In order to see the variation of the overall performance with respect to diﬀerent sets of reference signatures, we ran 25 tests using the proposed method with diﬀerent sets of 5 reference signatures, on the MCYT database. The mean EER for these tests was 10.89%, while standard deviation was 0.59. In fact, the worst performance was with the original set of references (genuine signatures [0–4]). The better performance with other reference sets can be explained by the fact that reference signatures collected over a wider time span better represent the time variation in the data. 0 5 10 15 20 25 30 Falserejectrate 0 10 20 30 40 50 60 70 80 90 False accept rate MCYT-100 (EER = 12.1%) SUSIG-Visual (EER = 6.2%) Figure 3: DET curves show how FAR (x-axis) and FRR (y-axis) values change according to changing acceptance threshold, for the tested databases. The proposed FFT system is very fast: it can process 4500 queries in the MCYT-100 database in 69 seconds of CPU time. 4.3. Eﬀects of Preprocessing Steps. The best results reported in Table 2 were obtained using few preprocessing steps, namely, pen-up duration encoding and drift and mean removal. Some of the other preprocessing steps used in previous work based on FFT [23, 25] were just not useful due to our normalized features (e.g., rotation and scale normal- ization), while resampling worsened results by removing discriminative information (30.02% versus 6.20% EER for the SUSIG database and 17.82% versus 12.11% EER for the MCYT database). On the other hand, removal of the drift (especially signiﬁcant in the x-component) was found to improveperformanceinbothourworkandinpreviouswork , by a few percent points. The eﬀects of drift and mean removal are most apparent when they are used together. Note that mean removal is normally not necessary, since translation invariance is provided when the ﬁrst Fourier coeﬃcient is discarded; however mean removal aﬀects the outcome due to zero padding. The proposed incorporation of the pen-up duration is also found to help increase performance (9.09% EER versus 6.20% EER for the SUSIG database). 4.4. Eﬀects of Distance Normalization. Normalization of the query distance, prior to using a ﬁxed threshold across all users, has been found to make a signiﬁcant diﬀerence on veriﬁcation performance, as shown in Table 4. Here, AvgN refers to dividing the distance between the query and the mean descriptor vector by the average distance of the reference signatures. This average is obtained by using a leave-1-out method whereby one of the reference signature EURASIP Journal on Advances in Signal Processing 9 Table 3: Eﬀects of various preprocessing steps on the best conﬁguration. The bold face shows the results of the proposed system, while the last column shows the results if resampling was added to the proposed preprocessing steps (drift and mean removal and pen-up duration incorporation when available). Dataset Feature Raw Drift Mean Drift + Mean Proposed = Drift + Mean + PenUp Proposed if resampled SUSIG-Visual y |x 8.18% 7.34% 11.52% 9.09% 6.20% 30.02% MCYT-100 y |x | p |az |al 20.31% 20.38% 13.51% 12.11% — 17.82% Table 4:Diﬀerent methodsfor user-dependent distance normaliza- tion using only the reference data. Dataset Feature AvgN MinN MaxN None MCYT-100 y |x | p |az |al 12.11% 13.2% 14.3% 21.5% SUSIG-Visual y |x 6.20% 8.1% 5.8% 14.1% is treated as query, while the others are used as reference, as described in Section 3.4.2. Similarly, MinN and MaxN refer to dividing the distance between the query and the mean descriptor vector by the minimum and maximum of the reference signature distances (again using the leave-one- out method), respectively. All three of these normalization methods are better than not doing any normalization at all. Notice that while AvgN gives the best results for the MCYT-100 dataset, MaxN has given the best results for the SUSIG database. This diﬀerence highlights an important aspect of the current work, which is the fact that the exact same system is used in testing both databases, without any adjustment. In all of the presented results, we use the AvgN normalization method. 4.5. Results of the Fusion with the DTW System. It has been shown in the last couple of years that the combination of several experts improves veriﬁcation performance in biometrics [7, 28, 34, 35]. Some of the results, especially as related to the work described here, are summarized in Section 4.6. In order to show that the proposed FFT system may com- plement an approach based on local features, we combined the FFT system with a slightly modiﬁed implementation of the DTW system described in . The distribution of the DTW and FFT scores in Figure 4 shows that the two systems’ scores show a loose correlation, which is an important factor in classiﬁer combination systems. The combination is done using the sum rule, after normalizing the scores of the two systems. The score normalization factor is selected separately for each database, so as to equalize the mean scores of the twosystems,ascomputedoverthereference signaturesinthat database. A better selection of the normalization factor can be made when training data is available. Note that using the sum rule with score normalization is equivalent to separating the genuine and forgery classes using a straight line with a ﬁxedslope,wherethe y-interceptisadjustedtoﬁndtheequal error rate. The results given in Table 5 show that the FFT system improves the performance of the DTW system signiﬁcantly, by 8% or 26% depending on the database. Furthermore, the 0 10 20 30 40 50 60 DTWscores 1 2 3 4 5 6 7 8 9 10 FFT scores Figure 4: The distribution of the DTW and FFT scores for the MCYT-100 database. improvement brings the EER rates to state-of-the-art levels given in Table 6 for both databases (3.03% for SUSIG and 7.22% for MCYT-100). The proposed FFT system is very fast: it can process 4500 queries in the MCYT-100 database in 69 seconds of CPU time. In comparison, the DTW system takes 36800 seconds for the same task, which corresponds to a factor of more than 500. Theoretically, the time complexity of the DTW system is O(N × M), where N and M are the lengths of the two signatures being compared, while that of the FFT is O(N logN) for a signature of length N. Hence, even though using the FFT system in addition to the DTW system results in negligeable time overhead, Figure 4 shows that the systems can also be called in a serial fashion to eliminate the more obvious forgeries using the FFT system and calling the DTW system only for the less certain cases. Using this test with a threshold of 4, the same reported results were obtained while gaining around 10% speed overall. The DTW approach is probably the most commonly used technique in online signature veriﬁcation, while quite successfuloverallandinparticularinaligningtwosignatures, the basic DTW approach has some shortcomings, such as assigning low distance scores to short dissimilar signatures. One such example is shown in Figure 5, along with all of the genuine signatures of the claimed user. As an approach using global features, the FFT-based system is expected to be useful in eliminating some of these errors, when used in fusion with DTW or other local approaches. 4.6. Comparison with Previous Work. Results of previous work tested on the MCYT database are given in Table 6 for comparison. Since SUSIG is a new database, we concentrated on previous work reporting results on the MCYT database. Even with this database, comparing diﬀerent results is 10 EURASIP Journal on Advances in Signal Processing Table 5: Results of the fusion of the FFT system with our Dynamic Time Warping system. Dataset y |xy |x | p |az |al DTW DTW + y |x DTW + y |x | p |az |al Improvement SUSIG-Visual 6.20% — 3.30% 3.03% — 8% MCYT-100 14.53% 12.11% 9.81% 7.8% 7.22% 26% Table 6: State-of-the-art results on the MCYT database using a priori normalization techniques. Unless otherwise indicated, all dimensions of the input signal are used. Reference Dataset Method Features Performance Garcia-Salicetti et al.  MCYT-280 HMM  5.73% HMM  8.39% String Matching  15.89% Fusion of [18, 31] 3.40% Faundez-Zanuy  MCYT-280 VQ 11.8% DTW 8.9% VQ-DTW 5.4% (DCF) Vivaracho-Pascual et al.  MCYT-280 Length normaliz./p-norm 6.8% (DCF) Nanni and Lumini  MCYT-100∗ SVM 100 global features 17.0% SVM-DTW  7.6% Nanni and Lumini  MCYT-100 Wavelet-DCT x, y x, y, az 11.4% Wavelet-DCT 9.8% Wavelet-DCT fused w/DTW, HMM, GM 5.2% Quan et al.  MCYT-100∗ STFT 7% This work MCYT-100 Proposed FFT x, y 12.11% DTW  9.81% FFT-DTW 7.22% diﬃcult due to varying experimental setups. In particular, we have (i) the subset of the MCYT database used: MCYT-280 is the test subset of the full database of 330 people where a 50-people portion is used for training, while MCYT-100 is the publicly available part consisting of 100 people and no allocated training subset; (ii) the number of reference signa- tures used (most systems use the ﬁrst 5 genuine signatures as suggested, while others use more, as necessitated by their veriﬁcation algorithm); (iii) number of available component signals used, such as coordinate sequence, pressure, and azimuth (not counting derived features); and (iv) whether a priorior a posteriori normalization is used for score normalization, as deﬁned in . Ingeneral,thehigherthenumberofreferences,thebetter one would expect the results to be, due to having more information about the genuine signatures of a user. Similarly, higher number of signal components normally give better results. Finally, score normalization aﬀects the performance signiﬁcantly, since the a posteriori normalization results are intendedtogivethebestpossibleresults,ifallgenuineand/or forger statistics in the database were known ahead of time. For this comparison, we tried to included recent results on theMCYTdatabase,using5referencesignaturesassuggested and a prioriscore normalization methods, to the best of our knowledge. Given the various factors aﬀecting performance and the diﬃculty in assessing the exact experimental setups of others’ work, an exact comparison of diﬀerent systems is not very easy. Nonetheless, we give the following as indicative results. The best results obtained with the MCYT-100 database is reported by Nanni and Lumini, with 5.2% EER using 3 measuredsignals(x, y,azimuth)usingfourexpertsincluding Wavelet, DTW and HMM approaches . In that work, the Wavelet based system itself achieves 9.8% EER. The other system developed by the same authors which uses Support Vector Machines (SVMs) with 100 global features obtains 17.0% on the MCYT-100 database (using a 20-people subset for training), while the combination of SVM and DTW (based on our DTW system used in the fusion part of this work ) achieves 7.6% . Quan et al. report 7% EER of usingwindowedFFTontheMCYT-100butusing15genuine signatures as reference (instead of 5 which is the suggested number). On the MCYT-280 database, Garcia-Salicetti et al. evalu- ates 3 individual systems in a study of complementarity; the individual systems’ performance are given as 5.73%, 8.39%, and 15.89%, while the best fusion system obtains 3.40% EER on skilled forgeries . Faundez-Zanuy reports 11.8% and 5.4% using Vector Quantization (VQ) and VQ combined with DTW respectively . However, instead of EER, they EURASIP Journal on Advances in Signal Processing 11 User: 0098 Query(red): f06 Figure 5: A forgery signature (shown on top) that was misclassiﬁed using the DTW system while it was correctly classiﬁed using the combined system. report the main results using the Detection Cost Function (DCF) with 5 genuine and 25 forgery signatures per person. Similarly, Vivaracho-Pascual et al. report a DCF of 6.8%, using the same experimental setup. The most apparent factor in these results is the eﬀect of classiﬁer combination. Classiﬁer combination or fusion systems are found to be useful in many pattern recognition problems, so the improvement of the results is not surprising and is parallelled in our current results as well. The other important factor aﬀecting performance is the dimensionality of the input signal. In some databases, x- andy-coordinates are the only available dimensions, while pressure, azimuth, and altitude are also available in others. Increasing the number of dimensions generally increases the veriﬁcation performance, as more relevant information is available to the classiﬁer. One interesting note is that the DTW appears as a component in each of the listed fusion systems. The performance of the proposed FFT system is lower than the state-of-the-art fusion systems, while it seems to be in par with single engine systems on the same database (12.11% versus 9.8% , 17.0% , and 9.81% with our DTW approach, on the MCYT database). Approaches using global features typically underperform compared to those using local features. On the other hand, global approaches are necessary in certain applications. Furthermore, due to their speed and complementarity, they are expected to be useful in fusion systems to increase the performance and/or the speed. We also have to underline the fact that when reporting results on a database, researchers typically report the results of the optimal set of features and algorithm steps, which introduces bias to the results. In fact, often a particular step of an algorithm improves the results on one database, while degrading it on another (e.g., diﬀerent distance normalization methods gave the best results in SUSIG and MCYT databases, as shown in Table 5). Therefore, the fact that our results are obtained by testing the same exact system on two diﬀerent databases with diﬀerent characteristics (e.g., signature types, sensors, measured signals, forgery skills) is important. As for comparison with previous work using FFT, the system developed by Lam et al.  is reported to have 2.5% error rate, however the dataset in their work is very small (8 genuine signatures of the same user and 152 forgeries provided by 19 forgers) and old, making a direct comparison impossible. Similarly, while the improvement of using windowed FFT, suggested by Quan et al.  is reasonable, their results are not readily comparable to ours: they report an EER of 7% on the MCYT-100, using 15 genuine signatures as reference instead of 5, presumably necessitated by their use of the Mahalanobis distance. As mentioned before, increased number of reference signatures are expected to increase performance and the resulting test set in their case is signiﬁcantly diﬀerent than ours. Furthermore, we have also shown that resampling step used in both of these works signiﬁcantly degrades veriﬁcation performance for the proposed method by removing some of the timing information which is useful in discriminating forgery and genuine signatures. 5. Future Work In the current system, we use only the magnitude of the Fourier coeﬃcients, discarding the phase information for simplicity, while phase information is actually a fundamental part of the signal. We expect that the use of the phase information can improve the system performance. Similarly, other extracted features, such as local velocity, can easily be used and would be expected to improve the system performance based on others’ work . Another improvement may be the use of windowed or Short Term Fourier Transform. The STFT aims to give more information about the timing as well as the frequency component of the signal, by breaking the input signal into a number of small segments by a windowing signal prior to the application of the Fourier transform. The size of the window used for this operation is an issue in general but for online signature veriﬁcation, separate strokes or high curvature points can be used for this purpose. An analysis of the errors shows that large portion of the errors is due to simple signatures, composed of simple or easilyreproducibletrajectories.Whilenotmuchmaybedone to reduce errors on these types of signatures, one could at least envision a system alerting users when they use simple signatures at enrollment time. 12 EURASIP Journal on Advances in Signal Processing 6. Summary and Discussions We presented a novel approach for online signature veriﬁca- tion using global features consisting of Fourier Descriptors that provide a compact and ﬁxed-length representation of an online signature. Our approach is signiﬁcantly diﬀerent in preprocessing, feature extraction, normalization and match- ing steps, compared to previous online signature veriﬁcation systems that are based on FFT. These steps are carefully designed to retain the full discriminatory information avail- able in the signature; in particular the incorporation of the timestamp information for representing pen-up durations is novel and had signiﬁcant eﬀects on performance. The proposed system is extensively tested using two large public databases, both in terms of overall performance and the eﬀects of individual preprocessing steps. The results are inferior to the best results obtained by fusion systems but the system shows potential as a stand-alone system to be used wherever ﬁxed-length representation is needed, and in complementing an approach based on local features. The latter is supported experimentally by the fact the combination of the proposed FFT system improved the results of our state-of-the-art DTW system, resulting in EER of 3.03% for the SUSIG database and 7.22% for the MCYT- 100 database. Furthermore, given the previously mentioned factors aﬀecting performance and the diﬃculty in assessing the exact experimental setups of others’ work, an exact comparison of EER results is not always meaningful. This is especially true since the proposed system is tested with exactly the same parameters on two diﬀerent databases with diﬀerent characteristics (e.g., signature types, sensors, measured signals, forgery skills). As for overall speed, the proposed system is very fast, about 500 times faster than a dynamic programming approach on the same database. The speed is thus one of the advantages of the proposed system and is especially important in fusion systems and identiﬁcation problems as well as quickly testing new algorithms or preprocessing steps. The main aspects of the developed FFT system can thus be summarized as follows: (i) it is very fast in training, feature extraction, and matching (about 2-3 orders of magnitude faster than the DTW system); (ii) it uses a ﬁxed-length feature vector comprised of global features of the signature, which is required in certain applications; (iii) its performance is lower than state-of-the-art results obtained by fusion systems; however its advantages and potential improvements make it a useful alter- native in online signature veriﬁcation, especially in complementing more complex but slower methods based on local features, such as the DTW or HMM approaches. Given its merits as a global approach and the suggested improvements, we believe that the proposed FFT-based system has potential as a stand-alone system but especially in complementing an approach based on local features. 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